Week 14: Apply 12
Eoin Hamell-Kelleher
2023-05-09
# Load packages
# Core
library(tidyverse)
library(tidyquant)
library(readr)
# Time series
library(lubridate)
library(tibbletime)
# modeling
library(broom)Goal
Examine how each asset contributes to portfolio standard deviation. This is to ensure that our risk is not concentrated in any one asset.
five stocks: “SPY”, “EFA”, “IJS”, “EEM”, “AGG” from 2012-12-31 to 2017-12-31
1 Import stock prices
symbols <- c("SPY", "EFA", "IJS", "EEM", "AGG")
prices <- tq_get(x = symbols,
get = "stock.prices",
from = "2012-12-31",
to = "2017-12-31")2 Convert prices to returns
asset_returns_tbl <- prices %>%
group_by(symbol) %>%
tq_transmute(select = adjusted,
mutate_fun = periodReturn,
period = "monthly",
type = "log") %>%
slice(-1) %>%
ungroup() %>%
set_names(c("asset", "date", "returns"))3 Component Contribution Step-by-Step
Refresh your memory on covariance with this video. Click this link Refresh your memory on matrix multiplication. Click this link
# Transform data into wide form
asset_returns_wide_tbl <- asset_returns_tbl %>%
pivot_wider(names_from = asset, values_from = returns) %>%
column_to_rownames(var = "date")
asset_returns_wide_tbl## AGG EEM EFA IJS SPY
## 2013-01-31 -0.0062311926 -0.0029353999 0.0366062698 0.052133346 4.992308e-02
## 2013-02-28 0.0058915620 -0.0231053950 -0.0129693922 0.016175396 1.267807e-02
## 2013-03-28 0.0009844680 -0.0102349089 0.0129693922 0.040258337 3.726807e-02
## 2013-04-30 0.0096391372 0.0120843648 0.0489677817 0.001222295 1.903008e-02
## 2013-05-31 -0.0202135319 -0.0494831030 -0.0306554141 0.041976213 2.333541e-02
## 2013-06-28 -0.0157783118 -0.0547284409 -0.0271446360 -0.001403290 -1.343500e-02
## 2013-07-31 0.0026875880 0.0131598108 0.0518603709 0.063541692 5.038616e-02
## 2013-08-30 -0.0082982190 -0.0257058733 -0.0197462980 -0.034743363 -3.045123e-02
## 2013-09-30 0.0111445437 0.0695889778 0.0753385007 0.063873430 3.115617e-02
## 2013-10-31 0.0082916286 0.0408611647 0.0320816277 0.034234073 4.526666e-02
## 2013-11-29 -0.0025100732 -0.0025941891 0.0054496983 0.041660963 2.920676e-02
## 2013-12-31 -0.0055833754 -0.0040741852 0.0215280708 0.012892396 2.559621e-02
## 2014-01-31 0.0152918839 -0.0903225241 -0.0534131976 -0.035775465 -3.588454e-02
## 2014-02-28 0.0037569263 0.0332203074 0.0595049505 0.045257656 4.451041e-02
## 2014-03-31 -0.0014818573 0.0380219842 -0.0046026026 0.013315236 8.261305e-03
## 2014-04-30 0.0081839040 0.0077726777 0.0165294199 -0.023184299 6.927368e-03
## 2014-05-30 0.0117214732 0.0290909599 0.0158284266 0.006205302 2.294127e-02
## 2014-06-30 -0.0005757341 0.0237340696 0.0091652299 0.037718885 2.043454e-02
## 2014-07-31 -0.0025121136 0.0135555467 -0.0263796551 -0.052009557 -1.352841e-02
## 2014-08-29 0.0114310679 0.0279047091 0.0018003039 0.043657827 3.870462e-02
## 2014-09-30 -0.0061679402 -0.0808566890 -0.0395981671 -0.061260322 -1.389244e-02
## 2014-10-31 0.0105846395 0.0140964530 -0.0026549510 0.068874624 2.327789e-02
## 2014-11-28 0.0065493559 -0.0155413204 0.0006253941 0.004773860 2.710130e-02
## 2014-12-31 0.0014747100 -0.0404419724 -0.0407468287 0.025295898 -2.539942e-03
## 2015-01-30 0.0203154948 -0.0068957467 0.0062265852 -0.054628216 -3.007704e-02
## 2015-02-27 -0.0089889384 0.0431361290 0.0614505787 0.056914681 5.468212e-02
## 2015-03-31 