Week 15: Code along 13
CHAPTER 11 Monte Carlo Simulation
Matthew White
2023-12-05
# 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$returns)
mean_port_return## [1] 0.005899135# Get standard deviation of portfolio returns
stddev_port_return <- sd(portfolio_returns_tbl$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] -0.0299050864 0.0033678905 0.0077432073 0.0210513591 -0.0216629953
## [6] 0.0342756411 -0.0079320879 -0.0362718196 -0.0127199366 -0.0102989012
## [11] 0.0119825986 -0.0477613602 0.0026051301 0.0566278782 -0.0158857757
## [16] -0.0181127460 -0.0259978233 0.0028979290 0.0521963785 -0.0056929972
## [21] 0.0086131448 -0.0037963236 0.0236597509 -0.0070240364 0.0047557775
## [26] -0.0165782915 0.0203927561 0.0057960283 0.0141696113 0.0164926249
## [31] -0.0589275062 0.0162491203 0.0418639836 0.0415027084 0.0124057726
## [36] 0.0115431816 -0.0320917090 0.0294515819 0.0188455709 -0.0023082963
## [41] 0.0336337471 -0.0430079470 0.0333632911 -0.0283726637 -0.0194320190
## [46] -0.0484805339 0.0129109363 -0.0445013114 0.0058648983 0.0131080393
## [51] 0.0296351325 0.0193771884 -0.0133506114 0.0073743306 0.0431830587
## [56] 0.0547957571 -0.0158879153 0.0102444642 -0.0076281661 0.0136255476
## [61] 0.0039750933 0.0226563117 -0.0096959242 -0.0022118471 0.0265731564
## [66] 0.0441542169 0.0260207480 0.0259838583 -0.0100751741 -0.0160938176
## [71] -0.0287249590 -0.0010443350 0.0344108090 0.0015272373 0.0201939045
## [76] 0.0210099516 -0.0471934920 -0.0117104227 0.0368107877 -0.0342510915
## [81] 0.0180272438 -0.0451360714 0.0302160764 -0.0103749623 0.0242504993
## [86] -0.0021991642 -0.0362615196 0.0358805702 0.0392561734 -0.0064209449
## [91] 0.0135473756 0.0155282632 -0.0038283083 -0.0357856662 -0.0182638275
## [96] 0.0072643386 0.0174044803 0.0402712773 0.0127269581 0.0316010554
## [101] 0.0328107610 0.0407107102 -0.0113971240 0.0497307128 0.0167752256
## [106] 0.0132019441 0.0349157939 0.0259235957 0.0415199843 -0.0058236765
## [111] 0.0056030080 0.0210411086 -0.0111166061 0.0127747640 0.0185302183
## [116] 0.0190383679 0.0012390111 -0.0084461945 0.0138674212 0.0004296047# 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 0.970
## 3 1.00
## 4 1.01
## 5 1.02
## 6 0.978
## 7 1.03
## 8 0.992
## 9 0.964
## 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 0.970
## 3 0.973
## 4 0.981
## 5 1.00
## 6 0.980
## 7 1.01
## 8 1.01
## 9 0.969
## 10 0.957
## # ℹ 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 6.8590766 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
## 2 101.
## 3 102.
## 4 102.
## 5 102.
## 6 105.
## 7 106.
## 8 105.
## 9 106.
## 10 106.
## # ℹ 231 more rows7 Running multiple simulations
# cretae 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(.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, 0.50, 0.75, 1)) %>%
round(2)## 0% 25% 50% 75% 100%
## 1.18 1.77 1.98 2.33 2.828 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 $1 over 120 months")
line plot with max, median, and min
# step 1 summarize data into max, median, and min of last value
sim_summary <- monte_carlo_sim_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
summarise(max = max(growth),
median = median(growth),
min = min(growth))
sim_summary## # A tibble: 1 × 3
## max median min
## <dbl> <dbl> <dbl>
## 1 2.82 1.98 1.18# step 2 plot
monte_carlo_sim_51 %>%
# filter for max, min, median sim
group_by(sim) %>%
filter(last(growth) == sim_summary$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 growth of $1 over 120 months",
subtitle = "max, med, and min sim")