Week 15: Code along 13
CHAPTER 11 Monte Carlo Simulation
Cohen Karr
2025-12-03
# 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$returns)
mean_port_return## [1] 0.005899138# Get standard deviation of portfolio returns
stddev_port_return <- sd(portfolio_returns_tbl$returns)
stddev_port_return## [1] 0.02347489# Construct a normal distribution
simulated_monthly_returns <- rnorm(120, mean_port_return, stddev_port_return)
simulated_monthly_returns## [1] 0.0069775087 -0.0154897058 -0.0059977151 0.0025217892 -0.0081421732
## [6] 0.0084578970 0.0211735309 0.0405471289 -0.0109948252 -0.0078324418
## [11] -0.0256458036 0.0288656664 -0.0017367493 -0.0154146137 0.0000136658
## [16] -0.0221349441 -0.0002418098 -0.0563077952 0.0039174109 -0.0094379799
## [21] -0.0128626024 -0.0457771892 0.0173119978 0.0434895551 0.0022352218
## [26] -0.0359336420 -0.0065279437 0.0044975671 -0.0726662013 0.0004031295
## [31] 0.0332074894 0.0168780058 0.0123575618 0.0151399076 0.0176034976
## [36] 0.0077914522 -0.0132105581 -0.0095540624 0.0664896144 -0.0130799296
## [41] -0.0037353963 0.0199008049 0.0222473695 0.0259483499 -0.0032350157
## [46] 0.0171537989 -0.0187041772 0.0096839265 -0.0026268889 0.0097938094
## [51] -0.0143975249 0.0395672467 0.0095279276 -0.0013105944 0.0124309310
## [56] -0.0375797172 0.0374763184 0.0117154569 0.0427175122 0.0467394463
## [61] -0.0409826029 0.0390838226 0.0192792364 0.0047267202 -0.0020350436
## [66] 0.0263789922 0.0079525369 -0.0251212622 0.0056485095 0.0070495775
## [71] 0.0291356170 0.0190560182 0.0005173322 0.0359818628 -0.0288524664
## [76] -0.0051734897 0.0150646617 -0.0383248403 0.0340324521 0.0290663489
## [81] -0.0302357742 0.0068731876 0.0238664885 -0.0153004638 0.0199604116
## [86] 0.0466228444 0.0097079617 0.0437602913 -0.0135064805 0.0233215408
## [91] 0.0292252188 -0.0070105745 0.0105092511 -0.0049905583 -0.0230262593
## [96] 0.0348154816 -0.0103336559 0.0030916911 -0.0012322308 -0.0563668267
## [101] -0.0116207610 -0.0164968557 -0.0117692651 -0.0034118931 0.0123061259
## [106] 0.0234626276 0.0115462353 -0.0330357875 0.0016673201 0.0004261275
## [111] -0.0326671992 0.0206161889 0.0298778343 0.0099034248 0.0064907749
## [116] 0.0077481206 -0.0228619039 -0.0458283106 0.0092079455 0.0307238731# 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 0.985
## 4 0.994
## 5 1.00
## 6 0.992
## 7 1.01
## 8 1.02
## 9 1.04
## 10 0.989
## # 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 0.991
## 4 0.985
## 5 0.988
## 6 0.980
## 7 0.988
## 8 1.01
## 9 1.05
## 10 1.04
## # ℹ 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 3.6328556 Simulation function
simulate_accumulation <- function(initial_value, N, mean_return, sd_return) {
# Add a dollar
simulated_returns_add_1 <- tibble(returns = c(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: 241 × 1
## growth
## <dbl>
## 1 1
## 2 0.987
## 3 0.983
## 4 0.980
## 5 0.979
## 6 0.981
## 7 0.992
## 8 1.01
## 9 1.02
## 10 1.03
## # ℹ 231 more rows7 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 $1 over 120 months")
# Step 1 Summarize data into max, median, and min of last value
sim_summary <- monte_carle_sim_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
summarise(max = max(growth),
median = median(growth),
min = min(growth))
sim_summary## # A tibble: 1 × 3
## max median min
## <dbl> <dbl> <dbl>
## 1 3.88 1.98 1.17# Step 2 Plot
monte_carle_sim_51 %>%
# Filter for max, median, min sim
group_by(sim) %>%
filter(last(growth) == sim_summary$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")