Week 15: Code along 13
CHAPTER 11 Monte Carlo Simulation
Griffin Crane
2022-12-13
# 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$returns)
mean_port_return## [1] 0.005899133# Get standard deviation of portfolio returns
stddev_port_return <- sd(portfolio_returns_tbl$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] 0.0100140708 0.0461517167 0.0099223818 0.0077907340 0.0138696758
## [6] -0.0044693399 0.0059158067 0.0140063631 0.0622668614 -0.0476367894
## [11] 0.0189471953 0.0133162241 0.0387409004 0.0025266587 -0.0236839772
## [16] 0.0015316647 0.0179850523 -0.0064272815 -0.0283184506 0.0629721274
## [21] 0.0098294981 0.0103389188 0.0530626648 0.0336748631 0.0176759974
## [26] 0.0370831160 0.0114848276 0.0316809829 0.0044146169 0.0302799475
## [31] 0.0150270922 -0.0125375799 -0.0158771741 0.0132784342 -0.0198553397
## [36] 0.0151955240 0.0249566693 -0.0305871978 0.0220584311 0.0256580626
## [41] 0.0052027045 0.0198478098 0.0135105196 0.0141714091 0.0470592001
## [46] 0.0124803316 -0.0181611304 0.0130902841 0.0099895677 0.0181120167
## [51] -0.0008802235 -0.0410763011 0.0079905316 0.0226767048 -0.0524410267
## [56] 0.0313970551 0.0069516716 0.0309762118 -0.0589663566 -0.0200633069
## [61] -0.0070760851 -0.0258452049 -0.0009969278 -0.0155550701 0.0209636064
## [66] -0.0127748364 0.0156925113 0.0357775019 0.0350286880 0.0396848344
## [71] -0.0047711221 0.0202800918 0.0209020089 -0.0089025186 -0.0445091580
## [76] -0.0364490250 0.0433436390 0.0155803227 0.0322516156 -0.0088799733
## [81] -0.0091810878 -0.0035626145 0.0186852541 -0.0146440107 0.0137742883
## [86] 0.0289732828 0.0392897099 -0.0303553266 0.0333647234 -0.0137803742
## [91] 0.0300355420 0.0096680052 -0.0173706969 -0.0255295734 0.0066483594
## [96] 0.0195662575 0.0224942819 0.0133412858 -0.0174088946 -0.0133178356
## [101] -0.0075239378 0.0338364782 -0.0267656733 -0.0415568793 0.0132909713
## [106] -0.0033008988 -0.0043084243 0.0383138677 0.0008053330 0.0423763474
## [111] -0.0002802744 0.0275881716 -0.0382321256 0.0285780149 0.0215897200
## [116] 0.0245591183 -0.0111696390 -0.0265800199 0.0372716308 0.0116024656# 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 1.01
## 3 1.05
## 4 1.01
## 5 1.01
## 6 1.01
## 7 0.996
## 8 1.01
## 9 1.01
## 10 1.06
## # … 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 1.01
## 3 1.06
## 4 1.07
## 5 1.08
## 6 1.09
## 7 1.09
## 8 1.09
## 9 1.11
## 10 1.18
## # … with 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 8.753046 Simulation function
simulate_accumulation <- function(init_value, N, mean, stdev) {
tibble(returns = c(init_value, 1 + rnorm(N, mean, stdev))) %>%
mutate(growth = accumulate(returns, function(x, y) x*y)) %>%
select(growth)
}
simulate_accumulation(1, 120, mean_port_return, stddev_port_return)## # A tibble: 121 × 1
## growth
## <dbl>
## 1 1
## 2 0.989
## 3 1.00
## 4 0.975
## 5 1.00
## 6 1.04
## 7 1.07
## 8 1.08
## 9 1.06
## 10 1.10
## # … with 111 more rows# Save the function
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
monte_carlo_sim_51 <- starts %>%
# Simulate
map_dfc(simulate_accumulation,
N = 120,
mean = mean_port_return,
stdev = stddev_port_return) %>%
# Add the column, month
mutate(month = seq(1:nrow(.))) %>%
# Arrange column names
select(month, everything()) %>%
set_names(c("month", names(starts))) %>%
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# Calculate the quantiles for simulated values
probs <- c(.005, .025, .25, .5, .75, .975, .995)
monte_carlo_sim_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
pull(growth) %>%
# Find the quantiles
quantile(probs = probs) %>%
round(2)## 0.5% 2.5% 25% 50% 75% 97.5% 99.5%
## 1.12 1.24 1.76 2.02 2.42 2.84 2.898 Visualizing simulations with ggplot
monte_carlo_sim_51 %>%
ggplot(aes(x = month, y = growth, col = sim)) +
geom_line() +
theme(legend.position = "none")
# Simplify the plot
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 2.90 2.02 1.09monte_carlo_sim_51 %>%
group_by(sim) %>%
filter(last(growth) == sim_summary$max |
last(growth) == sim_summary$median |
last(growth) == sim_summary$min) %>%
# Plot
ggplot(aes(month, growth, col = sim)) +
geom_line() +
theme() +
labs(title = "Simulating growth of $1 over 120 months")