0.0037405952 -0.0150863546 -0.0143888272 0.010156398 -1.583013e-02
## 2015-04-30 -0.0032328217 0.0662813686 0.0358164986 -0.018417676 9.785637e-03
## 2015-05-29 -0.0043837170 -0.0419110105 0.0019526917 0.007509721 1.277425e-02
## 2015-06-30 -0.0108256141 -0.0297466579 -0.0316787464 0.004171549 -2.052134e-02
## 2015-07-31 0.0085847607 -0.0651780755 0.0201143765 -0.027375551 2.233798e-02
## 2015-08-31 -0.0033640373 -0.0925124157 -0.0771523775 -0.047268237 -6.288659e-02
## 2015-09-30 0.0080814639 -0.0318249314 -0.0451949311 -0.038464356 -2.584702e-02
## 2015-10-30 0.0006855854 0.0618082004 0.0640259285 0.063589509 8.163469e-02
## 2015-11-30 -0.0038981858 -0.0255603267 -0.0075556977 0.024415169 3.648338e-03
## 2015-12-31 -0.0019192217 -0.0389471503 -0.0235951354 -0.052157113 -1.743338e-02
## 2016-01-29 0.0123300189 -0.0516369092 -0.0567578984 -0.060306770 -5.106871e-02
## 2016-02-29 0.0088313198 -0.0082114634 -0.0339138008 0.020605281 -8.261809e-04
## 2016-03-31 0.0087093580 0.1218790178 0.0637455859 0.089910304 6.510026e-02
## 2016-04-29 0.0025460805 0.0040792904 0.0219752057 0.021044191 3.933117e-03
## 2016-05-31 0.0001355689 -0.0376284622 -0.0008562266 0.004396887 1.686872e-02
## 2016-06-30 0.0191668473 0.0445822638 -0.0244913881 0.008292535 3.470050e-03
## 2016-07-29 0.0054297239 0.0524423143 0.0390001135 0.049348244 3.582163e-02
## 2016-08-31 -0.0021566404 0.0087983786 0.0053268421 0.011261140 1.197118e-03
## 2016-09-30 0.0005160925 0.0248730027 0.0132791871 0.008614605 5.781623e-05
## 2016-10-31 -0.0082051509 -0.0083122303 -0.0224037254 -0.038134764 -1.748910e-02
## 2016-11-30 -0.0259900675 -0.0451616881 -0.0179744481 0.125246299 3.617611e-02
## 2016-12-30 0.0025384292 -0.0025300265 0.0267030340 0.031491858 2.006916e-02
## 2017-01-31 0.0021258274 0.0644314713 0.0323818541 -0.012143733 1.773650e-02
## 2017-02-28 0.0064382233 0.0172577695 0.0118362426 0.013428467 3.853918e-02
## 2017-03-31 -0.0005530170 0.0361889948 0.0318058872 -0.006533198 1.249130e-03
## 2017-04-28 0.0090295199 0.0168663683 0.0239520280 0.005107994 9.877179e-03
## 2017-05-31 0.0068472999 0.0280599886 0.0348103459 -0.022862434 1.401426e-02
## 2017-06-30 -0.0001827013 0.0092237211 0.0029558685 0.029151717 6.354781e-03
## 2017-07-31 0.0033345293 0.0565944875 0.0261878299 0.007481667 2.034572e-02
## 2017-08-31 0.0093690314 0.0232437060 -0.0004482268 -0.027565040 2.913399e-03
## 2017-09-29 -0.0057320286 -0.0004461787 0.0233428412 0.082322018 1.994925e-02
## 2017-10-31 0.0009777077 0.0322784865 0.0166537060 0.005916103 2.329065e-02
## 2017-11-30 -0.0014839913 -0.0038969510 0.0068696192 0.036913076 3.010792e-02
## 2017-12-29 0.0047404255 0.0369254109 0.0133986185 -0.003731106 1.205508e-02# Covariance of asset returns
covariance_matrix <- cov(asset_returns_wide_tbl)
covariance_matrix## AGG EEM EFA IJS SPY
## AGG 7.398568e-05 0.0001042116 4.178371e-05 -7.811936e-05 -9.030653e-06
## EEM 1.042116e-04 0.0017547101 1.039016e-03 6.437751e-04 6.795436e-04
## EFA 4.178371e-05 0.0010390161 1.064236e-03 6.490284e-04 6.975400e-04
## IJS -7.811936e-05 0.0006437751 6.490284e-04 1.565449e-03 8.290247e-04
## SPY -9.030653e-06 0.0006795436 6.975400e-04 8.290247e-04 7.408288e-04# Standard deviation of portfolio
# Summarizes how much each asset's returns vary with those of other assets within the portfolio into a single number
w <- c(0.25, 0.25, 0.2, 0.2, 0.1)
