Week 15: Code along 13
CHAPTER 11 Monte Carlo Simulation
Julius Mondschein
2024-01-07
# 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$returns)
mean_port_return## [1] 0.005899132# Get standard deviation of portfolio returns
stddev_port_return <- sd(portfolio_returns_tbl$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] -0.0372485072 0.0084313707 -0.0035024084 -0.0012059103 0.0172982005
## [6] -0.0235756401 0.0212842337 -0.0164073328 -0.0099610253 -0.0012820396
## [11] -0.0308599544 0.0140926289 0.0207499480 -0.0335933492 0.0200368219
## [16] 0.0052025281 -0.0151865895 0.0236629394 -0.0136135738 0.0331710396
## [21] 0.0062522568 0.0098817588 0.0108856630 0.0203083895 0.0373138488
## [26] -0.0053366964 0.0042320178 0.0547158030 0.0011544625 -0.0177628093
## [31] 0.0343649802 -0.0047099723 -0.0158861514 0.0195499161 0.0077984698
## [36] -0.0201430215 0.0016718791 0.0046977422 0.0008006202 0.0487808175
## [41] 0.0051093450 -0.0382154592 0.0615375785 0.0271691804 0.0025095535
## [46] -0.0214472866 0.0292875150 0.0003772257 0.0248041348 0.0216295368
## [51] 0.0326274646 0.0170331967 -0.0160169182 -0.0124287888 0.0015582751
## [56] 0.0213757479 -0.0211232961 0.0320765407 0.0083012749 0.0159888424
## [61] 0.0052788697 0.0056672495 0.0134804531 -0.0086857591 0.0134056560
## [66] -0.0181009052 -0.0085845950 0.0126414969 -0.0135471541 -0.0027262541
## [71] 0.0468901762 0.0182526925 0.0080871960 -0.0211728488 0.0008637090
## [76] 0.0197285490 -0.0207526949 0.0017702112 0.0403479632 0.0046182204
## [81] -0.0057555440 0.0264548805 -0.0018998218 0.0207492827 0.0077647769
## [86] -0.0267498045 -0.0154537579 0.0199853477 0.0196604208 0.0109870816
## [91] 0.0262974311 -0.0369446999 0.0195743953 0.0315536409 0.0204661882
## [96] 0.0409485262 0.0202354408 -0.0456702587 0.0206717456 0.0005424448
## [101] -0.0479981379 0.0187526405 0.0132968124 0.0368498204 -0.0184294243
## [106] 0.0090299781 -0.0211662527 -0.0070592131 -0.0081609968 0.0389410582
## [111] 0.0238633950 -0.0050453508 0.0358106050 0.0242584592 0.0143841059
## [116] -0.0283416532 0.0093829176 0.0108898714 -0.0273601154 0.0059098121# 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 0.963
## 3 1.01
## 4 0.996
## 5 0.999
## 6 1.02
## 7 0.976
## 8 1.02
## 9 0.984
## 10 0.990
## # 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 0.963
## 3 0.971
## 4 0.967
## 5 0.966
## 6 0.983
## 7 0.960
## 8 0.980
## 9 0.964
## 10 0.955
## # ℹ 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 6.8970926 Simulation function
simulate_accmulation <- 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_accmulation(initial_value = 100, N = 240, mean_return = 0.005, sd_return = 0.01) %>%
tail()## # A tibble: 6 × 1
## growth
## <dbl>
## 1 457.
## 2 464.
## 3 464.
## 4 470.
## 5 475.
## 6 482.dump(list = c("simulate_accumulation"),
file = "../00_scripts/simulate_accumilation.R")Add a dollar
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(.x = .,
.f = ~simulate_accmulation(initial_value = .x,
N = 240,
mean_return = mean_port_return,
sd_return = stddev_port_return)) %>%
# Add column worth
mutate(month = 1:nrow(.)) %>%
select(month, everything()) %>%
# Rerange 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: 12,291 × 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 12,281 more rowsmonte_carlo_sim_51 %>%
group_by(sim) %>%
summarize(growth = last(growth)) %>%
ungroup() %>%
pull(growth) %>%
quantile(probs = c(0, 0.25, 0.5, 0.75, 1)) %>%
round(2)## 0% 25% 50% 75% 100%
## 1.71 3.22 4.25 5.19 14.098 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 1s over 20 months")
Line plot with max, median, min
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 14.1 4.25 1.71# Step 2
monte_carlo_sim_51 %>%
# Filter by max, median, min
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 1s over 20 months",
subtitle = "Maximum, Median, and Minimum Simulation")## $title
## [1] "Simulating growth of 1s over 20 months"
##
## $subtitle
## [1] "Maximum, Median, and Minimum Simulation"
##
## attr(,"class")
## [1] "labels"