Code along 13
CHAPTER 11 Monte Carlo Simulation
Sumaj Billin
2024-12-02
# 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] 0.0001387232 0.0081299344 0.0379596369 0.0180492660 0.0189740215
## [6] -0.0071479811 -0.0270392048 0.0251277621 0.0102185800 -0.0106558614
## [11] -0.0056770835 0.0035288532 -0.0226253314 0.0169871553 0.0071122698
## [16] -0.0063341981 -0.0215102848 -0.0138651303 -0.0164809533 -0.0261114440
## [21] -0.0035176910 -0.0205659895 0.0669980657 -0.0028909033 0.0096502480
## [26] 0.0170409615 0.0376642941 0.0268991790 0.0034209923 -0.0414054745
## [31] -0.0018277732 -0.0175842442 -0.0329363636 -0.0237413887 -0.0372949588
## [36] 0.0270634094 0.0117296755 -0.0220329110 0.0535085752 0.0226148781
## [41] 0.0484713186 -0.0203143882 0.0056468633 0.0238218828 0.0349931218
## [46] -0.0249688670 0.0207458576 0.0225768238 0.0016948944 0.0406467179
## [51] 0.0098957237 0.0364257538 0.0054075025 0.0045473718 0.0404504272
## [56] -0.0140963042 -0.0051915007 -0.0036449283 0.0078668509 -0.0145132021
## [61] 0.0048460327 0.0180496028 0.0179391543 -0.0022857752 0.0036713106
## [66] 0.0098486180 0.0186255727 -0.0057857306 -0.0307223191 0.0042683662
## [71] -0.0363815113 0.0295553667 -0.0062511201 0.0035780732 -0.0122837701
## [76] 0.0165094315 0.0191602156 -0.0366484600 0.0465420790 0.0267496928
## [81] 0.0209692784 0.0160173521 0.0279651605 0.0394851378 0.0314655065
## [86] 0.0489959410 0.0389422120 0.0034646415 -0.0338395057 0.0259524819
## [91] 0.0535224262 0.0106602294 -0.0059603493 0.0252214846 0.0203152340
## [96] -0.0047637121 -0.0625740172 -0.0027242670 0.0208521614 -0.0059899805
## [101] 0.0223038331 -0.0381313704 -0.0257015706 0.0102438039 -0.0115725287
## [106] 0.0022458967 0.0323221857 -0.0200326904 0.0041191566 -0.0039408642
## [111] 0.0089014319 0.0308477828 0.0241364153 0.0180812612 0.0030897358
## [116] -0.0287984270 -0.0065511975 0.0220289192 -0.0098819615 -0.0200596251# 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 1.01
## 4 1.04
## 5 1.02
## 6 1.02
## 7 0.993
## 8 0.973
## 9 1.03
## 10 1.01
## # 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 1.01
## 4 1.05
## 5 1.07
## 6 1.09
## 7 1.08
## 8 1.05
## 9 1.08
## 10 1.09
## # ℹ 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 6.3699056 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)) %>%
return(simulated_growth)
}
simulate_accumulation(initial_value = 100, N = 240, mean_return = 0.005, sd_return = 0.01) %>%
tail()## # A tibble: 6 × 2
## returns growth
## <dbl> <dbl>
## 1 1.00 296.
## 2 1.01 299.
## 3 1.00 300.
## 4 0.999 300.
## 5 1.00 300.
## 6 0.994 299.dump(list = c("simulate_accumulation"),
file = "../Desktop/PSU_FIN3100_FinancialAnalytics/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 1set.seed(1234)
# Simulate
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 the column, month
mutate(month = (1:nrow(.))) %>%
select(month, everything()) %>%
# Rearrange column names
# Transform to long form
pivot_longer(cols = -month, names_to = "sim", values_to = "growth")
monte_carlo_sim_51## # A tibble: 12,342 × 3
## month sim growth
## <int> <chr> <dbl>
## 1 1 returns...1 1
## 2 1 growth...2 1
## 3 1 returns...3 1
## 4 1 growth...4 1
## 5 1 returns...5 1
## 6 1 growth...6 1
## 7 1 returns...7 1
## 8 1 growth...8 1
## 9 1 returns...9 1
## 10 1 growth...10 1
## # ℹ 12,332 more rows# Find quantiles
monte_carlo_sim_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
pull(growth) %>%
# Find the quantiles
quantile(probs = c(0,0.25, 0.5, 0.75, 1)) %>%
round(2)## 0% 25% 50% 75% 100%
## 0.96 1.00 1.12 1.98 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 $1 over 120 months")
sim_summary <- monte_carlo_sim_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
summarise(max = max(growth),
median = median(growth),
min = min(growth))
sim_summary## # A tibble: 1 × 3
## max median min
## <dbl> <dbl> <dbl>
## 1 3.88 1.12 0.957monte_carlo_sim_51 %>%
group_by(sim) %>%
filter(last(growth) == sim_summary$max |
last(growth) == sim_summary$median |
last(growth) == sim_summary$min) %>%
ungroup() %>%
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 = "Maxixum, Median, and Minimun Simulation")