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Recent content in Finance on R ViewsHugo -- gohugo.ioen-usRStudio, Inc. All Rights Reserved.Rolling Origins and Fama French
https://rviews.rstudio.com/2018/12/26/rolling-origins-and-fama-french/
Wed, 26 Dec 2018 00:00:00 +0000https://rviews.rstudio.com/2018/12/26/rolling-origins-and-fama-french/
<p>Today, we continue our work on sampling so that we can run models on subsets of our data and then test the accuracy of the models on data not included in those subsets. In the machine learning prediction world, these two data sets are often called <code>training</code> data and <code>testing</code> data, but we’re not going to do any machine learning prediction today. We’ll stay with our good’ol Fama French regression models for the reasons explained last time: the goal is to explore a new method of sampling our data and I prefer to do that in the context of a familiar model and data set. In 2019, we’ll start expanding our horizons to different models and data, but for today it’s more of a tools exploration.</p>
<p>Loyal readers of this blog (if that’s you, thank you!) will remember that we undertook a similar project in the <a href="https://rviews.rstudio.com/2018/12/13/rsampling-fama-french/">previous post</a>, when we used k-fold cross-validation. Today, we will use rolling origin sampling of the data, which differs from k-fold cross-validation in the sense that with rolling origin we explicitly sample based on the dates of our observation. With rolling origin, our first sample starts on the first day, our second sample starts on the second day, our third sample starts on third day. Or, we could have the second sample start on the twentieth day, or we could have it start again on the first day and just include an extra day. Either way, we are aware of the order of our data when sampling. With k-fold cross-validation, we were randomly shuffling and then selecting observations. This distinction can be particularly important for economic time series where we believe that the order of our observations is important.</p>
<p>Let’s get to it.</p>
<p>First, we need our data and, as usual, we’ll import data for daily prices of 5 ETFs, convert them to returns (have a look <a href="http://www.reproduciblefinance.com/2017/09/25/asset-prices-to-log-returns/">here</a> for a refresher on that code flow), then import the 5 Fama French factor data and join it to our 5 ETF returns data. Here’s the code to make that happen (this code was covered in detail in <a href="http://www.reproduciblefinance.com/2018/06/07/fama-french-write-up-part-one/">this post</a>:</p>
<pre class="r"><code>library(tidyverse)
library(tidyquant)
library(rsample)
library(broom)
library(timetk)
symbols <- c("SPY", "EFA", "IJS", "EEM", "AGG")
# The prices object will hold our daily price data.
prices <-
getSymbols(symbols, src = 'yahoo',
from = "2012-12-31",
to = "2017-12-31",
auto.assign = TRUE,
warnings = FALSE) %>%
map(~Ad(get(.))) %>%
reduce(merge) %>%
`colnames<-`(symbols)
asset_returns_long <-
prices %>%
tk_tbl(preserve_index = TRUE, rename_index = "date") %>%
gather(asset, returns, -date) %>%
group_by(asset) %>%
mutate(returns = (log(returns) - log(lag(returns)))) %>%
na.omit()
factors_data_address <-
"http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/ftp/Global_5_Factors_Daily_CSV.zip"
factors_csv_name <- "Global_5_Factors_Daily.csv"
temp <- tempfile()
download.file(
# location of file to be downloaded
factors_data_address,
# where we want R to store that file
temp,
quiet = TRUE)
Global_5_Factors <-
read_csv(unz(temp, factors_csv_name), skip = 6 ) %>%
rename(date = X1, MKT = `Mkt-RF`) %>%
mutate(date = ymd(parse_date_time(date, "%Y%m%d")))%>%
mutate_if(is.numeric, funs(. / 100)) %>%
select(-RF)
data_joined_tidy <-
asset_returns_long %>%
left_join(Global_5_Factors, by = "date") %>%
na.omit()</code></pre>
<p>After running that code, we have an object called <code>data_joined_tidy</code>. It holds daily returns for 5 ETFs and the Fama French factors. Here’s a look at the first row for each ETF rows.</p>
<pre class="r"><code>data_joined_tidy %>%
slice(1)</code></pre>
<pre><code># A tibble: 5 x 8
# Groups: asset [5]
date asset returns MKT SMB HML RMW CMA
<date> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 2013-01-02 AGG -0.00117 0.0199 -0.0043 0.0028 -0.0028 -0.0023
2 2013-01-02 EEM 0.0194 0.0199 -0.0043 0.0028 -0.0028 -0.0023
3 2013-01-02 EFA 0.0154 0.0199 -0.0043 0.0028 -0.0028 -0.0023
4 2013-01-02 IJS 0.0271 0.0199 -0.0043 0.0028 -0.0028 -0.0023
5 2013-01-02 SPY 0.0253 0.0199 -0.0043 0.0028 -0.0028 -0.0023</code></pre>
<p>For today, let’s work with just the daily returns of EEM.</p>
<pre class="r"><code>eem_for_roll <- data_joined_tidy %>%
filter(asset == "EEM")</code></pre>
<p>We are going to regress those EEM daily returns on the Fama French factors and need a way to measure the accuracy of our various factor models. <a href="http://www.reproduciblefinance.com/2018/11/26/many-factor-models/">Previously</a>, we measured our models by looking at the adjusted R-squared, when the models were run on the entirety of the data. Last time, we use k-fold cross-validation to split the data into a bunch of subsets, then ran our model on the subsets and measured the accuracy against the holdouts from the subsets. Now, let’s use the <code>rolling_origin()</code> function from <code>rsample</code> to split our data into <code>analysis</code> and <code>assessment</code> sets in a time-aware way, then calculate RMSEs.</p>
<p>The <code>rolling_origin()</code> function needs a few arguments set. We first set <code>data</code> to be <code>eem_for_roll</code> Then, we assign <code>initial</code> to be <code>50</code> - this tells the function that the size of our first sample is 50 days. Our first chunk of <code>analysis</code> data will be the first 50 days of EEM returns. Next, we assign <code>assess</code> to be <code>5</code> - this tells the function that our <code>assessment</code> data is the 5 days of EEM returns following those first 50 days. Finally, we set <code>cumulative</code> to be <code>FALSE</code> - this tells the functions that each of splits is a 50 days. The first split is the first 50 days, starting on day 1 and ending on day 50. The next split starts on day 2 and ends on day 51. If we were to set <code>cumulative = TRUE</code>, the first split would be 50 days. The next split would be 51 days, the next split would be 52 days. And so on. The <code>analysis</code> split days would be accumulating. For now, we will leave it at <code>cumulative = FALSE</code>.</p>
<p>For that reason, we will append <code>_sliding</code> to the name of our object because the start of our window will slide each time.</p>
<pre class="r"><code>library(rsample)
roll_eem_sliding <-
rolling_origin(
data = eem_for_roll,
initial = 50,
assess = 10,
cumulative = FALSE
)</code></pre>
<p>Look at an individual split.</p>
<pre class="r"><code>one_eem_split <-
roll_eem_sliding$splits[[1]]
one_eem_split</code></pre>
<pre><code><50/10/1259></code></pre>
<p>That <code>50</code> is telling us there are 50 days or rows in the <code>analysis</code> set; that <code>10</code> is telling us that there are 10 rows in our <code>assessment</code> data - we’ll see how well our model predicts the return 5 days after the last observation in our data.</p>
<p>Here is the <code>analysis</code> subset of that split.</p>
<pre class="r"><code>analysis(one_eem_split) %>%
head()</code></pre>
<pre><code># A tibble: 6 x 8
# Groups: asset [1]
date asset returns MKT SMB HML RMW CMA
<date> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 2013-01-02 EEM 0.0194 0.0199 -0.0043 0.0028 -0.0028 -0.0023
2 2013-01-03 EEM -0.00710 -0.0021 0.00120 0.000600 0.0008 0.0013
3 2013-01-04 EEM 0.00200 0.0064 0.0011 0.0056 -0.0043 0.0036
4 2013-01-07 EEM -0.00759 -0.0014 0.00580 0 -0.0015 0.0001
5 2013-01-08 EEM -0.00900 -0.0027 0.0018 -0.00120 -0.0002 0.00120
6 2013-01-09 EEM 0.00428 0.0036 0.000300 0.0025 -0.0028 0.001 </code></pre>
<p>And the <code>assessment</code> subset - this is 10 rows.</p>
<pre class="r"><code>assessment(one_eem_split)</code></pre>
<pre><code># A tibble: 10 x 8
# Groups: asset [1]
date asset returns MKT SMB HML RMW CMA
<date> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 2013-03-15 EEM -0.00908 0.0025 0.006 -0.0001 0.0018 -0.0001
2 2013-03-18 EEM -0.0113 -0.0093 0.0021 -0.0039 0.0039 -0.001
3 2013-03-19 EEM -0.00712 -0.003 0.0015 -0.0023 0.0017 0.0028
4 2013-03-20 EEM 0.00570 0.0052 -0.003 0.0023 -0.002 0.002
5 2013-03-21 EEM -0.0102 -0.0037 0.0085 -0.0007 0.00120 0.0008
6 2013-03-22 EEM 0.00382 0.0033 -0.0042 -0.0023 0.0022 -0.0007
7 2013-03-25 EEM -0.000954 -0.0037 0.0036 -0.0032 0.0027 0.000600
8 2013-03-26 EEM 0.0142 0.0039 -0.0032 -0.0017 0.00120 -0.0017
9 2013-03-27 EEM 0.00376 -0.0016 0.0022 -0.0004 -0.0002 -0.0008
10 2013-03-28 EEM 0.00211 0.0033 -0.0022 -0.0031 0.000600 0.000600</code></pre>
<p>By way of comparison, here’s what the k-fold cross-validated data would look like.</p>
<pre class="r"><code>cved_eem <-
vfold_cv(eem_for_roll, v = 5)
assessment(cved_eem$splits[[1]]) %>%
head()</code></pre>
<pre><code># A tibble: 6 x 8
# Groups: asset [1]
date asset returns MKT SMB HML RMW CMA
<date> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 2013-01-04 EEM 0.00200 0.0064 0.0011 0.0056 -0.0043 0.0036
2 2013-01-14 EEM 0.00426 -0.000600 0.0002 0.0013 -0.001 0.0011
3 2013-01-31 EEM 0.00204 -0.0021 0.0038 0.000300 -0.0007 -0.0001
4 2013-02-04 EEM -0.0133 -0.0107 0.0059 -0.0027 0.0023 0.0004
5 2013-02-05 EEM 0.00114 0.0034 -0.0054 -0.0002 0.0008 -0.002
6 2013-02-06 EEM -0.00114 0.0019 0.0037 0.0022 -0.0018 0.0028</code></pre>
<p>Notice how the first date is not necessarily the first date in our data. In fact, if you run that code chunk a few times, the first date will be randomly selected. For me, it varied between January 2nd and January 15th.</p>
<p>Back to our <code>rolling_origin</code> data, we know that split 1 begins on day 1 and ends on day 50:</p>
<pre class="r"><code>analysis(roll_eem_sliding$splits[[1]]) %>%
slice(c(1,50))</code></pre>
<pre><code># A tibble: 2 x 8
# Groups: asset [1]
date asset returns MKT SMB HML RMW CMA
<date> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 2013-01-02 EEM 0.0194 0.0199 -0.0043 0.0028 -0.0028 -0.0023
2 2013-03-14 EEM 0.00418 0.0078 0.0021 0.0016 -0.002 0.0026</code></pre>
<p>And we know split 2 begins on day 2 and ends on day 51:</p>
<pre class="r"><code>analysis(roll_eem_sliding$splits[[2]]) %>%
slice(c(1,50))</code></pre>
<pre><code># A tibble: 2 x 8
# Groups: asset [1]
date asset returns MKT SMB HML RMW CMA
<date> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 2013-01-03 EEM -0.00710 -0.0021 0.00120 0.000600 0.0008 0.0013
2 2013-03-15 EEM -0.00908 0.0025 0.006 -0.0001 0.0018 -0.0001</code></pre>
<p>Now, we can start using our data to fit and then assess our model. The code flow is very similar to our previous post, but I’ll go ahead and run through it anyway.</p>
<p>First, let’s create a function that takes a split as an argument, fits a 3-factor Fama French model and calculates the root mean squared error. We will use the <code>rmse()</code> function from <code>yardstick</code> to measure our model. RMSE stands for root mean-squared error. It’s the sum of the squared differences between our fitted values and the actual values in the <code>assessment</code> data. A lower RMSE is better!</p>
<pre class="r"><code>library(yardstick)
ff_three_rmse <- function(split){
analysis_set <- analysis(split)
ff_model <- lm(returns ~ MKT + SMB + HML , data = analysis_set)
holdout <- assessment(split)
rmse <-
ff_model %>%
augment(newdata = holdout) %>%
rmse(returns, .fitted) %>%
pull(.estimate)
}</code></pre>
<p>Let’s pass our one split object <code>one_eem_split</code> to the function.</p>
<pre class="r"><code>ff_three_rmse(one_eem_split) %>%
print()</code></pre>
<pre><code>[1] 0.005311276</code></pre>
<p>Now, we can use <code>mutate()</code> and <code>map_dbl()</code> to pass all of our splits held in <code>roll_eem_sliding</code> to the function. This will return an <code>rmse</code> for this model when applied to each of our rolling origin splits. This takes a few seconds to run.</p>
<pre class="r"><code>rmse_one_model <-
roll_eem_sliding %>%
mutate(model = map_dbl(
.x = splits,
.f = ~(ff_three_rmse(.x))))
rmse_one_model %>%
head()</code></pre>
<pre><code># A tibble: 6 x 3
splits id model
* <list> <chr> <dbl>
1 <split [50/10]> Slice0001 0.00531
2 <split [50/10]> Slice0002 0.00621
3 <split [50/10]> Slice0003 0.00642
4 <split [50/10]> Slice0004 0.00647
5 <split [50/10]> Slice0005 0.00654
6 <split [50/10]> Slice0006 0.00649</code></pre>
<p>This is the same as we did last time. Now, let’s get functional. We will define three models that we wish to test with via our rolling origin splits and RMSE, then pass those models to one function.</p>
<p>First, we use <code>as.formula()</code> to define our three models.</p>
<pre class="r"><code>mod_form_1 <- as.formula(returns ~ MKT)
mod_form_2 <- as.formula(returns ~ MKT + SMB + HML)
mod_form_3 <- as.formula(returns ~ MKT + SMB + HML + RMW + CMA)</code></pre>
<p>We write one function that takes <code>split</code> as an argument, same as above, but also takes <code>formula</code> as an argument, so we can pass it different models. This gives us the flexibility to more easily define new models and pass them to <code>map</code> so I’ll append <code>_flex</code> to the name of this function.</p>
<pre class="r"><code>ff_rolling_flex <- function(split, formula) {
split_for_data <- analysis(split)
ff_model <- lm(formula, data = split_for_data)
holdout <- assessment(split)
rmse <-
ff_model %>%
augment(newdata = holdout) %>%
rmse(returns, .fitted) %>%
pull(.estimate)
}
rmse_model_1_flex <-
roll_eem_sliding %>%
mutate(model_1_rmse = map_dbl(roll_eem_sliding$splits,
ff_rolling_flex,
formula = mod_form_1))
rmse_model_1_flex %>%
head()</code></pre>
<pre><code># A tibble: 6 x 3
splits id model_1_rmse
* <list> <chr> <dbl>
1 <split [50/10]> Slice0001 0.00601
2 <split [50/10]> Slice0002 0.00548
3 <split [50/10]> Slice0003 0.00547
4 <split [50/10]> Slice0004 0.00556
5 <split [50/10]> Slice0005 0.00544
6 <split [50/10]> Slice0006 0.00539</code></pre>
<p>Again same as last time, we can run all of our models using the <code>mutate()</code> and <code>map_dbl</code> combination. Warning, this one takes a bit of time and compute to run - proceed with caution if you’re on a desktop.</p>
<pre class="r"><code>rmse_3_models <-
roll_eem_sliding %>%
mutate(model_1_rmse = map_dbl(roll_eem_sliding$splits,
ff_rolling_flex,
formula = mod_form_1),
model_2_rmse = map_dbl(roll_eem_sliding$splits,
ff_rolling_flex,
formula = mod_form_2),
model_3_rmse = map_dbl(roll_eem_sliding$splits,
ff_rolling_flex,
formula = mod_form_3))
rmse_3_models %>%
head()</code></pre>
<pre><code># A tibble: 6 x 5
splits id model_1_rmse model_2_rmse model_3_rmse
* <list> <chr> <dbl> <dbl> <dbl>
1 <split [50/10]> Slice0001 0.00601 0.00531 0.00496
2 <split [50/10]> Slice0002 0.00548 0.00621 0.00596
3 <split [50/10]> Slice0003 0.00547 0.00642 0.00617
4 <split [50/10]> Slice0004 0.00556 0.00647 0.00615
5 <split [50/10]> Slice0005 0.00544 0.00654 0.00613
6 <split [50/10]> Slice0006 0.00539 0.00649 0.00600</code></pre>
<p>Alright, we have our RMSE, from each of our 3 models, as applied to each of our splits.</p>
<p>Thus far, our substantive flow is very similar to our k-fold cross-validation work. But now, we can take advantage of the time-aware nature of our splits.</p>
<p>Let’s visualize how our RMSE has changed over different time-aware splits, for our various models. Remember, we know the exact start and end date for our <code>analysis</code> and <code>assessment</code> sets, so we can extract the date of, say the first observation in the <code>assessment</code> data and assign it to the split. We can consider this the date of that model run.</p>
<p>First, let’s create a function to extract the first date of each of our <code>assessment</code> sets.</p>
<pre class="r"><code>get_start_date <- function(x)
min(assessment(x)$date)</code></pre>
<p>Here’s how that works on our one split object.</p>
<pre class="r"><code>get_start_date(one_eem_split)</code></pre>
<pre><code>[1] "2013-03-15"</code></pre>
<p>That’s the first date in the assessment data:</p>
<pre class="r"><code>assessment(one_eem_split) %>%
select(date) %>%
slice(1)</code></pre>
<pre><code># A tibble: 1 x 2
# Groups: asset [1]
asset date
<chr> <date>
1 EEM 2013-03-15</code></pre>
<p>We want to add a column to our <code>results_3</code> object called <code>start_date</code>. We’ll use our usual <code>mutate()</code> and then <code>map()</code> flow to apply the <code>get_start_date()</code> function to each of our splits, but we’ll need to pipe the result to <code>reduce(c)</code> to coerce the result to a date. <code>map()</code> returns a list by default and we want a vector of dates.</p>
<pre class="r"><code>rmse_3_models_with_start_date <-
rmse_3_models %>%
mutate(start_date = map(roll_eem_sliding$splits, get_start_date) %>% reduce(c)) %>%
select(start_date, everything())
rmse_3_models_with_start_date %>%
head()</code></pre>
<pre><code># A tibble: 6 x 6
start_date splits id model_1_rmse model_2_rmse model_3_rmse
* <date> <list> <chr> <dbl> <dbl> <dbl>
1 2013-03-15 <split [50/10]> Slice… 0.00601 0.00531 0.00496
2 2013-03-18 <split [50/10]> Slice… 0.00548 0.00621 0.00596
3 2013-03-19 <split [50/10]> Slice… 0.00547 0.00642 0.00617
4 2013-03-20 <split [50/10]> Slice… 0.00556 0.00647 0.00615
5 2013-03-21 <split [50/10]> Slice… 0.00544 0.00654 0.00613
6 2013-03-22 <split [50/10]> Slice… 0.00539 0.00649 0.00600</code></pre>
<p>We can head to <code>ggplot</code> for some visualizing. I’d like to plot all of my RMSE’s in different colors and the best way to do that is to <code>gather()</code> this data to tidy format, with a column called <code>model</code> and a column called <code>value</code>. It’s necessary to coerce to a data frame first, using <code>as.data.frame()</code>.</p>
<pre class="r"><code>rmse_3_models_with_start_date %>%
as.data.frame() %>%
select(-splits, -id) %>%
gather(model, value, -start_date) %>%
head()</code></pre>
<pre><code> start_date model value
1 2013-03-15 model_1_rmse 0.006011868
2 2013-03-18 model_1_rmse 0.005483091
3 2013-03-19 model_1_rmse 0.005470834
4 2013-03-20 model_1_rmse 0.005557170
5 2013-03-21 model_1_rmse 0.005439921
6 2013-03-22 model_1_rmse 0.005391862</code></pre>
<p>Next, we can use some of our familiar <code>ggplot</code> methods to plot our RMSEs over time, and see if we notice this model degrading or improving in different periods.</p>
<pre class="r"><code>rmse_3_models_with_start_date %>%
as.data.frame() %>%
select(-splits, -id) %>%
gather(model, value, -start_date) %>%
ggplot(aes(x = start_date, y = value, color = model)) +
geom_point() +
facet_wrap(~model, nrow = 2) +
scale_x_date(breaks = scales::pretty_breaks(n = 10)) +
theme(axis.text.x = element_text(angle = 45, vjust = 1, hjust=1))</code></pre>
<p><img src="/post/2018-12-20-rolling-origins-and-fama-french_files/figure-html/unnamed-chunk-22-1.png" width="672" /></p>
<p>All of the models have a jump in RMSE, meaning they performed worse, around the end of 2017. We aren’t focused on the theory of these models today but if we were, this would be a good place to start investigating. This is just the beginning of our exploration of <code>rolling_origin</code>, but I love how it opens the door for ways to think about visualizing our models.</p>
<p>And finally, those same public service announcements from last time are still live, so I’ll mention them one more time.</p>
<p>Thanks to everyone who has checked out my new book! The price just got lowered for the holidays. See on <a href="https://www.amazon.com/Reproducible-Finance-Portfolio-Analysis-Chapman/dp/1138484032">Amazon</a> or on the <a href="https://www.crcpress.com/Reproducible-Finance-with-R-Code-Flows-and-Shiny-Apps-for-Portfolio-Analysis/Jr/p/book/9781138484030">CRC homepage</a> (okay, that was more of an announcement about my book).</p>
<p>Applications are open for the <a href="https://www.battlefin.com/">Battlefin</a> alternative data contest and RStudio is one of the tools you can use to analyze the data. Check it out <a href="www.alternativedatacombine.com">here</a>. They’ll announce 25 finalists in January who will get to compete for a cash prize and connect with some quant hedge funds. Go get’em!</p>
<p>A special thanks to <a href="https://www.linkedin.com/in/bruce-fox-98504612/">Bruce Fox</a> who suggested we might want to expand on the Monte Carlo simulation in <a href="https://www.amazon.com/Reproducible-Finance-Portfolio-Analysis-Chapman/dp/1138484032">the book</a> to take account of different distributions implied by historical returns and different market regimes that might arise. Today’s rolling origin framework will also lay the foundation, I hope, for implementing some of Bruce’s ideas in January.</p>
<p>Thanks for reading and see you next time.</p>
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Rsampling Fama French
https://rviews.rstudio.com/2018/12/13/rsampling-fama-french/
Thu, 13 Dec 2018 00:00:00 +0000https://rviews.rstudio.com/2018/12/13/rsampling-fama-french/
<p>Today we will continue our work on Fama French factor models, but more as a vehicle to explore some of the awesome stuff happening in the world of <a href="https://www.tidyverse.org/articles/2018/11/tidymodels-update-nov-18/">tidy models</a>. For new readers who want get familiar with Fama French before diving into this post, see <a href="https://rviews.rstudio.com/2018/04/11/introduction-to-fama-french/">here</a> where we covered importing and wrangling the data, <a href="https://rviews.rstudio.com/2018/05/10/rolling-fama-french/">here</a> where we covered rolling models and visualization, my most recent previous post <a href="https://rviews.rstudio.com/2018/11/19/many-factor-models/">here</a> where we covered managing many models, and if you’re into Shiny, <a href="http://www.reproduciblefinance.com/shiny/fama-french-three-factor/">this flexdashboard</a>.</p>
<p>Our goal today is to explore k-fold cross-validation via the <code>rsample</code> package, and a bit of model evaluation via the <code>yardstick</code> package. We started the model evaluation theme last time when we used <code>tidy()</code>, <code>glance()</code> and <code>augment()</code> from the <code>broom</code> package. In this post, we will use the <code>rmse()</code> function from <code>yardstick</code>, but our main focus will be on the <code>vfold_cv()</code> function from <code>rsample</code>. We are going to explore these tools in the context of linear regression and Fama French, which might seem weird since these tools would typically be employed in the realms of machine learning, classification, and the like. We’ll stay in the world of explanatory models via linear regression world for a few reasons.</p>
<p>First, and this is a personal preference, when getting to know a new package or methodology, I prefer to do so in a context that’s already familiar. I don’t want to learn about <code>rsample</code> whilst also getting to know a new data set and learning the complexities of some crazy machine learning model. Since Fama French is familiar from our previous work, we can focus on the new tools in <code>rsample</code> and <code>yardstick</code>. Second, factor models are important in finance, despite relying on good old linear regression. We won’t regret time spent on factor models, and we might even find creative new ways to deploy or visualize them.</p>
<p>The plan for today is take the same models that we ran in the last post, only this use k-fold cross validation and bootstrapping to try to assess the quality of those models.</p>
<p>For that reason, we’ll be working with the same data as we did previously. I won’t go through the logic again, but in short, we’ll import data for daily prices of five ETFs, convert them to returns (have a look <a href="http://www.reproduciblefinance.com/2017/09/25/asset-prices-to-log-returns/">here</a> for a refresher on that code flow), then import the five Fama French factor data and join it to our five ETF returns data. Here’s the code to make that happen:</p>
<pre class="r"><code>library(tidyverse)
library(broom)
library(tidyquant)
library(timetk)
symbols <- c("SPY", "EFA", "IJS", "EEM", "AGG")
