# 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 -0.0062310755 -0.0029353524  0.0366063075  0.052133749  4.992320e-02
## 2013-02-28  0.0058911741 -0.0231055673 -0.0129695710  0.016174668  1.267783e-02
## 2013-03-28  0.0009852591 -0.0102346689  0.0129695710  0.040258653  3.726810e-02
## 2013-04-30  0.0096390259  0.0120847883  0.0489677225  0.001222682  1.902988e-02
## 2013-05-31 -0.0202142584 -0.0494832470 -0.0306554215  0.041976135  2.333551e-02
## 2013-06-28 -0.0157781037 -0.0547287410 -0.0271445615 -0.001403003 -1.343447e-02
## 2013-07-31  0.0026874575  0.0131596369  0.0518603476  0.063541216  5.038602e-02
## 2013-08-30 -0.0082980847 -0.0257053596 -0.0197461995 -0.034743154 -3.045132e-02
## 2013-09-30  0.0111443367  0.0695886503  0.0753384920  0.063873584  3.115578e-02
## 2013-10-31  0.0082919009  0.0408612969  0.0320815538  0.034234064  4.526668e-02
## 2013-11-29 -0.0025093654 -0.0025942554  0.0054495429  0.041661124  2.920721e-02
## 2013-12-31 -0.0055831513 -0.0040742478  0.0215282903  0.012892297  2.559599e-02
## 2014-01-31  0.0152914271 -0.0903223719 -0.0534133799 -0.035775465 -3.588474e-02
## 2014-02-28  0.0037569207  0.0332202296  0.0595051622  0.045257498  4.451061e-02
## 2014-03-31 -0.0014817233  0.0380219391 -0.0046027981  0.013315035  8.261428e-03
## 2014-04-30  0.0081824926  0.0077726260  0.0165296002 -0.023184204  6.927077e-03
## 2014-05-30  0.0117220568  0.0290910789  0.0158281228  0.006205221  2.294123e-02
## 2014-06-30 -0.0005755446  0.0237341658  0.0091656325  0.037718731  2.043487e-02
## 2014-07-31 -0.0025123098  0.0135555715 -0.0263797036 -0.052009271 -1.352891e-02
## 2014-08-29  0.0114315357  0.0279044700  0.0018002702  0.043658092  3.870491e-02
## 2014-09-30 -0.0061674463 -0.0808569191 -0.0395984935 -0.061260743 -1.389251e-02
## 2014-10-31  0.0105837723  0.0140967943 -0.0026546525  0.068874975  2.327789e-02
## 2014-11-28  0.0065492659 -0.0155415250  0.0006251511  0.004773647  2.710135e-02
## 2014-12-31  0.0014752744 -0.0404417595 -0.0407468134  0.025295890 -2.539612e-03
## 2015-01-30  0.0203146985 -0.0068959149  0.0062265877 -0.054628016 -3.007726e-02
## 2015-02-27 -0.0089879255  0.0431359639  0.0614505401  0.056914632  5.468185e-02
## 2015-03-31  0.0037404838 -0.0150859040 -0.0143887072  0.010156304 -1.583014e-02
## 2015-04-30 -0.0032335966  0.0662809635  0.0358168090 -0.018417601  9.785906e-03
## 2015-05-29 -0.0043839472 -0.0419109109  0.0019524967  0.007509804  1.277446e-02
## 2015-06-30 -0.0108249851 -0.0297466142 -0.0316788252  0.004171383 -2.052144e-02
## 2015-07-31  