Week 15: Code along 13
CHAPTER 11 Monte Carlo Simulation
Eli Henschen
2025-12-1
# 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] 0.02347491# Construct a normal distribution
simulated_monthly_returns <- rnorm(120, mean_port_return, stddev_port_return)
simulated_monthly_returns## [1] 0.0012354300 -0.0276043532 -0.0191348511 0.0119943906 -0.0337410310
## [6] 0.0040928742 0.0639845743 0.0277532995 0.0348829409 0.0483646288
## [11] 0.0544258230 0.0071962263 -0.0124474568 -0.0043872349 -0.0339698587
## [16] 0.0062539086 0.0362377424 0.0115627988 0.0342190116 -0.0022231799
## [21] 0.0203059602 -0.0030028348 0.0205044061 -0.0164428996 -0.0036333678
## [26] 0.0016741734 -0.0250142360 -0.0403982449 -0.0044175098 0.0303997146
## [31] 0.0169852945 -0.0188144789 0.0112135035 0.0133380231 0.0332173183
## [36] -0.0358397353 0.0080817854 0.0158346221 0.0142491200 0.0049752076
## [41] -0.0097942116 -0.0210149315 -0.0056371051 -0.0013923750 0.0422018859
## [46] 0.0244607575 -0.0244194595 0.0035566834 -0.0528343574 0.0121331918
## [51] 0.0288072503 0.0401560631 0.0108777575 0.0205216479 0.0054051367
## [56] 0.0135271165 -0.0063019986 0.0045517127 0.0289575743 -0.0132281823
## [61] 0.0355019360 0.0016292691 0.0065859789 0.0050729700 0.0167869153
## [66] 0.0348044357 0.0131886197 0.0415715320 0.0152439457 0.0177551017
## [71] -0.0226765750 0.0053880151 0.0443281619 -0.0274180630 0.0293980518
## [76] 0.0036094609 -0.0191646358 -0.0242582534 -0.0059730013 -0.0133816127
## [81] -0.0043147197 -0.0052036244 0.0007965703 0.0406193170 -0.0079135947
## [86] -0.0276516440 -0.0239935289 0.0109454361 0.0218554963 0.0114029689
## [91] -0.0161509728 -0.0193252103 -0.0134523707 -0.0195673008 -0.0163932456
## [96] 0.0040943289 -0.0245249419 0.0207958850 0.0088742263 0.0479534822
## [101] -0.0175859871 0.0058341869 0.0181289406 -0.0575284958 0.0432883835
## [106] -0.0263193591 0.0314960058 0.0114473139 -0.0056186441 0.0144064749
## [111] -0.0350623462 0.0227339412 -0.0248318198 -0.0398352252 0.0038914708
## [116] -0.0232882752 0.0458376421 0.0036652479 -0.0051435651 0.0244694144# 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 1.00
## 3 0.972
## 4 0.981
## 5 1.01
## 6 0.966
## 7 1.00
## 8 1.06
## 9 1.03
## 10 1.03
## # 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 1.00
## 3 0.974
## 4 0.955
## 5 0.966
## 6 0.934
## 7 0.938
## 8 0.998
## 9 1.03
## 10 1.06
## # ℹ 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 4.643356 Simulation function
simulate_accumulation <- function(initial_value, N, mean_return, sd_return) {
# Add a dollar
simulated_returns_add_1 <- tibble(returns = c(initial_value, 1, 1 + rnorm(N, mean_return, sd_return)))
# Calculate the cumulative growth of a dollar
simulated_growth <- simulated_returns_add_1 %>%
mutate(growth = accumulate(returns, function(x, y) x*y)) %>%
select(growth)
return(simulated_growth)
}
simulate_accumulation(initial_value = 100, N = 240, mean_return = 0.005, sd_return = 0.01)## # A tibble: 242 × 1
## growth
## <dbl>
## 1 100
## 2 100
## 3 100.0
## 4 99.3
## 5 98.7
## 6 97.4
## 7 97.8
## 8 98.8
## 9 98.2
## 10 98.1
## # ℹ 232 more rowsdump(list = c("simulate_accumulation"),
file = "../00_scripts/simulate_accumulation.R")7 Running multiple simulations
# Create a vector of 1s as starting point
sims <- 51
starts<- rep(1, sims) %>%
set_names(paste0("sim", 1:sims))
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# for reproducible research
set.seed(1234)
monte_carlo_sim_51 <-starts %>%
# Simulate
map_dfc(.x = .,
.f = ~simulate_accumulation(initial_value = .x,
N= 120,
mean_return = mean_port_return,
sd_return = stddev_port_return)) %>%
# Add column month
mutate(month = 1:nrow(.)) %>%
select(month, everything()) %>%
# rearrange column names
set_names(c("month",names(starts))) %>%
# transform to long form
pivot_longer(cols = -month, names_to = "sim", values_to = "growth")
monte_carlo_sim_51## # A tibble: 6,222 × 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 6,212 more rows# find quantiles
monte_carlo_sim_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
pull(growth) %>%
quantile(probs = c(0, 0.25, 0.5,0.75,1)) %>%
round(2)## 0% 25% 50% 75% 100%
## 1.17 1.59 1.98 2.40 3.888 Visualizing simulations with ggplot
monte_carlo_sim_51 %>%
ggplot(aes(x = month, y = growth, color = sim)) +
geom_line() +
theme(legend.position = "none") +
theme(plot.title = element_text(hjust = 0.5))
labs(title = "Simulating growth of $1 over 120 months")## $title
## [1] "Simulating growth of $1 over 120 months"
##
## attr(,"class")
## [1] "labels"Line plot with max, median and min
# summarize data into max, median, and min of last value
sim_summary <- monte_carlo_sim_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
summarise(max =max(growth),
median = median(growth),
min = min(growth))
sim_summary## # A tibble: 1 × 3
## max median min
## <dbl> <dbl> <dbl>
## 1 3.88 1.98 1.17# Plot
monte_carlo_sim_51 %>%
# filter for max, median and min sim
group_by(sim) %>%
filter(last(growth) == sim_summary$max |
last(growth) == sim_summary$median |
last(growth) == sim_summary$min) %>%
ungroup() %>%
# plot
ggplot(aes(x = month, y = growth, color = sim)) +
geom_line() +
theme(legend.position = "none") +
theme(plot.title = element_text(hjust = 0.5)) +
theme(plot.subtitle = element_text(hjust = 0.5)) +
labs(title = "Simulating growth of $1 over 120 months",
subtitle = "Maximum, Median, and Minimum Simulation")