Week 15: Code along 13
CHAPTER 11 Monte Carlo Simulation
Thomas Hall
2025-06-25
# 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$returns)
mean_port_return## [1] 0.005899135# Get standard deviation of portfolio returns
stddev_port_return <- sd(portfolio_returns_tbl$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] 0.0014113267 0.0205473641 0.0122582077 0.0272985212 0.0226674872
## [6] 0.0079705309 -0.0473168376 0.0512617404 0.0049084374 0.0497020495
## [11] 0.0183265169 -0.0125334877 0.0220259369 0.0201033638 -0.0036715950
## [16] 0.0134146589 0.0026421533 0.0301157682 0.0006107447 0.0039786465
## [21] 0.0305592699 -0.0247108922 -0.0026691814 0.0459715579 0.0133974933
## [26] -0.0301510272 0.0340890378 0.0074549307 -0.0235651649 -0.0045577767
## [31] -0.0313963778 0.0272050062 0.0035815494 0.0291541946 -0.0241554486
## [36] 0.0183035606 -0.0303238350 0.0250044373 0.0173857414 -0.0337742374
## [41] -0.0265401444 0.0393633339 0.0423237254 -0.0156617309 0.0064399085
## [46] 0.0352548505 -0.0278953146 0.0483394477 0.0105746406 -0.0466200277
## [51] 0.0191799739 0.0067242467 -0.0012080813 -0.0080865317 0.0153332772
## [56] -0.0172155417 -0.0366513258 0.0392278256 0.0416894315 -0.0164324359
## [61] 0.0244484317 0.0019793416 0.0208014708 0.0408845633 -0.0079323293
## [66] -0.0269240873 -0.0129448850 0.0202906041 -0.0023145126 0.0504943184
## [71] -0.0087364410 -0.0188559460 0.0047147420 0.0028703071 0.0496722067
## [76] 0.0284963473 -0.0076544968 -0.0286014228 -0.0059332083 0.0207153221
## [81] 0.0004393355 0.0552115953 0.0093142273 0.0260910255 0.0441780103
## [86] -0.0068939606 -0.0130391422 -0.0081601741 0.0095324839 0.0389827046
## [91] 0.0116788535 0.0228769163 -0.0293016914 0.0385632546 0.0099557217
## [96] 0.0527374351 -0.0443575324 0.0173205142 0.0242232700 -0.0392051351
## [101] 0.0201618638 -0.0297374715 -0.0229670671 -0.0158000352 0.0277609740
## [106] 0.0021165035 0.0010825416 -0.0091921863 0.0043364952 0.0151618038
## [111] -0.0205102517 -0.0225871196 0.0203150681 0.0050438445 0.0270214534
## [116] -0.0214973204 0.0183635355 -0.0163034267 0.0138489520 0.0571984696# 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 1.02
## 4 1.01
## 5 1.03
## 6 1.02
## 7 1.01
## 8 0.953
## 9 1.05
## 10 1.00
## # 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 1.02
## 4 1.03
## 5 1.06
## 6 1.09
## 7 1.10
## 8 1.04
## 9 1.10
## 10 1.10
## # ℹ 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 8.1121066 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 + 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) %>%
tail()## # A tibble: 6 × 1
## growth
## <dbl>
## 1 287.
## 2 292.
## 3 291.
## 4 291.
## 5 291.
## 6 296.dump(list = c("simulate_accumulation"),
file = "../00_scripts/simulate_accumulation.R")7 Running multiple simulations
# Create a vector of 1s as a starting point
sims <- 100
starts <- rep(1, sims) %>%
set_names(paste0("sim", 1:sims))
starts## sim1 sim2 sim3 sim4 sim5 sim6 sim7 sim8 sim9 sim10 sim11
## 1 1 1 1 1 1 1 1 1 1 1
## sim12 sim13 sim14 sim15 sim16 sim17 sim18 sim19 sim20 sim21 sim22
## 1 1 1 1 1 1 1 1 1 1 1
## sim23 sim24 sim25 sim26 sim27 sim28 sim29 sim30 sim31 sim32 sim33
## 1 1 1 1 1 1 1 1 1 1 1
## sim34 sim35 sim36 sim37 sim38 sim39 sim40 sim41 sim42 sim43 sim44
## 1 1 1 1 1 1 1 1 1 1 1
## sim45 sim46 sim47 sim48 sim49 sim50 sim51 sim52 sim53 sim54 sim55
## 1 1 1 1 1 1 1 1 1 1 1
## sim56 sim57 sim58 sim59 sim60 sim61 sim62 sim63 sim64 sim65 sim66
## 1 1 1 1 1 1 1 1 1 1 1
## sim67 sim68 sim69 sim70 sim71 sim72 sim73 sim74 sim75 sim76 sim77
## 1 1 1 1 1 1 1 1 1 1 1
## sim78 sim79 sim80 sim81 sim82 sim83 sim84 sim85 sim86 sim87 sim88
## 1 1 1 1 1 1 1 1 1 1 1
## sim89 sim90 sim91 sim92 sim93 sim94 sim95 sim96 sim97 sim98 sim99
## 1 1 1 1 1 1 1 1 1 1 1
## sim100
## 1# Simulate
# for reporducible 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: 12,100 × 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 12,090 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%
## 1.09 1.74 2.00 2.39 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 $1 over 120 months")
Line plot with max, median, and min
# Step 1 Summarize data into max, median, and min of last value
sim_summary <- monte_carlo_sim_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
summarise(max = max(growth),
median = median(growth),
min = min(growth))
sim_summary## # A tibble: 1 × 3
## max median min
## <dbl> <dbl> <dbl>
## 1 3.88 2.00 1.09# Step 2 Plot
monte_carlo_sim_51 %>%
# Filter for max, median, and min sim
group_by(sim) %>%
filter(last(growth) == sim_summary$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")