Week 15: Code along 13
CHAPTER 11 Monte Carlo Simulation
Jacob Blades
2022-09-19
# 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$returns)
mean_port_return## [1] 0.005899134# Get standard deviation of portfolio returns
stddev_port_return <- sd(portfolio_returns_tbl$returns)
stddev_port_return## [1] 0.02347489# Construct a normal distribution
simulated_monthly_returns <- rnorm(120, mean_port_return, stddev_port_return)
simulated_monthly_returns## [1] 7.007167e-03 -1.120069e-02 -7.202954e-03 -2.603098e-02 2.296531e-02
## [6] 6.850456e-02 5.419975e-03 2.696210e-02 -7.458763e-03 7.972068e-03
## [11] -1.704925e-03 -1.112759e-02 1.406072e-03 -4.983600e-03 2.073622e-02
## [16] -1.162772e-02 2.340538e-02 1.158981e-02 2.062503e-03 2.448833e-02
## [21] 2.878508e-02 2.153319e-02 4.479353e-02 1.704621e-02 -6.451777e-03
## [26] 2.443331e-02 2.136770e-02 -5.425893e-03 2.830732e-03 -6.587222e-03
## [31] -1.370521e-02 4.456520e-02 5.286544e-03 -5.770919e-03 2.445509e-02
## [36] 2.245882e-02 4.231758e-02 -1.507591e-02 -2.203672e-02 3.029635e-02
## [41] 2.872408e-02 -3.404255e-02 5.941100e-03 -3.918016e-03 -1.620816e-02
## [46] -6.155788e-03 -1.160537e-02 7.034976e-03 3.269813e-02 3.046601e-02
## [51] 5.408162e-03 3.518736e-02 -6.419689e-03 2.446678e-02 2.675501e-02
## [56] -1.575817e-02 1.101946e-02 -1.708156e-02 -4.642671e-03 7.291315e-03
## [61] 1.378982e-02 -2.484471e-02 4.261425e-02 1.935466e-02 -3.555354e-02
## [66] 2.287178e-02 3.263694e-02 5.316300e-02 -1.766969e-02 -1.215023e-02
## [71] -8.584360e-03 2.073973e-02 -5.331672e-03 -9.219263e-03 -6.318276e-03
## [76] -1.262873e-02 -1.648913e-02 1.608701e-03 5.833732e-02 3.398761e-02
## [81] 3.522679e-03 -5.671290e-03 1.774308e-03 -4.005827e-02 2.273644e-02
## [86] -1.306014e-02 -2.226120e-02 -1.606420e-02 2.729944e-02 1.063171e-02
## [91] -7.147639e-02 4.944525e-02 1.825535e-02 9.641408e-03 2.946852e-02
## [96] 4.734130e-02 -1.873800e-02 1.808891e-02 -8.069022e-03 1.440447e-02
## [101] 9.882744e-03 4.691813e-02 -1.760688e-02 5.404864e-02 5.863402e-03
## [106] 4.545020e-02 -2.118388e-02 -1.177199e-02 3.538998e-02 -3.312908e-02
## [111] -1.530040e-02 1.235587e-02 -2.299402e-03 1.170941e-02 2.510317e-02
## [116] 7.571815e-05 -2.829545e-04 -9.985101e-03 -1.057095e-02 1.907445e-02# 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 1.01
## 3 0.989
## 4 0.993
## 5 0.974
## 6 1.02
## 7 1.07
## 8 1.01
## 9 1.03
## 10 0.993
## # 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 1.01
## 3 0.996
## 4 0.989
## 5 0.963
## 6 0.985
## 7 1.05
## 8 1.06
## 9 1.09
## 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] 8.4528166 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(100, N = 240, mean_return = 0.005, sd_return = 0.01) %>%
tail()## # A tibble: 6 × 1
## growth
## <dbl>
## 1 279.
## 2 275.
## 3 280.
## 4 279.
## 5 282.
## 6 280.7 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 1monte_carlo_sim_51 <- starts %>%
map_dfc(.x = .,
.f = ~simulate_accumulation(initial_value = .x,
N = 120,
mean_return = mean_port_return,
sd_return = stddev_port_return)) %>%
mutate(month = 1:nrow(.)) %>%
select(month, everything()) %>%
set_names(c("month", names(starts))) %>%
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 rowsmonte_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%
## 1.18 1.71 1.97 2.29 3.508 Visualizing simulations with ggplot
monte_carlo_sim_51 %>%
ggplot(aes(x = month, y = growth, col = sim)) +
geom_line() +
theme(legend.position = "none")
# Simplify the plot
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 3.50 1.97 1.18monte_carlo_sim_51 %>%
group_by(sim) %>%
filter(last(growth) == sim_summary$max |
last(growth) == sim_summary$median |
last(growth) == sim_summary$min) %>%
# Plot
ggplot(aes(month, growth, col = sim)) +
geom_line() +
theme()