Week 15: Code along 13
CHAPTER 11 Monte Carlo Simulation
Ty Bacci
2025-12-01
# 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] 0.0020940117 0.0228570910 -0.0070963721 0.0183441329 -0.0182350583
## [6] 0.0276234417 -0.0184474062 0.0579892545 -0.0087632095 0.0239184925
## [11] 0.0160238808 0.0157250915 0.0079238519 -0.0309010102 0.0014812254
## [16] 0.0083680526 0.0128329244 0.0536208392 0.0065303663 -0.0047701108
## [21] 0.0506378228 0.0059354298 0.0274007333 -0.0218340692 -0.0059995460
## [26] -0.0412812876 -0.0172286592 -0.0343622511 0.0084621600 -0.0204162998
## [31] 0.0385408314 -0.0430560729 -0.0298817852 -0.0036758774 0.0199058095
## [36] 0.0421447042 0.0421068984 0.0388077582 0.0060951811 0.0053984036
## [41] 0.0285410669 0.0493610404 0.0250534533 0.0422801466 -0.0253840544
## [46] -0.0194307697 0.0203089798 0.0102677866 0.0315583446 -0.0114898993
## [51] 0.0226706467 0.0215034894 0.0540411767 0.0462712044 -0.0014524207
## [56] 0.0517230302 -0.0049792435 0.0026381958 -0.0040007969 -0.0034067871
## [61] 0.0234335795 -0.0102803500 0.0178206792 0.0096772472 0.0547597564
## [66] 0.0122604041 0.0055302049 0.0339888246 -0.0229719401 0.0002291472
## [71] 0.0613162668 -0.0381254566 0.0177160518 0.0445365603 -0.0280828723
## [76] -0.0338055365 0.0248712121 0.0142280695 -0.0010815852 0.0031550375
## [81] 0.0027020684 -0.0230137744 0.0058272949 -0.0243084673 -0.0241863618
## [86] -0.0208660316 -0.0040433168 0.0232132294 0.0225929763 -0.0001793122
## [91] 0.0173014764 0.0479476650 0.0336937616 0.0154036435 0.0026168113
## [96] -0.0399152608 -0.0290865952 0.0279895754 0.0169056203 0.0061536808
## [101] -0.0255797445 0.0225921367 0.0183414203 0.0063846262 -0.0272113619
## [106] 0.0115692550 -0.0108051430 0.0326204837 0.0011177291 0.0154885515
## [111] -0.0178341072 0.0281841569 0.0455822611 0.0115582389 -0.0055552508
## [116] 0.0106403081 -0.0283947622 0.0288892583 0.0483511258 0.0161128732# 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 0.993
## 5 1.02
## 6 0.982
## 7 1.03
## 8 0.982
## 9 1.06
## 10 0.991
## # 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 1.02
## 5 1.04
## 6 1.02
## 7 1.05
## 8 1.03
## 9 1.09
## 10 1.08
## # ℹ 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 10.268756 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>
## 1 321.
## 2 321.
## 3 328.
## 4 333.
## 5 336.
## 6 340.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
set.seed(1234)
monte_carlo_sim51 <- 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 months
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_sim51## # 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_sim51 %>%
group_by(sim) %>%
summarize(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_sim51 %>%
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 $1 over 120 months")
## Line plot with max, median and min
# step 1, summarize data into max, med, min of last value
sim_summary <- monte_carlo_sim51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
summarise(max = max(growth),
median = median(growth),
min = min(growth))
sim_summary## # A tibble: 1 × 3
## max median min
## <dbl> <dbl> <dbl>
## 1 3.88 1.98 1.17# Step 2, plot
monte_carlo_sim51 %>%
group_by(sim) %>%
filter(last(growth) == sim_summary$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 = "Max, Median, minimum simulation")