Week 15: Code along 13
CHAPTER 11 Monte Carlo Simulation
Michael Malatesta”
2022-09-19
# 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$returns)
mean_port_return## [1] 0.005899137# Get standard deviation of portfolio returns
stddev_port_return <- sd(portfolio_returns_tbl$returns)
stddev_port_return## [1] 0.02347488# Construct a normal distribution
simulated_monthly_returns <- rnorm(120, mean_port_return, stddev_port_return)
simulated_monthly_returns## [1] 0.0071972837 0.0152584239 0.0555442113 -0.0122131659 -0.0081550523
## [6] -0.0323012840 -0.0279738386 -0.0219677052 -0.0128843951 0.0143678377
## [11] -0.0202477360 -0.0091797138 0.0128693642 0.0214067777 0.0067456869
## [16] 0.0114737292 -0.0448000271 0.0304533824 0.0121356315 0.0264024566
## [21] -0.0231949181 -0.0175752845 -0.0025489216 0.0168601739 -0.0210971830
## [26] -0.0103164349 0.0399784964 -0.0360110731 0.0099722576 0.0370944402
## [31] 0.0311619831 0.0005441281 -0.0001044430 -0.0365285520 -0.0039547015
## [36] -0.0148450783 -0.0133247504 0.0135216086 -0.0074041507 0.0174300745
## [41] -0.0433062969 -0.0229424428 0.0304365240 -0.0265971173 0.0179034009
## [46] -0.0040170866 0.0230504292 -0.0196068261 0.0399766991 0.0173382260
## [51] -0.0167761373 -0.0166662861 -0.0206760741 0.0164043391 0.0370529168
## [56] 0.0161256940 -0.0294839216 0.0360273828 0.0170673578 0.0159602487
## [61] 0.0055748578 -0.0246453586 0.0054421012 0.0362165465 0.0230904273
## [66] 0.0048237389 0.0322772953 0.0088869300 0.0263476772 -0.0114711743
## [71] -0.0052092776 0.0013362081 -0.0008085510 0.0186602444 0.0155807525
## [76] -0.0079392251 -0.0356329593 -0.0119109931 0.0009518493 0.0146316275
## [81] 0.0069309962 0.0151959413 0.0002243850 0.0192069098 0.0113655273
## [86] 0.0217664688 0.0131274642 0.0156581953 0.0045770612 0.0408392365
## [91] -0.0147687015 0.0033893863 0.0470891692 -0.0098972527 -0.0446925811
## [96] 0.0359533655 0.0137308053 -0.0005685398 -0.0140814234 0.0054645849
## [101] -0.0193270192 0.0342096965 0.0033805084 0.0259350509 0.0119494262
## [106] 0.0248708922 0.0122038835 -0.0163563339 0.0015663357 0.0309786907
## [111] -0.0395427418 -0.0140094871 0.0358935676 0.0133782704 0.0074263387
## [116] -0.0148034467 0.0150896064 -0.0072573500 0.0127632874 -0.0191391023# 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 1.02
## 4 1.06
## 5 0.988
## 6 0.992
## 7 0.968
## 8 0.972
## 9 0.978
## 10 0.987
## # 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 1.01
## 3 1.02
## 4 1.08
## 5 1.07
## 6 1.06
## 7 1.02
## 8 0.995
## 9 0.973
## 10 0.960
## # ℹ 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 4.162866 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 1.03
## 3 1.06
## 4 1.09
## 5 1.12
## 6 1.12
## 7 1.09
## 8 1.10
## 9 1.09
## 10 1.07
## # ℹ 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(paste("sim", 1:sims, sep = ""))
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
## # ℹ 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%
## 0.91 1.11 1.57 1.86 2.26 3.37 3.528 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 3.54 1.86 0.845monte_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()