# The prices object will hold our daily price data.
prices <-
getSymbols(symbols, src = 'yahoo',
from = "2012-12-31",
to = "2017-12-31",
auto.assign = TRUE,
warnings = FALSE) %>%
map(~Ad(get(.))) %>%
reduce(merge) %>%
`colnames<-`(symbols)
asset_returns_long <-
prices %>%
tk_tbl(preserve_index = TRUE, rename_index = "date") %>%
gather(asset, returns, -date) %>%
group_by(asset) %>%
mutate(returns = (log(returns) - log(lag(returns)))) %>%
na.omit()
factors_data_address <-
"http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/ftp/Global_5_Factors_Daily_CSV.zip"
factors_csv_name <- "Global_5_Factors_Daily.csv"
temp <- tempfile()
download.file(
# location of file to be downloaded
factors_data_address,
# where we want R to store that file
temp,
quiet = TRUE)
Global_5_Factors <-
read_csv(unz(temp, factors_csv_name), skip = 6 ) %>%
rename(date = X1, MKT = `Mkt-RF`) %>%
mutate(date = ymd(parse_date_time(date, "%Y%m%d")))%>%
mutate_if(is.numeric, funs(. / 100)) %>%
select(-RF)
data_joined_tidy <-
asset_returns_long %>%
left_join(Global_5_Factors, by = "date") %>%
na.omit()</code></pre>
<p>After running that code, we have an object called <code>data_joined_tidy</code>. It holds daily returns for 5 ETFs and the Fama French factors. Here’s a look at the first row for each ETF rows.</p>
<pre class="r"><code>data_joined_tidy %>%
slice(1)</code></pre>
<pre><code># A tibble: 5 x 8
# Groups: asset [5]
date asset returns MKT SMB HML RMW CMA
<date> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 2013-01-02 AGG -0.00117 0.0199 -0.0043 0.0028 -0.0028 -0.0023
2 2013-01-02 EEM 0.0194 0.0199 -0.0043 0.0028 -0.0028 -0.0023
3 2013-01-02 EFA 0.0154 0.0199 -0.0043 0.0028 -0.0028 -0.0023
4 2013-01-02 IJS 0.0271 0.0199 -0.0043 0.0028 -0.0028 -0.0023
5 2013-01-02 SPY 0.0253 0.0199 -0.0043 0.0028 -0.0028 -0.0023</code></pre>
<p>Let’s work with just one ETF for today and use <code>filter(asset == "AGG")</code> to shrink our data down to just that ETF.</p>
<pre class="r"><code>agg_ff_data <-
data_joined_tidy %>%
filter(asset == "AGG")</code></pre>
<p>Okay, we’re going to regress the daily returns of AGG on one factor, then three factors, then five factors, and we want to evaluate how well each model explains AGG’s returns. That means we need a way to test the model. Last time, we looked at the adjusted r-squared values when the model was run on the entirety of AGG returns. Today, we’ll evaluate the model using k-fold cross validation. That’s a pretty jargon-heavy phrase that isn’t part of the typical finance lexicon. Let’s start with the second part, <code>cross-validation</code>. Instead of running our model on the entire data set - all the daily returns of AGG - we’ll run it on just part of the data set, then test the results on the part that we did not use. Those different subsets of our original data are often called the training and the testing sets, though <code>rsample</code> calls them the <code>analysis</code> and <code>assessment</code> sets. We validate the model results by applying them to the <code>assessment</code> data and seeing how the model performed.</p>
<p>The <code>k-fold</code> bit refers to the fact that we’re not just dividing our data into training and testing subsets, we’re actually going to divide it into a bunch of groups, a <code>k</code> number of groups, or a <code>k</code> number of <code>folds</code>. One of those folds will be used as the validation set; the model will be fit on the other <code>k - 1</code> sets, and then tested on the validation set. We’re doing this with a linear model to see how well it explains the data; it’s typically used in machine learning to see how well a model predicts data (we’ll get there in 2019).<a href="#fn1" class="footnoteRef" id="fnref1"><sup>1</sup></a></p>
<p>If you’re like me, it will take a bit of tinkering to really grasp k-fold cross validation, but <code>rsample</code> as a great function for dividing our data into k-folds. If we wish to use five folds (the state of the art seems to be either five or ten folds), we call the <code>vfold_cv()</code> function, pass it our data object <code>agg_ff_data</code>, and set <code>v = 5</code>.</p>
<pre class="r"><code>library(rsample)
library(yardstick)
set.seed(752)
cved_ff<-
vfold_cv(agg_ff_data, v = 5)
cved_ff</code></pre>
<pre><code># 5-fold cross-validation
# A tibble: 5 x 2
splits id
<list> <chr>
1 <split [1K/252]> Fold1
2 <split [1K/252]> Fold2
3 <split [1K/252]> Fold3
4 <split [1K/252]> Fold4
5 <split [1K/251]> Fold5</code></pre>
<p>We have an object called <code>cved_ff</code>, with a column called <code>splits</code> and a column called <code>id</code>. Let’s peek at the first split.</p>
<pre class="r"><code>cved_ff$splits[[1]]</code></pre>
<pre><code><1007/252/1259></code></pre>
<p>Three numbers. The first, 1007, is telling us how many observations are in the <code>analysis</code>. Since we have five folds, we should have 80% (or 4/5) of our data in the <code>analysis</code> set. The second number, 252, is telling us how many observations are in the <code>assessment</code>, which is 20% of our original data. The third number, 1259, is the total number of observations in our original data.</p>
<p>Next, we want to apply a model to the <code>analysis</code> set of this k-folded data and test the results on the <code>assessment</code> set. Let’s start with one factor and run a simple linear model, <code>lm(returns ~ MKT)</code>.</p>
<p>We want to run it on <code>analysis(cved_ff$splits[[1]])</code> - the analysis set of out first split.</p>
<pre class="r"><code>ff_model_test <- lm(returns ~ MKT, data = analysis(cved_ff$splits[[1]]))
ff_model_test</code></pre>
<pre><code>
Call:
lm(formula = returns ~ MKT, data = analysis(cved_ff$splits[[1]]))
Coefficients:
(Intercept) MKT
0.0001025 -0.0265516 </code></pre>
<p>Nothing too crazy so far. Now we want to test on our assessment data. The first step is to add that data to the original set. We’ll use <code>augment()</code> for that task, and pass it <code>assessment(cved_ff$splits[[1]])</code></p>
<pre class="r"><code>ff_model_test %>%
augment(newdata = assessment(cved_ff$splits[[1]])) %>%
head() %>%
select(returns, .fitted)</code></pre>
<pre><code> returns .fitted
1 0.0009021065 1.183819e-04
2 0.0011726989 4.934779e-05
3 0.0010815505 1.157267e-04
4 -0.0024385815 -7.544460e-05
5 -0.0021715702 -8.341007e-05
6 0.0028159467 3.865527e-04</code></pre>
<p>We just added our fitted values to the <code>assessment</code> data, the subset of the data on which the model was not fit. How well did our model do when compare the fitted values to the data in the held out set?</p>
<p>We can use the <code>rmse()</code> function from <code>yardstick</code> to measure our model. RMSE stands for root mean-squared error. It’s the sum of the squared differences between our fitted values and the actual values in the <code>assessment</code> data. A lower RMSE is better!</p>
<pre class="r"><code>ff_model_test %>%
augment(newdata = assessment(cved_ff$splits[[1]])) %>%
rmse(returns, .fitted)</code></pre>
<pre><code># A tibble: 1 x 3
.metric .estimator .estimate
<chr> <chr> <dbl>
1 rmse standard 0.00208</code></pre>
<p>Now that we’ve done that piece by piece, let’s wrap the whole operation into one function. This function takes one argument, a <code>split</code>, and we’re going to use <code>pull()</code> so we can extract the raw number, instead of the entire <code>tibble</code> result.</p>
<pre class="r"><code>model_one <- function(split) {
split_for_model <- analysis(split)
ff_model <- lm(returns ~ MKT, data = split_for_model)
holdout <- assessment(split)
rmse <- ff_model %>%
augment(newdata = holdout) %>%
rmse(returns, .fitted) %>%
pull(.estimate)
}</code></pre>
<p>Now we pass it our first split.</p>
<pre class="r"><code>model_one(cved_ff$splits[[1]]) %>%
head()</code></pre>
<pre><code>[1] 0.002080324</code></pre>
<p>Now we want to apply that function to each of our five folds that are stored in <code>agg_cved_ff</code>. We do that with a combination of <code>mutate()</code> and <code>map_dbl()</code>. We use <code>map_dbl()</code> instead of <code>map</code> because we are returning a number here and there’s not a good reason to store that number in a list column.</p>
<pre class="r"><code> cved_ff %>%
mutate(rmse = map_dbl(cved_ff$splits, model_one))</code></pre>
<pre><code># 5-fold cross-validation
# A tibble: 5 x 3
splits id rmse
* <list> <chr> <dbl>
1 <split [1K/252]> Fold1 0.00208
2 <split [1K/252]> Fold2 0.00189
3 <split [1K/252]> Fold3 0.00201
4 <split [1K/252]> Fold4 0.00224
5 <split [1K/251]> Fold5 0.00190</code></pre>
<p>OK, we have five RMSE’s since we ran the model on five separate <code>analysis</code> fold sets and tested on five separate <code>assessment</code> fold sets. Let’s find the average RMSE by taking the <code>mean()</code> of the <code>rmse</code> column. That can help reduce noisiness that resulted from our random creation of those five folds.</p>
<pre class="r"><code>cved_ff %>%
mutate(rmse = map_dbl(cved_ff$splits, model_one)) %>%
summarise(mean_rse = mean(rmse)) </code></pre>
<pre><code># 5-fold cross-validation
# A tibble: 1 x 1
mean_rse
<dbl>
1 0.00202</code></pre>
<p>We now have the mean RMSE after running on our model, <code>lm(returns ~ MKT)</code>, on all five of our folds.</p>
<p>That process for finding the mean RMSE can be applied other models, as well. Let’s suppose we wish to find the mean RMSE for two other models: <code>lm(returns ~ MKT + SMB + HML)</code>, the Fama French three-factor model, and <code>lm(returns ~ MKT + SMB + HML + RMW + CMA</code>, the Fama French five-factor model. By comparing the mean RMSE’s, we can evaluate which model explained the returns of AGG better. Since we’re just adding more and more factors, the models can be expected to get more and more accurate but again, we are exploring the <code>rsample</code> machinery and creating a template where we can pop in whatever models we wish to compare.</p>
<p>First, let’s create two new functions, that follow the exact same code pattern as above but house the three-factor and five-factor models.</p>
<pre class="r"><code>model_two <- function(split) {
split_for_model <- analysis(split)
ff_model <- lm(returns ~ MKT + SMB + HML, data = split_for_model)
holdout <- assessment(split)
rmse <-
ff_model %>%
augment(newdata = holdout) %>%
rmse(returns, .fitted) %>%
pull(.estimate)
}
model_three <- function(split) {
split_for_model <- analysis(split)
ff_model <- lm(returns ~ MKT + SMB + HML + RMW + CMA, data = split_for_model)
holdout <- assessment(split)
rmse <-
ff_model %>%
augment(newdata = holdout) %>%
rmse(returns, .fitted) %>%
pull(.estimate)
}</code></pre>
<p>Now we pass those three models to the same <code>mutate()</code> with <code>map_dbl()</code> flow that we used with just one model. The result will be three new columns of RMSE’s, one for each of our three models applied to our five folds.</p>
<pre class="r"><code>cved_ff %>%
mutate(
rmse_model_1 = map_dbl(
splits,
model_one),
rmse_model_2 = map_dbl(
splits,
model_two),
rmse_model_3 = map_dbl(
splits,
model_three))</code></pre>
<pre><code># 5-fold cross-validation
# A tibble: 5 x 5
splits id rmse_model_1 rmse_model_2 rmse_model_3
* <list> <chr> <dbl> <dbl> <dbl>
1 <split [1K/252]> Fold1 0.00208 0.00211 0.00201
2 <split [1K/252]> Fold2 0.00189 0.00184 0.00178
3 <split [1K/252]> Fold3 0.00201 0.00195 0.00194
4 <split [1K/252]> Fold4 0.00224 0.00221 0.00213
5 <split [1K/251]> Fold5 0.00190 0.00183 0.00177</code></pre>
<p>We can also find the mean RMSE for each model.</p>
<pre class="r"><code>cved_ff %>%
mutate(
rmse_model_1 = map_dbl(
splits,
model_one),
rmse_model_2 = map_dbl(
splits,
model_two),
rmse_model_3 = map_dbl(
splits,
model_three)) %>%
summarise(mean_rmse_model_1 = mean(rmse_model_1),
mean_rmse_model_2 = mean(rmse_model_2),
mean_rmse_model_3 = mean(rmse_model_3))</code></pre>
<pre><code># 5-fold cross-validation
# A tibble: 1 x 3
mean_rmse_model_1 mean_rmse_model_2 mean_rmse_model_3
<dbl> <dbl> <dbl>
1 0.00202 0.00199 0.00192</code></pre>
<p>That code flow worked just fine, but we had to repeat ourselves when creating the functions for each model. Let’s toggle to a flow where we define three models - the ones that we wish to test with via cross-validation and RMSE - then pass those models to one function.</p>
<p>First, we use <code>as.formula()</code> to define our three models.</p>
<pre class="r"><code>mod_form_1 <- as.formula(returns ~ MKT)
mod_form_2 <- as.formula(returns ~ MKT + SMB + HML)
mod_form_3 <- as.formula(returns ~ MKT + SMB + HML + RMW + CMA)</code></pre>
<p>Now we write one function that takes <code>split</code> as an argument, same as above, but also takes <code>formula</code> as an argument, so we can pass it different models. This gives us the flexibility to more easily define new models and pass them to <code>map</code>, so I’ll append <code>_flex</code> to the name of this function.</p>
<pre class="r"><code>ff_rmse_models_flex <- function(split, formula) {
split_for_data <- analysis(split)
ff_model <- lm(formula, data = split_for_data)
holdout <- assessment(split)
rmse <-
ff_model %>%
augment(newdata = holdout) %>%
rmse(returns, .fitted) %>%
pull(.estimate)
}</code></pre>
<p>Now we use the same code flow as before, except we call <code>map_dbl()</code>, pass it our <code>cved_ff$splits</code> object, our new <code>flex</code> function called <code>ff_rmse_models_flex()</code>, and the model we wish to pass as the <code>formula</code> argument. First we pass it <code>mod_form_1</code>.</p>
<pre class="r"><code>cved_ff %>%
mutate(rmse_model_1 = map_dbl(cved_ff$splits,
ff_rmse_models_flex,
formula = mod_form_1))</code></pre>
<pre><code># 5-fold cross-validation
# A tibble: 5 x 3
splits id rmse_model_1
* <list> <chr> <dbl>
1 <split [1K/252]> Fold1 0.00208
2 <split [1K/252]> Fold2 0.00189
3 <split [1K/252]> Fold3 0.00201
4 <split [1K/252]> Fold4 0.00224
5 <split [1K/251]> Fold5 0.00190</code></pre>
<p>Now let’s pass it all three models and find the mean RMSE.</p>
<pre class="r"><code>cved_ff %>%
mutate(rmse_model_1 = map_dbl(cved_ff$splits,
ff_rmse_models_flex,
formula = mod_form_1),
rmse_model_2 = map_dbl(cved_ff$splits,
ff_rmse_models_flex,
formula = mod_form_2),
rmse_model_3 = map_dbl(cved_ff$splits,
ff_rmse_models_flex,
formula = mod_form_3)) %>%
summarise(mean_rmse_model_1 = mean(rmse_model_1),
mean_rmse_model_2 = mean(rmse_model_2),
mean_rmse_model_3 = mean(rmse_model_3))</code></pre>
<pre><code># 5-fold cross-validation
# A tibble: 1 x 3
mean_rmse_model_1 mean_rmse_model_2 mean_rmse_model_3
<dbl> <dbl> <dbl>
1 0.00202 0.00199 0.00192</code></pre>
<p>Alright, that code flow seems a bit more flexible than our original method of writing a function to assess each model. We didn’t do much hard thinking about functional form here, but hopefully this provides a template where you could assess more nuanced models. We’ll get into bootstrapping and time series work next week, then head to Shiny to ring in the New Year!</p>
<p>And, finally, a couple of public service announcements.</p>
<p>First, thanks to everyone who has checked out my new book! The price just got lowered for the holidays. See on <a href="https://www.amazon.com/Reproducible-Finance-Portfolio-Analysis-Chapman/dp/1138484032">Amazon</a> or on the <a href="https://www.crcpress.com/Reproducible-Finance-with-R-Code-Flows-and-Shiny-Apps-for-Portfolio-Analysis/Jr/p/book/9781138484030">CRC homepage</a> (okay, that was more of an announcement about my book).</p>
<p>Second, applications are open for the <a href="https://www.battlefin.com/">Battlefin</a> alternative data contest, and RStudio is one of the tools you can use to analyze the data. Check it out <a href="https://www.battlefin.com/adc">here</a>. In January, they’ll announce 25 finalists who will get to compete for a cash prize and connect with some quant hedge funds. Go get ‘em!</p>
<p>Thanks for reading and see you next time.</p>
<div class="footnotes">
<hr />
<ol>
<li id="fn1"><p>For more on cross-validation, see “An Introduction to Statistical Learning”, chapter 5. Available online here: <a href="http://www-bcf.usc.edu/~gareth/ISL/" class="uri">http://www-bcf.usc.edu/~gareth/ISL/</a>.<a href="#fnref1">↩</a></p></li>
</ol>
</div>
<script>window.location.href='https://rviews.rstudio.com/2018/12/13/rsampling-fama-french/';</script>
Reproducible Finance, the book! And a discount for our readers
https://rviews.rstudio.com/2018/10/29/reproducible-finance-the-book/
Mon, 29 Oct 2018 00:00:00 +0000https://rviews.rstudio.com/2018/10/29/reproducible-finance-the-book/
<p>I’m thrilled to announce the release of my new book <a href="https://www.crcpress.com/Reproducible-Finance-with-R-Code-Flows-and-Shiny-Apps-for-Portfolio-Analysis/Regenstein-Jr/p/book/9781138484030">Reproducible Finance with R: Code Flows and Shiny Apps for Portfolio Analysis</a>, which originated as a series of R Views posts in this space. The <a href="https://rviews.rstudio.com/2016/11/09/reproducible-finance-with-r-the-sharpe-ratio/">first post</a> was written way back in November of 2016 - thanks to all the readers who have supported us along the way!</p>
<p>If you are familiar with the R Views posts, then you probably have a pretty good sense for the book’s style, prose, and code approach, but I’d like to add a few quick words of background.</p>
<p>The book’s practical motivations are: (1) to introduce R to finance professionals, or aspiring finance professionals, who wish to move beyond Excel for their quantitative work, and (2) to introduce various finance coding paradigms to R coders.</p>
<p>The softer motivation is to demonstrate and emphasize readable, reusable, and reproducible R code, data visualizations, and Shiny dashboards. It will be very helpful to have some background in the R programming language <em>or</em> in finance, but the most important thing is a desire to learn about the landscape of R code and finance packages.</p>
<p>An overarching goal of the book is to introduce the three major R paradigms for portfolio analysis: <code>xts</code>, the <code>tidyverse</code>, and <code>tidyquant</code>. As a result, we will frequently run the same analysis using three different code flows.</p>
<p>If that ‘three-universe’ structure seems a bit unclear, have a quick look back at <a href="https://rviews.rstudio.com/2017/12/13/introduction-to-skewness/">this post on skewness</a> and <a href="https://rviews.rstudio.com/2018/01/04/introduction-to-kurtosis/">this post on kurtosis</a>, and you’ll notice that we solve the same task and get the same result with different code paths.</p>
<p>For example, if we had portfolio returns saved in a tibble object called <code>portfolio_returns_tq_rebalanced_monthly</code>, and an <code>xts</code> object called <code>portfolio_returns_xts_rebalanced_monthly</code>, and our goal was to find the Sharpe Ratio of portfolio returns, we would start in the <code>xts</code> world with <code>SharpeRatio()</code> from the <code>PerformanceAnalytics</code> package.</p>
<pre class="r"><code># define risk free rate variable
rfr <- .0003
sharpe_xts <-
SharpeRatio(portfolio_returns_xts_rebalanced_monthly,
Rf = rfr,
FUN = "StdDev") %>%
`colnames<-`("sharpe_xts")
sharpe_xts</code></pre>
<pre><code>## sharpe_xts
## StdDev Sharpe (Rf=0%, p=95%): 0.2748752</code></pre>
<p>We next would use the tidyverse and run our calculations in a piped flow:</p>
<pre class="r"><code>sharpe_tidyverse_byhand <-
portfolio_returns_tq_rebalanced_monthly %>%
summarise(sharpe_dplyr = mean(returns - rfr)/
sd(returns - rfr))
sharpe_tidyverse_byhand</code></pre>
<pre><code>## # A tibble: 1 x 1
## sharpe_dplyr
## <dbl>
## 1 0.275</code></pre>
<p>And then head to the <code>tidyquant</code> paradigm where we can apply the <code>SharpeRatio()</code> function to a tidy tibble.</p>
<pre class="r"><code>sharpe_tq <-
portfolio_returns_tq_rebalanced_monthly %>%
tq_performance(Ra = returns,
performance_fun = SharpeRatio,
Rf = rfr,
FUN = "StdDev") %>%
`colnames<-`("sharpe_tq")</code></pre>
<p>We can compare our three Sharpe objects and confirm consistent results.</p>
<pre class="r"><code>sharpe_tq %>%
mutate(tidy_sharpe = sharpe_tidyverse_byhand$sharpe_dplyr,
xts_sharpe = sharpe_xts)</code></pre>
<pre><code>## # A tibble: 1 x 3
## sharpe_tq tidy_sharpe xts_sharpe
## <dbl> <dbl> <dbl>
## 1 0.275 0.275 0.275</code></pre>
<p>We might be curious how the Sharpe-Ratio-to-standard-deviation ratio of our portfolio compares to those of the component ETFs and a <code>ggplot</code> scatter is a nice way to visualize that.</p>
<pre class="r"><code>asset_returns_long %>%
na.omit() %>%
summarise(stand_dev = sd(returns),
sharpe = mean(returns - rfr)/
sd(returns - rfr)) %>%
add_row(asset = "Portfolio",
stand_dev =
portfolio_sd_xts_builtin[1],
sharpe =
sharpe_tq$sharpe_tq) %>%
ggplot(aes(x = stand_dev,
y = sharpe,
color = asset)) +
geom_point(size = 2) +
geom_text(
aes(x =
sd(portfolio_returns_tq_rebalanced_monthly$returns),
y =
sharpe_tq$sharpe_tq + .02,
label = "Portfolio")) +
ylab("Sharpe Ratio") +
xlab("standard deviation") +
ggtitle("Sharpe Ratio versus Standard Deviation") +
# The next line centers the title
theme_update(plot.title = element_text(hjust = 0.5))</code></pre>
<div class="figure"><span id="fig:unnamed-chunk-5"></span>
<img src="/post/2018-10-22-reproducible-finance-the-book_files/figure-html/unnamed-chunk-5-1.png" alt="Sharpe versus Standard Deviation" width="672" />
<p class="caption">
Figure 1: Sharpe versus Standard Deviation
</p>
</div>
<p>Finally, we are ready to calculate and visualize the Sharpe Ratio of a custom portfolio with Shiny and a flexdashboard, like the one found <a href="http://www.reproduciblefinance.com/shiny/sharpe-ratio/">here</a>.</p>
<p>As in the above example, most tasks in the book end with data visualization and then Shiny (a few early readers have commented with happy surprise that all the charts and code are in full color in the book - thanks to CRC press for making that happen!). Data visualization and Shiny are heavily emphasized - more so than in other finance books - and that might seem unusual. After all, every day we hear about how the financial world is becoming more quantitatively driven as firms race towards faster, more powerful algorithms. Why emphasize good ol’ data visualization? I believe, and have observed first-hand, that the ability to communicate and tell the story of data in a compelling way is only going to become more crucial as the financial world becomes more complex. Investors, limited partners, portfolio managers, clients, risk managers - they might not want to read our code or see our data, but we still need to communicate to them the value of our work. Data visualization and Shiny dashboards are a great way to do that. By book’s end, a reader will have built a collection of live, functioning flexdashboards that can be the foundation for more complex apps in the future.</p>
<p>If you’ve read this far, good news! Between now and December 31, 2018, there’s a 20% discount on the book being run at <a href="https://crcpress.com/Reproducible-Finance-with-R-Code-Flows-and-Shiny-Apps-for-Portfolio-Analysis/Jr/p/book/9781138484030">CRC</a>, and if you don’t see it applied, readers can use discount code SS120 on the <a href="https://crcpress.com/Reproducible-Finance-with-R-Code-Flows-and-Shiny-Apps-for-Portfolio-Analysis/Jr/p/book/9781138484030">CRC website</a>. The book is also available on <a href="https://www.amazon.com/Reproducible-Finance-Portfolio-Analysis-Chapman/dp/1138484032">Amazon as Kindle or paperback</a> (but there’s only than 10 copies left as of right now).</p>
<p>Thanks so much for reading, and happy coding!</p>
<script>window.location.href='https://rviews.rstudio.com/2018/10/29/reproducible-finance-the-book/';</script>
Monte Carlo Part Two
https://rviews.rstudio.com/2018/06/13/monte-carlo-part-two/
Wed, 13 Jun 2018 00:00:00 +0000https://rviews.rstudio.com/2018/06/13/monte-carlo-part-two/
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<p>In a <a href="https://rviews.rstudio.com/2018/06/05/monte-carlo/">previous post</a>, we reviewed how to set up and run a Monte Carlo (MC) simulation of future portfolio returns and growth of a dollar. Today, we will run that simulation many, many, times and then visualize the results.</p>
