Week 13: Code along 12
CHAPTER 10 Component Contribution to Standard Deviation
Julius Mondschein
2024-06-25
# 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 -0.0062315646 -0.0029356841 0.0366066739 0.052133642 4.992290e-02
## 2013-02-28 0.0058908422 -0.0231051224 -0.0129697514 0.016175193 1.267827e-02
## 2013-03-28 0.0009851843 -0.0102350107 0.0129697514 0.040258133 3.726775e-02
## 2013-04-30 0.0096392076 0.0120849040 0.0489675366 0.001222682 1.903024e-02
## 2013-05-31 -0.0202139874 -0.0494834901 -0.0306556861 0.041976043 2.333539e-02
## 2013-06-28 -0.0157787936 -0.0547283223 -0.0271442917 -0.001402810 -1.343401e-02
## 2013-07-31 0.0026882309 0.0131596979 0.0518602560 0.063541216 5.038568e-02
## 2013-08-30 -0.0082976307 -0.0257054199 -0.0197461995 -0.034743342 -3.045155e-02
## 2013-09-30 0.0111433289 0.0695888236 0.0753385742 0.063873860 3.115612e-02
## 2013-10-31 0.0082919457 0.0408610601 0.0320815512 0.034233976 4.526647e-02
## 2013-11-29 -0.0025097366 -0.0025940275 0.0054497009 0.041661287 2.920742e-02
## 2013-12-31 -0.0055832250 -0.0040743613 0.0215278976 0.012892054 2.559580e-02
## 2014-01-31 0.0152916379 -0.0903227368 -0.0534132249 -0.035775134 -3.588403e-02
## 2014-02-28 0.0037568319 0.0332206618 0.0595049311 0.045257168 4.450970e-02
## 2014-03-31 -0.0014809240 0.0380217573 -0.0046024895 0.013315194 8.261529e-03
## 2014-04-30 0.0081829692 0.0077726260 0.0165294466 -0.023184041 6.927565e-03
## 2014-05-30 0.0117220476 0.0290913038 0.0158284238 0.006204980 2.294123e-02
## 2014-06-30 -0.0005761783 0.0237339410 0.0091654077 0.037718885 2.043468e-02
## 2014-07-31 -0.0025125411 0.0135555715 -0.0263797798 -0.052009344 -1.352872e-02
## 2014-08-29 0.0114309435 0.0279046807 0.0018004986 0.043658011 3.870508e-02
## 2014-09-30 -0.0061667580 -0.0808567872 -0.0395986457 -0.061260743 -1.389297e-02
## 2014-10-31 0.0105842622 0.0140964517 -0.0026547319 0.068874821 2.327798e-02
## 2014-11-28 0.0065487608 -0.0155412963 0.0006253892 0.004774032 2.710170e-02
## 2014-12-31 0.0014752336 -0.0404422264 -0.0407468895 0.025295810 -2.540050e-03
## 2015-01-30 0.0203148160 -0.0068956168 0.0062263408 -0.054628245 -3.007699e-02
## 2015-02-27 -0.0089882361 0.0431359040 0.0614508588 0.056914637 5.468201e-02
## 2015-03-31 0.0037408140 -0.0150861372 -0.0143889400 0.010156379 -1.583048e-02
## 2015-04-30 -0.0032332525 0.0662814149 0.0358166605 -0.018417601 9.786079e-03
## 2015-05-29 -0.0043838122 -0.0419111291 0.0019527235 0.007509804 1.277420e-02
## 2015-06-30 -0.0108252441 -0.0297462626 -0.0316789031 0.004171457 -2.052118e-02
## 2015-07-31 0.0085842190 -0.0651784226 0.0201146365 -0.027375578 2.233777e-02
## 2015-08-31 -0.0033635547 -0.0925124728 -0.0771525624 -0.047268108 -6.288700e-02
## 2015-09-30 0.0080816911 -0.0318249253 -0.0451949829 -0.038464955 -2.584690e-02
## 2015-10-30 0.0006853482 0.0618085079 0.0640258975 0.063589885 8.163484e-02
## 2015-11-30 -0.0038985034 -0.0255605362 -0.0075555551 0.024415308 3.648562e-03
## 2015-12-31 -0.0019191984 -0.0389471020 -0.0235951925 -0.052157028 -1.743339e-02
## 2016-01-29 0.0123301913 -0.0516366010 -0.0567577314 -0.060307023 -5.106879e-02
## 2016-02-29 0.0088316634 -0.0082117552 -0.0339140811 0.020605230 -8.264234e-04
## 2016-03-31 0.0087091057 0.1218791941 0.0637456938 0.089910353 6.510018e-02
## 2016-04-29 0.0025462449 0.0040791492 0.0219751842 0.021044297 3.933751e-03
## 2016-05-31 0.0001353738 -0.0376285768 -0.0008560460 0.004396951 1.686836e-02
## 2016-06-30 0.0191667117 0.0445824706 -0.0244914502 0.008292413 3.469734e-03
## 2016-07-29 0.0054299839 0.0524421527 0.0390000281 0.049348321 3.582182e-02
## 2016-08-31 -0.0021565153 0.0087984755 0.0053269456 0.011261077 1.196862e-03
## 2016-09-30 0.0005156207 0.0248728142 0.0132791585 0.008614789 5.819965e-05
## 2016-10-31 -0.0082047991 -0.0083120694 -0.0224036669 -0.038135178 -1.748908e-02
## 2016-11-30 -0.0259898089 -0.0451617686 -0.0179745666 0.125246893 3.617620e-02
## 2016-12-30 0.0025378974 -0.0025300740 0.0267028628 0.031491719 2.006896e-02
## 2017-01-31 0.0021266027 0.0644315155 0.0323819590 -0.012143844 1.773633e-02
## 2017-02-28 0.0064373304 0.0172578369 0.0118363924 0.013428546 3.853939e-02
## 2017-03-31 -0.0005525665 0.0361888681 0.0318057133 -0.006533076 1.249217e-03
## 2017-04-28 0.0090290606 0.0168662874 0.0239523650 0.005107970 9.877293e-03
## 2017-05-31 0.0068471886 0.0280600984 0.0348101105 -0.022862755 1.401400e-02
## 2017-06-30 -0.0001826005 0.0092236570 0.0029559265 0.029152067 6.354754e-03
## 2017-07-31 0.0033346330 0.0565946495 0.0261877281 0.007481197 2.034585e-02
## 2017-08-31 0.0093695214 0.0232435724 -0.0004482236 -0.027564532 2.913422e-03
## 2017-09-29 -0.0057323988 -0.0004460965 0.0233429133 0.082321617 1.994914e-02
## 2017-10-31 0.0009777967 0.0322784749 0.0166537293 0.005916156 2.329076e-02
## 2017-11-30 -0.0014840998 -0.0038970493 0.0068697707 0.036913336 3.010785e-02
## 2017-12-29 0.0047402862 0.0369256799 0.0133984037 -0.003731415 1.205500e-02# Covariance of asset returns
covariance_matrix <- cov(asset_returns_wide_tbl)
covariance_matrix## AGG EEM EFA IJS SPY
## AGG 7.398378e-05 0.0001042101 4.178048e-05 -7.812162e-05 -9.032562e-06
## EEM 1.042101e-04 0.0017547132 1.039018e-03 6.437737e-04 6.795432e-04
## EFA 4.178048e-05 0.0010390175 1.064239e-03 6.490303e-04 6.975407e-04
## IJS -7.812162e-05 0.0006437737 6.490303e-04 1.565453e-03 8.290258e-04
## SPY -9.032562e-06 0.0006795432 6.975407e-04 8.290258e-04 7.408282e-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,] 0.02347491# 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,] 0.0003874027 0.009257153 0.005815633 0.005684472 0.002330249rowSums(component_contribution)## [1] 0.02347491# 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")
asset_returns_wide_tbl## AGG EEM EFA IJS SPY
