Week 15: Code along 13
CHAPTER 11 Monte Carlo Simulation
Eli Pockette
2025-12-01
# Load packages
# Core
library(tidyverse)
library(tidyquant)
library(dplyr)
library(ggplot2)
# 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$returns)
mean_port_return## [1] 0.005899136# Get standard deviation of portfolio returns
stddev_port_return <- sd(portfolio_returns_tbl$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] 1.279338e-02 2.076291e-02 -4.128848e-02 -1.294067e-02 -1.118665e-02
## [6] 2.805612e-05 9.257051e-04 5.498149e-02 2.763578e-02 1.095363e-02
## [11] -6.035550e-03 -2.272842e-03 -1.443694e-02 7.057701e-04 5.416540e-02
## [16] 3.709707e-02 1.421126e-02 2.457024e-02 2.477235e-02 3.062962e-03
## [21] -3.209622e-03 -1.279667e-03 -3.569199e-03 -7.263769e-04 2.091303e-02
## [26] -2.405693e-02 2.526077e-02 -2.184379e-02 4.887952e-02 2.820898e-02
## [31] 2.371361e-02 -2.763539e-02 -4.716893e-03 6.196046e-02 2.797432e-02
## [36] -1.038325e-02 3.287429e-02 1.897072e-02 3.532723e-02 6.921543e-03
## [41] -2.698595e-02 1.826841e-02 7.043678e-03 3.237274e-02 1.053779e-02
## [46] 1.767959e-02 1.779833e-02 6.323776e-02 -1.056960e-02 3.675662e-02
## [51] 9.130217e-03 -4.175674e-02 2.961281e-02 -3.713731e-02 2.550750e-02
## [56] 5.564386e-04 1.356971e-02 -2.831279e-03 -5.291432e-03 1.150384e-02
## [61] 5.005799e-03 -8.133086e-03 -2.293851e-02 -1.647883e-02 1.780108e-02
## [66] 4.830121e-02 -1.260705e-02 6.956828e-03 -5.019948e-03 -1.531033e-02
## [71] 2.369405e-03 -3.968775e-02 2.411455e-02 -1.199358e-02 3.366768e-02
## [76] -2.127768e-02 1.817668e-02 4.062960e-02 -1.883542e-02 -1.435464e-02
## [81] -2.668624e-03 5.698905e-02 3.831565e-02 2.608445e-02 -1.917191e-02
## [86] 3.102527e-02 3.973461e-02 1.876187e-02 -3.303407e-03 8.428273e-03
## [91] 4.020452e-04 1.442096e-02 1.740599e-02 -4.608842e-03 -9.239876e-03
## [96] -1.710835e-03 1.575349e-02 -8.234121e-04 1.762643e-03 6.009784e-03
## [101] 2.463029e-02 -3.387958e-03 5.796381e-02 8.362120e-03 -2.239126e-02
## [106] -7.987742e-03 -3.635177e-03 4.621286e-02 1.619648e-02 -8.400347e-04
## [111] 1.769757e-02 5.350393e-02 4.783240e-03 2.332598e-02 -1.967999e-02
## [116] 1.624986e-02 9.053295e-03 2.454552e-02 3.967212e-02 2.492642e-02# 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 1.01
## 3 1.02
## 4 0.959
## 5 0.987
## 6 0.989
## 7 1.00
## 8 1.00
## 9 1.05
## 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 1.01
## 3 1.03
## 4 0.991
## 5 0.978
## 6 0.967
## 7 0.967
## 8 0.968
## 9 1.02
## 10 1.05
## # ℹ 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 11.825466 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>
## 1 294.
## 2 298.
## 3 298.
## 4 300.
## 5 305.
## 6 307.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 <- 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")
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# 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 $1 over 120 months")
Line plot with max, median, and min
# Step 1 Summarize data into max, median, min of lat value
sim_summary <- monte_carlo_sim_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
summarise(max = max(growth),
median = median(growth),
min = min(growth))
sim_summary## # A tibble: 1 × 3
## max median min
## <dbl> <dbl> <dbl>
## 1 3.88 1.98 1.17# Step 2 Plot
monte_carlo_sim_51 %>%
group_by(sim) %>%
filter(last(growth) == sim_summary$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")## $title
## [1] "Simulating growth of $1 over 120 months"
##
## $subtitle
## [1] "Maximum, Median and Minimum Simulation"
##
## attr(,"class")
## [1] "labels"