<p>Our ultimate goal is to build a Shiny app that allows an end user to build a custom portfolio, simulate returns, and visualize the results. If you just can’t wait, a link to the final Shiny app is available <a href="http://www.reproduciblefinance.com/shiny/monte-carlo-simulation/">here</a>.</p>
<p>This post builds off the work we did previously. I won’t go through the logic again, but the code for building a portfolio, calculating returns, mean and standard deviation of returns, and using them for a simulation is here:</p>
<pre class="r"><code>library(tidyquant)
library(tidyverse)
library(timetk)
library(broom)
library(highcharter)
symbols <- c("SPY","EFA", "IJS", "EEM","AGG")
prices <-
getSymbols(symbols, src = 'yahoo',
from = "2012-12-31",
to = "2017-12-31",
auto.assign = TRUE, warnings = FALSE) %>%
map(~Ad(get(.))) %>%
reduce(merge) %>%
`colnames<-`(symbols)
w <- c(0.25, 0.25, 0.20, 0.20, 0.10)
asset_returns_long <-
prices %>%
to.monthly(indexAt = "lastof", OHLC = FALSE) %>%
tk_tbl(preserve_index = TRUE, rename_index = "date") %>%
gather(asset, returns, -date) %>%
group_by(asset) %>%
mutate(returns = (log(returns) - log(lag(returns)))) %>%
na.omit()
portfolio_returns_tq_rebalanced_monthly <-
asset_returns_long %>%
tq_portfolio(assets_col = asset,
returns_col = returns,
weights = w,
col_rename = "returns",
rebalance_on = "months")
mean_port_return <-
mean(portfolio_returns_tq_rebalanced_monthly$returns)
stddev_port_return <-
sd(portfolio_returns_tq_rebalanced_monthly$returns)
simulation_accum_1 <- function(init_value, N, mean, stdev) {
tibble(c(init_value, 1 + rnorm(N, mean, stdev))) %>%
`colnames<-`("returns") %>%
mutate(growth =
accumulate(returns,
function(x, y) x * y)) %>%
select(growth)
}</code></pre>
<p>That code allows us to run one simulation of the growth of a dollar over the next 10 years, with the <code>simulation_accum_1()</code> that we built for that purpose. Today, we will review how to run 51 simulations, though we could choose any number (and our Shiny application allows an end user to do just that).</p>
<p>First, we need an empty matrix with 51 columns, an initial value of $1, and intuitive column names.</p>
<p>We will use the <code>rep()</code> function to create 51 columns with a 1 as the value, and <code>set_names()</code> to name each column with the appropriate simulation number.</p>
<pre class="r"><code>sims <- 51
starts <-
rep(1, sims) %>%
set_names(paste("sim", 1:sims, sep = ""))</code></pre>
<p>Take a peek at <code>starts</code> to see what we just created, and how it can house our simulations.</p>
<pre class="r"><code>head(starts)</code></pre>
<pre><code>sim1 sim2 sim3 sim4 sim5 sim6
1 1 1 1 1 1 </code></pre>
<pre class="r"><code>tail(starts)</code></pre>
<pre><code>sim46 sim47 sim48 sim49 sim50 sim51
1 1 1 1 1 1 </code></pre>
<p>51 columns, with a value of 1 in one row. This is where we will store the results of the 51 simulations.</p>
<p>Now, we want to apply <code>simulation_accum_1</code> to each of the 51 columns of the <code>starts</code> matrix, and we will do that using the <code>map_dfc()</code> function from the <code>purrr</code> package.</p>
<p><code>map_dfc()</code> takes a vector - in this case, the columns of <code>starts</code> - and applies a function to it. By appending <code>dfc()</code> to the <code>map_</code> function, we are asking the function to store each of its results as the column of a data frame (<code>map_df()</code> does the same thing, but stores results in the rows of a data frame). After running the code below, we will have a data frame with 51 columns, one for each of our simulations.</p>
<p>We also need to choose how many months to simulate (the N argument to our simulation function) and supply the distribution parameters as we did before. We do not supply the <code>init_value</code> argument because the <code>init_value</code> is 1, that same 1 that is already in the 51 columns.</p>
<pre class="r"><code>monte_carlo_sim_51 <-
map_dfc(starts,
simulation_accum_1,
N = 120,
mean = mean_port_return,
stdev = stddev_port_return)
tail(monte_carlo_sim_51 %>% select(growth1, growth2,
growth49, growth50), 3)</code></pre>
<pre><code># A tibble: 3 x 4
growth1 growth2 growth49 growth50
<dbl> <dbl> <dbl> <dbl>
1 1.81 1.38 2.32 1.80
2 1.84 1.37 2.38 1.84
3 1.82 1.33 2.39 1.82</code></pre>
<p>Have a look at the results. We now have 51 simulations of the growth of a dollar, and we simulated that growth over 120 months; however, the results are missing a piece that we need for visualization, namely a <code>month</code> column.</p>
<p>Let’s add that <code>month</code> column with <code>mutate()</code> and give it the same number of rows as our data frame. These are months out into the future. We will use <code>mutate(month = seq(1:nrow(.)))</code>, and then clean up the column names. <code>nrow()</code> is equal to the number of rows in our object. If we were to change to 130 simulations, that would generate 130 rows, and <code>nrow()</code> would be equal to 130.</p>
<pre class="r"><code>monte_carlo_sim_51 <-
monte_carlo_sim_51 %>%
mutate(month = seq(1:nrow(.))) %>%
select(month, everything()) %>%
`colnames<-`(c("month", names(starts))) %>%
mutate_all(funs(round(., 2)))
tail(monte_carlo_sim_51 %>% select(month, sim1, sim2,
sim49, sim50), 3)</code></pre>
<pre><code># A tibble: 3 x 5
month sim1 sim2 sim49 sim50
<dbl> <dbl> <dbl> <dbl> <dbl>
1 119 2.16 1.81 1.46 2.32
2 120 2.28 1.84 1.46 2.38
3 121 2.26 1.82 1.46 2.39</code></pre>
<p>We have accomplished our goal of running 51 simulations, and could head to data visualization now, but let’s explore an alternative method using the the <code>rerun()</code> function from <code>purrr</code>. As its name implies, this function will “rerun” another function, and we stipulate how many times to do that by setting <code>.n = number of times to rerun</code>. For example to run the <code>simulation_accum_1</code> function 5 times, we would set the following:</p>
<pre class="r"><code>monte_carlo_rerun_5 <-
rerun(.n = 5,
simulation_accum_1(1,
120,
mean_port_return,
stddev_port_return))</code></pre>
<p>That returned a list of 5 data frames, or 5 simulations. We can look at the first few rows of each data frame by using <code>map(..., head)</code>.</p>
<pre class="r"><code>map(monte_carlo_rerun_5, head)</code></pre>
<pre><code>[[1]]
# A tibble: 6 x 1
growth
<dbl>
1 1
2 0.983
3 0.965
4 0.946
5 0.967
6 0.962
[[2]]
# A tibble: 6 x 1
growth
<dbl>
1 1
2 0.980
3 0.975
4 0.964
5 0.969
6 0.914
[[3]]
# A tibble: 6 x 1
growth
<dbl>
1 1
2 1.03
3 0.997
4 0.979
5 1.04
6 1.04
[[4]]
# A tibble: 6 x 1
growth
<dbl>
1 1
2 0.974
3 0.962
4 0.943
5 0.942
6 0.963
[[5]]
# A tibble: 6 x 1
growth
<dbl>
1 1
2 0.990
3 1.02
4 1.06
5 1.12
6 1.13 </code></pre>
<p>Let’s consolidate that list of data frames to one <code>tibble</code>. We start by collapsing to vectors with <code>simplify_all()</code>, then add nicer names with the <code>names()</code> function and finally coerce to tibble with <code>as_tibble()</code>. Let’s run it 51 times to match our previous results.</p>
<pre class="r"><code>reruns <- 51
monte_carlo_rerun_51 <-
rerun(.n = reruns,
simulation_accum_1(1,
120,
mean_port_return,
stddev_port_return)) %>%
simplify_all() %>%
`names<-`(paste("sim", 1:reruns, sep = " ")) %>%
as_tibble() %>%
mutate(month = seq(1:nrow(.))) %>%
select(month, everything())
tail(monte_carlo_rerun_51 %>% select(`sim 1`, `sim 2`,
`sim 49`, `sim 50`), 3)</code></pre>
<pre><code># A tibble: 3 x 4
`sim 1` `sim 2` `sim 49` `sim 50`
<dbl> <dbl> <dbl> <dbl>
1 1.99 1.97 3.66 2.20
2 2.00 1.86 3.68 2.25
3 2.02 1.95 3.78 2.18</code></pre>
<p>Now we have two objects holding the results of 51 simulations, <code>monte_carlo_rerun_51</code> and <code>monte_carlo_sim_51</code>.</p>
<p>Each has 51 columns of simulations and 1 column of months. Note that we have 121 rows because we started with an initial value of 1, and then simulated returns over 120 months.</p>
<p>Now let’s get to visualization with <code>ggplot()</code> and visualize the results in <code>monte_carlo_sim_51</code>. The same code flows for visualization would also apply to <code>monte_carlo_rerun_51</code>, but we will run them for only <code>monte_carlo_sim_51</code> here.</p>
<p>We start with a chart of all 51 simulations, and assign a different color to each one by setting <code>ggplot(aes(x = month, y = growth, color = sim))</code>. <code>ggplot()</code> will automatically generate a legend for all 51 time series, but that gets quite crowded. We will suppress the legend with <code>theme(legend.position = "none")</code>.</p>
<pre class="r"><code>monte_carlo_sim_51 %>%
gather(sim, growth, -month) %>%
group_by(sim) %>%
ggplot(aes(x = month, y = growth, color = sim)) +
geom_line() +
theme(legend.position="none")</code></pre>
<p><img src="/post/2018-06-12-monte-carlo-part-2_files/figure-html/unnamed-chunk-9-1.png" width="672" /></p>
<p>We can check the minimum, maximum and median simulation with the <code>summarise()</code> function here.</p>
<pre class="r"><code>sim_summary <-
monte_carlo_sim_51 %>%
gather(sim, growth, -month) %>%
group_by(sim) %>%
summarise(final = last(growth)) %>%
summarise(
max = max(final),
min = min(final),
median = median(final))
sim_summary</code></pre>
<pre><code># A tibble: 1 x 3
max min median
<dbl> <dbl> <dbl>
1 4.81 1.31 2.3</code></pre>
<p>We can clean up our original visualization by including only the max, min and median that were just calculated.</p>
<pre class="r"><code>monte_carlo_sim_51 %>%
gather(sim, growth, -month) %>%
group_by(sim) %>%
filter(
last(growth) == sim_summary$max ||
last(growth) == sim_summary$median ||
last(growth) == sim_summary$min) %>%
ggplot(aes(x = month, y = growth)) +
geom_line(aes(color = sim)) </code></pre>
<p><img src="/post/2018-06-12-monte-carlo-part-2_files/figure-html/unnamed-chunk-11-1.png" width="672" /></p>
<p>Now let’s port our results over to <code>highcharter</code>, but in a major departure from our usual code flow, we will pass a tidy <code>tibble</code> instead of an <code>xts</code> object.</p>
<p>Our first step is to convert the data from wide to long tidy format with the <code>gather()</code> function.</p>
<pre class="r"><code>mc_gathered <-
monte_carlo_sim_51 %>%
gather(sim, growth, -month) %>%
group_by(sim)
head(mc_gathered)</code></pre>
<pre><code># A tibble: 6 x 3
# Groups: sim [1]
month sim growth
<dbl> <chr> <dbl>
1 1 sim1 1
2 2 sim1 0.99
3 3 sim1 1.01
4 4 sim1 1.06
5 5 sim1 1.08
6 6 sim1 1.1 </code></pre>
<p>We can now pass this <code>tibble</code> directly to the <code>hchart()</code> function, specify the type of chart as <code>line</code>, and then work with a similar grammar to <code>ggplot()</code>. The difference is we use <code>hcaes</code>, which stands for <code>highcharter aesthetic</code>, instead of <code>aes</code>.</p>
<pre class="r"><code># This takes a few seconds to run
hchart(mc_gathered,
type = 'line',
hcaes(y = growth,
x = month,
group = sim)) %>%
hc_title(text = "51 Simulations") %>%
hc_xAxis(title = list(text = "months")) %>%
hc_yAxis(title = list(text = "dollar growth"),
labels = list(format = "${value}")) %>%
hc_add_theme(hc_theme_flat()) %>%
hc_exporting(enabled = TRUE) %>%
hc_legend(enabled = FALSE)</code></pre>
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<p>We just plotted 51 lines in <code>highcharter</code> using a tidy <code>tibble</code>. For tidy data fans out there, this is a big deal because we can stay in the grammar of the tidyverse but also use <code>highcharter</code>.</p>
<p>Very similar to what we did with <code>ggplot</code>, let’s isolate the maximum, minimum and median simulations and save them to an object called <code>mc_max_med_min</code>.</p>
<pre class="r"><code>mc_max_med_min <-
mc_gathered %>%
filter(
last(growth) == sim_summary$max ||
last(growth) == sim_summary$median ||
last(growth) == sim_summary$min) %>%
group_by(sim)</code></pre>
<p>Now we pass that filtered object to <code>hchart()</code>.</p>
<pre class="r"><code>hchart(mc_max_med_min,
type = 'line',
hcaes(y = growth,
x = month,
group = sim)) %>%
hc_title(text = "Min, Max, Median Simulations") %>%
hc_xAxis(title = list(text = "months")) %>%
hc_yAxis(title = list(text = "dollar growth"),
labels = list(format = "${value}")) %>%
hc_add_theme(hc_theme_flat()) %>%
hc_exporting(enabled = TRUE) %>%
hc_legend(enabled = FALSE)</code></pre>
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<p>That concludes our visualization of Monte Carlo simulations.</p>
<script>window.location.href='https://rviews.rstudio.com/2018/06/13/monte-carlo-part-two/';</script>
Monte Carlo
https://rviews.rstudio.com/2018/06/05/monte-carlo/
Tue, 05 Jun 2018 00:00:00 +0000https://rviews.rstudio.com/2018/06/05/monte-carlo/
<p>Today, we change gears from our previous work on <a href="https://rviews.rstudio.com/2018/05/10/rolling-fama-french/">Fama French</a> and run a Monte Carlo (MC) simulation of future portfolio returns. Monte Carlo relies on repeated, random sampling. We will sample based on two parameters: mean and standard deviation of portfolio returns. Our long-term goal (long-term == over the next two or three blog posts) is to build a Shiny app that allows an end user to build a custom portfolio, simulate returns and visualize the results. If you just can’t wait, a link to that final Shiny app is <a href="http://www.reproduciblefinance.com/shiny/monte-carlo-simulation/">here</a>.</p>
<div class="figure">
<img src="/post/2018-06-07-Monte-Carlo_files/MC.png" />
</div>
<p>Let’s get to it.</p>
<p>Devoted readers won’t be surprised that we will be simulating the returns of our usual portfolio, which consists of:</p>
<pre><code>+ SPY (S&P 500 fund) weighted 25%
+ EFA (a non-US equities fund) weighted 25%
+ IJS (a small-cap value fund) weighted 20%
+ EEM (an emerging-mkts fund) weighted 20%
+ AGG (a bond fund) weighted 10%</code></pre>
<p>Before we can simulate that portfolio, we need to calculate portfolio monthly returns, which was covered in my previous post, <a href="https://rviews.rstudio.com/2017/10/11/from-asset-to-portfolio-returns/">Introduction to Portfolio Returns</a>.</p>
<p>I won’t go through the logic again, but the code is here:</p>
<pre class="r"><code>library(tidyquant)
library(tidyverse)
library(timetk)
library(broom)
symbols <- c("SPY","EFA", "IJS", "EEM","AGG")
prices <-
getSymbols(symbols, src = 'yahoo',
from = "2012-12-31",
to = "2017-12-31",
auto.assign = TRUE, warnings = FALSE) %>%
map(~Ad(get(.))) %>%
reduce(merge) %>%
`colnames<-`(symbols)
w <- c(0.25, 0.25, 0.20, 0.20, 0.10)
asset_returns_long <-
prices %>%
to.monthly(indexAt = "lastof", OHLC = FALSE) %>%
tk_tbl(preserve_index = TRUE, rename_index = "date") %>%
gather(asset, returns, -date) %>%
group_by(asset) %>%
mutate(returns = (log(returns) - log(lag(returns)))) %>%
na.omit()
portfolio_returns_tq_rebalanced_monthly <-
asset_returns_long %>%
tq_portfolio(assets_col = asset,
returns_col = returns,
weights = w,
col_rename = "returns",
rebalance_on = "months")</code></pre>
<p>We will be working with the data object <code>portfolio_returns_tq_rebalanced_monthly</code> and we first find the mean and <a href="http://www.reproduciblefinance.com/code/standard-deviation/">standard deviation</a> of returns.</p>
<p>We will name those variables <code>mean_port_return</code> and <code>stddev_port_return</code>.</p>
<pre class="r"><code>mean_port_return <-
mean(portfolio_returns_tq_rebalanced_monthly$returns)
stddev_port_return <-
sd(portfolio_returns_tq_rebalanced_monthly$returns)</code></pre>
<p>Then we use the <code>rnorm()</code> function to sample from a distribution with mean equal to <code>mean_port_return</code> and standard deviation equal to <code>stddev_port_return</code>. That is the crucial random sampling that underpins this exercise.</p>
<p>We also must decide how many draws to pull from this distribution, meaning how many monthly returns we will simulate. 120 months is 10 years, and that feels like a good amount of time.</p>
<pre class="r"><code>simulated_monthly_returns <- rnorm(120,
mean_port_return,
stddev_port_return)</code></pre>
<p>Have a quick look at the simulated monthly returns.</p>
<pre class="r"><code>head(simulated_monthly_returns)</code></pre>
<pre><code>[1] 0.05216143 0.02271485 -0.04271307 0.04811250 -0.05575058 0.06006096</code></pre>
<pre class="r"><code>tail(simulated_monthly_returns)</code></pre>
<pre><code>[1] 0.024794729 0.008539198 -0.018852629 -0.002656127 -0.025583337
[6] 0.004412755</code></pre>
<p>Next, we calculate how a dollar would have grown given those random monthly returns. We first add a 1 to each of our monthly returns, because we start with $1.</p>
<pre class="r"><code>simulated_returns_add_1 <-
tibble(c(1, 1 + simulated_monthly_returns)) %>%
`colnames<-`("returns")
head(simulated_returns_add_1)</code></pre>
<pre><code># A tibble: 6 x 1
returns
<dbl>
1 1
2 1.05
3 1.02
4 0.957
5 1.05
6 0.944</code></pre>
<p>That data is now ready to be converted into the cumulative growth of a dollar. We can use either <code>accumulate()</code> from <code>purrr</code> or <code>cumprod()</code>. Let’s use both of them with <code>mutate()</code> and confirm consistent, reasonable results.</p>
<pre class="r"><code>simulated_growth <-
simulated_returns_add_1 %>%
mutate(growth1 = accumulate(returns, function(x, y) x * y),
growth2 = accumulate(returns, `*`),
growth3 = cumprod(returns)) %>%
select(-returns)
tail(simulated_growth)</code></pre>
<pre><code># A tibble: 6 x 3
growth1 growth2 growth3
<dbl> <dbl> <dbl>
1 2.09 2.09 2.09
2 2.11 2.11 2.11
3 2.07 2.07 2.07
4 2.06 2.06 2.06
5 2.01 2.01 2.01
6 2.02 2.02 2.02</code></pre>
<p>We just ran three simulations of dollar growth over 120 months. We passed in the same monthly returns, and that’s why we got three equivalent results.</p>
<p>Are they reasonable? What compound annual growth rate (CAGR) is implied by this simulation?</p>
<pre class="r"><code>cagr <-
((simulated_growth$growth1[nrow(simulated_growth)]^
(1/10)) - 1) * 100
cagr <- round(cagr, 2)</code></pre>
<p>This simulation implies an annual compounded growth of 7.26%. That seems reasonable given our actual returns have all been taken from a raging bull market. Remember, the above code is a simulation based on sampling from a normal distribution. If you re-run this code on your own, you will get a different result.</p>
<p>If we feel good about this first simulation, we can run several more to get a sense for how they are distributed. Before we do that, let’s create several different functions that could run the same simulation.</p>
<div id="several-simulation-functions" class="section level2">
<h2>Several Simulation Functions</h2>
<p>Let’s build three simulation functions that incorporate the <code>accumulate()</code> and <code>cumprod()</code> workflows above. We have confirmed they give consistent results so it’s a matter of stylistic preference as to which one is chosen in the end. Perhaps you feel that one is more flexible or extensible, or fits better with your team’s code flows.</p>
<p>Each of the below functions needs four arguments: N for the number of months to simulate (we chose 120 above), <code>init_value</code> for the starting value (we used $1 above), and the mean-standard deviation pair to create draws from a normal distribution. We <em>choose</em> N and <code>init_value</code>, and derive the mean-standard deviation pair from our portfolio monthly returns.</p>
<p>Here is our first growth simulation function using <code>accumulate()</code>.</p>
<pre class="r"><code>simulation_accum_1 <- function(init_value, N, mean, stdev) {
tibble(c(init_value, 1 + rnorm(N, mean, stdev))) %>%
`colnames<-`("returns") %>%
mutate(growth =
accumulate(returns,
function(x, y) x * y)) %>%
select(growth)
}</code></pre>
<p>Almost identical, here is the second simulation function using <code>accumulate()</code>.</p>
<pre class="r"><code>simulation_accum_2 <- function(init_value, N, mean, stdev) {
tibble(c(init_value, 1 + rnorm(N, mean, stdev))) %>%
`colnames<-`("returns") %>%
mutate(growth = accumulate(returns, `*`)) %>%
select(growth)
}</code></pre>
<p>Finally, here is a simulation function using <code>cumprod()</code>.</p>
<pre class="r"><code>simulation_cumprod <- function(init_value, N, mean, stdev) {
tibble(c(init_value, 1 + rnorm(N, mean, stdev))) %>%
`colnames<-`("returns") %>%
mutate(growth = cumprod(returns)) %>%
select(growth)
}</code></pre>
<p>Here is a function that uses all three methods, in case we want a fast way to re-confirm consistency.</p>
<pre class="r"><code>simulation_confirm_all <- function(init_value, N, mean, stdev) {
tibble(c(init_value, 1 + rnorm(N, mean, stdev))) %>%
`colnames<-`("returns") %>%
mutate(growth1 = accumulate(returns, function(x, y) x * y),
growth2 = accumulate(returns, `*`),
growth3 = cumprod(returns)) %>%
select(-returns)
}</code></pre>
<p>Let’s test that <code>confirm_all()</code> function with an <code>init_value</code> of 1, N of 120, and our parameters.</p>
<pre class="r"><code>simulation_confirm_all_test <-
simulation_confirm_all(1, 120,
mean_port_return, stddev_port_return)
tail(simulation_confirm_all_test)</code></pre>
<pre><code># A tibble: 6 x 3
growth1 growth2 growth3
<dbl> <dbl> <dbl>
1 2.26 2.26 2.26
2 2.22 2.22 2.22
3 2.17 2.17 2.17
4 2.21 2.21 2.21
5 2.20 2.20 2.20
6 2.23 2.23 2.23</code></pre>
<p>That’s all for today. Next time, we will explore methods for running more than one simulation with the above functions and then visualizing the results. See you then.</p>
</div>
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Rolling Fama French
https://rviews.rstudio.com/2018/05/10/rolling-fama-french/
Thu, 10 May 2018 00:00:00 +0000https://rviews.rstudio.com/2018/05/10/rolling-fama-french/
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<p>In a <a href="https://rviews.rstudio.com/2018/04/11/introduction-to-fama-french/">previous post</a>, we reviewed how to import the Fama French 3-Factor data, wrangle that data, and then regress our portfolio returns on the factors. Please have a look at that previous post, as the following work builds upon it. For more background on Fama French, see the original article published in <em>The Journal of Financial Economics</em>, <a href="https://www.sciencedirect.com/science/article/pii/0304405X93900235">Common risk factors in the returns on stocks and bonds</a>.</p>
<p>Today, we will explore the rolling Fama French model and the explanatory power of the 3 factors in different time periods. In the financial world, we often look at rolling means, standard deviations and models to make sure we haven’t missed anything unusual, risky, or concerning during different market or economic regimes. Our portfolio returns history is for the years 2013 through 2017, which is rather a short history, but there still might a be a 24-month period where the Fama French factors were particularly strong, weak, or meaningless. We would like to unearth and hypothesize about what explains them or their future likelihood.</p>
<p>We will be working with our usual portfolio consisting of:</p>
<pre><code>+ SPY (S&P500 fund) weighted 25%
+ EFA (a non-US equities fund) weighted 25%
+ IJS (a small-cap value fund) weighted 20%
+ EEM (an emerging-mkts fund) weighted 20%
+ AGG (a bond fund) weighted 10%</code></pre>
<p>Before we can run a Fama French model for that portfolio, we need to find portfolio monthly returns, which was covered in <a href="https://rviews.rstudio.com/2017/10/11/from-asset-to-portfolio-returns/">this post</a>. I won’t go through the logic again but the code is here:</p>
<pre class="r"><code>library(tidyquant)
library(tidyverse)
library(timetk)
symbols <- c("SPY","EFA", "IJS", "EEM","AGG")
prices <-
getSymbols(symbols, src = 'yahoo',
from = "2012-12-31",
to = "2017-12-31",
auto.assign = TRUE, warnings = FALSE) %>%
map(~Ad(get(.))) %>%
reduce(merge) %>%
`colnames<-`(symbols)
w <- c(0.25, 0.25, 0.20, 0.20, 0.10)
asset_returns_long <-
prices %>%
to.monthly(indexAt = "lastof", OHLC = FALSE) %>%
tk_tbl(preserve_index = TRUE, rename_index = "date") %>%
