Week 15: Code along 13
CHAPTER 11 Monte Carlo Simulation
Ben Brown
2022-1126
# Load packages
# Core
library(tidyverse)
library(tidyquant)
# time series
library(timetk)Goal
Simulate future portfolio returns
five stocks: “SPY”, “EFA”, “IJS”, “EEM”, “AGG”
market: “SPY”
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 Assign a weight to each asset
# symbols
symbols <- asset_returns_tbl %>% distinct(asset) %>% pull()
symbols## [1] "AGG" "EEM" "EFA" "IJS" "SPY"# weights
weights <- c(0.25, 0.25, 0.2, 0.2, 0.1)
weights## [1] 0.25 0.25 0.20 0.20 0.10w_tbl <- tibble(symbols, weights)
w_tbl## # A tibble: 5 × 2
## symbols weights
## <chr> <dbl>
## 1 AGG 0.25
## 2 EEM 0.25
## 3 EFA 0.2
## 4 IJS 0.2
## 5 SPY 0.14 Build a portfolio
# ?tq_portfolio
portfolio_returns_tbl <- asset_returns_tbl %>%
tq_portfolio(assets_col = asset,
returns_col = returns,
weights = w_tbl,
rebalance_on = "months",
col_rename = "returns")
portfolio_returns_tbl## # A tibble: 60 × 2
## date returns
## <date> <dbl>
## 1 2013-01-31 0.0204
## 2 2013-02-28 -0.00239
## 3 2013-03-28 0.0121
## 4 2013-04-30 0.0174
## 5 2013-05-31 -0.0128
## 6 2013-06-28 -0.0247
## 7 2013-07-31 0.0321
## 8 2013-08-30 -0.0224
## 9 2013-09-30 0.0511
## 10 2013-10-31 0.0301
## # ℹ 50 more rows5 Simulating growth of a dollar
# Get mean portfolio return
mean_port_return <- mean(portfolio_returns_tbl$returns)
mean_port_return## [1] 0.005899137# Get standard deviation of portfolio returns
stddev_port_return <- sd(portfolio_returns_tbl$returns)
stddev_port_return## [1] 0.02347493# Construct a normal distribution
simulated_monthly_returns <- rnorm(120, mean_port_return, stddev_port_return)
simulated_monthly_returns## [1] -0.0559751398 0.0042695559 0.0024663238 0.0690526124 0.0077275230
## [6] 0.0337830573 0.0204873530 -0.0174490722 0.0076342030 0.0046675256
## [11] -0.0209806773 0.0289227597 -0.0085544857 0.0065065245 0.0317326919
## [16] -0.0299939382 0.0262992023 -0.0056126010 0.0196433582 0.0175785189
## [21] -0.0513377401 -0.0143780514 -0.0025770540 0.0478345280 0.0261175306
## [26] 0.0042139296 -0.0017726341 -0.0115058313 0.0285354374 0.0252726183
## [31] -0.0073354009 0.0261706014 0.0308811371 -0.0029921118 0.0287064122
## [36] -0.0271305800 0.0388790861 0.0168876780 0.0241759352 -0.0011158874
## [41] 0.0408434936 -0.0006312985 -0.0200045126 -0.0183771127 0.0352769326
## [46] -0.0103810821 0.0173342838 0.0657639602 -0.0430892682 -0.0374320815
## [51] 0.0592537519 0.0061242917 0.0095476051 0.0284464856 -0.0102958761
## [56] -0.0159580354 0.0302198928 0.0099134736 0.0168939595 -0.0107067857
## [61] 0.0134431296 -0.0160833913 0.0544649733 0.0276381950 0.0027287523
## [66] -0.0040052434 -0.0071478275 0.0170586660 -0.0008986999 -0.0111738274
## [71] -0.0133961559 0.0241322544 0.0208976266 0.0295062527 -0.0082693061
## [76] 0.0177055539 -0.0609833033 0.0543252280 -0.0133118864 -0.0023020386
## [81] 0.0277275868 -0.0037723751 -0.0277676993 -0.0174878515 0.0293616416
## [86] -0.0095832880 0.0289427942 -0.0186744304 0.0424063415 -0.0024902834
## [91] -0.0365773935 -0.0425778261 -0.0213059419 0.0329144880 0.0482666373
## [96] 0.0230367398 0.0364055911 0.0151930986 -0.0258446268 0.0144033006
## [101] 0.0054513585 -0.0190017676 -0.0185911002 0.0399724629 0.0551836013
## [106] 0.0168824947 -0.0484101519 0.0200933030 0.0180807362 0.0315230342
## [111] 0.0132899348 0.0031077370 0.0065688231 0.0089169128 0.0091039892
## [116] 0.0414133840 -0.0008940590 -0.0151558117 -0.0111520763 -0.0034301717# Add a dollar
simulated_returns_add_1 <- tibble(returns = c(1, 1 + simulated_monthly_returns))
