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$returns)
mean_port_return## [1] 0.005899133# Get standard deviation of portfolio returns
stddev_port_return <- sd(portfolio_returns_tbl$returns)
stddev_port_return## [1] 0.02347492# Construct a normal distribution
simulated_monthly_returns <- rnorm(120, mean_port_return, stddev_port_return)
simulated_monthly_returns## [1] 5.664494e-02 2.132702e-03 -2.819401e-03 6.207251e-02 -1.558037e-02
## [6] 2.621292e-02 4.017841e-02 4.366369e-03 -1.276966e-02 8.539063e-03
## [11] 2.725477e-02 3.077369e-02 -3.939597e-03 2.939894e-02 -4.300034e-03
## [16] 2.805713e-02 -2.078411e-02 5.715396e-03 4.276805e-02 4.718893e-02
## [21] 1.523678e-02 -2.392857e-02 1.880352e-02 -3.654756e-02 1.410652e-02
## [26] -2.782078e-02 2.431442e-02 -1.165357e-02 3.884026e-03 2.922365e-02
## [31] 3.604176e-03 3.001632e-02 5.495354e-03 1.536899e-02 -2.284113e-02
## [36] 6.089556e-02 -4.098606e-03 2.828056e-02 3.558082e-05 2.733682e-02
## [41] -8.398606e-03 9.590824e-03 -1.399627e-02 1.626116e-02 1.190950e-02
## [46] -2.529055e-02 -4.364196e-02 2.994095e-02 8.581562e-03 2.142634e-02
## [51] 2.882006e-04 -2.251603e-02 3.025642e-02 9.858350e-03 8.867978e-03
## [56] 1.363421e-02 -1.322459e-02 -1.564851e-02 4.775154e-02 -7.990845e-03
## [61] 2.060054e-02 -4.677652e-03 3.784065e-02 -1.948780e-02 -2.638636e-03
## [66] 1.972103e-03 -3.444016e-02 3.308583e-03 4.293697e-02 3.588417e-02
## [71] 7.762206e-03 -1.587815e-02 -1.391481e-02 -1.401328e-02 -1.932965e-02
## [76] 4.997728e-02 -3.405985e-03 3.576179e-02 1.807531e-02 -5.566100e-03
## [81] -1.239962e-02 -8.423063e-03 -5.816009e-03 1.175545e-02 -2.612649e-02
## [86] -1.983313e-02 -1.741944e-02 8.723943e-03 9.455142e-03 2.729623e-02
## [91] -2.380087e-02 5.631010e-03 2.375116e-02 -2.657895e-02 3.351867e-03
## [96] 2.133084e-02 3.397449e-02 5.930209e-02 -6.013563e-03 -8.423081e-03
## [101] -1.718189e-02 3.488058e-02 -2.462510e-03 1.762661e-02 -2.195097e-02
## [106] 2.599786e-02 5.381167e-03 -3.508837e-03 2.538609e-02 3.388541e-03
## [111] 1.220423e-02 3.996229e-02 8.301722e-03 2.467910e-02 3.814107e-02
## [116] 4.793166e-02 5.408995e-03 -3.033133e-02 2.989583e-05 1.841303e-02# 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 1.06
## 3 1.00
## 4 0.997
## 5 1.06
## 6 0.984
## 7 1.03
## 8 1.04
## 9 1.00
## 10 0.987
## # … 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 1.06
## 3 1.06
## 4 1.06
## 5 1.12
## 6 1.10
## 7 1.13
## 8 1.18
## 9 1.18
## 10 1.17
## # … with 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 9.7018736 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(120, 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>
## 1 166.
## 2 166.
## 3 165.
## 4 166.
## 5 168.
## 6 169.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
# for reproducible results
set.seed(1234)
monte_carlo_sim_51 <- starts %>%
# Stimulate
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
## # … with 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, 0.5, 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_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 $1 over 120 months")
Line plot with max, median, and min
# Step 1 Summarize data into max, median, and min of last value
sim_summary <- monte_carlo_sim_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
summarise(max = max(growth),
median = median(growth),
min = min(growth))
sim_summary## # A tibble: 1 × 3
## max median min
## <dbl> <dbl> <dbl>
## 1 3.88 1.98 1.17# Step 2 Plot
monte_carlo_sim_51 %>%
# Filter for max, median, and min
group_by(sim) %>%
filter(last(growth) == sim_summary$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 $1 over 120 months", subtitle = "Maximum, Median, and Minimum Simulation")