gather(asset, returns, -date) %>%
group_by(asset) %>%
mutate(returns = (log(returns) - log(lag(returns)))) %>%
na.omit()
portfolio_returns_tq_rebalanced_monthly <-
asset_returns_long %>%
tq_portfolio(assets_col = asset,
returns_col = returns,
weights = w,
col_rename = "returns",
rebalance_on = "months")</code></pre>
<p>We also need to import the Fama French factors and combine them into one object with our portfolio returns. We painstakingly covered this in the <a href="https://rviews.rstudio.com/2018/04/11/introduction-to-fama-french/">previous post</a> and the code for doing so is here:</p>
<pre class="r"><code>temp <- tempfile()
base <-
"http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/ftp/"
factor <-
"Global_3_Factors"
format<-
"_CSV.zip"
full_url <-
glue(base,
factor,
format,
sep ="")
download.file(
full_url,
temp,
quiet = TRUE)
Global_3_Factors <-
read_csv(unz(temp, "Global_3_Factors.csv"),
skip = 6) %>%
rename(date = X1) %>%
mutate_at(vars(-date), as.numeric) %>%
mutate(date =
rollback(ymd(parse_date_time(date, "%Y%m") + months(1)))) %>%
filter(date >=
first(portfolio_returns_tq_rebalanced_monthly$date) & date <=
last(portfolio_returns_tq_rebalanced_monthly$date))
ff_portfolio_returns <-
portfolio_returns_tq_rebalanced_monthly %>%
left_join(Global_3_Factors, by = "date") %>%
mutate(MKT_RF = Global_3_Factors$`Mkt-RF`/100,
SMB = Global_3_Factors$SMB/100,
HML = Global_3_Factors$HML/100,
RF = Global_3_Factors$RF/100,
R_excess = round(returns - RF, 4))</code></pre>
<p>We now have one data frame <code>ff_portfolio_returns</code> that holds our Fama French factors and portfolio returns. Let’s get to the rolling analysis.</p>
<p>We first define a rolling model with the <code>rollify()</code> function from <code>tibbletime</code>. However, instead of wrapping an existing function, such as <code>kurtosis()</code> or <code>skewness()</code>, we will pass in our linear Fama French model.</p>
<pre class="r"><code># Choose a 24-month rolling window
window <- 24
library(tibbletime)
# define a rolling ff model with tibbletime
rolling_lm <-
rollify(.f = function(R_excess, MKT_RF, SMB, HML) {
lm(R_excess ~ MKT_RF + SMB + HML)
}, window = window, unlist = FALSE)</code></pre>
<p>Next, we pass columns from <code>ff_portfolio_returns</code> to the rolling function model.</p>
<pre class="r"><code>rolling_ff_betas <-
ff_portfolio_returns %>%
mutate(rolling_ff =
rolling_lm(R_excess,
MKT_RF,
SMB,
HML)) %>%
slice(-1:-23) %>%
select(date, rolling_ff)
head(rolling_ff_betas, 3)</code></pre>
<pre><code># A tibble: 3 x 2
date rolling_ff
<date> <list>
1 2014-12-31 <S3: lm>
2 2015-01-31 <S3: lm>
3 2015-02-28 <S3: lm> </code></pre>
<p>We now have a new data frame called <code>rolling_ff_betas</code>, in which the column <code>rolling_ff</code> holds an S3 object of our model results. We can <code>tidy()</code> that column with <code>map(rolling_ff, tidy)</code> and then <code>unnest()</code> the results, very similar to <a href="https://rviews.rstudio.com/2018/02/08/capm-beta/">our CAPM</a> work, except we have more than one independent variable.</p>
<pre class="r"><code>rolling_ff_betas <-
ff_portfolio_returns %>%
mutate(rolling_ff =
rolling_lm(R_excess,
MKT_RF,
SMB,
HML)) %>%
mutate(tidied = map(rolling_ff,
tidy,
conf.int = T)) %>%
unnest(tidied) %>%
slice(-1:-23) %>%
select(date, term, estimate, conf.low, conf.high) %>%
filter(term != "(Intercept)") %>%
rename(beta = estimate, factor = term) %>%
group_by(factor)
head(rolling_ff_betas, 3)</code></pre>
<pre><code># A tibble: 3 x 5
# Groups: factor [3]
date factor beta conf.low conf.high
<date> <chr> <dbl> <dbl> <dbl>
1 2014-12-31 MKT_RF 0.931 0.784 1.08
2 2014-12-31 SMB -0.0130 -0.278 0.252
3 2014-12-31 HML -0.160 -0.459 0.139</code></pre>
<p>We now have rolling betas and confidence intervals for each of our 3 factors. Let’s apply the same code logic and extract the rolling R-squared for our model. The only difference is we call <code>glance()</code> instead of <code>tidy()</code>.</p>
<pre class="r"><code>rolling_ff_rsquared <-
ff_portfolio_returns %>%
mutate(rolling_ff =
rolling_lm(R_excess,
MKT_RF,
SMB,
HML)) %>%
slice(-1:-23) %>%
mutate(glanced = map(rolling_ff,
glance)) %>%
unnest(glanced) %>%
select(date, r.squared, adj.r.squared, p.value)
head(rolling_ff_rsquared, 3)</code></pre>
<pre><code># A tibble: 3 x 4
date r.squared adj.r.squared p.value
<date> <dbl> <dbl> <dbl>
1 2014-12-31 0.898 0.883 4.22e-10
2 2015-01-31 0.914 0.901 8.22e-11
3 2015-02-28 0.919 0.907 4.19e-11</code></pre>
<p>We have extracted rolling factor betas and the rolling model R-squared, now let’s visualize.</p>
<div id="visualizing-rolling-fama-french" class="section level2">
<h2>Visualizing Rolling Fama French</h2>
<p>We start by charting the rolling factor betas with <code>ggplot()</code>. This gives us an view into how the explanatory power of each factor has changed over time.</p>
<pre class="r"><code>rolling_ff_betas %>%
ggplot(aes(x = date,
y = beta,
color = factor)) +
geom_line() +
labs(title= "24-Month Rolling FF Factor Betas",
x = "rolling betas") +
scale_x_date(breaks = scales::pretty_breaks(n = 10)) +
theme_minimal() +
theme(plot.title = element_text(hjust = 0.5),
axis.text.x = element_text(angle = 90)) </code></pre>
<p><img src="/post/2018-05-09-rolling-fama-french_files/figure-html/unnamed-chunk-7-1.png" width="672" /></p>
<p>The rolling factor beta chart reveals some interesting trends. Both SMB and HML have hovered around zero, while the MKT factor has hovered around 1. That’s consistent with our plot of betas with confidence intervals from last time.</p>
<p>Next, let’s visualize the rolling R-squared with <code>highcharter</code>.</p>
<p>We first convert <code>rolling_ff_rsquared</code> to <code>xts</code>, using the <code>tk_xts()</code> function.</p>
<pre class="r"><code>rolling_ff_rsquared_xts <-
rolling_ff_rsquared %>%
tk_xts(date_var = date, silent = TRUE)</code></pre>
<p>Then pass the <code>xts</code> object to a <code>highchart(type = "stock")</code> code flow, adding the rolling R-squared time series with <code>hc_add_series(rolling_ff_rsquared_xts$r.squared...)</code>.</p>
<pre class="r"><code>highchart(type = "stock") %>%
hc_add_series(rolling_ff_rsquared_xts$r.squared,
color = "cornflowerblue",
name = "r-squared") %>%
hc_title(text = "Rolling FF 3-Factor R-Squared") %>%
hc_add_theme(hc_theme_flat()) %>%
hc_navigator(enabled = FALSE) %>%
hc_scrollbar(enabled = FALSE) %>%
hc_exporting(enabled = TRUE)</code></pre>
<div id="htmlwidget-1" style="width:100%;height:500px;" class="highchart html-widget"></div>
<script type="application/json" data-for="htmlwidget-1">{"x":{"hc_opts":{"title":{"text":"Rolling FF 3-Factor R-Squared"},"yAxis":{"title":{"text":null}},"credits":{"enabled":false},"exporting":{"enabled":true},"plotOptions":{"series":{"turboThreshold":0},"treemap":{"layoutAlgorithm":"squarified"},"bubble":{"minSize":5,"maxSize":25}},"annotationsOptions":{"enabledButtons":false},"tooltip":{"delayForDisplay":10},"series":[{"data":[[1419984000000,0.898193874076082],[1422662400000,0.913613669331354],[1425081600000,0.919257460405196],[1427760000000,0.919053841151536],[1430352000000,0.920192764383095],[1433030400000,0.920671410263961],[1435622400000,0.919801526746092],[1438300800000,0.908279914990913],[1440979200000,0.92707755328544],[1443571200000,0.916184001290169],[1446249600000,0.926114696965536],[1448841600000,0.925254318145437],[1451520000000,0.923190960293236],[1454198400000,0.929828565005735],[1456704000000,0.922988355712527],[1459382400000,0.938348757232282],[1461974400000,0.937402062865212],[1464652800000,0.936268502221349],[1467244800000,0.912164668775577],[1469923200000,0.916567108467131],[1472601600000,0.918047866998973],[1475193600000,0.917334780359066],[1477872000000,0.938055499197554],[1480464000000,0.939201416001601],[1483142400000,0.938828531422516],[1485820800000,0.939091105917217],[1488240000000,0.934435224922171],[1490918400000,0.934421254157578],[1493510400000,0.934758653807486],[1496188800000,0.936641187924107],[1498780800000,0.934909707340084],[1501459200000,0.943913578310692],[1504137600000,0.932829172819464],[1506729600000,0.926037292248958],[1509408000000,0.911850534203785],[1512000000000,0.910661077924351],[1514678400000,0.916587636494134]],"color":"cornflowerblue","name":"r-squared"}],"navigator":{"enabled":false},"scrollbar":{"enabled":false}},"theme":{"colors":["#f1c40f","#2ecc71","#9b59b6","#e74c3c","#34495e","#3498db","#1abc9c","#f39c12","#d35400"],"chart":{"backgroundColor":"#ECF0F1"},"xAxis":{"gridLineDashStyle":"Dash","gridLineWidth":1,"gridLineColor":"#BDC3C7","lineColor":"#BDC3C7","minorGridLineColor":"#BDC3C7","tickColor":"#BDC3C7","tickWidth":1},"yAxis":{"gridLineDashStyle":"Dash","gridLineColor":"#BDC3C7","lineColor":"#BDC3C7","minorGridLineColor":"#BDC3C7","tickColor":"#BDC3C7","tickWidth":1},"legendBackgroundColor":"rgba(0, 0, 0, 0.5)","background2":"#505053","dataLabelsColor":"#B0B0B3","textColor":"#34495e","contrastTextColor":"#F0F0F3","maskColor":"rgba(255,255,255,0.3)"},"conf_opts":{"global":{"Date":null,"VMLRadialGradientURL":"http =//code.highcharts.com/list(version)/gfx/vml-radial-gradient.png","canvasToolsURL":"http =//code.highcharts.com/list(version)/modules/canvas-tools.js","getTimezoneOffset":null,"timezoneOffset":0,"useUTC":true},"lang":{"contextButtonTitle":"Chart context menu","decimalPoint":".","downloadJPEG":"Download JPEG image","downloadPDF":"Download PDF document","downloadPNG":"Download PNG image","downloadSVG":"Download SVG vector image","drillUpText":"Back to {series.name}","invalidDate":null,"loading":"Loading...","months":["January","February","March","April","May","June","July","August","September","October","November","December"],"noData":"No data to display","numericSymbols":["k","M","G","T","P","E"],"printChart":"Print chart","resetZoom":"Reset zoom","resetZoomTitle":"Reset zoom level 1:1","shortMonths":["Jan","Feb","Mar","Apr","May","Jun","Jul","Aug","Sep","Oct","Nov","Dec"],"thousandsSep":" ","weekdays":["Sunday","Monday","Tuesday","Wednesday","Thursday","Friday","Saturday"]}},"type":"stock","fonts":[],"debug":false},"evals":[],"jsHooks":[]}</script>
<p>That chart looks choppy, but the R-squared never really left the range between .9 and .95. We can tweak the minimum and maximum y-axis values for some perspective.</p>
<pre class="r"><code>highchart(type = "stock") %>%
hc_add_series(rolling_ff_rsquared_xts$r.squared,
color = "cornflowerblue",
name = "r-squared") %>%
hc_title(text = "Rolling FF 3-Factor R-Squared") %>%
hc_yAxis( max = 2, min = 0) %>%
hc_add_theme(hc_theme_flat()) %>%
hc_navigator(enabled = FALSE) %>%
hc_scrollbar(enabled = FALSE) %>%
hc_exporting(enabled = TRUE)</code></pre>
<div id="htmlwidget-2" style="width:100%;height:500px;" class="highchart html-widget"></div>
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<p>Ah, when the y-axis is zoomed out a bit, our R-squared looks consistently near 1 for the life of the portfolio.</p>
<p>That’s all for today. Thanks and see you next time!</p>
</div>
<script>window.location.href='https://rviews.rstudio.com/2018/05/10/rolling-fama-french/';</script>
Introduction to Fama French
https://rviews.rstudio.com/2018/04/11/introduction-to-fama-french/
Wed, 11 Apr 2018 00:00:00 +0000https://rviews.rstudio.com/2018/04/11/introduction-to-fama-french/
<p>In two previous posts, we <a href="https://rviews.rstudio.com/2018/02/08/capm-beta/">calculated</a> and then <a href="https://rviews.rstudio.com/2018/03/02/capm-and-visualization/">visualized</a> the CAPM beta of a portfolio by fitting a simple linear model.</p>
<p>Today, we move beyond CAPM’s simple linear regression and explore the Fama French (FF) multi-factor model of equity risk/return. For more background, have a look at the original article published in <em>The Journal Financial Economics</em>, <a href="https://www.sciencedirect.com/science/article/pii/0304405X93900235">Common risk factors in the returns on stocks and bonds</a>.</p>
<p>The FF model extends CAPM by regressing portfolio returns on several variables, in addition to market returns. From a general data science point of view, FF extends CAPM’s simple linear regression, where we had one independent variable, to a multiple linear regression, where we have numerous independent variables.</p>
<p>We are going to look at the FF 3-factor model, which tests the explanatory power of (1) market returns (same as CAPM), (2) firm size (small versus big) and (3) firm value (book to market ratio). The <code>firm value</code> factor is labeled as <code>HML</code> in FF, which stands for high-minus-low and refers to a firm’s book-to-market ratio. When we regress portfolio returns on the <code>HML</code> factor, we are investigating how much of the returns are the result of including stocks with a high book-to-market ratio (sometimes called the <code>value premium</code>, because high book-to-market stocks are called value stocks).</p>
<p>A large portion of this post covers importing data from the FF website and wrangling it for use with our portfolio returns. We will see that wrangling the data is conceptually easy to understand but practically time-consuming to implement. However, mashing together data from disparate sources is a necessary skill for anyone in industry that has data streams from different vendors and wants to get creative about how to use them. Once the data is wrangled, fitting the model is not time-consuming.</p>
<p>Today, we will be working with our usual portfolio consisting of:</p>
<pre><code>+ SPY (S&P500 fund) weighted 25%
+ EFA (a non-US equities fund) weighted 25%
+ IJS (a small-cap value fund) weighted 20%
+ EEM (an emerging-mkts fund) weighted 20%
+ AGG (a bond fund) weighted 10%</code></pre>
<p>Before we can calculate beta for that portfolio, we need to find portfolio monthly returns, which was covered in my previous post, <a href="https://rviews.rstudio.com/2017/10/11/from-asset-to-portfolio-returns/">Introduction to Portfolio Returns</a>. I won’t go through the logic again, but the code is here:</p>
<pre class="r"><code>library(tidyquant)
library(tidyverse)
library(timetk)
library(broom)
library(glue)
symbols <- c("SPY","EFA", "IJS", "EEM","AGG")
prices <-
getSymbols(symbols, src = 'yahoo',
from = "2012-12-31",
to = "2017-12-31",
auto.assign = TRUE, warnings = FALSE) %>%
map(~Ad(get(.))) %>%
reduce(merge) %>%
`colnames<-`(symbols)
w <- c(0.25, 0.25, 0.20, 0.20, 0.10)
asset_returns_long <-
prices %>%
to.monthly(indexAt = "lastof", OHLC = FALSE) %>%
tk_tbl(preserve_index = TRUE, rename_index = "date") %>%
gather(asset, returns, -date) %>%
group_by(asset) %>%
mutate(returns = (log(returns) - log(lag(returns)))) %>%
na.omit()
portfolio_returns_tq_rebalanced_monthly <-
asset_returns_long %>%
tq_portfolio(assets_col = asset,
returns_col = returns,
weights = w,
col_rename = "returns",
rebalance_on = "months")</code></pre>
<p>We will be working with one object of portfolio returns:</p>
<pre><code>+ portfolio_returns_tq_rebalanced_monthly</code></pre>
<p>Let’s get to it.</p>
<div id="importing-and-wrangling-the-fama-french-factors" class="section level3 unnumbered">
<h3>Importing and Wrangling the Fama French Factors</h3>
<p>Our first task is to get the FF data and, fortunately, FF make their factor data available on the internet. We will document each step for importing and cleaning this data, to an extent that might be overkill. Frustrating for us now, but a time-saver later when we need to update this model or extend to the 5-factor case.</p>
<p>Have a look at the <a href="http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html">FF website</a>. The data are packaged as zip files, so we will need to do a bit more than call <code>read_csv()</code>. Let’s use the <code>tempfile()</code> function from base R to create a variable called <code>temp</code>. This is where we will put the zipped file.</p>
<pre class="r"><code>temp <- tempfile()</code></pre>
<p>R has created a temporary file called <code>temp</code> that will be cleaned up when we exit this session. Download 3-factor zip with <a href="%22http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/ftp/Global_3_Factors_CSV.zip%22">this link</a>. We want to pass that to <code>download.file()</code> and store the result in <code>temp</code>.</p>
<p>First, though, we will break that string into three pieces: <code>base</code>, <code>factor</code> and <code>format</code> - this is not necessary for today’s task, but it will come in handy if we want to build a Shiny application to let a user choose a factor from the FF website, or if we just want to re-run this analysis with a different set of FF factors. We will then <code>glue()</code> those together and save the string as <code>full_url</code>.</p>
<pre class="r"><code>base <-
"http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/ftp/"
factor <-
"Global_3_Factors"
format<-
"_CSV.zip"
full_url <-
glue(base,
factor,
format,
sep ="")</code></pre>
<p>Now we pass <code>full_url</code> to <code>download.file()</code>.</p>
<pre class="r"><code>download.file(
full_url,
temp,
quiet = TRUE)</code></pre>
<p>Finally, we can read the csv file using <code>read_csv()</code> after unzipping that data with the <code>unz()</code> function.</p>
<pre class="r"><code>Global_3_Factors <-
read_csv(unz(temp,
"Global_3_Factors.csv"))
head(Global_3_Factors) </code></pre>
<pre><code>## # A tibble: 6 x 1
## `This file was created using the 201802 Bloomberg database.`
## <chr>
## 1 Missing data are indicated by -99.99.
## 2 <NA>
## 3 199007
## 4 199008
## 5 199009
## 6 199010</code></pre>
<p>We have imported the dataset, but we do not see any factors, just a column with weirdly formatted dates</p>
<p>When this occurs, it <em>often</em> can be fixed by skipping a certain number of rows that contain metadata. Have a look at what happens if we skip 6 rows.</p>
<pre class="r"><code>Global_3_Factors <-
read_csv(unz(temp,
"Global_3_Factors.csv"),
skip = 6)
head(Global_3_Factors)</code></pre>
<pre><code>## # A tibble: 6 x 5
## X1 `Mkt-RF` SMB HML RF
## <chr> <chr> <chr> <chr> <chr>
## 1 199007 0.86 0.77 -0.25 0.68
## 2 199008 -10.82 -1.60 0.60 0.66
## 3 199009 -11.98 1.23 0.81 0.60
## 4 199010 9.57 -7.39 -4.25 0.68
## 5 199011 -3.86 1.22 1.14 0.57
## 6 199012 1.10 -0.79 -1.60 0.60</code></pre>
<p>This is what were were expecting, 5 columns: one called <code>X1</code> that holds the weirdly formatted dates, then <code>Mkt-Rf</code> for the market returns above the risk-free rate, <code>SMB</code> for the size factor, <code>HML</code> for the value factor, and <code>RF</code> for the risk-free rate.</p>
<p>However, the data have been coerced to a character format - look at the class of each column:</p>
<pre class="r"><code>map(Global_3_Factors, class)</code></pre>
<pre><code>## $X1
## [1] "character"
##
## $`Mkt-RF`
## [1] "character"
##
## $SMB
## [1] "character"
##
## $HML
## [1] "character"
##
## $RF
## [1] "character"</code></pre>
<p>We have two options for coercing those columns to the right format. First, we can do so upon import, by supplying the argument <code>col_types = cols(col_name = col_double(),...</code> for each numeric column.</p>
<pre class="r"><code>Global_3_Factors <-
read_csv(unz(temp,
"Global_3_Factors.csv"),
skip = 6,
col_types = cols(
`Mkt-RF` = col_double(),
SMB = col_double(),
HML = col_double(),
RF = col_double()))
head(Global_3_Factors)</code></pre>
<pre><code>## # A tibble: 6 x 5
## X1 `Mkt-RF` SMB HML RF
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 199007 0.860 0.770 -0.250 0.680
## 2 199008 -10.8 -1.60 0.600 0.660
## 3 199009 -12.0 1.23 0.810 0.600
## 4 199010 9.57 -7.39 -4.25 0.680
## 5 199011 -3.86 1.22 1.14 0.570
## 6 199012 1.10 -0.790 -1.60 0.600</code></pre>
<p>That works well, but it’s specific to the FF 3-factor set with those specific column names. If we imported a different FF factor set, we would need to specify different column names.</p>
<p>As an alternate approach, the code chunk below converts the columns to numeric after import, but is more general. It can be applied to other FF factor collections.</p>
<p>To do this, we rename the <code>X1</code> column to <code>date</code>, and then use the <code>dplyr</code> verb <code>mutate_at(vars(-date), as.numeric)</code> to change our column formats to numeric. The <code>vars()</code> function operates like the <code>select()</code> function in that we can tell it to operate on all columns except the <code>date</code> column by putting a negative sign in front of <code>date</code>. This column coercion flow is more flexible in that it would work for different FF factor sets.</p>
<pre class="r"><code>Global_3_Factors <-
read_csv(unz(temp,
"Global_3_Factors.csv"),
skip = 6) %>%
rename(date = X1) %>%
mutate_at(vars(-date), as.numeric)
head(Global_3_Factors)</code></pre>
<pre><code>## # A tibble: 6 x 5
## date `Mkt-RF` SMB HML RF
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 199007 0.860 0.770 -0.250 0.680
## 2 199008 -10.8 -1.60 0.600 0.660
## 3 199009 -12.0 1.23 0.810 0.600
## 4 199010 9.57 -7.39 -4.25 0.680
## 5 199011 -3.86 1.22 1.14 0.570
## 6 199012 1.10 -0.790 -1.60 0.600</code></pre>
<p>We now have numeric data for our factors and the date column has a better label, but the wrong format.</p>
<p>We can use the <code>lubridate</code> package to parse that date string into a nicer date format. We will use the <code>parse_date_time()</code> function, and call the <code>ymd()</code> function to make sure the end result is in a date format. Again, when working with data from a new source, the date and, indeed, any column can come in many formats.</p>
<pre class="r"><code>Global_3_Factors <-
read_csv(unz(temp, "Global_3_Factors.csv"),
skip = 6) %>%
rename(date = X1) %>%
mutate_at(vars(-date), as.numeric) %>%
mutate(date =
ymd(parse_date_time(date, "%Y%m")))
head(Global_3_Factors)</code></pre>
<pre><code>## # A tibble: 6 x 5
## date `Mkt-RF` SMB HML RF
## <date> <dbl> <dbl> <dbl> <dbl>
## 1 1990-07-01 0.860 0.770 -0.250 0.680
## 2 1990-08-01 -10.8 -1.60 0.600 0.660
## 3 1990-09-01 -12.0 1.23 0.810 0.600
## 4 1990-10-01 9.57 -7.39 -4.25 0.680
## 5 1990-11-01 -3.86 1.22 1.14 0.570
## 6 1990-12-01 1.10 -0.790 -1.60 0.600</code></pre>
<p>The date format looks good, and that matters because we want to trim the factor data that the FF dates match our portfolio dates. However, notice that FF uses the first of the month and our portfolio returns use the last of the month. Again, it’s <code>lubridate</code> to the rescue with the <code>rollback()</code> function. This will roll monthly dates back to the last day of the previous month. The first date in our FF data is “1990-07-01”. Let’s roll it back.</p>
<pre class="r"><code>Global_3_Factors %>%
select(date) %>%
mutate(date = lubridate::rollback(date)) %>%
head(1)</code></pre>
<pre><code>## # A tibble: 1 x 1
## date
## <date>
## 1 1990-06-30</code></pre>
<p>If we want to reset our dates to the last of the month, we need to add one first, then rollback.</p>
<pre class="r"><code>Global_3_Factors %>%
select(date) %>%
mutate(date = lubridate::rollback(date + months(1))) %>%
head(1)</code></pre>
<pre><code>## # A tibble: 1 x 1
## date
## <date>
## 1 1990-07-31</code></pre>
<p>There are other ways we could have gotten around this issue - most notably, way back in the beginning, we could have indexed our portfolio returns to <code>indexAt = firstof</code> - but it was a good chance to introduce the <code>rollback()</code> function, and we will not always have that option. Sometimes two data sets are thrown at us and we have to wrangle them from there.</p>
<p>Finally, we want only the FF factor data that aligns with our portfolio data, so we <code>filter()</code> by the <code>first()</code> and <code>last()</code> date in our portfolio returns object.</p>
<pre class="r"><code>Global_3_Factors <-
read_csv(unz(temp, "Global_3_Factors.csv"),
skip = 6) %>%
rename(date = X1) %>%
mutate_at(vars(-date), as.numeric) %>%
mutate(date =
rollback(ymd(parse_date_time(date, "%Y%m") + months(1)))) %>%
filter(date >=
first(portfolio_returns_tq_rebalanced_monthly$date) & date <=
last(portfolio_returns_tq_rebalanced_monthly$date))
