Week 15: Code along 13
CHAPTER 11 Monte Carlo Simulation
Ryan Berg
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$returns)
mean_port_return## [1] 0.005899137# Get standard deviation of portfolio returns
stddev_port_return <- sd(portfolio_returns_tbl$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] 8.963905e-05 -2.885415e-03 3.370368e-02 -2.391953e-04 -2.103136e-02
## [6] 4.593694e-03 4.209680e-02 2.398950e-02 -4.444444e-03 -1.685896e-03
## [11] 1.272998e-02 2.656181e-02 -2.821826e-02 -1.327751e-02 2.817012e-02
## [16] -1.261704e-02 -5.867640e-04 -4.340838e-03 4.285321e-03 -2.554305e-02
## [21] 3.108442e-02 -9.162811e-03 -4.880504e-03 -4.015087e-03 -2.294966e-02
## [26] 2.358376e-02 1.370565e-02 -3.667026e-03 2.781322e-03 -3.368377e-03
## [31] 4.199597e-02 -1.972256e-02 4.537257e-02 1.777547e-02 -1.288692e-02
## [36] -2.224000e-02 2.533090e-02 2.403984e-02 1.570693e-02 -2.779338e-02
## [41] -2.292618e-02 -2.901363e-02 -2.245595e-02 6.587289e-02 -2.577883e-03
## [46] -2.114908e-02 -1.604064e-02 -4.075098e-02 3.979813e-03 1.171160e-02
## [51] -2.266663e-02 -2.352823e-02 -5.503401e-03 1.110844e-02 2.577765e-02
## [56] -5.513459e-03 1.490712e-02 -7.597913e-03 -9.945841e-03 -3.676245e-02
## [61] -3.884468e-03 5.486852e-03 -3.016147e-02 4.997055e-02 4.577521e-02
## [66] -8.479940e-03 -1.382276e-02 -1.774832e-02 -1.052739e-02 -6.304042e-02
## [71] -2.926472e-03 2.112300e-02 -1.724644e-02 -3.930923e-03 2.419861e-02
## [76] 1.453845e-02 4.479324e-02 3.946367e-02 1.575172e-02 -6.378120e-03
## [81] 1.453841e-02 2.356904e-02 5.358455e-02 1.565901e-02 5.287907e-02
## [86] -5.937413e-03 7.907624e-03 -7.948448e-03 -6.293166e-03 1.573346e-02
## [91] 3.997506e-02 -5.518209e-03 1.137082e-03 -1.443962e-03 2.598927e-03
## [96] -3.509668e-02 3.466241e-02 -9.268443e-03 4.138497e-02 3.096520e-02
## [101] -7.823942e-03 -1.511371e-02 1.817567e-02 -2.583663e-02 4.596328e-02
## [106] 2.983865e-02 1.023710e-02 -2.108578e-03 3.327309e-02 -3.268235e-02
## [111] 1.777931e-02 -1.224527e-02 -1.698345e-02 1.416890e-02 9.958738e-04
## [116] 1.334033e-03 1.663847e-02 2.012296e-02 4.822744e-03 3.660317e-03# 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 0.997
## 4 1.03
## 5 1.00
## 6 0.979
## 7 1.00
## 8 1.04
## 9 1.02
## 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 1.00
## 3 0.997
## 4 1.03
## 5 1.03
## 6 1.01
## 7 1.01
## 8 1.06
## 9 1.08
## 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] 4.6637786 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>
## 1 340.
## 2 339.
## 3 340.
## 4 346.
## 5 347.
## 6 351.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
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%
## 1.02 1.85 2.15 2.48 4.338 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 $1 over 120 months")
Line plot with max, median, and min
# Step 1 Summarize data into max, median, and min of last value
sim_summary <- monte_carlo_sim_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
summarise(max = max(growth),
median = median(growth),
min = min(growth))
sim_summary## # A tibble: 1 × 3
## max median min
## <dbl> <dbl> <dbl>
## 1 4.33 2.15 1.02# Step 2 Plot
monte_carlo_sim_51 %>%
group_by(sim) %>%
filter(last(growth) == sim_summary$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",
subtitile = "Maximum, Median, and Minimum Simulation")