Week 15: Code along 13
CHAPTER 11 Monte Carlo Simulation
James Hannigan
2024-12-01
# 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] 0.02347491# Construct a normal distribution
simulated_monthly_returns <- rnorm(120, mean_port_return, stddev_port_return)
simulated_monthly_returns## [1] 0.0039167337 0.0152799998 -0.0473314630 0.0716699970 0.0052219677
## [6] -0.0251875575 -0.0510532032 0.0245705616 -0.0233689170 -0.0326671965
## [11] 0.0259470417 -0.0011522722 0.0363629263 -0.0215825435 -0.0088422006
## [16] -0.0273937587 0.0228322976 -0.0419046694 0.0164107806 -0.0031307406
## [21] -0.0024306333 -0.0346436150 0.0097713341 -0.0222829000 0.0116501893
## [26] -0.0126562244 0.0085557960 -0.0066025100 0.0116449593 0.0181394317
## [31] -0.0152831345 -0.0144656194 0.0260247610 -0.0325762295 0.0375498170
## [36] -0.0501897956 0.0117077203 0.0256731582 0.0121526795 -0.0038640863
## [41] 0.0237493684 0.0454070068 0.0121794966 -0.0074552256 0.0086673130
## [46] 0.0187234034 -0.0004201395 0.0222385695 0.0452880616 0.0117684094
## [51] -0.0061486654 -0.0093419384 0.0205938249 0.0453265844 -0.0328245410
## [56] -0.0244497974 0.0314070124 0.0258534772 -0.0003902100 0.0040814764
## [61] 0.0079316444 -0.0182450535 -0.0158781315 -0.0361917850 -0.0244904266
## [66] -0.0433729297 -0.0234328974 0.0321838758 0.0183256054 0.0182928590
## [71] 0.0431681844 0.0489987831 0.0077516933 -0.0175699697 0.0121395006
## [76] -0.0126114193 0.0216217442 0.0050135003 -0.0003325347 0.0071162110
## [81] 0.0338984985 0.0730272201 -0.0048989184 0.0222190485 -0.0093835459
## [86] 0.0226067872 0.0318119455 -0.0123931403 -0.0133357132 0.0393334476
## [91] -0.0114292871 0.0372978697 0.0172340008 -0.0100339659 -0.0175613116
## [96] -0.0097400606 0.0137387669 -0.0096011806 0.0405620546 0.0030917413
## [101] 0.0042371090 -0.0285517937 0.0110049778 -0.0496999358 -0.0018095183
## [106] -0.0282906305 -0.0068977718 0.0459560992 -0.0006753488 0.0844580517
## [111] 0.0260640687 -0.0029106438 0.0298843413 -0.0138746624 0.0086558472
## [116] 0.0149629441 -0.0171687000 -0.0124443511 -0.0059885644 0.0386018192# 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 0.953
## 5 1.07
## 6 1.01
## 7 0.975
## 8 0.949
## 9 1.02
## 10 0.977
## # 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 1.02
## 4 0.971
## 5 1.04
## 6 1.05
## 7 1.02
## 8 0.968
## 9 0.991
## 10 0.968
## # ℹ 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 4.8232396 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>
## 1 347.
## 2 348.
## 3 348.
## 4 353.
## 5 353.
## 6 360.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()) %>%
#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%
## 1.17 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 growht of $1 over 120 months")
Line plot with max, median, and min
# Step 1 summarize data into max, median, and min of last value
sim_summary <- monte_carlo_sim_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
summarise(max = max(growth),median = median(growth), min = min(growth))
sim_summary## # A tibble: 1 × 3
## max median min
## <dbl> <dbl> <dbl>
## 1 3.88 1.98 1.17#Step 2 plot
monte_carlo_sim_51 %>%
# Filter for max,median, and min sim
group_by(sim) %>%
filter(last(growth) == sim_summary$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 growht of $1 over 120 months",
subtitle = "Maximum, Median, and Minimum Simulation")