0.0085847731 -0.0651781336  0.0201144823 -0.027375350  2.233786e-02
## 2015-08-31 -0.0033640153 -0.0925122045 -0.0771524861 -0.047268341 -6.288674e-02
## 2015-09-30  0.0080815874 -0.0318249895 -0.0451948966 -0.038464708 -2.584708e-02
## 2015-10-30  0.0006853688  0.0618080999  0.0640258921  0.063589484  8.163492e-02
## 2015-11-30 -0.0038982389 -0.0255603382 -0.0075557175  0.024415390  3.648391e-03
## 2015-12-31 -0.0019190911 -0.0389471047 -0.0235951944 -0.052156956 -1.743365e-02
## 2016-01-29  0.0123306487 -0.0516366048 -0.0567578247 -0.060307028 -5.106863e-02
## 2016-02-29  0.0088314709 -0.0082116050 -0.0339139044  0.020605483 -8.261495e-04
## 2016-03-31  0.0087081049  0.1218789852  0.0637457796  0.089910338  6.510026e-02
## 2016-04-29  0.0025468022  0.0040792827  0.0219751824  0.021044069  3.933324e-03
## 2016-05-31  0.0001350113 -0.0376285078 -0.0008562138  0.004397100  1.686870e-02
## 2016-06-30  0.0191668088  0.0445824016 -0.0244912802  0.008292486  3.469734e-03
## 2016-07-29  0.0054296826  0.0524420901  0.0390001903  0.049348103  3.582198e-02
## 2016-08-31 -0.0021559298  0.0087987244  0.0053266973  0.011261286  1.197184e-03
## 2016-09-30  0.0005156516  0.0248726884  0.0132790773  0.008614581  5.779714e-05
## 2016-10-31 -0.0082047357 -0.0083120078 -0.0224035026 -0.038134965 -1.748924e-02
## 2016-11-30 -0.0259901428 -0.0451617629 -0.0179746496  0.125246181  3.617613e-02
## 2016-12-30  0.0025382843 -0.0025304580  0.0267030275  0.031491920  2.006896e-02
## 2017-01-31  0.0021263055  0.0644317116  0.0323817146 -0.012143600  1.773656e-02
## 2017-02-28  0.0064377107  0.0172577789  0.0118364721  0.013428548  3.853916e-02
## 2017-03-31 -0.0005532638  0.0361891001  0.0318057133 -0.006533138  1.249290e-03
## 2017-04-28  0.0090299641  0.0168662855  0.0239523650  0.005108093  9.877076e-03
## 2017-05-31  0.0068468803  0.0280598776  0.0348099666 -0.022862819  1.401443e-02
## 2017-06-30 -0.0001824719  0.0092236580  0.0029561422  0.029152071  6.354540e-03
## 2017-07-31  0.0033342260  0.0565947574  0.0261877263  0.007481198  2.034599e-02
## 2017-08-31  0.0093691094  0.0232437716 -0.0004484333 -0.027564411  2.913352e-03
## 2017-09-29 -0.0057317982 -0.0004461961  0.0233429166  0.082321389  1.994913e-02
## 2017-10-31  0.0009776176  0.0322782788  0.0166537987  0.005916157  2.329050e-02
## 2017-11-30 -0.0014838399 -0.0038965654  0.0068698379  0.036913454  3.010817e-02
## 2017-12-29  0.0047402118  0.0369251992  0.0133983378 -0.003731086  1.205507e-02
# Covariance of asset returns
covariance_matrix <- cov(asset_returns_wide_tbl)

