Week 15: Code along 13
CHAPTER 11 Monte Carlo Simulation
Blake Yaffe
2024-12-11
# 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] 0.02347493# Construct a normal distribution
simulated_monthly_returns <- rnorm(120, mean_port_return, stddev_port_return)
simulated_monthly_returns## [1] 0.0265816156 0.0013107839 -0.0261058131 -0.0108457655 0.0067555323
## [6] -0.0207605222 0.0299589072 -0.0024424685 -0.0036324955 0.0190464884
## [11] -0.0155519651 -0.0040203595 0.0176249141 0.0333390310 0.0126542211
## [16] 0.0129097803 -0.0239291670 -0.0123151188 0.0179247834 -0.0409623753
## [21] -0.0262170808 0.0002591953 0.0364926555 0.0300578000 -0.0036937954
## [26] 0.0328496130 0.0076124511 0.0031944233 -0.0077112121 0.0116355395
## [31] 0.0037282185 0.0265107010 -0.0075094362 0.0213215962 -0.0191457337
## [36] -0.0336690337 -0.0073564725 0.0084562246 0.0070061849 -0.0277615294
## [41] -0.0253944778 0.0084432854 0.0024096278 0.0539599113 -0.0630820401
## [46] 0.0144921160 0.0171038420 0.0485123891 -0.0421474688 -0.0171279795
## [51] 0.0226238955 0.0123184886 0.0097904972 -0.0177384452 0.0269300205
## [56] 0.0161246732 -0.0377969173 0.0157352099 0.0112946560 -0.0163627362
## [61] -0.0248205618 0.0409210896 0.0218647314 -0.0138455483 -0.0156480955
## [66] 0.0374964642 -0.0086434394 0.0119887112 0.0449801435 0.0130170927
## [71] 0.0065817087 0.0154316493 -0.0013746457 0.0063771015 0.0174766880
## [76] -0.0379961230 -0.0062746475 -0.0073222225 -0.0219965988 0.0148046412
## [81] 0.0430911735 0.0016345432 0.0155517831 -0.0014543427 0.0021890190
## [86] -0.0211384913 -0.0592610206 0.0143243419 -0.0063313462 0.0180610069
## [91] -0.0025471748 0.0168840863 0.0043244878 0.0683806573 0.0030512231
## [96] -0.0119812648 0.0266368697 -0.0108117963 0.0191762224 -0.0170062519
## [101] 0.0306292803 0.0146458865 -0.0525286531 0.0185251623 0.0179060779
## [106] 0.0000275934 -0.0190550800 0.0081446495 -0.0069963219 -0.0048090603
## [111] 0.0238085332 -0.0066392854 -0.0026348632 0.0064101761 0.0203596654
## [116] -0.0344428672 0.0170345027 0.0374919100 0.0095938171 0.0019926565# 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 1.03
## 3 1.00
## 4 0.974
## 5 0.989
## 6 1.01
## 7 0.979
## 8 1.03
## 9 0.998
## 10 0.996
## # 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 1.03
## 3 1.03
## 4 1.00
## 5 0.990
## 6 0.997
## 7 0.976
## 8 1.01
## 9 1.00
## 10 0.999
## # ℹ 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 3.6298656 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>
## 1 382.
## 2 385.
## 3 395.
## 4 403.
## 5 413.
## 6 414.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 $1 over 120 months")
Line plot with max, median, and min
# Step 1 summarize data into max, median, and min of last value
sim_summary <- monte_carlo_sim_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
summarise(max = max(growth),median = median(growth), min = min(growth))
sim_summary## # A tibble: 1 × 3
## max median min
## <dbl> <dbl> <dbl>
## 1 3.88 1.98 1.17#Step 2 plot
monte_carlo_sim_51 %>%
# Filter for max,median, and min sim
group_by(sim) %>%
filter(last(growth) == sim_summary$max |
last(growth) == sim_summary$median |
last(growth) == sim_summary$min) %>%
ungroup() %>%
# Plot
ggplot(aes(x = month, y = growth, 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 growht of $1 over 120 months",
subtitle = "Maximum, Median, and Minimum Simulation")