head(Global_3_Factors, 3)</code></pre>
<pre><code>## # A tibble: 3 x 5
## date `Mkt-RF` SMB HML RF
## <date> <dbl> <dbl> <dbl> <dbl>
## 1 2013-01-31 5.46 0.140 2.01 0.
## 2 2013-02-28 0.100 0.330 -0.780 0.
## 3 2013-03-31 2.29 0.830 -2.03 0.</code></pre>
<pre class="r"><code>tail(Global_3_Factors, 3)</code></pre>
<pre><code>## # A tibble: 3 x 5
## date `Mkt-RF` SMB HML RF
## <date> <dbl> <dbl> <dbl> <dbl>
## 1 2017-10-31 1.80 -0.850 -0.950 0.0900
## 2 2017-11-30 1.93 -0.680 -0.260 0.0800
## 3 2017-12-31 1.38 0.940 0.140 0.0900</code></pre>
<p>All that work enables us to merge these data objects together with <code>left_join(...by = "date")</code>. We also convert the FF data to decimal and create a new column called <code>R_excess</code> to hold our returns above the risk-free rate.</p>
<pre class="r"><code>ff_portfolio_returns <-
portfolio_returns_tq_rebalanced_monthly %>%
left_join(Global_3_Factors, by = "date") %>%
mutate(MKT_RF = Global_3_Factors$`Mkt-RF`/100,
SMB = Global_3_Factors$SMB/100,
HML = Global_3_Factors$HML/100,
RF = Global_3_Factors$RF/100,
R_excess = round(returns - RF, 4))
head(ff_portfolio_returns, 4)</code></pre>
<pre><code>## # A tibble: 4 x 8
## date returns `Mkt-RF` SMB HML RF MKT_RF R_excess
## <date> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 2013-01-31 0.0308 5.46 0.00140 0.0201 0. 0.0546 0.0308
## 2 2013-02-28 -0.000870 0.100 0.00330 -0.00780 0. 0.00100 -0.000900
## 3 2013-03-31 0.0187 2.29 0.00830 -0.0203 0. 0.0229 0.0187
## 4 2013-04-30 0.0206 3.02 -0.0121 0.00960 0. 0.0302 0.0206</code></pre>
<p>We now we have one object with our portfolio returns and FF factors, and can proceed to the simplest part of our exercise from a coding perspective, and the only part that our bosses/colleagues/clients/investors will care about: the modeling and visualization.</p>
<p>Luckily, we can copy/paste the flow from our <a href="https://rviews.rstudio.com/2018/02/08/capm-beta/">CAPM work</a>, now that we have the data in a nice format. CAPM used simple linear regression, whereas FF uses multiple regression with many independent variables. Accordingly, our 3-factor FF equation is <code>lm(R_excess ~ MKT_RF + SMB + HML</code>.</p>
<p>We will make one addition to the CAPM code flow, which is to include the 95% confidence interval for our coefficients. We do that by setting <code>tidy(model, conf.int = T, conf.level = .95)</code>.</p>
<pre class="r"><code>ff_dplyr_byhand <-
ff_portfolio_returns %>%
do(model =
lm(R_excess ~ MKT_RF + SMB + HML,
data = .)) %>%
tidy(model, conf.int = T, conf.level = .95)
ff_dplyr_byhand %>%
mutate_if(is.numeric, funs(round(., 3))) %>%
select(-statistic)</code></pre>
<pre><code>## term estimate std.error p.value conf.low conf.high
## 1 (Intercept) -0.001 0.001 0.195 -0.004 0.001
## 2 MKT_RF 0.894 0.036 0.000 0.823 0.965
## 3 SMB 0.056 0.076 0.458 -0.095 0.208
## 4 HML 0.030 0.061 0.629 -0.092 0.151</code></pre>
<p>Our model object now contains a <code>conf.high</code> and <code>conf.low</code> column to hold our confidence interval min and max values.</p>
<p>We can pipe these results to <code>ggplot()</code> and create a scatter of coefficients with confidence intervals. I don’t want to plot the intercept so will filter it out of the code flow.</p>
<p>We add the confidence intervals with <code>geom_errorbar(aes(ymin = conf.low, ymax = conf.high))</code>.</p>
<pre class="r"><code>ff_dplyr_byhand %>%
mutate_if(is.numeric, funs(round(., 3))) %>%
filter(term != "(Intercept)") %>%
ggplot(aes(x = term, y = estimate, shape = term, color = term)) +
geom_point() +
geom_errorbar(aes(ymin = conf.low, ymax = conf.high)) +
labs(title = "FF 3-Factor Coefficients for Our Portfolio",
subtitle = "nothing in this post is investment advice",
x = "",
y = "coefficient",
caption = "data source: Fama French website and yahoo! Finance") +
theme_minimal() +
theme(plot.title = element_text(hjust = 0.5),
plot.subtitle = element_text(hjust = 0.5),
plot.caption = element_text(hjust = 0))</code></pre>
<p><img src="/post/2018-04-10-introduction-to-fama-french_files/figure-html/unnamed-chunk-16-1.png" width="672" /></p>
<p>The results here are predictable because, as with CAPM, we are regressing a portfolio that contains the market on 3 factors, one of which is the market. Thus, the market factor dominates this model and the other two factors contain zero in their confidence bands.</p>
<p>Next time we will complicate this work by calculating FF factor coefficients for multiple funds/portfolios, examining rolling R squared’s and visualizing the results. See you then!</p>
</div>
<script>window.location.href='https://rviews.rstudio.com/2018/04/11/introduction-to-fama-french/';</script>
Visualizing the Capital Asset Pricing Model
https://rviews.rstudio.com/2018/03/02/capm-and-visualization/
Fri, 02 Mar 2018 00:00:00 +0000https://rviews.rstudio.com/2018/03/02/capm-and-visualization/
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<p>In a <a href="https://rviews.rstudio.com/2018/02/08/capm-beta/">previous post</a>, we covered how to calculate CAPM beta for our usual portfolio consisting of:</p>
<pre><code>+ SPY (S&P500 fund) weighted 25%
+ EFA (a non-US equities fund) weighted 25%
+ IJS (a small-cap value fund) weighted 20%
+ EEM (an emerging-mkts fund) weighted 20%
+ AGG (a bond fund) weighted 10%</code></pre>
<p>Today, we will move on to visualizing the CAPM beta and explore some <code>ggplot</code> and <code>highcharter</code> functionality, along with the <code>broom</code> package.</p>
<p>Before we can do any of this CAPM work, we need to calculate the portfolio returns, covered in <a href="https://rviews.rstudio.com/2017/10/11/from-asset-to-portfolio-returns/">this post</a>, and then calculate the CAPM beta for the portfolio and the individual assets covered in <a href="https://rviews.rstudio.com/2018/02/08/capm-beta/">this post</a>.</p>
<p>I will not present that code or logic again but we will utilize four data objects from that previous work:</p>
<pre><code>+ portfolio_returns_tq_rebalanced_monthly (a tibble of portfolio monthly returns)
+ market_returns_tidy (a tibble of SP500 monthly returns)
+ beta_dplyr_byhand (a tibble of market betas for our 5 individual assets)
+ asset_returns_long (a tibble of returns for our 5 individual assets)</code></pre>
<p>Let’s get to it.</p>
<div id="visualizing-the-relationship-between-portfolio-returns-risk-and-market-returns" class="section level3 unnumbered">
<h3>Visualizing the Relationship between Portfolio Returns, Risk and Market Returns</h3>
<p>The CAPM beta number is telling us about the linear relationship between our portfolio returns and the market returns. It’s also telling us about the riskiness of our portfolio - how volatile the portfolio is relative to the market. Before we get to beta itself, let’s take a look at expected monthly returns of our assets scattered against monthly risk of our individual assets.</p>
<pre class="r"><code>library(tidyquant)
library(tidyverse)
library(timetk)
library(tibbletime)
library(scales)
# This theme_update will center your ggplot titles
theme_update(plot.title = element_text(hjust = 0.5))
asset_returns_long %>%
group_by(asset) %>%
summarise(expected_return = mean(returns),
stand_dev = sd(returns)) %>%
ggplot(aes(x = stand_dev, y = expected_return, color = asset)) +
geom_point(size = 2) +
ylab("expected return") +
xlab("standard deviation") +
ggtitle("Expected Monthly Returns v. Risk") +
scale_y_continuous(label = function(x){ paste0(x, "%")}) </code></pre>
<p><img src="/post/2018-02-28-capm-and-visualization_files/figure-html/unnamed-chunk-1-1.png" width="672" /></p>
<p>Where does our portfolio fit on this scatter plot? Let’s add it to the <code>ggplot()</code> flow with <code>geom_point(aes(x = sd(portfolio_returns_tq_rebalanced_monthly$returns), y = mean(portfolio_returns_tq_rebalanced_monthly$returns)), color = "cornflowerblue", size = 3)</code>.</p>
<pre class="r"><code>asset_returns_long %>%
group_by(asset) %>%
summarise(expected_return = mean(returns),
stand_dev = sd(returns)) %>%
ggplot(aes(x = stand_dev, y = expected_return, color = asset)) +
geom_point(size = 2) +
geom_point(aes(x = sd(portfolio_returns_tq_rebalanced_monthly$returns),
y = mean(portfolio_returns_tq_rebalanced_monthly$returns)),
color = "cornflowerblue",
size = 3) +
geom_text(
aes(x = sd(portfolio_returns_tq_rebalanced_monthly$returns) * 1.09,
y = mean(portfolio_returns_tq_rebalanced_monthly$returns),
label = "portfolio")) +
ylab("expected return") +
xlab("standard deviation") +
ggtitle("Expected Monthly Returns v. Risk") +
scale_y_continuous(labels = function(x){ paste0(x, "%")}) </code></pre>
<p><img src="/post/2018-02-28-capm-and-visualization_files/figure-html/unnamed-chunk-2-1.png" width="672" /></p>
<p>Our portfolio return/risk looks all right, though the SP500 has a higher expected return for just a bit more risk. It’s been tough to beat the market the last five years. EEM and EFA have a higher risk and lower expected return (no rational investor wants that!) and IJS has a higher risk and a higher expected return (some rational investors do want that!).</p>
<p>In general, the scatter is providing some return-risk context for our portfolio. It’s not directly part of CAPM, but I like to start here to get in the return-risk mindset.</p>
<p>Next, let’s turn to CAPM more directly and visualize the relationship between our portfolio and the market with a scatter plot of market returns on the x-axis and portfolio returns on the y-axis. First, we will add the market returns to our portfolio tibble by calling <code>mutate(market_returns = market_returns_tidy$returns)</code>. Then, we set our x- and y-axis with <code>ggplot(aes(x = market_returns, y = returns))</code>.</p>
<pre class="r"><code>portfolio_returns_tq_rebalanced_monthly %>%
mutate(market_returns = market_returns_tidy$returns) %>%
ggplot(aes(x = market_returns, y = returns)) +
geom_point(color = "cornflowerblue") +
ylab("portfolio returns") +
xlab("market returns") +
ggtitle("Scatterplot of portfolio returns v. market returns")</code></pre>
<p><img src="/post/2018-02-28-capm-and-visualization_files/figure-html/unnamed-chunk-3-1.png" width="672" /></p>
<p>This scatter plot is communicating the same strong linear relationship as our numeric beta calculation from the previous post. We can add a simple regression line to it with <code>geom_smooth(method = "lm", se = FALSE, color = "green", size = .5)</code>.</p>
<pre class="r"><code>portfolio_returns_tq_rebalanced_monthly %>%
mutate(market_returns = market_returns_tidy$returns) %>%
ggplot(aes(x = market_returns, y = returns)) +
geom_point(color = "cornflowerblue") +
geom_smooth(method = "lm", se = FALSE, color = "green", size = .5) +
ylab("portfolio returns") +
xlab("market returns") +
ggtitle("Scatterplot with regression line")</code></pre>
<p><img src="/post/2018-02-28-capm-and-visualization_files/figure-html/unnamed-chunk-4-1.png" width="672" /></p>
<p>The green line is produced by the call to <code>geom_smooth(method = 'lm')</code>. Under the hood, <code>ggplot</code> fits a linear model of the relationship between market returns and portfolio returns. The slope of that green line is the CAPM beta that we calculated earlier. To confirm that we can add a line to the scatter that has a slope equal to our beta calculation and a y-intercept equal to what I labeled as alpha in the <code>beta_dplyr_byhand</code> object.</p>
<p>To add the line, we invoke <code>geom_abline(aes(intercept = beta_dplyr_byhand$estimate[1], slope = beta_dplyr_byhand$estimate[2]), color = "purple")</code>.</p>
<pre class="r"><code>portfolio_returns_tq_rebalanced_monthly %>%
mutate(market_returns = market_returns_tidy$returns) %>%
ggplot(aes(x = market_returns, y = returns)) +
geom_point(color = "cornflowerblue") +
geom_abline(aes(
intercept = beta_dplyr_byhand$estimate[1],
slope = beta_dplyr_byhand$estimate[2]),
color = "purple",
size = .5) +
ylab("portfolio returns") +
xlab("market returns") +
ggtitle("Scatterplot with hand calculated slope")</code></pre>
<p><img src="/post/2018-02-28-capm-and-visualization_files/figure-html/unnamed-chunk-5-1.png" width="672" /></p>
<p>We can plot both lines simultaneously to confirm to ourselves that they are the same - they should be right on top of each other but the purple line, our manual <code>abline</code>, extends into infinity so, we should see it start where the green line ends.</p>
<pre class="r"><code>portfolio_returns_tq_rebalanced_monthly %>%
mutate(market_returns = market_returns_tidy$returns) %>%
ggplot(aes(x = market_returns, y = returns)) +
geom_point(color = "cornflowerblue") +
geom_abline(aes(
intercept = beta_dplyr_byhand$estimate[1],
slope = beta_dplyr_byhand$estimate[2]),
color = "purple",
size = .5) +
geom_smooth(method = "lm", se = FALSE, color = "green", size = .5) +
ylab("portfolio returns") +
xlab("market returns") +
ggtitle("Compare CAPM beta line to regression line")</code></pre>
<p><img src="/post/2018-02-28-capm-and-visualization_files/figure-html/unnamed-chunk-6-1.png" width="672" /></p>
<p>All right, that seems to visually confirm (or strongly support) that the fitted line calculated by <code>ggplot</code> and <code>geom_smooth()</code> has a slope equal to the beta we calculated ourselves. Why did we go through this exercise? Well, CAPM beta is a bit “jargony”. Since we need to map that jargon over to the world of linear modeling, it’s a useful practice to consider how jargon reduces to data science concepts. This isn’t a particularly complicated bit of jargon, but it’s good practice to get in the habit of reducing jargon.</p>
</div>
<div id="a-bit-more-on-linear-regression-augmenting-our-data" class="section level3 unnumbered">
<h3>A Bit More on Linear Regression: Augmenting Our Data</h3>
<p>Before concluding our analysis of CAPM beta, let’s explore the <code>augment()</code> function from <code>broom</code> and how it helps to create a few more interesting visualizations.</p>
<p>The code chunk below starts with model results from <code>lm(returns ~ market_returns_tidy$returns...)</code>, which is regressing our portfolio returns on the market returns. We store the results in a list-column called called <code>model</code>. Next, we call <code>augment(model)</code> which will add predicted values to the original data set and return a tibble.</p>
<p>Those predicted values will be in the <code>.fitted</code> column. For some reason, the <code>date</code> column gets dropped. It’s nice to have this for visualizations so we will add it back in with <code>mutate(date = portfolio_returns_tq_rebalanced_monthly$date)</code>.</p>
<pre class="r"><code>library(broom)
portfolio_model_augmented <-
portfolio_returns_tq_rebalanced_monthly %>%
do(model = lm(returns ~ market_returns_tidy$returns, data = .))%>%
augment(model) %>%
mutate(date = portfolio_returns_tq_rebalanced_monthly$date)
head(portfolio_model_augmented)</code></pre>
<pre><code>## returns market_returns_tidy.returns .fitted .se.fit
## 1 -0.0008696132 0.01267837 0.008294282 0.001431288
## 2 0.0186624378 0.03726809 0.030451319 0.001984645
## 3 0.0206248830 0.01903021 0.014017731 0.001485410
## 4 -0.0053529692 0.02333503 0.017896670 0.001563417
## 5 -0.0229487590 -0.01343411 -0.015234859 0.001953853
## 6 0.0411705787 0.05038580 0.042271276 0.002521506
## .resid .hat .sigma .cooksd .std.resid date
## 1 -0.009163896 0.01698211 0.01101184 0.006116973 -0.8415272 2013-02-28
## 2 -0.011788881 0.03265148 0.01096452 0.020099701 -1.0913142 2013-03-28
## 3 0.006607152 0.01829069 0.01104500 0.003434000 0.6071438 2013-04-30
## 4 -0.023249640 0.02026222 0.01062704 0.047294116 -2.1386022 2013-05-31
## 5 -0.007713900 0.03164618 0.01103127 0.008323550 -0.7137165 2013-06-28
## 6 -0.001100697 0.05270569 0.01107986 0.000294938 -0.1029661 2013-07-31</code></pre>
<p>Let’s use <code>ggplot()</code> to see how well the fitted return values match the actual return values.</p>
<pre class="r"><code>portfolio_model_augmented %>%
ggplot(aes(x = date)) +
geom_line(aes(y = returns, color = "actual returns")) +
geom_line(aes(y = .fitted, color = "fitted returns")) +
scale_colour_manual("",
values = c("fitted returns" = "green",
"actual returns" = "cornflowerblue")) +
xlab("date") +
ggtitle("Fitted versus actual returns")</code></pre>
<p><img src="/post/2018-02-28-capm-and-visualization_files/figure-html/unnamed-chunk-8-1.png" width="672" /></p>