simulated_returns_add_1## # A tibble: 121 × 1
## returns
## <dbl>
## 1 1
## 2 0.944
## 3 1.00
## 4 1.00
## 5 1.07
## 6 1.01
## 7 1.03
## 8 1.02
## 9 0.983
## 10 1.01
## # ℹ 111 more rows# Calculate the cumulative growth of a dollar
simulated_growth <- simulated_returns_add_1 %>%
mutate(growth = accumulate(returns, function(x, y) x*y)) %>%
select(growth)
simulated_growth## # A tibble: 121 × 1
## growth
## <dbl>
## 1 1
## 2 0.944
## 3 0.948
## 4 0.950
## 5 1.02
## 6 1.02
## 7 1.06
## 8 1.08
## 9 1.06
## 10 1.07
## # ℹ 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 8.3180746 Simulation function
simulate_accumulation <- function(init_value, N, mean, stdev) {
tibble(returns = c(init_value, 1 + rnorm(N, mean, stdev))) %>%
mutate(growth = accumulate(returns, function(x, y) x*y)) %>%
select(growth)
}
simulate_accumulation(1, 120, mean_port_return, stddev_port_return)## # A tibble: 121 × 1
## growth
## <dbl>
## 1 1
## 2 0.987
## 3 0.997
## 4 0.977
## 5 0.996
## 6 1.02
## 7 1.02
## 8 1.05
## 9 1.04
## 10 1.05
## # ℹ 111 more rows7 Running multiple simulations
# Create a vector of 1s as a starting point
sims <- 51
starts <- rep(1, sims) %>%
set_names(paste("sim", 1:sims, sep = ""))
starts## sim1 sim2 sim3 sim4 sim5 sim6 sim7 sim8 sim9 sim10 sim11 sim12 sim13
## 1 1 1 1 1 1 1 1 1 1 1 1 1
## sim14 sim15 sim16 sim17 sim18 sim19 sim20 sim21 sim22 sim23 sim24 sim25 sim26
## 1 1 1 1 1 1 1 1 1 1 1 1 1
## sim27 sim28 sim29 sim30 sim31 sim32 sim33 sim34 sim35 sim36 sim37 sim38 sim39
## 1 1 1 1 1 1 1 1 1 1 1 1 1
## sim40 sim41 sim42 sim43 sim44 sim45 sim46 sim47 sim48 sim49 sim50 sim51
## 1 1 1 1 1 1 1 1 1 1 1 1# Simulate
monte_carlo_sim_51 <- starts %>%
# Simulate
map_dfc(simulate_accumulation,
N = 120,
mean = mean_port_return,
stdev = stddev_port_return) %>%
# Add the column, month
mutate(month = seq(1:nrow(.))) %>%
# Arrange column names
select(month, everything()) %>%
set_names(c("month", names(starts))) %>%
pivot_longer(cols = -month, names_to = "sim", values_to = "growth")
monte_carlo_sim_51## # A tibble: 6,171 × 3
## month sim growth
## <int> <chr> <dbl>
## 1 1 sim1 1
## 2 1 sim2 1
## 3 1 sim3 1
## 4 1 sim4 1
## 5 1 sim5 1
## 6 1 sim6 1
## 7 1 sim7 1
## 8 1 sim8 1
## 9 1 sim9 1
## 10 1 sim10 1
## # ℹ 6,161 more rows# Calculate the quantiles for simulated values
probs <- c(.005, .025, .25, .5, .75, .975, .995)
monte_carlo_sim_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
pull(growth) %>%
# Find the quantiles
quantile(probs = probs) %>%
round(2)## 0.5% 2.5% 25% 50% 75% 97.5% 99.5%
## 1.00 1.26 1.71 2.07 2.30 3.48 4.248 Visualizing simulations with ggplot
monte_carlo_sim_51 %>%
ggplot(aes(x = month, y = growth, col = sim)) +
geom_line() +
theme(legend.position = "none")
# Simplify the plot
sim_summary <- monte_carlo_sim_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
summarise(max = max(growth),
median = median(growth),
min = min(growth))
sim_summary## # A tibble: 1 × 3
## max median min
## <dbl> <dbl> <dbl>
## 1 4.43 2.07 0.915monte_carlo_sim_51 %>%
group_by(sim) %>%
filter(last(growth) == sim_summary$max |
last(growth) == sim_summary$median |
last(growth) == sim_summary$min) %>%
# Plot
ggplot(aes(month, growth, col = sim)) +
geom_line() +
theme()