## 2013-01-31 -0.0062315646 -0.0029356841 0.0366066739 0.052133642 4.992290e-02
## 2013-02-28 0.0058908422 -0.0231051224 -0.0129697514 0.016175193 1.267827e-02
## 2013-03-28 0.0009851843 -0.0102350107 0.0129697514 0.040258133 3.726775e-02
## 2013-04-30 0.0096392076 0.0120849040 0.0489675366 0.001222682 1.903024e-02
## 2013-05-31 -0.0202139874 -0.0494834901 -0.0306556861 0.041976043 2.333539e-02
## 2013-06-28 -0.0157787936 -0.0547283223 -0.0271442917 -0.001402810 -1.343401e-02
## 2013-07-31 0.0026882309 0.0131596979 0.0518602560 0.063541216 5.038568e-02
## 2013-08-30 -0.0082976307 -0.0257054199 -0.0197461995 -0.034743342 -3.045155e-02
## 2013-09-30 0.0111433289 0.0695888236 0.0753385742 0.063873860 3.115612e-02
## 2013-10-31 0.0082919457 0.0408610601 0.0320815512 0.034233976 4.526647e-02
## 2013-11-29 -0.0025097366 -0.0025940275 0.0054497009 0.041661287 2.920742e-02
## 2013-12-31 -0.0055832250 -0.0040743613 0.0215278976 0.012892054 2.559580e-02
## 2014-01-31 0.0152916379 -0.0903227368 -0.0534132249 -0.035775134 -3.588403e-02
## 2014-02-28 0.0037568319 0.0332206618 0.0595049311 0.045257168 4.450970e-02
## 2014-03-31 -0.0014809240 0.0380217573 -0.0046024895 0.013315194 8.261529e-03
## 2014-04-30 0.0081829692 0.0077726260 0.0165294466 -0.023184041 6.927565e-03
## 2014-05-30 0.0117220476 0.0290913038 0.0158284238 0.006204980 2.294123e-02
## 2014-06-30 -0.0005761783 0.0237339410 0.0091654077 0.037718885 2.043468e-02
## 2014-07-31 -0.0025125411 0.0135555715 -0.0263797798 -0.052009344 -1.352872e-02
## 2014-08-29 0.0114309435 0.0279046807 0.0018004986 0.043658011 3.870508e-02
## 2014-09-30 -0.0061667580 -0.0808567872 -0.0395986457 -0.061260743 -1.389297e-02
## 2014-10-31 0.0105842622 0.0140964517 -0.0026547319 0.068874821 2.327798e-02
## 2014-11-28 0.0065487608 -0.0155412963 0.0006253892 0.004774032 2.710170e-02
## 2014-12-31 0.0014752336 -0.0404422264 -0.0407468895 0.025295810 -2.540050e-03
## 2015-01-30 0.0203148160 -0.0068956168 0.0062263408 -0.054628245 -3.007699e-02
## 2015-02-27 -0.0089882361 0.0431359040 0.0614508588 0.056914637 5.468201e-02
## 2015-03-31 0.0037408140 -0.0150861372 -0.0143889400 0.010156379 -1.583048e-02
## 2015-04-30 -0.0032332525 0.0662814149 0.0358166605 -0.018417601 9.786079e-03
## 2015-05-29 -0.0043838122 -0.0419111291 0.0019527235 0.007509804 1.277420e-02
## 2015-06-30 -0.0108252441 -0.0297462626 -0.0316789031 0.004171457 -2.052118e-02
## 2015-07-31 0.0085842190 -0.0651784226 0.0201146365 -0.027375578 2.233777e-02
## 2015-08-31 -0.0033635547 -0.0925124728 -0.0771525624 -0.047268108 -6.288700e-02
## 2015-09-30 0.0080816911 -0.0318249253 -0.0451949829 -0.038464955 -2.584690e-02
## 2015-10-30 0.0006853482 0.0618085079 0.0640258975 0.063589885 8.163484e-02
## 2015-11-30 -0.0038985034 -0.0255605362 -0.0075555551 0.024415308 3.648562e-03
## 2015-12-31 -0.0019191984 -0.0389471020 -0.0235951925 -0.052157028 -1.743339e-02
## 2016-01-29 0.0123301913 -0.0516366010 -0.0567577314 -0.060307023 -5.106879e-02
## 2016-02-29 0.0088316634 -0.0082117552 -0.0339140811 0.020605230 -8.264234e-04
## 2016-03-31 0.0087091057 0.1218791941 0.0637456938 0.089910353 6.510018e-02
## 2016-04-29 0.0025462449 0.0040791492 0.0219751842 0.021044297 3.933751e-03
## 2016-05-31 0.0001353738 -0.0376285768 -0.0008560460 0.004396951 1.686836e-02
## 2016-06-30 0.0191667117 0.0445824706 -0.0244914502 0.008292413 3.469734e-03
## 2016-07-29 0.0054299839 0.0524421527 0.0390000281 0.049348321 3.582182e-02
## 2016-08-31 -0.0021565153 0.0087984755 0.0053269456 0.011261077 