Week 15: Code along 13
CHAPTER 11 Monte Carlo Simulation
Nahomy Palma
2024-07-01
# 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$returns)
mean_port_return## [1] 0.005899134# Get standard deviation of portfolio returns
stddev_port_return <- sd(portfolio_returns_tbl$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] 6.974145e-03 8.538236e-03 -5.808555e-03 2.738124e-03 4.354513e-03
## [6] 2.097564e-02 -3.841319e-02 3.577688e-04 6.652539e-03 -6.746341e-05
## [11] -4.419140e-02 1.294551e-02 -1.110889e-02 -2.016121e-03 -4.163990e-03
## [16] 2.541247e-02 5.787496e-02 5.492442e-02 -1.634170e-02 -2.124204e-02
## [21] 4.320859e-02 -2.413924e-02 -1.560925e-03 2.451332e-03 2.233458e-03
## [26] 4.618001e-02 2.526199e-03 -2.950761e-02 -9.543316e-03 -1.055635e-02
## [31] 3.315853e-02 9.271288e-03 -8.667125e-03 1.722596e-02 -1.468165e-02
## [36] 1.797250e-02 -4.057371e-02 -4.344889e-03 -1.884246e-02 -7.156892e-03
## [41] -2.275767e-02 -8.497907e-04 2.675245e-04 7.053581e-03 5.002542e-03
## [46] -2.570873e-03 -5.072571e-03 4.181385e-02 3.596904e-03 5.430902e-03
## [51] 5.109403e-02 5.853906e-03 1.429263e-02 1.968459e-02 1.985146e-02
## [56] 1.275222e-02 -1.919631e-02 1.295584e-02 2.969887e-03 1.269025e-02
## [61] 7.749453e-03 4.330975e-02 -6.671164e-04 2.303246e-02 4.018117e-02
## [66] 2.384675e-02 4.594322e-02 8.111335e-03 -3.074907e-02 -1.118137e-02
## [71] -1.436026e-02 5.849087e-04 1.179983e-02 8.068807e-03 -2.053563e-02
## [76] 1.821962e-02 4.000420e-02 -2.298509e-02 -6.623858e-03 1.625599e-02
## [81] -7.321027e-03 1.351371e-02 8.185274e-03 -3.732352e-02 -1.288350e-03
## [86] 1.725338e-02 6.763745e-03 -2.273976e-02 -3.380796e-03 9.873144e-03
## [91] 3.295038e-02 -2.217683e-02 1.441425e-02 -6.208592e-03 2.588550e-02
## [96] -2.485227e-02 -1.613282e-02 1.597335e-02 3.665541e-02 -1.495197e-02
## [101] -1.850927e-02 1.227045e-02 6.678391e-03 -1.073004e-03 4.000223e-03
## [106] -1.207431e-02 5.115393e-03 -2.174259e-02 1.200306e-02 2.842179e-02
## [111] 1.181359e-02 -2.691019e-03 -1.073504e-02 -1.859325e-02 -1.961643e-02
## [116] 5.371232e-02 -1.620601e-02 -1.723250e-02 1.396390e-02 -4.752731e-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 0.994
## 5 1.00
## 6 1.00
## 7 1.02
## 8 0.962
## 9 1.00
## 10 1.01
## # 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 1.02
## 4 1.01
## 5 1.01
## 6 1.02
## 7 1.04
## 8 0.998
## 9 0.999
## 10 1.01
## # ℹ 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 4.19096 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>
## 1 463.
## 2 464.
## 3 467.
## 4 471.
## 5 470.
## 6 475.dump(list = c("simulate_accumulation"),
file = "../00_scripts/simulate_accumulatio.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 research
set.seed(1234)
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 a 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, 0.5, 0.75, 1)) %>%
round(2)## 0% 25% 50% 75% 100%
## 1.17 1.59 1.98 2.40 3.888 Visualizing simulations with ggplot
Line Plot of Simulations
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 $1 over 120 Months")
Line Plot with Max, Median, and Min
# Step 1 Summarize data into max, median, min of last value
sim_summary <- monte_carlo_sim_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
summarise(max = max(growth),
median = median(growth),
min = min(growth))
sim_summary## # A tibble: 1 × 3
## max median min
## <dbl> <dbl> <dbl>
## 1 3.88 1.98 1.17# Step 2 Plot
monte_carlo_sim_51 %>%
# Filter for max, median, min sim
group_by(sim) %>%
filter(last(growth) == sim_summary$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 = "Max, Median, Minimum Simulation")