Week 15: Code along 13
CHAPTER 11 Monte Carlo Simulation
Ashley Trogler
2024-06-24
# 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] 0.02347492# Construct a normal distribution
simulated_monthly_returns <- rnorm(120, mean_port_return, stddev_port_return)
simulated_monthly_returns## [1] 0.0245913356 0.0051049667 -0.0108268006 0.0228288200 0.0193465077
## [6] 0.0177450754 0.0252659357 -0.0328379650 0.0446191212 0.0102866005
## [11] 0.0470745122 0.0130757785 0.0185489812 0.0073051731 0.0022028364
## [16] 0.0122468799 0.0146968929 -0.0156907004 -0.0276144382 0.0195822758
## [21] 0.0215833875 -0.0078972443 -0.0175969867 0.0073526665 0.0010416342
## [26] 0.0087938219 0.0340006428 -0.0186052161 0.0013075687 -0.0124702019
## [31] -0.0200586143 -0.0037804463 0.0171417087 0.0060892769 0.0147800349
## [36] 0.0331768195 0.0189022301 0.0166722826 0.0422529005 -0.0023586561
## [41] -0.0122237522 0.0193353339 0.0270070220 0.0133695098 0.0014569060
## [46] 0.0037669706 -0.0424861822 0.0278996968 0.0247508214 0.0331318828
## [51] 0.0189387631 0.0052916946 0.0005247323 0.0013377710 -0.0397395788
## [56] -0.0598187505 0.0206869620 0.0331290756 0.0243084242 0.0083264536
## [61] 0.0014507881 -0.0460828635 -0.0430405928 0.0037381807 0.0050065918
## [66] 0.0304525619 0.0609603862 0.0081772251 0.0240862442 0.0197496313
## [71] 0.0209709731 -0.0079051247 0.0063759270 0.0172814165 0.0056820767
## [76] 0.0044667281 -0.0054503867 0.0240722375 0.0205470389 -0.0329578765
## [81] 0.0139783349 -0.0288087050 -0.0041352893 -0.0495899865 -0.0439337761
## [86] 0.0116881177 0.0159006522 -0.0098440360 -0.0045533020 -0.0129073692
## [91] 0.0078829197 -0.0303584545 0.0261045869 0.0093600095 -0.0078524297
## [96] 0.0497549550 0.0313243385 0.0032187412 -0.0040406309 -0.0063115178
## [101] 0.0397533791 0.0337550632 0.0077766900 0.0140149419 0.0339458101
## [106] 0.0537001003 0.0147348895 0.0104140165 0.0358953725 -0.0029362120
## [111] 0.0374822796 0.0381927467 -0.0142330923 -0.0347303951 0.0183580749
## [116] 0.0170123145 -0.0213667867 -0.0133910041 -0.0277218377 0.0567301776# 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 1.02
## 3 1.01
## 4 0.989
## 5 1.02
## 6 1.02
## 7 1.02
## 8 1.03
## 9 0.967
## 10 1.04
## # 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 1.02
## 3 1.03
## 4 1.02
## 5 1.04
## 6 1.06
## 7 1.08
## 8 1.11
## 9 1.07
## 10 1.12
## # ℹ 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 8.4624146 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>
## 1 336.
## 2 334.
## 3 333.
## 4 337.
## 5 333.
## 6 334.dump(list = c("simulate_accumulation"),
file = "../00_scripts/simulate_accumulation.R")7 Running multiple simulations
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 name
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, 0.50, 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 $1 Over 120 Months")
Line plot with max, median, min
# Step 1: summarize data into max, median, and min of last value
sim_summary <- monte_carlo_sim_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
summarise(max = max(growth),
median = median(growth),
min = min(growth))
sim_summary## # A tibble: 1 × 3
## max median min
## <dbl> <dbl> <dbl>
## 1 3.88 1.98 1.17# Step 2 Plot
monte_carlo_sim_51 %>%
group_by(sim) %>%
filter(last(growth) == sim_summary$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")