<p>Those monthly returns and fitted values seem to track well. Let’s convert both actual returns and fitted returns to the growth of a dollar and run the same comparison. This isn’t a traditional way to visualize actual versus fitted, but it’s still useful.</p>
<pre class="r"><code>portfolio_model_augmented %>%
mutate(actual_growth = cumprod(1 + returns),
fitted_growth = cumprod(1 + .fitted)) %>%
ggplot(aes(x = date)) +
geom_line(aes(y = actual_growth, color = "actual growth")) +
geom_line(aes(y = fitted_growth, color = "fitted growth")) +
xlab("date") +
ylab("actual and fitted growth") +
ggtitle("Growth of a dollar: actual v. fitted") +
scale_x_date(breaks = pretty_breaks(n = 8)) +
scale_y_continuous(labels = dollar) +
scale_colour_manual("",
values = c("fitted growth" = "green",
"actual growth" = "cornflowerblue")) </code></pre>
<p><img src="/post/2018-02-28-capm-and-visualization_files/figure-html/unnamed-chunk-9-1.png" width="672" /></p>
<p>Our fitted growth tracks our actual growth well, though the actual growth is lower than predicted for most of the five year history.</p>
</div>
<div id="to-highcharter" class="section level3">
<h3>To Highcharter!</h3>
<p>A nice side benefit of <code>augment()</code> is that it allows us to create an interesting <code>highcharter</code> object that replicates our scatter + regression <code>ggplot</code> from earlier.</p>
<p>First, let’s build the base scatter plot of portfolio returns, which are housed in <code>portfolio_model_augmented$returns</code>, against market returns, which are housed in <code>portfolio_model_augmented$market_returns_tidy.returns</code>.</p>
<pre class="r"><code>library(highcharter)
highchart() %>%
hc_title(text = "Portfolio v. Market Returns") %>%
hc_add_series_scatter(round(portfolio_model_augmented$returns, 4),
round(portfolio_model_augmented$market_returns_tidy.returns, 4)) %>%
hc_xAxis(title = list(text = "Market Returns")) %>%
hc_yAxis(title = list(text = "Portfolio Returns"))</code></pre>
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<p>That looks good; but hover over one of the points. If you’re like me, you will desperately wish that the date of the observation were being displayed. Let’s add that date display functionality.</p>
<p>First, we need to supply the date observations, so we will add a <code>date</code> variable with <code>hc_add_series_scatter(..., date = portfolio_returns_tq_rebalanced_monthly$date)</code>. Then, we want the tool tip to pick up and display that variable. That is done with <code>hc_tooltip(formatter = JS("function(){return ('port return: ' + this.y + ' <br> mkt return: ' + this.x + ' <br> date: ' + this.point.date)}"))</code>. We are creating a custom tool tip function to pick up the date. Run the code chunk below and hover over a point.</p>
<pre class="r"><code>highchart() %>%
hc_title(text = "Portfolio v. Market Returns") %>%
hc_add_series_scatter(round(portfolio_model_augmented$returns, 4),
round(portfolio_model_augmented$market_returns_tidy.returns, 4),
date = portfolio_model_augmented$date) %>%
hc_xAxis(title = list(text = "Market Returns")) %>%
hc_yAxis(title = list(text = "Portfolio Returns")) %>%
hc_tooltip(formatter = JS("function(){
return ('port return: ' + this.y + ' <br> mkt return: ' + this.x +
' <br> date: ' + this.point.date)}"))</code></pre>
<div id="htmlwidget-2" style="width:100%;height:500px;" class="highchart html-widget"></div>
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<p>I was curious about the most negative reading in the bottom left, and this new tool tip makes it easy to see that it occurred in August of 2015.</p>
<p>Finally, let’s add the regression line.</p>
<p>To do that, we need to supply x and y coordinates to <code>highcharter</code> and specify that we want to add a line instead of more scatter points. We have the x and y coordinates for our fitted regression line because we added them with the <code>augment()</code> function. The x’s are the market returns and the y’s are the fitted values. We add this element to our code flow with <code>hc_add_series(portfolio_model_augmented, type = "line", hcaes(x = market_returns_tidy.returns, y = .fitted), name = "CAPM Beta = Regression Slope")</code></p>
<pre class="r"><code>highchart() %>%
hc_title(text = "Scatter with Regression Line") %>%
hc_add_series(portfolio_model_augmented,
type = "scatter",
hcaes(x = round(market_returns_tidy.returns, 4),
y = round(returns, 4),
date = date),
name = "Returns") %>%
hc_add_series(portfolio_model_augmented,
type = "line",
hcaes(x = market_returns_tidy.returns, y = .fitted),
name = "CAPM Beta = Regression Slope") %>%
hc_xAxis(title = list(text = "Market Returns")) %>%
hc_yAxis(title = list(text = "Portfolio Returns")) %>%
hc_tooltip(formatter = JS("function(){
return ('port return: ' + this.y + ' <br> mkt return: ' + this.x +
' <br> date: ' + this.point.date)}"))</code></pre>
<div id="htmlwidget-3" style="width:100%;height:500px;" class="highchart html-widget"></div>
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<p>That’s all for today and thanks for reading.</p>
</div>
<script>window.location.href='https://rviews.rstudio.com/2018/03/02/capm-and-visualization/';</script>
Calculating Beta in the Capital Asset Pricing Model
https://rviews.rstudio.com/2018/02/08/capm-beta/
Thu, 08 Feb 2018 00:00:00 +0000https://rviews.rstudio.com/2018/02/08/capm-beta/
<p>Today we will continue our portfolio fun by calculating the CAPM beta of our portfolio returns. That will entail fitting a linear model and, when we get to visualization next time, considering the meaning of our results from the perspective of asset returns.</p>
<p>By way of brief background, the Capital Asset Pricing Model (CAPM) is a model, created by William Sharpe, that estimates the return of an asset based on the return of the market and the asset’s linear relationship to the return of the market. That linear relationship is the stock’s beta coefficient, or just good ol’ beta.</p>
<p>CAPM was introduced back in 1964, garnered a Nobel for its creator, and, like many ephocally important theories, has been widely used, updated, criticized, debunked, revived, re-debunked, etc. Fama and French have written that CAPM “is the centerpiece of MBA investment courses. Indeed, it is often the only asset pricing model taught in these courses…[u]nfortunately, the empirical record of the model is poor.”<a href="#fn1" class="footnoteRef" id="fnref1"><sup>1</sup></a></p>
<p>With that, we will forge ahead with our analysis because calculating CAPM betas can serve as a nice template for more complex models in a team’s work and sometimes it’s a good idea to start with a simple model, even if it hasn’t stood up to empirical rigor. Plus, it might have been questioned by future research, but it’s still an iconic model that we should learn and love.</p>
<p>We are going to focus on one particular aspect of CAPM: beta. Beta, as we noted above, is the beta coefficient of an asset that results from regressing the returns of that asset on market returns. It captures the linear relationship between the asset/portfolio and the market. For our purposes, it’s a good vehicle for exploring reproducible flows for modeling or regressing our portfolio returns on the market returns. Even if your team dislikes CAPM in favor of more nuanced models, these code flows can serve as a good base for the building of those more complex models.</p>
<p>We are going to be calculating beta in several ways: by-hand (for illustrative purposes), in the <code>xts</code> world with <code>PerformanceAnalytics</code>, in the tidyverse with <code>dplyr</code>, and in the <code>tidyquant</code> world. These seem to be the most popular paradigms for doing financial time series work, and even within a team there can be differing preferences. I don’t think everyone needs to grind through their work using each paradigm, but I do think it’s helpful to be fluent, or, at least, conversant, in the various worlds. If you’re a tidyverse type of person but need to collaborate with an <code>xts</code> or <code>tidyquant</code> enthusiast, it will help if each of you is familiar with the three universes (though at some point ya just have to choose a code flow and get stuff done).</p>
<p>We will be working with and calculating beta for our usual portfolio consisting of:</p>
<pre><code>+ SPY (S&P500 fund) weighted 25%
+ EFA (a non-US equities fund) weighted 25%
+ IJS (a small-cap value fund) weighted 20%
+ EEM (an emerging-mkts fund) weighted 20%
+ AGG (a bond fund) weighted 10%</code></pre>
<p>Before we can calculate beta for that portfolio, we need to find portfolio monthly returns, which was covered in <a href="https://rviews.rstudio.com/2017/10/11/from-asset-to-portfolio-returns/">this post</a>.</p>
<p>I won’t go through the logic again but the code is here:</p>
<pre class="r"><code>library(tidyquant)
library(tidyverse)
library(timetk)
library(tibbletime)
library(broom)
symbols <- c("SPY","EFA", "IJS", "EEM","AGG")
prices <-
getSymbols(symbols, src = 'yahoo',
from = "2013-01-01",
to = "2017-12-31",
auto.assign = TRUE, warnings = FALSE) %>%
map(~Ad(get(.))) %>%
reduce(merge) %>%
`colnames<-`(symbols)
prices_monthly <- to.monthly(prices, indexAt = "last", OHLC = FALSE)
asset_returns_xts <- na.omit(Return.calculate(prices_monthly, method = "log"))
w <- c(0.25, 0.25, 0.20, 0.20, 0.10)
portfolio_returns_xts_rebalanced_monthly <-
Return.portfolio(asset_returns_xts, weights = w, rebalance_on = "months") %>%
`colnames<-`("returns")
asset_returns_long <-
prices %>%
to.monthly(indexAt = "last", OHLC = FALSE) %>%
tk_tbl(preserve_index = TRUE, rename_index = "date") %>%
gather(asset, returns, -date) %>%
group_by(asset) %>%
mutate(returns = (log(returns) - log(lag(returns)))) %>%
na.omit()
portfolio_returns_tq_rebalanced_monthly <-
asset_returns_long %>%
tq_portfolio(assets_col = asset,
returns_col = returns,
weights = w,
col_rename = "returns",
rebalance_on = "months")</code></pre>
<p>We will be working with two objects of portfolio returns and one object of our individual asset returns:</p>
<pre><code>+ portfolio_returns_xts_rebalanced_monthly (an xts of monthly returns)
+ portfolio_returns_tq_rebalanced_monthly (a tibble of monthly returns)
+ asset_returns_long (a tidy tibble of monthly returns for those 5 assets above)</code></pre>
<p>Let’s get to it.</p>
<div id="capm-and-market-returns" class="section level3 unnumbered">
<h3>CAPM and Market Returns</h3>
<p>Our first step is to make a choice about which asset to use as a proxy for the market return, and we will go with the SPY ETF, effectively treating the S&P 500 as the market. That’s going to make our calculations substantively uninteresting because (1) SPY is 25% of our portfolio and (2) we have chosen assets and a time period (2013 - 2017) in which correlations with SPY have been high. It will offer one benefit in the way of a sanity check, which I’ll note below. With those caveats in mind, feel free to choose a different asset for the market return and try to reproduce this work, or construct a different portfolio that does not include SPY.</p>
<p>Let’s calculate our market return for SPY and save it as <code>market_return_xts</code>. Note the start date is “2013-01-01” and the end date is “2017-12-31”, so we will be working with five years of returns.</p>
<pre class="r"><code>spy_monthly_xts <-
getSymbols("SPY",
src = 'yahoo',
from = "2013-01-01",
to = "2017-12-31",
auto.assign = TRUE,
warnings = FALSE) %>%
map(~Ad(get(.))) %>%
reduce(merge) %>%
`colnames<-`("SPY") %>%
to.monthly(indexAt = "last", OHLC = FALSE)
market_returns_xts <-
Return.calculate(spy_monthly_xts, method = "log") %>%
na.omit()</code></pre>
<p>We will also want a <code>data.frame</code> object of market returns, and will convert the <code>xts</code> object using <code>tk_tbl(preserve_index = TRUE, rename_index = "date")</code> from the <code>timetk</code> package.</p>
<pre class="r"><code>market_returns_tidy <-
market_returns_xts %>%
tk_tbl(preserve_index = TRUE, rename_index = "date") %>%
na.omit() %>%
select(date, returns = SPY)
head(market_returns_tidy)</code></pre>
<pre><code>## # A tibble: 6 x 2
## date returns
## <date> <dbl>
## 1 2013-02-28 0.01267837
## 2 2013-03-28 0.03726809
## 3 2013-04-30 0.01903021
## 4 2013-05-31 0.02333503
## 5 2013-06-28 -0.01343411
## 6 2013-07-31 0.05038580</code></pre>
<p>We have a <code>market_returns_tidy</code> object. Let’s make sure it’s periodicity aligns perfectly with our portfolio returns periodicity</p>
<pre class="r"><code>portfolio_returns_tq_rebalanced_monthly %>%
mutate(market_returns = market_returns_tidy$returns) %>%
head()</code></pre>
<pre><code>## # A tibble: 6 x 3
## date returns market_returns
## <date> <dbl> <dbl>
## 1 2013-02-28 -0.0008696129 0.01267837
## 2 2013-03-28 0.0186624381 0.03726809
## 3 2013-04-30 0.0206248856 0.01903021
## 4 2013-05-31 -0.0053529694 0.02333503
## 5 2013-06-28 -0.0229487618 -0.01343411
## 6 2013-07-31 0.0411705818 0.05038580</code></pre>
<p>Note that if the periodicities did not align, <code>mutate()</code> would have thrown an error in the code chunk above.</p>
</div>
<div id="calculating-capm-beta" class="section level3 unnumbered">
<h3>Calculating CAPM Beta</h3>
<p>There are several R code flows to calculate portfolio beta but first let’s have a look at the equation.</p>
<pre class="r"><code>$${\beta}_{portfolio} = cov(R_p, R_m)/\sigma_m $$</code></pre>
<p><span class="math display">\[{\beta}_{portfolio} = cov(R_p, R_m)/\sigma_m \]</span></p>
<p>Portfolio beta is equal to the covariance of the portfolio returns and market returns, divided by the variance of market returns.</p>
<p>We can calculate the numerator, or covariance of portfolio and market returns, with <code>cov(portfolio_returns_xts_rebalanced_monthly, market_returns_tidy$returns)</code> and the denominator with <code>var(market_return$returns)</code>.</p>
<p>Our portfolio beta is equal to:</p>
<pre class="r"><code>cov(portfolio_returns_xts_rebalanced_monthly,market_returns_tidy$returns)/var(market_returns_tidy$returns)</code></pre>
<pre><code>## [,1]
## returns 0.9010689</code></pre>
<p>That beta is quite near to 1 as we were expecting - after all, SPY is a big part of this portfolio.</p>
<p>We can also calculate portfolio beta by finding the beta of each of our assets and then multiplying by asset weights. That is, another equation for portfolio beta is the weighted sum of the asset betas:</p>
<pre class="r"><code>$${\beta}_{portfolio} ={\sum_{i=1}^n}W _i~{\beta}_i $$</code></pre>
<p><span class="math display">\[{\beta}_{portfolio} ={\sum_{i=1}^n}W _i~{\beta}_i \]</span></p>
<p>To use that method with R, we first find the beta for each of our assets, and this gives us an opportunity to introduce a code flow for running regression analysis.</p>
<p>We need to regress each of our individual asset returns on the market return. We could do that for asset 1 with <code>lm(asset_return_1 ~ market_returns_tidy$returns)</code>, and then again for asset 2 with <code>lm(asset_return_2 ~ market_returns_tidy$returns)</code>, etc. for all five of our assets. But if we had a 50-asset portfolio, that would be impractical. Instead let’s write a code flow and use <code>map()</code> to regress all of our assets and calculate betas with one call.</p>
<p>We will start with our <code>asset_returns_long</code> tidy data frame and will then run <code>nest(-asset)</code>.</p>
<pre class="r"><code>beta_assets <-
asset_returns_long %>%
na.omit() %>%
nest(-asset)
beta_assets</code></pre>
<pre><code>## # A tibble: 5 x 2
## asset data
## <chr> <list>
## 1 SPY <tibble [59 x 2]>
## 2 EFA <tibble [59 x 2]>
## 3 IJS <tibble [59 x 2]>
## 4 EEM <tibble [59 x 2]>
## 5 AGG <tibble [59 x 2]></code></pre>
<p>That <code>nest(-asset)</code> changed our data frame so that there are two columns: one called <code>asset</code> that holds our asset name and one called <code>data</code> that holds a list of returns for each asset. We have now ‘nested’ a list of returns within a column.</p>
<p>Now we can use <code>map()</code> to apply a function to each of those nested lists and store the results in a new column via the <code>mutate()</code> function. The whole piped command is <code>mutate(model = map(data, ~ lm(returns ~ market_returns_tidy$returns, data = .)))</code></p>
<pre class="r"><code>beta_assets <-
asset_returns_long %>%
na.omit() %>%
nest(-asset) %>%
mutate(model = map(data, ~ lm(returns ~ market_returns_tidy$returns, data = .)))
