Week 15: Code along 13
CHAPTER 11 Monte Carlo Simulation
Job Boonstoppel
2023-06-23
# 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] 0.02347491# Construct a normal distribution
simulated_monthly_returns <- rnorm(120, mean_port_return, stddev_port_return)
simulated_monthly_returns## [1] -0.0016775750 0.0138970417 0.0029126456 0.0376453644 -0.0142245577
## [6] 0.0224203622 -0.0148264137 0.0062142863 -0.0158679152 0.0268565161
## [11] 0.0129852250 -0.0017522349 -0.0036084845 -0.0083523787 0.0017060156
## [16] 0.0083238261 -0.0001713892 0.0236575403 -0.0128064541 -0.0149282433
## [21] -0.0186260060 0.0302158616 0.0453590154 -0.0161391759 -0.0237522059
## [26] 0.0056446421 0.0068269986 -0.0308165243 0.0001290838 -0.0003184444
## [31] -0.0321084897 -0.0208347533 -0.0218863925 0.0133792117 0.0027966597
## [36] -0.0136773252 0.0384138034 0.0302759160 -0.0223968952 0.0269258022
## [41] 0.0152536822 0.0226238069 0.0057193335 -0.0044932627 -0.0048593298
## [46] 0.0442728766 0.0192122835 -0.0059947381 0.0141853224 0.0013889597
## [51] 0.0215258630 0.0330552859 -0.0084221980 0.0182877185 0.0017405035
## [56] 0.0235325322 -0.0076930558 -0.0620698345 0.0368732387 -0.0250408190
## [61] -0.0119861351 0.0145834128 -0.0089296127 -0.0060538260 -0.0270494992
## [66] 0.0199351475 0.0307418797 0.0195796865 0.0328244617 0.0144723199
## [71] -0.0220074361 -0.0039690023 -0.0368621534 -0.0052776641 -0.0076680655
## [76] 0.0432789400 0.0245475078 0.0268016702 -0.0425128909 0.0055774490
## [81] -0.0162376399 -0.0192604651 -0.0358617945 0.0158777061 -0.0271767610
## [86] 0.0277708716 0.0155193570 0.0240617858 0.0200719078 0.0125596237
## [91] 0.0130413955 0.0024764072 0.0294205800 -0.0168479570 0.0320475506
## [96] 0.0374142436 0.0242916608 0.0132741196 0.0002914924 -0.0219789571
## [101] -0.0023049695 0.0449419090 -0.0175430033 0.0309167022 0.0052818565
## [106] 0.0173470114 0.0057436659 -0.0210494624 -0.0124545228 -0.0041573000
## [111] 0.0121756918 0.0115540322 -0.0123859891 0.0128031284 0.0626475654
## [116] 0.0396207500 -0.0083686450 -0.0240974410 -0.0204741459 -0.0250085440# 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 0.998
## 3 1.01
## 4 1.00
## 5 1.04
## 6 0.986
## 7 1.02
## 8 0.985
## 9 1.01
## 10 0.984
## # 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 0.998
## 3 1.01
## 4 1.02
## 5 1.05
## 6 1.04
## 7 1.06
## 8 1.05
## 9 1.05
## 10 1.04
## # ℹ 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 4.7342226 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>
## 1 382.
## 2 388.
## 3 393.
## 4 398.
## 5 408.
## 6 415.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_carle_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")
# Find quantiles
monte_carle_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_carle_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 $1 over 120 months")
Line plot with max, median, and min
# Step 1 Summarize data into max, median, and min of last value
sim_summary <- monte_carle_sim_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
summarise(max = max(growth),
median = median(growth),
min = min(growth))
sim_summary## # A tibble: 1 × 3
## max median min
## <dbl> <dbl> <dbl>
## 1 3.88 1.98 1.17# Step 2 Plot
monte_carle_sim_51 %>%
# Filter for max, median, min sim
group_by(sim) %>%
filter(last(growth) == sim_summary$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 Mimimum Simulation")