Week 15: Code along 13
CHAPTER 11 Monte Carlo Simulation
Daniel Lee
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
## # … with 50 more rows5 Simulating growth of a dollar
# Get mean portfolio return
mean_port_return <- mean(portfolio_returns_tbl$returns)
mean_port_return## [1] 0.005899135# Get standard deviation of portfolio returns
stddev_port_return <- sd(portfolio_returns_tbl$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] 0.0225199704 0.0107473523 -0.0038386176 0.0088959134 -0.0353409028
## [6] -0.0030316433 0.0206778370 -0.0305801885 0.0204916488 0.0270848109
## [11] 0.0205368241 -0.0032615061 0.0689762127 -0.0197852743 0.0075069343
## [16] 0.0337089567 -0.0262115314 0.0078358592 -0.0150381461 0.0504132473
## [21] -0.0186496474 0.0094648220 0.0180720002 0.0155729011 -0.0133546466
## [26] -0.0166994608 0.0087151603 -0.0047995200 0.0050528945 0.0157166092
## [31] 0.0447474706 -0.0316019417 -0.0080288095 -0.0204332476 0.0059083035
## [36] -0.0187277217 0.0079360587 -0.0027183435 0.0366738397 -0.0085531811
## [41] 0.0128917864 0.0209872678 0.0030733140 -0.0032351678 0.0332871433
## [46] 0.0159993681 0.0188189728 0.0484802597 0.0153242914 0.0262918147
## [51] -0.0171575603 -0.0033354854 0.0103581225 0.0304762491 -0.0152122881
## [56] -0.0580541069 -0.0097712707 0.0275814722 0.0032812518 -0.0158682361
## [61] -0.0128493537 0.0327385955 -0.0034115498 -0.0167558135 0.0347903897
## [66] 0.0094198984 0.0325807414 -0.0002980885 0.0168958808 -0.0266485485
## [71] -0.0120078789 0.0002253205 -0.0258001506 -0.0102507630 0.0027145939
## [76] 0.0110852556 0.0197639563 0.0085135948 0.0234149645 -0.0038197012
## [81] -0.0200295692 0.0351108207 0.0327029500 -0.0361168061 0.0345421607
## [86] -0.0091409720 0.0172083225 0.0114414394 0.0150447467 0.0195285833
## [91] -0.0326407617 -0.0034507344 0.0222230048 -0.0019536749 0.0103553315
## [96] 0.0249750329 -0.0005334054 0.0335842820 -0.0245768270 0.0010109243
## [101] 0.0094683355 0.0151625724 0.0060211179 0.0465021765 0.0087273000
## [106] 0.0078938122 0.0572253732 0.0104754927 -0.0066482114 -0.0001146956
## [111] 0.0186112063 0.0260949771 0.0285702490 0.0581543423 0.0229394414
## [116] -0.0091967956 0.0219292176 0.0095632883 -0.0289171035 0.0141049246# 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 1.02
## 3 1.01
## 4 0.996
## 5 1.01
## 6 0.965
## 7 0.997
## 8 1.02
## 9 0.969
## 10 1.02
## # … 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 1.02
## 3 1.03
## 4 1.03
## 5 1.04
## 6 1.00
## 7 0.999
## 8 1.02
## 9 0.988
## 10 1.01
## # … with 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 8.6013066 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) %>% tail()## # A tibble: 6 × 1
## growth
## <dbl>
## 1 251.
## 2 248.
## 3 248.
## 4 253.
## 5 252.
## 6 253.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
set.seed(1234)
monte_carlo_sim51 <- 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 months
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_sim51## # 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# find quantiles
monte_carlo_sim51 %>%
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%
## 1.17 1.59 1.98 2.40 3.888 Visualizing simulations with ggplot
monte_carlo_sim51 %>%
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 $1 over 120 months")
Line plot with max, median and min
# step 1, summarize data into max, med, min of last value
sim_summary <- monte_carlo_sim51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
summarise(max = max(growth),
median = median(growth),
min = min(growth))
sim_summary## # A tibble: 1 × 3
## max median min
## <dbl> <dbl> <dbl>
## 1 3.88 1.98 1.17# Step 2, plot
monte_carlo_sim51 %>%
group_by(sim) %>%
filter(last(growth) == sim_summary$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, Median, minimum simulation")