beta_assets</code></pre>
<pre><code>## # A tibble: 5 x 3
## asset data model
## <chr> <list> <list>
## 1 SPY <tibble [59 x 2]> <S3: lm>
## 2 EFA <tibble [59 x 2]> <S3: lm>
## 3 IJS <tibble [59 x 2]> <S3: lm>
## 4 EEM <tibble [59 x 2]> <S3: lm>
## 5 AGG <tibble [59 x 2]> <S3: lm></code></pre>
<p>We now have three columns: <code>asset</code> which we had before, <code>data</code> which we had before, and <code>model</code> which we just added. The <code>model</code> column holds the results of the regression <code>lm(returns ~ market_returns_tidy$returns, data = .)</code> that we ran for each of our assets. Those results are a beta and an intercept for each of our assets, but they are not in a great format for presentation to others, or even readability by ourselves.</p>
<p>Let’s tidy up our results with the <code>tidy()</code> function from the <code>broom</code> package. We want to apply that function to our model column and will use the <code>mutate()</code> and <code>map()</code> combination again. The complete call is to <code>mutate(model = map(model, tidy))</code>.</p>
<pre class="r"><code>beta_assets <-
asset_returns_long %>%
na.omit() %>%
nest(-asset) %>%
mutate(model = map(data, ~ lm(returns ~ market_returns_tidy$returns, data = .))) %>%
mutate(model = map(model, tidy))
beta_assets</code></pre>
<pre><code>## # A tibble: 5 x 3
## asset data model
## <chr> <list> <list>
## 1 SPY <tibble [59 x 2]> <data.frame [2 x 5]>
## 2 EFA <tibble [59 x 2]> <data.frame [2 x 5]>
## 3 IJS <tibble [59 x 2]> <data.frame [2 x 5]>
## 4 EEM <tibble [59 x 2]> <data.frame [2 x 5]>
## 5 AGG <tibble [59 x 2]> <data.frame [2 x 5]></code></pre>
<p>We are getting close now, but the <code>model</code> column holds nested data frames. Have a look and see that they are nicely formatted data frames:</p>
<pre class="r"><code>beta_assets$model</code></pre>
<pre><code>## [[1]]
## term estimate std.error statistic
## 1 (Intercept) 1.806734e-18 1.136381e-18 1.589902e+00
## 2 market_returns_tidy$returns 1.000000e+00 3.899949e-17 2.564136e+16
## p.value
## 1 0.1173886
## 2 0.0000000
##
## [[2]]
## term estimate std.error statistic
## 1 (Intercept) -0.005427739 0.002908978 -1.865858
## 2 market_returns_tidy$returns 0.945476441 0.099833320 9.470550
## p.value
## 1 6.720983e-02
## 2 2.656258e-13
##
## [[3]]
## term estimate std.error statistic
## 1 (Intercept) -0.001693293 0.003639218 -0.4652905
## 2 market_returns_tidy$returns 1.120583127 0.124894444 8.9722416
## p.value
## 1 6.434963e-01
## 2 1.713903e-12
##
## [[4]]
## term estimate std.error statistic
## 1 (Intercept) -0.00811518 0.004785237 -1.695878
## 2 market_returns_tidy$returns 0.95562574 0.164224722 5.819013
## p.value
## 1 9.536495e-02
## 2 2.841106e-07
##
## [[5]]
## term estimate std.error statistic
## 1 (Intercept) 0.001888304 0.001230331 1.5347933
## 2 market_returns_tidy$returns -0.005419543 0.042223776 -0.1283529
## p.value
## 1 0.1303671
## 2 0.8983215</code></pre>
<p>Still, I don’t like to end up with nested data frames, so let’s <code>unnest()</code> that <code>model</code> column.</p>
<pre class="r"><code>beta_assets <-
asset_returns_long %>%
na.omit() %>%
nest(-asset) %>%
mutate(model = map(data, ~ lm(returns ~ market_returns_tidy$returns, data = .))) %>%
mutate(model = map(model, tidy)) %>%
unnest(model)
beta_assets</code></pre>
<pre><code>## # A tibble: 10 x 6
## asset term estimate std.error
## <chr> <chr> <dbl> <dbl>
## 1 SPY (Intercept) 1.806734e-18 1.136381e-18
## 2 SPY market_returns_tidy$returns 1.000000e+00 3.899949e-17
## 3 EFA (Intercept) -5.427739e-03 2.908978e-03
## 4 EFA market_returns_tidy$returns 9.454764e-01 9.983332e-02
## 5 IJS (Intercept) -1.693293e-03 3.639218e-03
## 6 IJS market_returns_tidy$returns 1.120583e+00 1.248944e-01
## 7 EEM (Intercept) -8.115180e-03 4.785237e-03
## 8 EEM market_returns_tidy$returns 9.556257e-01 1.642247e-01
## 9 AGG (Intercept) 1.888304e-03 1.230331e-03
## 10 AGG market_returns_tidy$returns -5.419543e-03 4.222378e-02
## # ... with 2 more variables: statistic <dbl>, p.value <dbl></code></pre>
<p>Now that looks human-readable and presentable. We will do one further cleanup and get rid of the intercept results, since we are isolating the betas.</p>
<pre class="r"><code>beta_assets <-
asset_returns_long %>%
na.omit() %>%
nest(-asset) %>%
mutate(model = map(data, ~ lm(returns ~ market_returns_tidy$returns, data = .))) %>%
unnest(model %>% map(tidy)) %>%
filter(term == "market_returns_tidy$returns") %>%
select(-term)
beta_assets</code></pre>
<pre><code>## # A tibble: 5 x 5
## asset estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 SPY 1.000000000 3.899949e-17 2.564136e+16 0.000000e+00
## 2 EFA 0.945476441 9.983332e-02 9.470550e+00 2.656258e-13
## 3 IJS 1.120583127 1.248944e-01 8.972242e+00 1.713903e-12
## 4 EEM 0.955625743 1.642247e-01 5.819013e+00 2.841106e-07
## 5 AGG -0.005419543 4.222378e-02 -1.283529e-01 8.983215e-01</code></pre>
<p>A quick sanity check on those asset betas should reveal that SPY has beta of 1 with itself.</p>
<pre class="r"><code>beta_assets %>% select(asset, estimate) %>% filter(asset == "SPY")</code></pre>
<pre><code>## # A tibble: 1 x 2
## asset estimate
## <chr> <dbl>
## 1 SPY 1</code></pre>
<p>Now let’s see how our combination of these assets leads to a portfolio beta.</p>
<p>Let’s assign portfolio weights as we chose above.</p>
<pre class="r"><code>w <- c(0.25, 0.25, 0.20, 0.20, 0.10)</code></pre>
<p>Now we can use those weights to get our portfolio beta, based on the betas of the individual assets.</p>
<pre class="r"><code>beta_byhand <-
w[1] * beta_assets$estimate[1] +
w[2] * beta_assets$estimate[2] +
w[3] * beta_assets$estimate[3] +
w[4] * beta_assets$estimate[4] +
w[5] * beta_assets$estimate[5]
beta_byhand</code></pre>
<pre><code>## [1] 0.9010689</code></pre>
<p>That beta is the same as we calculated above using the covariance/variance method, and now we know the the covariance of portfolio returns and market returns divided by the variance of market returns is equal to the weighted estimates we got by regressing each asset’s return on market returns.</p>
</div>
<div id="calculating-capm-beta-in-the-xts-world" class="section level3 unnumbered">
<h3>Calculating CAPM Beta in the <code>xts</code> World</h3>
<p>We can make things even more efficient, of course, with built-in functions. Let’s go to the <code>xts</code> world and use the built-in <code>CAPM.beta()</code> function from <code>PerformanceAnalytics</code>. That function takes two arguments: the returns for the portfolio (or any asset) whose beta we wish to calculate, and the market returns. Our function will look like <code>CAPM.beta(portfolio_returns_xts_rebalanced_monthly, mkt_return_xts)</code>.</p>
<pre class="r"><code>beta_builtin_xts <- CAPM.beta(portfolio_returns_xts_rebalanced_monthly, market_returns_xts)
beta_builtin_xts</code></pre>
<pre><code>## [1] 0.9010689</code></pre>
</div>
<div id="calculating-capm-beta-in-the-tidyverse" class="section level3 unnumbered">
<h3>Calculating CAPM Beta in the Tidyverse</h3>
<p>We will run that same function through a <code>dplyr</code> and <code>tidyquant</code> code flow to stay in the tidy world.</p>
<p>First we’ll use <code>dplyr</code> to grab our portfolio beta. We’ll return to this flow later for some visualization, but for now will extract the portfolio beta.</p>
<p>To calculate the beta, we call <code>do(model = lm(returns ~ market_returns_tidy$returns, data = .))</code>. Then we head back to the <code>broom</code> package and use the <code>tidy()</code> function to make our model results a little easier on the eyes.</p>
<pre class="r"><code>beta_dplyr_byhand <-
portfolio_returns_tq_rebalanced_monthly %>%
do(model = lm(returns ~ market_returns_tidy$returns, data = .)) %>%
tidy(model) %>%
mutate(term = c("alpha", "beta"))
beta_dplyr_byhand</code></pre>
<pre><code>## term estimate std.error statistic p.value
## 1 alpha -0.003129799 0.00155617 -2.011219 4.903980e-02
## 2 beta 0.901068930 0.05340627 16.871969 7.855042e-24</code></pre>
</div>
<div id="calculating-capm-beta-in-the-tidyquant-world" class="section level3 unnumbered">
<h3>Calculating CAPM Beta in the Tidyquant World</h3>
<p>Let’s use one more flow with built-in functions, this time using <code>tidyquant</code> and the <code>tq_performance()</code> function. This will allow us to apply the <code>CAPM.beta()</code> function from <code>PerformanceAnalytics</code> to a data frame.</p>
<pre class="r"><code>beta_builtin_tq <-
portfolio_returns_tq_rebalanced_monthly %>%
mutate(market_return = market_returns_tidy$returns) %>%
na.omit() %>%
tq_performance(Ra = returns,
Rb = market_return,
performance_fun = CAPM.beta) %>%
`colnames<-`("beta_tq")</code></pre>
<p>Let’s take a quick look at our four beta calculations.</p>
<pre class="r"><code>beta_byhand</code></pre>
<pre><code>## [1] 0.9010689</code></pre>
<pre class="r"><code>beta_builtin_xts</code></pre>
<pre><code>## [1] 0.9010689</code></pre>
<pre class="r"><code>beta_dplyr_byhand$estimate[2]</code></pre>
<pre><code>## [1] 0.9010689</code></pre>
<pre class="r"><code>beta_builtin_tq$beta_tq</code></pre>
<pre><code>## [1] 0.9010689</code></pre>
<p>Consistent results and a beta near 1 as we were expecting, since our portfolio has a 25% allocation to the S&P 500. We’re less concerned with numbers than we are with the various code flows used to get here. Next time we’ll do some visualizing - see you then!</p>
</div>
<div class="footnotes">
<hr />
<ol>
<li id="fn1"><p>The Capital Asset Pricing Model: Theory and Evidence Eugene F. Fama and Kenneth R. French, The Capital Asset Pricing Model: Theory and Evidence, The Journal of Economic Perspectives, Vol. 18, No. 3 (Summer, 2004), pp. 25-46<a href="#fnref1">↩</a></p></li>
</ol>
</div>
<script>window.location.href='https://rviews.rstudio.com/2018/02/08/capm-beta/';</script>
Introduction to Kurtosis
https://rviews.rstudio.com/2018/01/04/introduction-to-kurtosis/
Thu, 04 Jan 2018 00:00:00 +0000https://rviews.rstudio.com/2018/01/04/introduction-to-kurtosis/
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<p>Happy 2018 and welcome to our first reproducible finance post of the year! What better way to ring in a new beginning than pondering/calculating/visualizing returns distributions.</p>
<p>We ended 2017 by tackling <a href="https://rviews.rstudio.com/2017/12/13/introduction-to-skewness/">skewness</a>, and we will begin 2018 by tackling kurtosis.</p>
<p>Kurtosis is a measure of the degree to which portfolio returns appear in the tails of our distribution. A normal distribution has a kurtosis of 3, which follows from the fact that a normal distribution does have some of its mass in its tails. A distribution with a kurtosis greater than 3 has more returns out in its tails than the normal, and one with kurtosis less than 3 has fewer returns in its tails than the normal. That matters to investors because more bad returns out in tails means that our portfolio might be at risk of a rare but huge downside. The terminology is a bit confusing. Negative kurtosis is considered less risky because it has fewer returns out in the tails. Negative == less risky? We’re not used to that in finance.</p>
<p>Kurtosis is often has the word ‘excess’ appended to its description, as in ‘negative excess kurtosis’ or ‘positive excess kurtosis’. That ‘excess’ is in comparison to a normal distribution kurtosis of 3. A distribution with negative excess kurtosis equal to -1 has an actual kurtosis of 2.</p>
<p>Enough with the faux investopedia entry, let’s get to the calculations, R code and visualizations.</p>
<p>Here’s the equation for excess kurtosis. Note that we subtract 3 at the end:</p>
<p><span class="math display">\[Kurtosis=\sum_{t=1}^n (x_i-\overline{x})^4/n \bigg/ (\sum_{t=1}^n (x_i-\overline{x})^2/n)^{2}-3 \]</span></p>
<p>By way of reminder, we will be working with our usual portfolio consisting of:</p>
<pre><code>+ SPY (S&P500 fund) weighted 25%
+ EFA (a non-US equities fund) weighted 25%
+ IJS (a small-cap value fund) weighted 20%
+ EEM (an emerging-mkts fund) weighted 20%
+ AGG (a bond fund) weighted 10%</code></pre>
<p>Before we can calculate kurtosis, we need to find portfolio monthly returns, which was covered in <a href="https://rviews.rstudio.com/2017/10/11/from-asset-to-portfolio-returns/">this post</a>.</p>
<p>Building off that previous work, we will be working with two objects of portfolio returns:</p>
<pre><code>+ portfolio_returns_xts_rebalanced_monthly (an xts of monthly returns)
+ portfolio_returns_tq_rebalanced_monthly (a tibble of monthly returns)</code></pre>
<p>Now we are going to test our past self’s <a href="https://rviews.rstudio.com/2017/12/13/introduction-to-skewness/">work on skewness</a>, and reuse that code flow to expedite the kurtosis work. The logic will remain the same, but we will call different built-in functions and different by-hand calculations.</p>
<p>For the xts world, we use the <code>kurtosis()</code> function instead of the <code>skewness()</code> function.</p>
<pre class="r"><code>kurt_xts <- kurtosis(portfolio_returns_xts_rebalanced_monthly$returns)
kurt_xts</code></pre>
<pre><code>## [1] 0.5267736</code></pre>
<p>For tidy, we have the same piped flow and use the formula for kurtosis for our by-hand calculations. Our by-hand result is labeled with <code>kurt_byhand</code>, and involves quite a few parentheticals to map it back to the kurtosis equation above.</p>
<pre class="r"><code>kurt_tidy <-
portfolio_returns_tq_rebalanced_monthly %>%
summarise(
kurt_builtin = kurtosis(returns),
kurt_byhand =
((sum((returns - mean(returns))^4)/length(returns))/
((sum((returns - mean(returns))^2)/length(returns))^2)) - 3) %>%
select(kurt_builtin, kurt_byhand)</code></pre>
<p>Let’s confirm that we have consistent calculations.</p>
<pre class="r"><code>kurt_xts</code></pre>
<pre><code>## [1] 0.5267736</code></pre>
<pre class="r"><code>kurt_tidy$kurt_builtin</code></pre>
<pre><code>## [1] 0.5267736</code></pre>
<pre class="r"><code>kurt_tidy$kurt_byhand</code></pre>
<pre><code>## [1] 0.5267736</code></pre>
<p>We have consistent results from <code>xts</code> and the tidy built-in/by-hand worlds, and we were able to reuse our code from above to shorten the development time here. dd</p>
<p>Let’s do the same with the visualizations and head straight for a density plot, starting with the same <code>portfolio_density_plot</code>.</p>
<pre class="r"><code>portfolio_density_plot <-
portfolio_returns_tq_rebalanced_monthly %>%
ggplot(aes(x = returns)) +
stat_density(geom = "line", alpha = 1, colour = "cornflowerblue")
portfolio_density_plot</code></pre>
<p><img src="/post/2018-01-03-introduction-to-kurtosis_files/figure-html/unnamed-chunk-4-1.png" width="672" /></p>
<p>We are interested in <em>both</em> tails for kurtosis, so let’s shade at 2 standard deviations above and below the mean return (for our skewness work, we only shaded the negative tail).</p>
<pre class="r"><code>mean <- mean(portfolio_returns_tq_rebalanced_monthly$returns)
sd_pos <- mean + (2 * sd(portfolio_returns_tq_rebalanced_monthly$returns))
sd_neg <- mean - (2 * sd(portfolio_returns_tq_rebalanced_monthly$returns))
sd_pos_shaded_area <-
ggplot_build(portfolio_density_plot)$data[[1]] %>%
filter(x > sd_pos )
sd_neg_shaded_area <-
ggplot_build(portfolio_density_plot)$data[[1]] %>%
filter(x < sd_neg)
portfolio_density_plot <-
portfolio_density_plot +
geom_area(data = sd_pos_shaded_area, aes(x = x, y = y), fill="pink", alpha = 0.5) +
geom_area(data = sd_neg_shaded_area, aes(x = x, y = y), fill="pink", alpha = 0.5) +
scale_x_continuous(breaks = scales::pretty_breaks(n = 10))
portfolio_density_plot</code></pre>
<p><img src="/post/2018-01-03-introduction-to-kurtosis_files/figure-html/unnamed-chunk-5-1.png" width="672" /></p>
<p>That density chart is a good look at the mass in both tails, where we have defined ‘tail’ as being two standard deviations away from the mean. We can add a line for the mean, as did in our skewness visualization, with the following.</p>
<pre class="r"><code>mean <- mean(portfolio_returns_tq_rebalanced_monthly$returns)
mean_line_data <-
ggplot_build(portfolio_density_plot)$data[[1]] %>%
filter(x <= mean)
portfolio_density_plot +
geom_segment(data = mean_line_data, aes(x = mean, y = 0, xend = mean, yend = density),
color = "black", linetype = "dotted") +
annotate(geom = "text", x = mean, y = 5, label = "mean",
fontface = "plain", angle = 90, alpha = .8, vjust = 1.75)</code></pre>
<p><img src="/post/2018-01-03-introduction-to-kurtosis_files/figure-html/unnamed-chunk-6-1.png" width="672" /></p>
<p>Finally, we can calculate and chart the rolling kurtosis with the same logic as we did for skewness. The only difference is that here we call <code>fun = kurtosis</code> instead of <code>fun = skewness</code>.</p>
<pre class="r"><code>window <- 6
rolling_kurt_xts <- na.omit(apply.rolling(portfolio_returns_xts_rebalanced_monthly, window,
fun = kurtosis))</code></pre>
<p>Now we pop that <code>xts</code> object into <code>highcharter</code> for a visualization.</p>
<pre class="r"><code>highchart(type = "stock") %>%
hc_title(text = "Rolling Kurt") %>%
hc_add_series(rolling_kurt_xts, name = "Rolling kurtosis", color = "cornflowerblue") %>%
hc_yAxis(title = list(text = "kurtosis"),
opposite = FALSE,
max = .03,
min = -.03) %>%
hc_navigator(enabled = FALSE) %>%
hc_scrollbar(enabled = FALSE) </code></pre>
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<p>We didn’t cover this before, but what if we wanted to use <code>ggplot</code> for the rolling kurtosis? We could convert that <code>xts</code> object to a tibble with <code>tk_tbl()</code> from the <code>timetk</code> package, and then pipe straight to <code>ggplot</code>.</p>
<pre class="r"><code>rolling_kurt_xts %>%
tk_tbl(preserve_index = TRUE, rename_index = "date") %>%
rename(rolling_kurtosis = calcs) %>%
ggplot(aes(x = date, y = rolling_kurtosis)) +
geom_line(color = "cornflowerblue") +
xlab("date") +
ylab("rolling kurtosis") +
ggtitle("Rolling Kurtosis")</code></pre>
<p><img src="/post/2018-01-03-introduction-to-kurtosis_files/figure-html/unnamed-chunk-9-1.png" width="672" /></p>