sd_portfolio <- sqrt(t(w) %*% covariance_matrix %*% w)
sd_portfolio## [,1]
## [1,] 0.02347492# Component contribution
# Similar to the formula for sd_portfolio
# Mathematical trick to summarize the same, sd_portfolio, by asset instead of a single number
component_contribution <- (t(w) %*% covariance_matrix * w) / sd_portfolio[1,1]
component_contribution## AGG EEM EFA IJS SPY
## [1,] 0.0003874256 0.009257147 0.005815628 0.005684467 0.002330249rowSums(component_contribution)## [1] 0.02347492# Component contribution in percentage
component_percentages <- (component_contribution / sd_portfolio[1,1]) %>%
round(3) %>%
as_tibble()
component_percentages## # A tibble: 1 × 5
## AGG EEM EFA IJS SPY
## <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 0.017 0.394 0.248 0.242 0.099component_percentages %>%
as_tibble() %>%
gather(key = "asset", value = "contribution")## # A tibble: 5 × 2
## asset contribution
## <chr> <dbl>
## 1 AGG 0.017
## 2 EEM 0.394
## 3 EFA 0.248
## 4 IJS 0.242
## 5 SPY 0.0994 Component Contribution with a Custom Function
# Transform data into wide form
asset_returns_wide_tbl <- asset_returns_tbl %>%
pivot_wider(names_from = asset, values_from = returns) %>%
column_to_rownames(var = "date")
# Custom function
calculate_component_contribution <- function(asset_returns_wide_tbl, w) {
# Covariance of asset returns
covariance_matrix <- cov(asset_returns_wide_tbl)
# Standard deviation of portfolio
sd_portfolio <- sqrt(t(w) %*% covariance_matrix %*% w)
# Component contribution
component_contribution <- (t(w) %*% covariance_matrix * w) / sd_portfolio[1,1]
# Component contribution in percentage
component_percentages <- (component_contribution / sd_portfolio[1,1]) %>%
round(3) %>%
as_tibble()
return(component_percentages)
}
asset_returns_wide_tbl %>% calculate_component_contribution(w = c(0.25,0.25,0.2,0.2,0.1))## # A tibble: 1 × 5
## AGG EEM EFA IJS SPY
## <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 0.017 0.394 0.248 0.242 0.0995 Visualizing Component Contribution
# Figure 10.1 Contribution to Standard Deviation ----
asset_returns_wide_tbl %>%
calculate_component_contribution(w = c(0.25,0.25,0.2,0.2,0.1)) %>%
gather(key = "asset", value = "contribution") %>%
ggplot(aes(asset, contribution)) +
geom_col(fill = "cornflowerblue") +
theme(plot.title = element_text(hjust = 0.5)) +
scale_y_continuous(labels = scales::percent_format(accuracy = 1)) +
labs(title = "Percent Contribution to Portfolio Standard Deviation",
y = "Percent Contribution to Risk",
x = NULL)
# Figure 10.2 Weight versus Contribution ----
asset_returns_wide_tbl %>%
calculate_component_contribution(w = c(0.25,0.25,0.2,0.2,0.1)) %>%
gather(key = "asset", value = "contribution") %>%
add_column(weights = c(0.25,0.25,0.2,0.2,0.1)) %>%
pivot_longer(cols = c(contribution, weights), names_to = "type", values_to = "value") %>%
ggplot(aes(asset, value, fill = type)) +
geom_col(position = "dodge") +
theme(plot.title = element_text(hjust = 0.5)) +
scale_y_continuous(labels = scales::percent_format(accuracy = 1)) +
theme_tq() +
scale_fill_tq() +
labs(title = "Percent Contribution to Volatility",
y = "percent",
x = "asset") 
6 Rolling Component Contribution
calculate_comp_contrib_by_window <- function(asset_returns_wide_tbl,
