Week 15: Code along 13
CHAPTER 11 Monte Carlo Simulation
Nils Skogestig
2025-11-28
# 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$returns)
mean_port_return## [1] 0.005899136# Get standard deviation of portfolio returns
stddev_port_return <- sd(portfolio_returns_tbl$returns)
stddev_port_return## [1] 0.02347488# Construct a normal distribution
simulated_monthly_returns <- rnorm(120, mean_port_return, stddev_port_return)
simulated_monthly_returns## [1] 0.0015976533 -0.0017668981 0.0268258501 0.0238477813 0.0110655765
## [6] -0.0236647290 -0.0109129598 0.0107999180 -0.0289094158 0.0197409565
## [11] 0.0186234289 0.0187890136 0.0318992688 0.0291597563 -0.0092005989
## [16] -0.0158755018 0.0058973089 0.0170979246 -0.0443283971 -0.0146922882
## [21] 0.0408860652 0.0189169710 0.0271089795 -0.0213919592 0.0354092889
## [26] 0.0182220546 0.0170311007 0.0130123682 -0.0379274088 0.0176976690
## [31] -0.0170840515 0.0069091822 -0.0026301256 0.0056904551 -0.0109740928
## [36] 0.0148289998 0.0124591489 -0.0182570840 -0.0111114300 -0.0161679198
## [41] -0.0070378026 0.0143928224 0.0425919693 0.0037389726 0.0359098670
## [46] 0.0355913452 0.0024392454 -0.0250652622 -0.0266226004 -0.0053436255
## [51] 0.0303090016 0.0272475301 0.0250026158 0.0015368348 0.0453848852
## [56] -0.0163448139 0.0158831326 -0.0226370046 -0.0022720456 -0.0260592335
## [61] -0.0049306058 0.0347596524 0.0296421730 0.0183017278 -0.0058281373
## [66] 0.0353999198 -0.0044287132 -0.0151949374 0.0246972647 0.0461600953
## [71] 0.0069488283 -0.0050109103 -0.0384229665 0.0156549402 -0.0513632051
## [76] 0.0083525152 0.0164167279 0.0320340008 0.0323209636 -0.0198679709
## [81] -0.0127094091 -0.0461905725 0.0525810596 0.0063942155 0.0135220608
## [86] -0.0096547717 -0.0045809154 0.0049514620 0.0201418677 0.0705818752
## [91] -0.0269135319 0.0225419390 0.0013138136 -0.0282503344 -0.0212529955
## [96] 0.0032174596 -0.0103568843 0.0126949218 -0.0001235707 0.0004501185
## [101] 0.0201221139 0.0370747903 -0.0288810798 -0.0137720858 0.0149262764
## [106] 0.0472218583 -0.0003391493 0.0040415705 0.0328646571 -0.0148798694
## [111] -0.0099180515 0.0401291195 0.0439952426 -0.0068095952 -0.0226550751
## [116] -0.0068453242 0.0485296298 0.0044507566 -0.0284374011 0.0496497579# 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 0.998
## 4 1.03
## 5 1.02
## 6 1.01
## 7 0.976
## 8 0.989
## 9 1.01
## 10 0.971
## # 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 1.00
## 4 1.03
## 5 1.05
## 6 1.06
## 7 1.04
## 8 1.03
## 9 1.04
## 10 1.01
## # ℹ 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 7.0940716 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>
## 1 379.
## 2 378.
## 3 387.
## 4 388.
## 5 394.
## 6 395.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_carle_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_carle_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_carle_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 $1 over 120 months")
Line plot with max, median, and min
# Step 1 Summarize data into max, median, and min of last value
sim_summary <- monte_carle_sim_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
summarise(max = max(growth),
median = median(growth),
min = min(growth))
sim_summary## # A tibble: 1 × 3
## max median min
## <dbl> <dbl> <dbl>
## 1 3.88 1.98 1.17# Step 2 Plot
monte_carle_sim_51 %>%
# Filter for max, median, min sim
group_by(sim) %>%
filter(last(growth) == sim_summary$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 Mimimum Simulation")