Week 15: Code along 13
CHAPTER 11 Monte Carlo Simulation
Brady Martin
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] 0.02347494# Construct a normal distribution
simulated_monthly_returns <- rnorm(120, mean_port_return, stddev_port_return)
simulated_monthly_returns## [1] 0.019108037 0.007423440 0.009775975 0.009323253 0.023016764
## [6] 0.027944517 -0.021411589 -0.013848056 0.009664662 0.018798717
## [11] -0.024264755 0.003162282 -0.012159254 -0.015019843 -0.017254009
## [16] -0.033001893 0.070737612 0.029007727 -0.029390103 -0.004867164
## [21] -0.002045550 -0.022718264 0.005183583 0.023926223 0.042053403
## [26] -0.023118202 -0.018294120 0.016663981 -0.002744558 0.018942200
## [31] -0.005063453 0.025098735 -0.015566362 -0.032104471 -0.050802982
## [36] 0.009649763 0.042125999 0.025999348 0.003634232 -0.008833498
## [41] 0.016489280 -0.018124346 -0.010444461 0.002014509 0.011658999
## [46] 0.026276775 0.019359216 0.016620352 0.032024406 -0.049469445
## [51] -0.018006612 0.026084405 -0.034121363 -0.062140111 -0.027874403
## [56] -0.001603079 -0.020374121 -0.001125365 0.014890987 0.002972947
## [61] 0.027162987 0.020626758 0.041586454 -0.019272559 0.006046916
## [66] 0.008499250 0.029939680 -0.019565223 -0.031894033 -0.039127682
## [71] -0.007734399 0.001649552 -0.012093632 0.003875013 0.002173515
## [76] -0.039761078 0.013510432 -0.023931265 0.053688774 0.041493144
## [81] 0.016194581 -0.002191179 -0.040451858 0.023884436 0.010812286
## [86] -0.028705940 0.007609486 -0.005462884 -0.013162204 0.052818488
## [91] 0.014609561 0.008210942 0.037284119 -0.033421896 0.013076243
## [96] 0.011502859 -0.030722318 0.019566556 0.010416363 -0.019802734
## [101] -0.029519058 -0.013993371 0.002410873 0.009742294 0.049381101
## [106] 0.030844337 0.021343618 0.038385122 0.047813361 0.005784428
## [111] 0.039075125 0.035836623 0.018959347 -0.017838847 0.026554395
## [116] 0.023689540 0.027201912 0.002485848 0.022552413 -0.002562977# 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 1.02
## 3 1.01
## 4 1.01
## 5 1.01
## 6 1.02
## 7 1.03
## 8 0.979
## 9 0.986
## 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 1.02
## 3 1.03
## 4 1.04
## 5 1.05
## 6 1.07
## 7 1.10
## 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] 4.3150166 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>
## 1 353.
## 2 350.
## 3 353.
## 4 364.
## 5 361.
## 6 362.7 Running multiple simulations
# Create a Vector of 1s as a 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# Simulate
# 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")
# 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, colour = sim)) +
geom_line() +
theme(legend.position = "none") +
theme(plot.title = element_text(hjust = 0.5)) +
labs(title = "Simulating Growth of $1 Over 120 Months")
Line Plot With Max, Median, and Min
# Step 1 Summarize Data into Max, Median, and Min of Last Value
sim_summary <- monte_carlo_sim_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
summarise(max = max(growth),
median = median(growth),
min = min(growth))
sim_summary## # A tibble: 1 × 3
## max median min
## <dbl> <dbl> <dbl>
## 1 3.88 1.98 1.17# Step 2 Plot
monte_carlo_sim_51 %>%
# Filter for Max, Median, and Min Sim
group_by(sim) %>%
filter(last(growth) == sim_summary$max |
last(growth) == sim_summary$median |
last(growth) == sim_summary$min) %>%
ungroup() %>%
# Plot
ggplot(aes(x = month, y = growth, colour = 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")## $title
## [1] "Simulating Growth of $1 Over 120 Months"
##
## $subtitle
## [1] "Maximum, Median, and Minimum Simulation"
##
## attr(,"class")
## [1] "labels"