covariance_matrix
##               AGG          EEM          EFA           IJS           SPY
## AGG  7.398395e-05 0.0001042081 0.0000417814 -7.811944e-05 -9.032206e-06
## EEM  1.042081e-04 0.0017547073 0.0010390155  6.437744e-04  6.795426e-04
## EFA  4.178140e-05 0.0010390155 0.0010642378  6.490302e-04  6.975410e-04
## IJS -7.811944e-05 0.0006437744 0.0006490302  1.565449e-03  8.290271e-04
## SPY -9.032206e-06 0.0006795426 0.0006975410  8.290271e-04  7.408313e-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,] 0.0234749
# 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,] 0.0003874051 0.009257134 0.005815633 0.005684474 0.002330252
rowSums(component_contribution)
## [1] 0.0234749
# 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 0.099
component_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          0.099

4 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 -0.0062310755 -0.0029353524  0.0366063075  0.052133749  4.992320e-02
## 2013-02-28  0.0058911741 -0.0231055673 -0.0129695710  0.016174668  1.267783e-02
## 2013-03-28  0.0009852591 -0.0102346689  0.0129695710  0.040258653  3.726810e-02
## 2013-04-30  0.0096390259  0.0120847883  0.0489677225  0.001222682  1.902988e-02
## 2013-05-31 -0.0202142584 -0.0494832470 -0.0306554215  0.041976135  2.333551e-02
## 2013-06-28 -0.0157781037 -0.0547287410 -0.0271445615 -0.001403003 -1.343447e-02
## 2013-07-31  0.0026874575  0.0131596369  0.0518603476  0.063541216  5.038602e-02
## 2013-08-30 -0.0082980847 -0.0257053596 -0.0197461995 -0.034743154 -3.045132e-02
## 2013-09-30  0.0111443367  0.0695886503  0.0753384920  0.063873584  3.115578e-02
## 2013-10-31  0.0082919009  0.0408612969  0.0320815538  0.034234064  4.526668e-02
## 2013-11-29 -0.0025093654 -0.0025942554  0.0054495429  0.041661124  2.920721e-02
## 2013-12-31 -0.0055831513 -0.0040742478  0.0215282903  0.012892297  2.559599e-02
## 2014-01-31  0.0152914271 -0.0903223719 -0.0534133799 -0.035775465 -3.588474e-02
## 2014-02-28  0.0037569207  0.0332202296  0.0595051622  0.045257498  4.451061e-02
## 2014-03-31 -0.0014817233  0.0380219391 -0.0046027981  0.013315035  8.261428e-03
## 2014-04-30  0.0081824926  0.0077726260  0.0165296002 -0.023184204  6.927077e-03
## 2014-05-30  0.0117220568  0.0290910789  0.0158281228  0.006205221  2.294123e-02
## 2014-06-30 -0.0005755446  0.0237341658  0.0091656325  0.037718731  2.043487e-02
## 2014-07-31 -0.0025123098  0.0135555715 -0.0263797036 -0.052009271 -1.352891e-02
## 2014-08-29  0.0114315357  0.0279044700  0.0018002702  0.043658092  3.870491e-02
## 2014-09-30 -0.0061674463 -0.0808569191 -0.0395984935 -0.061260743 -1.389251e-02
## 2014-10-31  0.0105837723  0.0140967943 -0.0026546525  0.068874975  2.327789e-02
## 2014-11-28  0.0065492659 -0.0155415250  0.0006251511  0.004773647  2.710135e-02
## 2014-12-31  0.0014752744 -0.0404417595 -0.0407468134  0.025295890 -2.539612e-03
## 2015-01-30  0.0203146985 -0.0068959149  0.0062265877 -0.054628016 -3.007726e-02
## 2015-02-27 -0.0089879255  0.0431359639  0.0614505401  0.056914632  5.468185e-02
## 2015-03-31  0.0037404838 -0.0150859040 -0.0143887072  0.010156304 -1.583014e-02
## 2015-04-30 -0.0032335966  0.0662809635  0.0358168090 -0.018417601  9.785906e-03
## 2015-05-29 -0.0043839472 -0.0419109109  0.0019524967  0.007509804  1.277446e-02
## 2015-06-30 -0.0108249851 -0.0297466142 -0.0316788252  0.004171383 -2.052144e-02
## 2015-07-31  0.0085847731 -0.0651781336  0.0201144823 -0.027375350  2.233786e-02
## 2015-08-31 -0.0033640153 -0.0925122045 -0.0771524861 -0.047268341 -6.288674e-02
## 2015-09-30  0.0080815874 -0.0318249895 -0.0451948966 -0.038464708 -2.584708e-02
## 2015-10-30  0.0006853688  0.0618080999  0.0640258921  0.063589484  8.163492e-02
## 2015-11-30 -0.0038982389 -0.0255603382 -0.0075557175  0.024415390  3.648391e-03
## 2015-12-31 -0.0019190911 -0.0389471047 -0.0235951944 -0.052156956 -1.743365e-02
## 2016-01-29  0.0123306487 -0.0516366048 -0.0567578247 -0.060307028 -5.106863e-02
## 2016-02-29  0.0088314709 -0.0082116050 -0.0339139044  0.020605483 -8.261495e-04
## 2016-03-31  0.0087081049  0.1218789852  0.0637457796  0.089910338  6.510026e-02
## 2016-04-29  0.0025468022  0.0040792827  0.0219751824  0.021044069  3.933324e-03
## 2016-05-31  0.0001350113 -0.0376285078 -0.0008562138  0.004397100  1.686870e-02
## 2016-06-30  0.0191668088  0.0445824016 -0.0244912802  0.008292486  3.469734e-03
## 2016-07-29  0.0054296826  0.0524420901  0.0390001903  0.049348103  3.582198e-02
## 2016-08-31 -0.0021559298  0.0087987244  0.0053266973  0.011261286  1.197184e-03
## 2016-09-30  0.0005156516  0.0248726884  0.0132790773  0.008614581  5.779714e-05
## 2016-10-31 -0.0082047357 -0.0083120078 -0.0224035026 -0.038134965 -1.748924e-02
## 2016-11-30 -0.0259901428 -0.0451617629 -0.0179746496  0.125246181  3.617613e-02
## 2016-12-30  0.0025382843 -0.0025304580  0.0267030275  0.031491920  2.006896e-02
## 2017-01-31  0.0021263055  0.0644317116  0.0323817146 -0.012143600  1.773656e-02
## 2017-02-28  0.0064377107  0.0172577789  0.0118364721  0.013428548  3.853916e-02
## 2017-03-31 -0.0005532638  0.0361891001  0.0318057133 -0.006533138  1.249290e-03
## 2017-04-28  0.0090299641  0.0168662855  0.0239523650  0.005108093  9.877076e-03
## 2017-05-31  0.0068468803  0.0280598776  0.0348099666 -0.022862819  1.401443e-02
## 2017-06-30 -0.0001824719  0.0092236580  0.0029561422  0.029152071  6.354540e-03
## 2017-07-31  0.0033342260  0.0565947574  0.0261877263  0.007481198  2.034599e-02
## 2017-08-31  0.0093691094  0.0232437716 -0.0004484333 -0.027564411  2.913352e-03
## 2017-09-29 -0.0057317982 -0.0004461961  0.0233429166  0.082321389  1.994913e-02
## 2017-10-31  0.0009776176  0.0322782788  0.0166537987  0.005916157  2.329050e-02
## 2017-11-30 -0.0014838399 -0.0038965654  0.0068698379  0.036913454  3.010817e-02
## 2017-12-29  0.0047402118  0.0369251992  0.0133983378 -0.003731086  1.205507e-02
calculate_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, .20, .20, .10)) 
## # A tibble: 1 × 5
##     AGG   EEM   EFA   IJS   SPY
##   <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 0.017 0.394 0.248 0.242 0.099

5 Visualizing Component Contribution

Colum Chart of Compound Contribution

plot_data <- asset_returns_wide_tbl %>% 
    
    calculate_component_contribution(w = c(.25, .25, .2, .2, .1)) %>% 
    
    # Transform to long from 
    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 = "Percent Contribution to Portfolio Votality")

6 Rolling Component Contribution

Colum Chart of Compound Contribution

plot_data <- asset_returns_wide_tbl %>% 
    
    calculate_component_contribution(w = c(.25, .25, .2, .2, .1)) %>% 
    
    # Transform to long from 
    pivot_longer(cols = everything(), names_to = "Asset", values_to = "Contribution")  %>% 

    #Add weights 
    add_column(weight = c(.25, .25, .2, .2, .1 )) %>% 
    
    # Transform to log 
    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 = "Percent Contribution to Portfolio Votality and Weight")