Week 15: Code along 13
CHAPTER 11 Monte Carlo Simulation
Reed Wilson
2023-04-30
# 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
## # … with 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] 0.02347491# Construct a normal distribution
simulated_monthly_returns <- rnorm(120, mean_port_return, stddev_port_return)
simulated_monthly_returns## [1] 2.041733e-02 1.015888e-02 -2.336634e-02 -1.471578e-02 -1.351180e-02
## [6] 1.010533e-02 -4.756080e-02 4.416466e-02 3.427588e-02 3.471348e-02
## [11] 8.515783e-03 8.083455e-03 1.317834e-02 2.909665e-02 -1.886589e-02
## [16] 4.991103e-02 1.354253e-02 -3.156233e-03 2.355579e-02 -8.218881e-04
## [21] -7.996795e-03 2.118552e-02 5.255245e-02 3.549528e-03 2.073503e-02
## [26] -2.414747e-02 -1.935889e-03 -8.747973e-04 -1.325908e-02 1.164116e-02
## [31] -9.814345e-03 6.226363e-02 1.744488e-02 2.489199e-02 -5.716319e-03
## [36] 4.558888e-02 3.016181e-02 2.160827e-02 2.744917e-02 3.538369e-02
## [41] -2.234071e-02 -8.604302e-03 -3.383897e-03 -9.684692e-04 3.197252e-02
## [46] 1.861759e-02 4.982971e-02 -1.691538e-02 3.179543e-02 -1.977228e-02
## [51] 6.136934e-03 4.988448e-03 -9.181008e-04 3.201019e-02 4.251780e-03
## [56] -1.479484e-02 -4.177462e-03 3.183675e-02 7.078307e-04 4.170986e-03
## [61] -4.119657e-03 -9.848269e-03 -2.312064e-03 2.821311e-02 1.971863e-02
## [66] 1.890609e-02 2.291630e-02 2.505892e-02 -5.974224e-04 1.800173e-02
## [71] 1.659016e-02 3.105200e-02 -1.124934e-02 -3.431179e-02 -9.372067e-03
## [76] 2.246292e-02 -2.238927e-02 1.057084e-02 1.706422e-02 -1.245153e-02
## [81] -1.770593e-02 -8.388476e-03 3.103151e-02 1.555918e-02 6.605948e-03
## [86] -1.592640e-02 -2.228123e-03 -1.479251e-02 3.155207e-02 3.436638e-02
## [91] 1.554073e-02 3.056112e-02 -3.348582e-02 2.312346e-03 7.584747e-05
## [96] -4.246650e-02 9.677393e-03 3.963584e-02 -8.521081e-03 -2.030325e-02
## [101] 2.827347e-02 -3.736899e-03 3.120848e-02 1.968878e-02 -1.640779e-03
## [106] 1.728500e-02 1.637242e-02 -3.979846e-02 -4.006375e-03 -5.645349e-03
## [111] -6.122304e-02 2.645772e-02 -1.519119e-02 -7.304426e-03 1.094772e-02
## [116] 3.943661e-02 -1.458934e-03 7.734013e-03 3.637767e-02 7.424480e-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 1.02
## 3 1.01
## 4 0.977
## 5 0.985
## 6 0.986
## 7 1.01
## 8 0.952
## 9 1.04
## 10 1.03
## # … with 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 1.01
## 5 0.992
## 6 0.978
## 7 0.988
## 8 0.941
## 9 0.983
## 10 1.02
## # … with 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 8.9605496 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>
## 1 248.
## 2 252.
## 3 253.
## 4 258.
## 5 260.
## 6 260.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())%>%
# 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
## # … with 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 $1 over 120 months")
Line Plot with max, medium, and min
# Step 1: Summarize data into max, median and min if last value
sim_summary <- monte_carlo_sim_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
summarise(max = max(growth),
median = median(growth),
min = min(growth))
sim_summary## # A tibble: 1 × 3
## max median min
## <dbl> <dbl> <dbl>
## 1 3.88 1.98 1.17# Step 2 Plot
monte_carlo_sim_51 %>%
# Filter for max, median, and min sim
group_by(sim) %>%
filter(last(growth) == sim_summary$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")