Week 14: Code along 12
CHAPTER 10 Component Contribution to Standard Deviation
Cameron Whitty
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# Time series
library(lubridate)
library(tibbletime)##
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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 -0.0062310457 -0.0029351463 0.0366064100 0.052132546 4.992329e-02
## 2013-02-28 0.0058914525 -0.0231052317 -0.0129694742 0.016175779 1.267773e-02
## 2013-03-28 0.0009848214 -0.0102349968 0.0129694742 0.040258197 3.726801e-02
## 2013-04-30 0.0096390312 0.0120844287 0.0489675324 0.001222230 1.903041e-02
## 2013-05-31 -0.0202136539 -0.0494832365 -0.0306555956 0.041976501 2.333560e-02
## 2013-06-28 -0.0157790397 -0.0547278361 -0.0271443442 -0.001402804 -1.343466e-02
## 2013-07-31 0.0026884008 0.0131593482 0.0518603060 0.063541236 5.038593e-02
## 2013-08-30 -0.0082985786 -0.0257056421 -0.0197463333 -0.034743386 -3.045122e-02
## 2013-09-30 0.0111446788 0.0695884725 0.0753386957 0.063873442 3.115603e-02
## 2013-10-31 0.0082914918 0.0408613118 0.0320814903 0.034234260 4.526629e-02
## 2013-11-29 -0.0025098286 -0.0025938430 0.0054498453 0.041661079 2.920721e-02
## 2013-12-31 -0.0055835536 -0.0040742395 0.0215279258 0.012892132 2.559605e-02
## 2014-01-31 0.0152918846 -0.0903226652 -0.0534132683 -0.035775228 -3.588445e-02
## 2014-02-28 0.0037569150 0.0332206105 0.0595050030 0.045257347 4.451024e-02
## 2014-03-31 -0.0014820057 0.0380216985 -0.0046024932 0.013315634 8.261100e-03
## 2014-04-30 0.0081840460 0.0077727328 0.0165292640 -0.023184430 6.927655e-03
## 2014-05-30 0.0117211727 0.0290911922 0.0158284448 0.006205040 2.294107e-02
## 2014-06-30 -0.0005756182 0.0237339422 0.0091654929 0.037719053 2.043455e-02
## 2014-07-31 -0.0025121221 0.0135553537 -0.0263798113 -0.052009905 -1.352841e-02
## 2014-08-29 0.0114309487 0.0279045056 0.0018005551 0.043657957 3.870468e-02
## 2014-09-30 -0.0061674243 -0.0808564550 -0.0395986151 -0.061260494 -1.389227e-02
## 2014-10-31 0.0105844785 0.0140964574 -0.0026548517 0.068874901 2.327789e-02
## 2014-11-28 0.0065492958 -0.0155409752 0.0006252476 0.004773888 2.710122e-02
## 2014-12-31 0.0014740779 -0.0404424139 -0.0407466475 0.025295754 -2.539736e-03
## 2015-01-30 0.0203159766 -0.0068955122 0.0062265356 -0.054628047 -3.007710e-02
## 2015-02-27 -0.0089887997 0.0431360190 0.0614505830 0.056914767 5.468205e-02
## 2015-03-31 0.0037406417 -0.0150860615 -0.0143887121 0.010156465 -1.583014e-02
## 2015-04-30 -0.0032334442 0.0662812245 0.0358164204 -0.018417877 9.785609e-03
## 2015-05-29 -0.0043832917 -0.0419112674 0.0019524781 0.007510015 1.277408e-02
## 2015-06-30 -0.0108256289 -0.0297465285 -0.0316786893 0.004171131 -2.052115e-02
## 2015-07-31 0.0085845606 -0.0651781464 0.0201144184 -0.027375352 2.233793e-02
## 2015-08-31 -0.0033631090 -0.0925124034 -0.0771523474 -0.047268277 -6.288654e-02
## 2015-09-30 0.0080809270 -0.0318248882 -0.0451948496 -0.038464672 -2.584735e-02
## 2015-10-30 0.0006852829 0.0618083473 0.0640260786 0.063589765 8.163534e-02
## 2015-11-30 -0.0038979023 -0.0255603330 -0.0075559791 0.024415115 3.647987e-03
## 2015-12-31 -0.0019189286 -0.0389473501 -0.0235951381 -0.052156928 -1.743350e-02
## 2016-01-29 0.0123300847 -0.0516365505 -0.0567577056 -0.060307072 -5.106870e-02
## 2016-02-29 0.0088317389 -0.0082116920 -0.0339138310 0.020605390 -8.259655e-04
## 2016-03-31 0.0087081490 0.1218789684 0.0637457506 0.089910462 6.510024e-02
## 2016-04-29 0.0025469287 0.0040795146 0.0219748413 0.021044242 3.933169e-03
## 2016-05-31 0.0001353220 -0.0376285978 -0.0008559120 0.004397103 1.686868e-02
## 2016-06-30 0.0191658779 0.0445823535 -0.0244916138 0.008292095 3.469675e-03
## 2016-07-29 0.0054302887 0.0524420342 0.0390004125 0.049348287 3.582216e-02
## 2016-08-31 -0.0021563460 0.0087986629 0.0053266743 0.011261133 1.196729e-03
## 2016-09-30 0.0005163148 0.0248728547 0.0132791635 0.008614736 5.794373e-05
## 2016-10-31 -0.0082058670 -0.0083120763 -0.0224034948 -0.038134742 -1.748914e-02
## 2016-11-30 -0.0259892240 -0.0451618807 -0.0179747460 0.125246070 3.617624e-02
## 2016-12-30 0.0025379634 -0.0025300489 0.0267031596 0.031492126 2.006906e-02
## 2017-01-31 0.0021263418 0.0644312942 0.0323817418 -0.012143956 1.773633e-02
## 2017-02-28 0.0064373565 0.0172578991 0.0118364314 0.013428709 3.853926e-02
## 2017-03-31 -0.0005527710 0.0361890944 0.0318056710 -0.006532682 1.249196e-03
## 2017-04-28 0.0090296250 0.0168664077 0.0239523293 0.005107513 9.877302e-03
## 2017-05-31 0.0068472518 0.0280598995 0.0348102158 -0.022862866 1.401427e-02
## 2017-06-30 -0.0001826277 0.0092236606 0.0029556298 0.029151834 6.354434e-03
## 2017-07-31 0.0033343818 0.0565946395 0.0261881019 0.007481467 2.034608e-02
## 2017-08-31 0.0093693372 0.0232436118 -0.0004484313 -0.027564590 2.913287e-03
## 2017-09-29 -0.0057325031 -0.0004461574 0.0233428849 0.082321917 1.994914e-02
## 2017-10-31 0.0009780204 0.0322785999 0.0166536521 0.005916098 2.329075e-02
## 2017-11-30 -0.0014838784 -0.0038969189 0.0068700998 0.036913032 3.010802e-02
## 2017-12-29 0.0047400524 0.0369254159 0.0133983425 -0.003731078 1.205501e-02# Covariance of asset returns
covariance_matrix <- cov(asset_returns_wide_tbl)
covariance_matrix## AGG EEM EFA IJS SPY
## AGG 7.398462e-05 0.0001042070 4.178323e-05 -0.0000781197 -9.032465e-06
## EEM 1.042070e-04 0.0017547072 1.039016e-03 0.0006437726 6.795422e-04
## EFA 4.178323e-05 0.0010390161 1.064237e-03 0.0006490294 6.975411e-04
## IJS -7.811970e-05 0.0006437726 6.490294e-04 0.0015654498 8.290255e-04
## SPY -9.032465e-06 0.0006795422 6.975411e-04 0.0008290255 7.408302e-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,] 0.02347489# 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,] 0.0003874072 0.009257129 0.005815637 0.00568447 0.00233025rowSums(component_contribution)## [1] 0.02347489# 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")