<p>Interestingly, this portfolio has displayed slight positive rolling excess kurtosis for most of its life, except during the last half of 2015 through early 2016.</p>
<p>That’s all for today. Our work on kurtosis was made a lot more efficient by our work on skewness - so let’s thank our 2017 selves for constructing a reproducible and reusable code flow! See you next time.</p>
<script>window.location.href='https://rviews.rstudio.com/2018/01/04/introduction-to-kurtosis/';</script>
Introduction to Skewness
https://rviews.rstudio.com/2017/12/13/introduction-to-skewness/
Wed, 13 Dec 2017 00:00:00 +0000https://rviews.rstudio.com/2017/12/13/introduction-to-skewness/
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<p>In previous posts <a href="https://rviews.rstudio.com/2017/07/12/introduction-to-volatility/">here</a>, <a href="https://rviews.rstudio.com/2017/07/18/introduction-to-rolling-volatility/">here</a>, and <a href="https://rviews.rstudio.com/2017/07/21/visualizing-portfolio-volatility/">here</a>, we spent quite a bit of time on portfolio volatility, using the standard deviation of returns as a proxy for volatility. Today we will begin to a two-part series on additional statistics that aid our understanding of return dispersion: skewness and kurtosis. Beyond being fancy words and required vocabulary for CFA level 1, these two concepts are both important and fascinating for lovers of returns distributions. For today, we will focus on skewness.</p>
<p>Skewness is the degree to which returns are asymmetric around the mean. Since a normal distribution is symmetric around the mean, skewness can be taken as one measure of how returns are not distributed normally. Why does skewness matter? If portfolio returns are right, or positively, skewed, it implies numerous small negative returns and a few large positive returns. If portfolio returns are left, or negatively, skewed, it implies numerous small positive returns and few large negative returns. The phrase “large negative returns” should trigger Pavlovian sweating for investors, even if it’s preceded by a diminutive modifier like “just a few”. For a portfolio manager, a negatively skewed distribution of returns implies a portfolio at risk of rare but large losses. This makes us nervous and is a bit like saying, “I’m healthy, except for my occasional massive heart attack.”</p>
<p>Let’s get to it.</p>
<p>First, have a look at one equation for skewness:</p>
<p><span class="math display">\[Skew=\sum_{t=1}^n (x_i-\overline{x})^3/n \bigg/ (\sum_{t=1}^n (x_i-\overline{x})^2/n)^{3/2}\]</span></p>
<p>Skew has important substantive implications for risk, and is also a concept that lends itself to data visualization. In fact, I find the visualizations of skewness more illuminating than the numbers themselves (though the numbers are what matter in the end). In this section, we will cover how to calculate skewness using <code>xts</code> and <code>tidyverse</code> methods, how to calculate rolling skewness, and how to create several data visualizations as pedagogical aids. We will be working with our usual portfolio consisting of:</p>
<pre><code>+ SPY (S&P500 fund) weighted 25%
+ EFA (a non-US equities fund) weighted 25%
+ IJS (a small-cap value fund) weighted 20%
+ EEM (an emerging-mkts fund) weighted 20%
+ AGG (a bond fund) weighted 10%</code></pre>
<p>Before we can calculate the skewness, we need to find portfolio monthly returns, which was covered in <a href="https://rviews.rstudio.com/2017/10/11/from-asset-to-portfolio-returns/">this post</a>.</p>
<p>Building off that previous work, we will be working with two objects of portfolio returns:</p>
<pre><code>+ portfolio_returns_xts_rebalanced_monthly (an xts of monthly returns)
+ portfolio_returns_tq_rebalanced_monthly (a tibble of monthly returns)</code></pre>
<p>Let’s begin in the <code>xts</code> world and make use of the <code>skewness()</code> function from <code>PerformanceAnalytics</code>.</p>
<pre class="r"><code>library(PerformanceAnalytics)
skew_xts <- skewness(portfolio_returns_xts_rebalanced_monthly$returns)
skew_xts</code></pre>
<pre><code>## [1] -0.1710568</code></pre>
<p>Our portfolio is relatively balanced, and a slight negative skewness of -0.1710568 is unsurprising and unworrisome. However, that final number could be omitting important information and we will resist the temptation to stop there. For example, is that slight negative skew being caused by one very large negative monthly return? If so, what happened? Or is it caused by several medium-sized negative returns? What caused those? Were they consecutive? Are they seasonal? We need to investigate further.</p>
<p>Before doing so and having fun with data visualization, let’s explore the <code>tidyverse</code> methods and confirm consistent results.</p>
<p>We will make use of the same <code>skewness()</code> function, but because we are using a tibble, we use <code>summarise()</code> as well and call <code>summarise(skew = skewness(returns)</code>. It’s not necessary, but we are also going to run this calculation by hand, the same as we have done with standard deviation. Feel free to delete the by-hand section from your code should this be ported to enterprise scripts, but keep in mind that there is a benefit to forcing ourselves and loved ones to write out equations: it emphasizes what those nice built-in functions are doing under the hood. If a client, customer or risk officer were ever to drill into our skewness calculations, it would be nice to have a super-firm grasp on the equation.</p>
<pre class="r"><code>library(tidyverse)
library(tidyquant)
skew_tidy <-
portfolio_returns_tq_rebalanced_monthly %>%
summarise(skew_builtin = skewness(returns),
skew_byhand =
(sum((returns - mean(returns))^3)/length(returns))/
((sum((returns - mean(returns))^2)/length(returns)))^(3/2)) %>%
select(skew_builtin, skew_byhand)</code></pre>
<p>Let’s confirm that we have consistent calculations.</p>
<pre class="r"><code>skew_xts</code></pre>
<pre><code>## [1] -0.1710568</code></pre>
<pre class="r"><code>skew_tidy$skew_builtin</code></pre>
<pre><code>## [1] -0.1710568</code></pre>
<pre class="r"><code>skew_tidy$skew_byhand</code></pre>
<pre><code>## [1] -0.1710568</code></pre>
<p>The results are consistent using xts and our <code>tidyverse</code>, by-hand methods. Again, though, that singular number -0.1710568 does not fully illuminate the riskiness or distribution of this portfolio. To dig deeper, let’s first visualize the density of returns with <code>stat_density</code> from <code>ggplot2</code>.</p>
<pre class="r"><code>portfolio_density_plot <-
portfolio_returns_tq_rebalanced_monthly %>%
ggplot(aes(x = returns)) +
stat_density(geom = "line", alpha = 1, colour = "cornflowerblue")
portfolio_density_plot</code></pre>
<p><img src="/post/2017-12-13-introduction-to-skewness_files/figure-html/unnamed-chunk-4-1.png" width="672" /></p>
<p>The slight negative skew is a bit more evident here. It would be nice to shade the area that falls below some threshold again, and let’s go with the mean return. To do that, let’s create an object called <code>shaded_area</code> using <code>ggplot_build(portfolio_density_plot)$data[[1]] %>% filter(x < mean(portfolio_returns_tq_rebalanced_monthly$returns))</code>. That snippet will take our original <code>ggplot</code> object and create a new object filtered for x values less than mean return. Then we use <code>geom_area</code> to add the shaded area to <code>portfolio_density_plot</code>.</p>
<pre class="r"><code>shaded_area_data <-
ggplot_build(portfolio_density_plot)$data[[1]] %>%
filter(x < mean(portfolio_returns_tq_rebalanced_monthly$returns))
portfolio_density_plot_shaded <-
portfolio_density_plot +
geom_area(data = shaded_area_data, aes(x = x, y = y), fill="pink", alpha = 0.5)
portfolio_density_plot_shaded</code></pre>
<p><img src="/post/2017-12-13-introduction-to-skewness_files/figure-html/unnamed-chunk-5-1.png" width="672" /></p>
<p>The shaded area highlights the mass of returns that fall below the mean. Let’s add a vertical line at the mean and median, and some explanatory labels. This will help to emphasize that negative skew indicates a mean less than the median.</p>
<p>First, create variables for mean and median so that we can add a vertical line.</p>
<pre class="r"><code>median <- median(portfolio_returns_tq_rebalanced_monthly$returns)
mean <- mean(portfolio_returns_tq_rebalanced_monthly$returns)</code></pre>
<p>We want the vertical lines to just touch the density plot so we once again use a call to <code>ggplot_build(portfolio_density_plot)$data[[1]]</code>.</p>
<pre class="r"><code>median_line_data <-
ggplot_build(portfolio_density_plot)$data[[1]] %>%
filter(x <= median)</code></pre>
<p>Now we can start adding aesthetics to the latest iteration of our graph, which is stored in the object <code>portfolio_density_plot_shaded</code>.</p>
<pre class="r"><code>portfolio_density_plot_shaded +
geom_segment(aes(x = 0, y = 1.9, xend = -.045, yend = 1.9),
arrow = arrow(length = unit(0.5, "cm")), size = .05) +
annotate(geom = "text", x = -.02, y = .1, label = "returns <= mean",
fontface = "plain", alpha = .8, vjust = -1) +
geom_segment(data = shaded_area_data, aes(x = mean, y = 0, xend = mean, yend = density),
color = "red", linetype = "dotted") +
annotate(geom = "text", x = mean, y = 5, label = "mean", color = "red",
fontface = "plain", angle = 90, alpha = .8, vjust = -1.75) +
geom_segment(data = median_line_data, aes(x = median, y = 0, xend = median, yend = density),
color = "black", linetype = "dotted") +
annotate(geom = "text", x = median, y = 5, label = "median",
fontface = "plain", angle = 90, alpha = .8, vjust = 1.75) +
ggtitle("Density Plot Illustrating Skewness")</code></pre>
<p><img src="/post/2017-12-13-introduction-to-skewness_files/figure-html/unnamed-chunk-8-1.png" width="672" /></p>
<p>We added quite a bit to the chart, possibly too much, but it’s better to be over-inclusive now to test different variants. We can delete any of those features when using this chart later, or refer back to these lines of code should we ever want to reuse some of the aesthetics.</p>
<p>At this point, we have calculated the skewness of this portfolio throughout its history, and done so using three methods. We have also created an explanatory visualization.</p>
<p>Similar to the portfolio standard deviation, though, our work is not complete until we look at rolling skewness. Perhaps the first two years of the portfolio were positive skewed, and last two were negative skewed but the overall skewness is slightly negative. We would like to understand how the skewness has changed over time, and in different economic and market regimes. To do so, we calculate and visualize the rolling skewness over time.</p>
<p>In the xts world, calculating rolling skewness is almost identical to calculating rolling standard deviation, except we call the <code>skewness()</code> function instead of <code>StdDev()</code>. Since this is a rolling calculation, we need a window of time for each skewness; here, we will use a six-month window.</p>
<pre class="r"><code>window <- 6
rolling_skew_xts <- na.omit(rollapply(portfolio_returns_xts_rebalanced_monthly, window,
function(x) skewness(x)))</code></pre>
<p>Now we pop that <code>xts</code> object into <code>highcharter</code> for a visualization. Let’s make sure our y-axis range is large enough to capture the nature of the rolling skewness fluctuations by setting the range to between 3 and -3 with <code>hc_yAxis(..., max = 3, min = -3)</code>. I find that if we keep the range from 1 to -1, it makes most rolling skews look like a roller coaster.</p>
<pre class="r"><code>library(highcharter)
highchart(type = "stock") %>%
hc_title(text = "Rolling") %>%
hc_add_series(rolling_skew_xts, name = "Rolling skewness", color = "cornflowerblue") %>%
hc_yAxis(title = list(text = "skewness"),
opposite = FALSE,
max = 3,
min = -3) %>%
hc_navigator(enabled = FALSE) %>%
hc_scrollbar(enabled = FALSE) </code></pre>
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<p>For completeness of methods, we can calculate rolling skewness in a <code>tibble</code> and then use <code>ggplot</code>.</p>
<p>We will make use of <code>rollapply()</code> from within <code>tq_mutate</code> in <code>tidyquant</code>.</p>
<pre class="r"><code>rolling_skew_tidy <-
portfolio_returns_tq_rebalanced_monthly %>%
tq_mutate(select = returns,
mutate_fun = rollapply,
width = window,
FUN = skewness,
col_rename = "skew")</code></pre>
<p><code>rolling_skew_tidy</code> is ready for <code>ggplot</code>. <code>ggplot</code> is not purpose-built for time series plotting, but we can set <code>aes(x = date, y = skew)</code> to make the x-axis our date values.</p>
<pre class="r"><code>library(scales)
theme_update(plot.title = element_text(hjust = 0.5))
rolling_skew_tidy %>%
ggplot(aes(x = date, y = skew)) +
geom_line(color = "cornflowerblue") +
ggtitle("Rolling Skew with ggplot") +
ylab(paste("Rolling", window, "month skewness", sep = " ")) +
scale_y_continuous(limits = c(-3, 3), breaks = pretty_breaks(n = 8)) +
scale_x_date(breaks = pretty_breaks(n = 8))</code></pre>
<p><img src="/post/2017-12-13-introduction-to-skewness_files/figure-html/unnamed-chunk-12-1.png" width="672" /></p>
<p>The rolling charts are quite illuminating and show that the six-month-interval skewness has been positive for about half the lifetime of this portfolio. Today, the overall skewness is negative, but the rolling skewness in mid-2016 was positive and greater than 1. It took a huge plunge starting at the end of 2016, and the lowest reading was -1.65 in March of 2017, most likely caused by one or two very large negative returns when the market was worried about the US election. We can see those worries start to abate as the rolling skewness becomes more positive throughout 2017.</p>
<p>That’s all for today. Thanks for reading and see you next time when we tackle kurtosis.</p>
<script>window.location.href='https://rviews.rstudio.com/2017/12/13/introduction-to-skewness/';</script>
Introduction to Visualizing Asset Returns
https://rviews.rstudio.com/2017/11/09/introduction-to-visualizing-asset-returns/
Thu, 09 Nov 2017 00:00:00 +0000https://rviews.rstudio.com/2017/11/09/introduction-to-visualizing-asset-returns/
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<p>In a <a href="https://rviews.rstudio.com/2017/10/11/from-asset-to-portfolio-returns/">previous post</a>, we reviewed how to import daily prices, build a portfolio, and calculate portfolio returns. Today, we will visualize the returns of our individual assets that ultimately get mashed into a portfolio. The motivation here is to make sure we have scrutinized our assets before they get into our portfolio, because once the portfolio has been constructed, it is tempting to keep the analysis at the portfolio level.</p>
<p>By way of a quick reminder, our ultimate portfolio consists of the following.</p>
<pre><code>+ SPY (S&P 500 fund) weighted 25%
+ EFA (a non-US equities fund) weighted 25%
+ IJS (a small-cap value fund) weighted 20%
+ EEM (an emerging-mkts fund) weighted 20%
+ AGG (a bond fund) weighted 10%</code></pre>
<p>Let’s load up our packages.</p>
<pre class="r"><code>library(tidyverse)
library(tidyquant)
library(timetk)
library(tibbletime)
library(highcharter)</code></pre>
<p>To get our objects into the global environment, we use the next code chunk, which should look familiar from the previous post: we will create one <code>xts</code> object and one <code>tibble</code>, in long/tidy format, of monthly log returns.</p>
<pre class="r"><code># The symbols vector holds our tickers.
symbols <- c("SPY","EFA", "IJS", "EEM","AGG")
prices <-
getSymbols(symbols, src = 'yahoo', from = "2005-01-01",
auto.assign = TRUE, warnings = FALSE) %>%
map(~Ad(get(.))) %>%
reduce(merge) %>%
`colnames<-`(symbols)
# XTS method
prices_monthly <- to.monthly(prices, indexAt = "last", OHLC = FALSE)
asset_returns_xts <- na.omit(Return.calculate(prices_monthly, method = "log"))
# Tidyverse method, to long, tidy format
asset_returns_long <-
prices %>%
to.monthly(indexAt = "last", OHLC = FALSE) %>%
tk_tbl(preserve_index = TRUE, rename_index = "date") %>%
gather(asset, returns, -date) %>%
group_by(asset) %>%
mutate(returns = (log(returns) - log(lag(returns))))</code></pre>
<p>We now have two objects holding monthly log returns, <code>asset_returns_xts</code> and <code>asset_returns_long</code>. First, let’s use <code>highcharter</code> to visualize the <code>xts</code> formatted returns.</p>
<p>Highcharter is fantastic for visualizing a time series or many time series. First, we set <code>highchart(type = "stock")</code> to get a nice time series line. Then we add each of our series to the highcharter code flow. In this case, we’ll add our columns from the xts object.</p>
<pre class="r"><code>highchart(type = "stock") %>%
hc_title(text = "Monthly Log Returns") %>%
hc_add_series(asset_returns_xts$SPY,
name = names(asset_returns_xts$SPY)) %>%
hc_add_series(asset_returns_xts$EFA,
name = names(asset_returns_xts$EFA)) %>%
hc_add_series(asset_returns_xts$IJS,
name = names(asset_returns_xts$IJS)) %>%
hc_add_theme(hc_theme_flat()) %>%
hc_navigator(enabled = FALSE) %>%
hc_scrollbar(enabled = FALSE)</code></pre>
<<<<<<< HEAD
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<p>Take a look at the chart. It has a line for the monthly log returns of 3 of our ETFs (and in my opinion it’s already starting to get crowded). We might be able to pull some useful intuition from this chart. Perhaps one of our ETFs remained stable during the 2008 financial crisis, or had an era of consistently negative/positive returns. Highcharter is great for plotting time series line charts.</p>
<p>Highcharter does have the capacity for histogram making. One method is to first call the base function <code>hist</code> on the data along with the arguments for breaks and <code>plot = FALSE</code>. Then we can call <code>hchart</code> on that object.</p>
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