# 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

# 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 -0.0062314447 -0.0029358074  0.0366064445  0.052133371  4.992292e-02
## 2013-02-28  0.0058912352 -0.0231050865 -0.0129695790  0.016175735  1.267868e-02
## 2013-03-28  0.0009851786 -0.0102348139  0.0129695790  0.040258026  3.726727e-02
## 2013-04-30  0.0096393156  0.0120847573  0.0489674772  0.001222315  1.903041e-02
## 2013-05-31 -0.0202141304 -0.0494836985 -0.0306553180  0.041976135  2.333540e-02
## 2013-06-28 -0.0157781614 -0.0547282506 -0.0271446453 -0.001402794 -1.343474e-02
## 2013-07-31  0.0026876946  0.0131597901  0.0518604521  0.063541364  5.038651e-02
## 2013-08-30 -0.0082977621 -0.0257054581 -0.0197463254 -0.034743527 -3.045138e-02
## 2013-09-30  0.0111431216  0.0695887396  0.0753384250  0.063873903  3.115587e-02
## 2013-10-31  0.0082926095  0.0408611570  0.0320815933  0.034233867  4.526626e-02
## 2013-11-29 -0.0025098782 -0.0025942099  0.0054496821  0.041661011  2.920724e-02
## 2013-12-31 -0.0055836075 -0.0040741683  0.0215280270  0.012892173  2.559627e-02
## 2014-01-31  0.0152917331 -0.0903226388 -0.0534133204 -0.035775201 -3.588492e-02
## 2014-02-28  0.0037569414  0.0332206284  0.0595050544  0.045257270  4.451074e-02
## 2014-03-31 -0.0014813577  0.0380218532 -0.0046025089  0.013315560  8.261279e-03
## 2014-04-30  0.0081829001  0.0077725282  0.0165293759 -0.023184184  6.927490e-03
## 2014-05-30  0.0117217321  0.0290910804  0.0158282324  0.006205094  2.294125e-02
## 2014-06-30 -0.0005753327  0.0237341897  0.0091655973  0.037718813  2.043407e-02
## 2014-07-31 -0.0025123680  0.0135553791 -0.0263796564 -0.052009482 -1.352856e-02
## 2014-08-29  0.0114307645  0.0279048146  0.0018004100  0.043657934  3.870474e-02
## 2014-09-30 -0.0061672464 -0.0808566437 -0.0395984610 -0.061260759 -1.389195e-02
## 2014-10-31  0.0105842754  0.0140964700 -0.0026548295  0.068874977  2.327766e-02
## 2014-11-28  0.0065491443 -0.0155413060  0.0006251507  0.004773810  2.710114e-02
## 2014-12-31  0.0014748859 -0.0404424143 -0.0407467370  0.025295778 -2.539361e-03
## 2015-01-30  0.0203147836 -0.0068955068  0.0062266885 -0.054628026 -3.007728e-02
## 2015-02-27 -0.0089878307  0.0431360303  0.0614504843  0.056914546  5.468184e-02
## 2015-03-31  0.0037398990 -0.0150864600 -0.0143887809  0.010156422 -1.583022e-02
## 2015-04-30 -0.0032329198  0.0662815357  0.0358165271 -0.018417764  9.786009e-03
## 2015-05-29 -0.0043832358 -0.0419107653  0.0019525568  0.007509947  1.277413e-02
## 2015-06-30 -0.0108254491 -0.0297471701 -0.0316786729  0.004171401 -2.052151e-02
## 2015-07-31  0.0085851245 -0.0651778348  0.0201145268 -0.027375637  2.233812e-02
## 2015-08-31 -0.0033645885 -0.0925122992 -0.0771525052 -0.047268287 -6.288664e-02
## 2015-09-30  0.0080816582 -0.0318250341 -0.0451948440 -0.038464602 -2.584730e-02
## 2015-10-30  0.0006852809  0.0618083383  0.0640259413  0.063589762  8.163515e-02
## 2015-11-30 -0.0038980491 -0.0255603460 -0.0075558689  0.024415008  3.648358e-03
## 2015-12-31 -0.0019192890 -0.0389470954 -0.0235949512 -0.052156904 -1.743350e-02
## 