Week 15: Code along 13
CHAPTER 11 Monte Carlo Simulation
Kenton Quaglieri
2024-12-01
# 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] 0.02347488# Construct a normal distribution
simulated_monthly_returns <- rnorm(120, mean_port_return, stddev_port_return)
simulated_monthly_returns## [1] -7.404312e-03 -4.126167e-03 -2.499467e-02 -2.376373e-03 3.270812e-02
## [6] -1.804371e-02 4.972078e-03 3.419272e-02 -8.055983e-03 9.933403e-03
## [11] -2.564137e-02 2.020817e-02 -8.744405e-03 -1.659414e-02 1.806786e-02
## [16] 2.594064e-02 1.958632e-02 -3.585763e-02 -1.434735e-02 4.507499e-03
## [21] 1.953403e-03 -6.848481e-03 -8.398770e-03 1.979438e-02 -7.809989e-03
## [26] -3.244919e-02 -3.006293e-02 7.503909e-03 2.878774e-02 1.773027e-02
## [31] 3.027485e-02 -4.146180e-03 -8.450976e-03 5.221664e-03 -1.403036e-02
## [36] -1.589187e-02 1.318464e-02 3.192075e-03 -7.987499e-04 -9.404603e-03
## [41] 6.746221e-03 -1.736502e-02 -1.233971e-02 -2.832421e-02 2.651379e-02
## [46] 2.781808e-02 -3.108868e-02 8.231438e-03 2.458958e-02 4.046298e-02
## [51] 7.042105e-03 3.213908e-03 3.914599e-02 -1.667563e-02 -1.252818e-02
## [56] -5.349285e-02 -1.609663e-02 -1.923026e-03 -3.800598e-02 1.095990e-02
## [61] -3.490615e-02 1.257192e-02 -4.306719e-02 1.927766e-02 -8.008520e-03
## [66] 2.515030e-02 3.479629e-03 1.221008e-02 -2.686625e-03 -4.123383e-04
## [71] 8.090473e-03 3.055581e-02 1.189211e-02 2.179588e-02 2.747454e-02
## [76] 2.329867e-02 1.541788e-02 1.487564e-02 -7.993791e-03 3.187934e-02
## [81] 2.379701e-02 -9.600298e-03 5.240225e-02 -2.852886e-02 -6.811218e-03
## [86] 3.560985e-02 3.479368e-02 2.889219e-02 3.737281e-02 -1.581318e-02
## [91] -8.362821e-03 -2.832078e-02 -2.353193e-03 1.734717e-02 2.801721e-02
## [96] 2.499535e-02 -3.643451e-02 9.412313e-03 -3.715787e-02 -1.286341e-02
## [101] 2.305565e-04 4.627692e-02 -2.688407e-02 6.347432e-02 2.711783e-02
## [106] 1.686786e-02 -6.176202e-03 1.492534e-02 3.189133e-02 -1.643044e-03
## [111] 1.762592e-02 3.096437e-02 -4.058624e-03 2.821928e-02 2.790025e-02
## [116] -3.227222e-02 -2.187199e-02 4.962936e-02 2.403338e-02 -9.440486e-05# 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 0.993
## 3 0.996
## 4 0.975
## 5 0.998
## 6 1.03
## 7 0.982
## 8 1.00
## 9 1.03
## 10 0.992
## # 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 0.993
## 3 0.989
## 4 0.964
## 5 0.962
## 6 0.993
## 7 0.975
## 8 0.980
## 9 1.01
## 10 1.01
## # ℹ 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 4.9239996 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>
## 1 289.
## 2 290.
## 3 301.
## 4 311.
## 5 314.
## 6 312.dump(list = c("simulate_accumulation"),
file = "../00_Data/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 growht 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")