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
## # ℹ 50 more rows5 Simulating growth of a dollar
# Get mean portfolio return
mean_port_return <- mean(portfolio_returns_tbl$returns)
mean_port_return## [1] 0.005899136# Get standard deviation of portfolio returns
stddev_port_return <- sd(portfolio_returns_tbl$returns)
stddev_port_return## [1] 0.02347493# Construct a normal distribution
simulated_monthly_returns <- rnorm(120, mean_port_return, stddev_port_return)
simulated_monthly_returns## [1] 1.325865e-02 1.424862e-02 7.470928e-03 3.690035e-03 3.058941e-02
## [6] 3.397114e-02 -1.762997e-02 -2.216122e-02 4.857209e-02 -2.033147e-02
## [11] -1.113413e-02 -2.708836e-02 8.770986e-03 4.482595e-02 4.838984e-03
## [16] 4.166168e-02 1.534562e-02 1.102349e-02 1.780844e-02 1.015336e-03
## [21] 3.314248e-02 1.709804e-02 2.240913e-02 -7.849796e-03 4.084634e-03
## [26] 1.233024e-02 4.406866e-02 3.459559e-02 8.422172e-03 2.685599e-02
## [31] 1.489639e-02 -7.199866e-03 -4.378060e-03 -3.204202e-03 9.214047e-03
## [36] -4.923617e-02 5.744560e-02 -2.092909e-02 4.082445e-02 2.515612e-02
## [41] 2.888590e-02 -2.168239e-02 -3.876620e-03 2.325367e-02 1.199379e-02
## [46] 2.257274e-03 -1.383982e-02 -8.781516e-03 -4.085591e-03 3.819135e-02
## [51] -3.235508e-02 1.431811e-02 1.747262e-02 1.833183e-02 -1.000901e-02
## [56] 3.224292e-02 -8.157968e-03 3.222447e-03 -1.157854e-02 -5.089829e-02
## [61] -1.713747e-02 7.260416e-02 -1.102216e-02 6.252727e-02 -1.339567e-03
## [66] -5.622235e-02 -7.209396e-03 -1.315803e-03 6.173669e-02 -1.874933e-02
## [71] -3.446263e-03 3.580277e-02 -2.099631e-02 5.251997e-02 1.488338e-02
## [76] 1.251279e-02 2.112237e-02 -3.321124e-02 -2.564261e-02 1.543575e-02
## [81] 1.533278e-05 8.026437e-03 1.236333e-02 4.311670e-02 -9.191289e-03
## [86] -1.945302e-03 2.794414e-02 8.934254e-03 7.634465e-03 2.047365e-02
## [91] -3.545953e-02 2.783733e-02 -1.848034e-02 4.260797e-02 4.720416e-02
## [96] 2.412467e-03 -5.614803e-03 -1.713190e-03 -2.791021e-02 -1.026426e-02
## [101] -3.107204e-02 7.706796e-03 1.011281e-02 -2.675055e-02 1.968697e-02
## [106] -9.974694e-03 1.060285e-02 1.531650e-02 2.070715e-02 -2.227356e-02
## [111] -4.972545e-03 1.808254e-02 7.481565e-03 8.234668e-03 -3.837347e-03
## [116] -3.663747e-02 4.282376e-03 1.981018e-02 2.645996e-02 -5.416325e-03# 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 1.01
## 4 1.01
## 5 1.00
## 6 1.03
## 7 1.03
## 8 0.982
## 9 0.978
## 10 1.05
## # 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 1.03
## 4 1.04
## 5 1.04
## 6 1.07
## 7 1.11
## 8 1.09
## 9 1.06
## 10 1.12
## # ℹ 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 7.911916 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 = 0.05, sd_return = 0.01) %>%
tail()## # A tibble: 6 × 1
## growth
## <dbl>
## 1 10679709.
## 2 11282997.
## 3 11809197.
## 4 12378377.
## 5 12953503.
## 6 13392187.7 Running multiple simulations
# Create a vector of 1s as a starting point
sims <- 51
starts <- rep(1, sims) %>%
set_names (paste("sim", 1:sims))
starts## sim 1 sim 2 sim 3 sim 4 sim 5 sim 6 sim 7 sim 8 sim 9 sim 10 sim 11
## 1 1 1 1 1 1 1 1 1 1 1
## sim 12 sim 13 sim 14 sim 15 sim 16 sim 17 sim 18 sim 19 sim 20 sim 21 sim 22
## 1 1 1 1 1 1 1 1 1 1 1
## sim 23 sim 24 sim 25 sim 26 sim 27 sim 28 sim 29 sim 30 sim 31 sim 32 sim 33
## 1 1 1 1 1 1 1 1 1 1 1
## sim 34 sim 35 sim 36 sim 37 sim 38 sim 39 sim 40 sim 41 sim 42 sim 43 sim 44
## 1 1 1 1 1 1 1 1 1 1 1
## sim 45 sim 46 sim 47 sim 48 sim 49 sim 50 sim 51
## 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 sim 1 1
## 2 1 sim 2 1
## 3 1 sim 3 1
## 4 1 sim 4 1
## 5 1 sim 5 1
## 6 1 sim 6 1
## 7 1 sim 7 1
## 8 1 sim 8 1
## 9 1 sim 9 1
## 10 1 sim 10 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, 0.5, 0.75, 1)) %>%
round(2)## 0% 25% 50% 75% 100%
## 1.09 1.63 1.95 2.22 3.128 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 $1 over 120 months")
Line plot with max, median, and min
# Step 1: Summarize data into max, median, and min of last value
sim_summary <- monte_carlo_sim_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
summarise(max = max(growth),
median = median(growth),
min = min(growth))
sim_summary## # A tibble: 1 × 3
## max median min
## <dbl> <dbl> <dbl>
## 1 3.12 1.95 1.09# Step 2: Plot
monte_carlo_sim_51 %>%
# Filter for max, median, and min sim
group_by(sim) %>%
filter(last(growth) == sim_summary$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 = "Maximum, Median, and Minimum Simulation")