2016-01-29  0.0123299645 -0.0516366428 -0.0567578043 -0.060306861 -5.106882e-02
## 2016-02-29  0.0088315667 -0.0082117316 -0.0339140803  0.020605401 -8.260311e-04
## 2016-03-31  0.0087093526  0.1218790109  0.0637460342  0.089910359  6.509993e-02
## 2016-04-29  0.0025459438  0.0040792910  0.0219748194  0.021044188  3.933497e-03
## 2016-05-31  0.0001360958 -0.0376287589 -0.0008561906  0.004397029  1.686854e-02
## 2016-06-30  0.0191661128  0.0445825552 -0.0244914120  0.008292354  3.469884e-03
## 2016-07-29  0.0054294404  0.0524424851  0.0390002525  0.049348286  3.582208e-02
## 2016-08-31 -0.0021560705  0.0087982188  0.0053267535  0.011261075  1.196632e-03
## 2016-09-30  0.0005158869  0.0248729848  0.0132790432  0.008614744  5.791795e-05
## 2016-10-31 -0.0082047578 -0.0083122584 -0.0224035120 -0.038134683 -1.748892e-02
## 2016-11-30 -0.0259894961 -0.0451617326 -0.0179744554  0.125246397  3.617609e-02
## 2016-12-30  0.0025374568 -0.0025301331  0.0267028261  0.031491513  2.006900e-02
## 2017-01-31  0.0021261847  0.0644313570  0.0323818299 -0.012143914  1.773636e-02
## 2017-02-28  0.0064378176  0.0172580800  0.0118364549  0.013428928  3.853927e-02
## 2017-03-31 -0.0005525506  0.0361887280  0.0318059025 -0.006533295  1.249073e-03
## 2017-04-28  0.0090290028  0.0168665520  0.0239521045  0.005107904  9.877278e-03
## 2017-05-31  0.0068473361  0.0280598440  0.0348103271 -0.022862479  1.401442e-02
## 2017-06-30 -0.0001829040  0.0092239356  0.0029559201  0.029151582  6.354619e-03
## 2017-07-31  0.0033349668  0.0565944167  0.0261878629  0.007481756  2.034574e-02
## 2017-08-31  0.0093688884  0.0232439051 -0.0004484447 -0.027564686  2.913657e-03
## 2017-09-29 -0.0057322923 -0.0004463692  0.0233428366  0.082321616  1.994895e-02
## 2017-10-31  0.0009781554  0.0322785151  0.0166537655  0.005916133  2.329071e-02
## 2017-11-30 -0.0014838960 -0.0038970436  0.0068699320  0.036913275  3.010819e-02
## 2017-12-29  0.0047400570  0.0369254672  0.0133981663 -0.003731260  1.205512e-02
# Covariance of asset returns
covariance_matrix <- cov(asset_returns_wide_tbl)

covariance_matrix
##               AGG          EEM          EFA           IJS           SPY
## AGG  7.398278e-05 0.0001042106 4.178322e-05 -7.811911e-05 -9.029489e-06
## EEM  1.042106e-04 0.0017547113 1.039018e-03  6.437738e-04  6.795430e-04
## EFA  4.178322e-05 0.0010390179 1.064237e-03  6.490294e-04  6.975414e-04
## IJS -7.811911e-05 0.0006437738 6.490294e-04  1.565449e-03  8.290241e-04
## SPY -9.029489e-06 0.0006795430 6.975414e-04  8.290241e-04  7.408301e-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,] 0.0003874159 0.009257148 0.005815635 0.005684466 0.002330251
rowSums(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 0.099
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")

# 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 0.099

5 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 0.088
asset_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 0.089
dump(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
## # ℹ 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."))