start = 1,
window = 24,
weights) {
# 1 Define start date
start_date <- rownames(asset_returns_wide_tbl)[start]
# 2 Define end date
end_date <- rownames(asset_returns_wide_tbl)[start + window]
# 3 Subset df
df_subset <- asset_returns_wide_tbl %>%
rownames_to_column(var = "date") %>%
filter(date >= start_date & date < end_date) %>%
column_to_rownames(var = "date")
# 4 Calculate component contribution
component_percentages <-df_subset %>%
calculate_component_contribution(w = weights)
# 5 Add end date to df
component_percentages %>%
mutate(date = ymd(end_date)) %>%
select(date, everything())
}
# Check the custom function
asset_returns_wide_tbl %>% calculate_comp_contrib_by_window(start = 1, window = 24,
w = c(0.25,0.25,0.2,0.2,0.1))## # A tibble: 1 × 6
## date AGG EEM EFA IJS SPY
## <date> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 2015-01-30 0.039 0.372 0.256 0.245 0.088asset_returns_wide_tbl %>% calculate_comp_contrib_by_window(start = 2, window = 24,
w = c(0.25,0.25,0.2,0.2,0.1))## # A tibble: 1 × 6
## date AGG EEM EFA IJS SPY
## <date> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 2015-02-27 0.036 0.374 0.249 0.252 0.089dump(list = c("calculate_component_contribution",
"calculate_comp_contrib_by_window"),
file = "../00_scripts/calculate_comp_contrib_to_portfolio_volatility.R")# Iterate the custom function
w <- c(0.25,0.25,0.2,0.2,0.1)
window <- 24
rolling_comp_contrib_tbl <- 1:(nrow(asset_returns_wide_tbl) - window) %>%
map_df(.x = ., .f = ~calculate_comp_contrib_by_window(asset_returns_wide_tbl,
start = .x,
weights = w,
window = window))
rolling_comp_contrib_tbl## # A tibble: 36 × 6
## date AGG EEM EFA IJS SPY
## <date> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 2015-01-30 0.039 0.372 0.256 0.245 0.088
## 2 2015-02-27 0.036 0.374 0.249 0.252 0.089
## 3 2015-03-31 0.027 0.37 0.255 0.255 0.092
## 4 2015-04-30 0.027 0.372 0.256 0.252 0.093
## 5 2015-05-29 0.024 0.385 0.254 0.246 0.092
## 6 2015-06-30 0.018 0.383 0.248 0.257 0.094
## 7 2015-07-31 0.014 0.375 0.253 0.261 0.097
## 8 2015-08-31 0.013 0.404 0.237 0.257 0.09
## 9 2015-09-30 0.012 0.407 0.248 0.238 0.094
## 10 2015-10-30 0.003 0.405 0.244 0.243 0.105
## # … with 26 more rows# Figure 10.3 Component Contribution ggplot ----
rolling_comp_contrib_tbl %>%
# Transform data to long form
pivot_longer(cols = -date, names_to = "asset", values_to = "contribution") %>%
# Plot
ggplot(aes(date, contribution, color = asset)) +
geom_line() +
scale_x_date(breaks = scales::pretty_breaks(n = 7)) +
scale_y_continuous(labels = scales::percent_format()) +
annotate(geom = "text",
x = as.Date("2016-07-01"),
y = 0.03,
color = "red", size = 5,
label = str_glue("AGG dips below zero sometimes, indicating
it reduces the portfolio volatility."))
# Figure 10.4 Stacked Component Contribution ggplot ----
rolling_comp_contrib_tbl %>%
# Transform data to long form
pivot_longer(cols = -date, names_to = "asset", values_to = "contribution") %>%
# Plot
ggplot(aes(date, contribution, fill = asset)) +
geom_area() +
scale_x_date(breaks = scales::pretty_breaks(n = 7)) +
scale_y_continuous(labels = scales::percent_format()) +
annotate(geom = "text",
x = as.Date("2016-07-01"),
y = 0.08,
color = "red", size = 5,
label = str_glue("AGG dips below zero sometimes, indicating
it reduces the portfolio volatility."))