Week 15: Code along 13
CHAPTER 11 Monte Carlo Simulation
Shea Clark
2024-12-02
# 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$returns)
mean_port_return## [1] 0.005899131# Get standard deviation of portfolio returns
stddev_port_return <- sd(portfolio_returns_tbl$returns)
stddev_port_return## [1] 0.0234749# Construct a normal distribution
simulated_monthly_returns <- rnorm(120, mean_port_return, stddev_port_return)
simulated_monthly_returns## [1] 0.0358037296 0.0084130986 0.0186964768 0.0232645765 0.0259720998
## [6] 0.0226594995 0.0434874181 0.0231453317 -0.0000118343 0.0526167614
## [11] -0.0079387038 0.0246508589 0.0365478553 -0.0179397336 0.0084788102
## [16] -0.0389297052 0.0278468380 0.0322199529 0.0222404190 0.0125077751
## [21] -0.0100109127 -0.0389032042 0.0233627406 0.0086675375 -0.0081453351
## [26] 0.0261267538 0.0063981866 0.0300584852 -0.0079713108 0.0419331386
## [31] 0.0145460208 0.0225586183 -0.0081455826 0.0080862251 0.0140375121
## [36] 0.0071111133 0.0036736840 0.0034707593 0.0047263863 0.0406961659
## [41] 0.0281883840 0.0077207500 0.0297605728 0.0100716534 0.0179450851
## [46] 0.0077710379 -0.0205531564 -0.0126424815 -0.0272227629 0.0359920393
## [51] 0.0013555850 -0.0117309902 0.0439884082 -0.0197669145 -0.0012235356
## [56] -0.0179988103 -0.0002462248 0.0417033691 -0.0003797219 0.0331731864
## [61] -0.0106987549 -0.0110039350 0.0041585266 0.0014082403 0.0075440182
## [66] 0.0174287271 0.0023770082 0.0278225423 0.0120487406 -0.0203079574
## [71] -0.0219170024 -0.0027425288 -0.0016927786 0.0057079021 0.0418184583
## [76] -0.0014543905 0.0314747430 0.0117733458 0.0007422958 0.0525921357
## [81] -0.0135492672 -0.0067569836 0.0358786352 -0.0215889276 -0.0323567483
## [86] 0.0095748719 -0.0291881035 0.0109574534 0.0288606632 0.0197171850
## [91] 0.0197639913 -0.0089961184 0.0006961572 0.0088885416 -0.0046653999
## [96] -0.0280420168 0.0361120106 0.0087574235 -0.0006197894 -0.0339941874
## [101] 0.0087486798 0.0029585880 -0.0109213203 0.0188440915 0.0246218524
## [106] -0.0264221348 0.0058046615 0.0530256382 0.0271589196 0.0237504954
## [111] -0.0365801452 -0.0118120293 0.0099939808 0.0134509624 -0.0030397116
## [116] 0.0062697607 0.0454753318 0.0350752663 -0.0141065845 0.0115090820# 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 1.04
## 3 1.01
## 4 1.02
## 5 1.02
## 6 1.03
## 7 1.02
## 8 1.04
## 9 1.02
## 10 1.00
## # 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 1.04
## 3 1.04
## 4 1.06
## 5 1.09
## 6 1.12
## 7 1.14
## 8 1.19
## 9 1.22
## 10 1.22
## # ℹ 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 10.310286 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>
## 1 326.
## 2 323.
## 3 325.
## 4 331.
## 5 332.
## 6 330.7 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())%>%
# Rename 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 $1 over 120 months")
# Step 1: Summarize data into max, median and min if last value
sim_summary <- monte_carlo_sim_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
summarise(max = max(growth),
median = median(growth),
min = min(growth))
sim_summary## # A tibble: 1 × 3
## max median min
## <dbl> <dbl> <dbl>
## 1 3.88 1.98 1.17 # Step 2 Plot
monte_carlo_sim_51 %>%
# Filter for max, median, and min sim
group_by(sim) %>%
filter(last(growth) == sim_summary$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 growth of $1 over 120 months",
subtitle = "Maximum, Median, and Minimum Simulation")