Week 15: Code along 13
CHAPTER 11 Monte Carlo Simulation
Colin Tracy
2022-12-06
# 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$returns)
mean_port_return## [1] 0.005899132# Get standard deviation of portfolio returns
stddev_port_return <- sd(portfolio_returns_tbl$returns)
stddev_port_return## [1] 0.0234749# Construct a normal distribution
simulated_monthly_returns <- rnorm(120, mean_port_return, stddev_port_return)
simulated_monthly_returns## [1] -0.0224060227 0.0467079557 0.0138235451 -0.0248294986 0.0074083372
## [6] 0.0134103831 -0.0276161327 -0.0065369192 0.0520205030 0.0164091252
## [11] -0.0319839591 0.0024085624 0.0060227911 0.0203443195 0.0295612012
## [16] 0.0285239925 0.0286264135 0.0259891741 0.0225096091 -0.0220537374
## [21] 0.0053839611 -0.0032398669 -0.0129644567 0.0157794546 0.0199522374
## [26] 0.0524803660 -0.0106409768 0.0103812783 0.0233760741 0.0851841262
## [31] -0.0094118370 -0.0443599262 0.0202732562 0.0081960691 0.0240720354
## [36] 0.0096010555 0.0555998714 0.0147843783 0.0312738932 0.0332369480
## [41] 0.0365601114 0.0522057346 0.0195431153 -0.0122693191 -0.0086934595
## [46] -0.0042411686 0.0410711553 0.0010361835 0.0158137912 -0.0074207525
## [51] -0.0121978958 -0.0086308050 0.0263358282 0.0077567333 0.0240096639
## [56] 0.0231367395 0.0013727543 0.0015472913 0.0081982515 -0.0094059860
## [61] 0.0478214933 -0.0093457839 0.0033597860 0.0296100412 0.0038382618
## [66] 0.0014605512 0.0262004402 -0.0104886389 0.0054840314 -0.0279039386
## [71] -0.0044484974 -0.0469392326 -0.0073534613 0.0025277250 -0.0275115955
## [76] -0.0126069897 0.0127570676 0.0158925511 0.0205464673 0.0247459333
## [81] 0.0143856205 -0.0313565818 -0.0446839218 -0.0310233534 0.0295996197
## [86] 0.0274178594 0.0101115862 -0.0061947217 0.0559373834 0.0460901092
## [91] 0.0159039257 0.0301422597 0.0141636988 0.0144127928 -0.0270316880
## [96] 0.0630224752 0.0055953433 -0.0253804206 0.0138869290 -0.0177508434
## [101] 0.0208511139 -0.0148661105 0.0167069674 -0.0033803572 0.0461280751
## [106] 0.0096982306 0.0165489334 -0.0277649084 0.0015069807 -0.0172864693
## [111] 0.0224990537 0.0109801829 -0.0007487827 -0.0157926602 -0.0145445059
## [116] -0.0092090062 -0.0142431084 0.0157769552 -0.0086526198 0.0019162467# 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 0.978
## 3 1.05
## 4 1.01
## 5 0.975
## 6 1.01
## 7 1.01
## 8 0.972
## 9 0.993
## 10 1.05
## # … 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 0.978
## 3 1.02
## 4 1.04
## 5 1.01
## 6 1.02
## 7 1.03
## 8 1.00
## 9 0.998
## 10 1.05
## # … with 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 9.4869476 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) ## # A tibble: 241 × 1
## growth
## <dbl>
## 1 100
## 2 98.7
## 3 97.8
## 4 97.9
## 5 96.8
## 6 97.3
## 7 97.9
## 8 98.4
## 9 99.1
## 10 101.
## # … with 231 more rowsdump(list = c("simulate_accumulation"),
file = "../00_scripts/accumulate_accumulation.R")7 Running multiple simulations
# Create a vector of 1's 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
set.seed(1234)
monte_carlo_simulation_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 rows
set_names(c("month", names(starts))) %>%
# Transform to long form
pivot_longer(cols = -month, names_to = "sim", values_to = "growth")
monte_carlo_simulation_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_simulation_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_simulation_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 $1 over 120 months")
Line plot with max, median, min
# Step 1 summarize data into max, median, and min of last value
sim_summary <- monte_carlo_simulation_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
summarise(max = max(growth),
median = median(growth),
min = min(growth))
sim_summary## # A tibble: 1 × 3
## max median min
## <dbl> <dbl> <dbl>
## 1 3.88 1.98 1.17# STep 2 plot
monte_carlo_simulation_51 %>%
# Filter for max, median and min sim
group_by(sim) %>%
filter(last(growth) == sim_summary$max |
last(growth) == sim_summary$min |
last(growth) == sim_summary$median) %>%
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 = "Max, Median, and Minimum Simulation")