1.196862e-03
## 2016-09-30 0.0005156207 0.0248728142 0.0132791585 0.008614789 5.819965e-05
## 2016-10-31 -0.0082047991 -0.0083120694 -0.0224036669 -0.038135178 -1.748908e-02
## 2016-11-30 -0.0259898089 -0.0451617686 -0.0179745666 0.125246893 3.617620e-02
## 2016-12-30 0.0025378974 -0.0025300740 0.0267028628 0.031491719 2.006896e-02
## 2017-01-31 0.0021266027 0.0644315155 0.0323819590 -0.012143844 1.773633e-02
## 2017-02-28 0.0064373304 0.0172578369 0.0118363924 0.013428546 3.853939e-02
## 2017-03-31 -0.0005525665 0.0361888681 0.0318057133 -0.006533076 1.249217e-03
## 2017-04-28 0.0090290606 0.0168662874 0.0239523650 0.005107970 9.877293e-03
## 2017-05-31 0.0068471886 0.0280600984 0.0348101105 -0.022862755 1.401400e-02
## 2017-06-30 -0.0001826005 0.0092236570 0.0029559265 0.029152067 6.354754e-03
## 2017-07-31 0.0033346330 0.0565946495 0.0261877281 0.007481197 2.034585e-02
## 2017-08-31 0.0093695214 0.0232435724 -0.0004482236 -0.027564532 2.913422e-03
## 2017-09-29 -0.0057323988 -0.0004460965 0.0233429133 0.082321617 1.994914e-02
## 2017-10-31 0.0009777967 0.0322784749 0.0166537293 0.005916156 2.329076e-02
## 2017-11-30 -0.0014840998 -0.0038970493 0.0068697707 0.036913336 3.010785e-02
## 2017-12-29 0.0047402862 0.0369256799 0.0133984037 -0.003731415 1.205500e-02calculate_component_contribution <- function(.data, w) {
# Covariance of asset returns
covariance_matrix <- cov(.data)
# Standard deviation of portfolio
# Summarizes how much each asset's returns vary with those of other assets within the portfolio into a single number
sd_portfolio <- sqrt(t(w) %*% covariance_matrix %*% w)
# 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
# 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(.25, .25, .2, .2, .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
Column Chart of Component Contribution
plot_data <- asset_returns_wide_tbl %>%
calculate_component_contribution(w = c(.25, .25, .2, .2, .1)) %>%
# transform to long form
pivot_longer(cols = everything(), names_to = "Asset", values_to = "Contribution")
plot_data %>%
ggplot(aes(x = Asset, y = Contribution)) +
geom_col(fill = "cornflowerblue") +
scale_y_continuous(labels = scales::percent_format(accuracy = 1)) +
theme(plot.title = element_text(hjust = 0.5)) +
labs(title = "Precent Contribution to Portfolio Volatility")
Column Chart of Component Contribution and Weight
plot_data <- asset_returns_wide_tbl %>%
calculate_component_contribution(w = c(.25, .25, .2, .2, .1)) %>%
# transform to long form
pivot_longer(cols = everything(), names_to = "Asset", values_to = "Contribution") %>%
# Add weight
add_column(weight = c(.25, .25, .2, .2, .1)) %>%
# transform to long
pivot_longer(cols = c(Contribution, weight), names_to = "type", values_to = "value")
plot_data %>%
ggplot(aes(x = Asset, y = value, fill = type)) +
geom_col(position = "dodge") +
scale_y_continuous(labels = scales::percent_format(accuracy = 1)) +
scale_fill_tq() +
theme(plot.title = element_text(hjust = 0.5)) +
theme_tq() +
labs(title = "Precent Contribution to Portfolio Volatility and Weight",
x = "Precent",
y = NULL)