Week 15: Code along 13
CHAPTER 11 Monte Carlo Simulation
Joseph Hansen
2023-12-05
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# 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$returns)
mean_port_return## [1] 0.005899133# Get standard deviation of portfolio returns
stddev_port_return <- sd(portfolio_returns_tbl$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] 0.0114258681 0.0382204239 -0.0327464044 0.0362028087 0.0134447230
## [6] -0.0023213363 0.0060137016 -0.0167954209 0.0425813199 0.0300550038
## [11] 0.0173470552 -0.0095464868 -0.0059179706 -0.0290869250 -0.0054840910
## [16] -0.0068652078 0.0326681124 0.0537718484 0.0419981094 0.0047883589
## [21] -0.0038703182 -0.0042418807 -0.0205813093 0.0029588779 0.0027546932
## [26] -0.0120350412 0.0006638152 -0.0194534360 -0.0250117129 -0.0539896060
## [31] 0.0394259278 0.0341779785 -0.0364026092 0.0237996577 -0.0071055181
## [36] -0.0184080707 -0.0128783803 0.0147247595 -0.0062263176 -0.0379655438
## [41] 0.0200088139 0.0198888707 0.0226550008 -0.0137237317 0.0376171097
## [46] -0.0241751903 0.0246517867 0.0060210429 0.0232537915 0.0291010443
## [51] 0.0081046421 -0.0107670691 -0.0361710909 0.0138997206 -0.0056007904
## [56] 0.0223870692 -0.0126979905 0.0216782456 -0.0168644829 0.0318989711
## [61] 0.0143959797 0.0006612105 -0.0133141216 -0.0129633740 0.0256893657
## [66] -0.0069821754 0.0110394276 0.0253962559 -0.0189605002 0.0201396622
## [71] -0.0064277370 0.0209232267 0.0017429690 0.0545311231 0.0102038149
## [76] 0.0372330055 -0.0131167022 0.0408809879 -0.0030146316 0.0052509411
## [81] 0.0079214321 0.0497065541 -0.0021762148 0.0052544299 0.0125272997
## [86] -0.0026622164 -0.0250234961 -0.0034958886 0.0117939263 0.0338583139
## [91] 0.0300727570 0.0392499814 -0.0031818308 0.0131226782 -0.0615470410
## [96] -0.0087704465 0.0096502229 0.0073355171 0.0002587300 0.0170258596
## [101] 0.0469912284 0.0128003716 -0.0072841121 -0.0016885715 0.0172123552
## [106] -0.0249545272 0.0239513834 -0.0452690269 0.0115380136 0.0061912647
## [111] 0.0408395723 0.0359430468 0.0041093335 -0.0240606063 0.0229568255
## [116] 0.0214661279 0.0106916914 0.0278639704 0.0079549381 0.0401534256# 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 1.04
## 4 0.967
## 5 1.04
## 6 1.01
## 7 0.998
## 8 1.01
## 9 0.983
## 10 1.04
## # 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 1.05
## 4 1.02
## 5 1.05
## 6 1.07
## 7 1.06
## 8 1.07
## 9 1.05
## 10 1.10
## # ℹ 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 7.8774486 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
## 2 102.
## 3 104.
## 4 104.
## 5 103.
## 6 105.
## 7 107.
## 8 108.
## 9 107.
## 10 107.
## # 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%
## 1.05 1.72 1.96 2.29 2.968 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 $1 over 120 months")
line plot with max, median, and min
# step 1 summarize data into max, median, and min of last value
sim_summary <- monte_carlo_sim_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
summarise(max = max(growth),
median = median(growth),
min = min(growth))
sim_summary## # A tibble: 1 × 3
## max median min
## <dbl> <dbl> <dbl>
## 1 2.96 1.96 1.05# step 2 plot
monte_carlo_sim_51 %>%
# filter for max, min, median sim
group_by(sim) %>%
filter(last(growth) == sim_summary$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")