Week 15: Code along 13
CHAPTER 11 Monte Carlo Simulation
Olivia Salesky
2022-12-6
# 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] 0.02347492# Construct a normal distribution
simulated_monthly_returns <- rnorm(120, mean_port_return, stddev_port_return)
simulated_monthly_returns## [1] 0.0038550518 0.0243762979 0.0214240024 0.0097243223 0.0114835416
## [6] -0.0092281202 -0.0073875961 -0.0315693828 -0.0171630967 -0.0051628927
## [11] -0.0236521504 0.0212067065 0.0073666282 -0.0529281417 0.0008082608
## [16] -0.0055114311 0.0101626747 0.0462538255 0.0254503890 -0.0354372131
## [21] -0.0056685148 -0.0345083704 -0.0156671737 -0.0158944268 0.0456761393
## [26] 0.0176476381 -0.0094090273 -0.0012297812 0.0402000071 0.0179914270
## [31] -0.0038455352 -0.0095483680 0.0257507086 -0.0036481248 -0.0295156672
## [36] 0.0524314235 0.0029734683 0.0413482117 0.0284570061 0.0135943714
## [41] 0.0146823324 0.0136036326 -0.0132574310 0.0333288370 0.0295253976
## [46] 0.0074325666 0.0222287472 -0.0092281684 0.0296308198 -0.0069221688
## [51] 0.0415319981 0.0148489439 0.0112547002 0.0174011999 0.0445615905
## [56] -0.0110698616 0.0233158382 -0.0108159610 0.0236778517 0.0185114196
## [61] -0.0121375785 -0.0125591917 0.0057456949 0.0211936597 -0.0107992018
## [66] 0.0100963541 -0.0110074219 0.0042146640 -0.0075514788 -0.0031524253
## [71] 0.0064596917 -0.0138300260 0.0294023984 -0.0242370175 0.0087809377
## [76] 0.0150408908 0.0333175685 0.0283259691 0.0004295439 0.0016344303
## [81] -0.0241790693 -0.0144148048 0.0305218615 0.0122162268 0.0066753813
## [86] -0.0057776368 -0.0336120406 -0.0056099418 -0.0075086403 -0.0322269841
## [91] -0.0025238247 -0.0525258057 0.0034121260 -0.0221562425 0.0283576600
## [96] 0.0244806494 0.0185548337 0.0254765071 0.0054019942 0.0280959285
## [101] -0.0089298860 0.0044055372 0.0266900270 0.0037643504 0.0327975343
## [106] -0.0072292087 -0.0193931811 -0.0035571415 0.0412650770 0.0427313747
## [111] -0.0090495612 0.0038490423 -0.0157032190 0.0037279872 0.0276778354
## [116] -0.0171107917 0.0744868169 0.0295906397 0.0244177559 0.0704004170# 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 1.02
## 4 1.02
## 5 1.01
## 6 1.01
## 7 0.991
## 8 0.993
## 9 0.968
## 10 0.983
## # … 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 1.00
## 3 1.03
## 4 1.05
## 5 1.06
## 6 1.07
## 7 1.06
## 8 1.05
## 9 1.02
## 10 1.00
## # … with 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 7.9840486 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>
## 1 281.
## 2 284.
## 3 288.
## 4 291.
## 5 298.
## 6 293.7 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
## # … 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 $1 over 120 months")
Line plot with max, median, and min
# Step 1 Summarize data into maximum, median, and minimum of last value
sim_summary <- monte_carlo_sim_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
summarise(max = max(growth),
median = median(growth),
min = min(growth))
sim_summary## # A tibble: 1 × 3
## max median min
## <dbl> <dbl> <dbl>
## 1 3.88 1.98 1.17# Step 2 Plot
monte_carlo_sim_51 %>%
# Filter for max, median, and min sim
group_by(sim) %>%
filter(last(growth) == sim_summary$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 growth of $1 over 120 months",
subtitle = "Maximum, Median, and Minimum Simulation")