Week 15: Code along 13
CHAPTER 11 Monte Carlo Simulation
Chris Bupreau
2022-12-06
# 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$returns)
mean_port_return## [1] 0.005899132# Get standard deviation of portfolio returns
stddev_port_return <- sd(portfolio_returns_tbl$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] 1.896632e-02 1.217248e-02 1.227409e-02 1.972233e-02 1.822330e-02
## [6] -4.635837e-02 1.717932e-02 -3.935831e-02 -1.719429e-02 3.526530e-02
## [11] -3.429227e-03 -4.488644e-02 2.528200e-03 2.176082e-03 7.100882e-03
## [16] -9.948643e-03 3.245354e-02 -4.977288e-02 5.950456e-04 2.094340e-02
## [21] 8.083819e-03 -2.763552e-02 2.793377e-02 5.809844e-03 1.014520e-02
## [26] 1.538351e-02 2.566174e-02 2.423042e-02 -3.445707e-02 -1.067754e-02
## [31] -1.184030e-02 1.172586e-02 -3.708744e-02 1.255980e-02 -4.232571e-03
## [36] 1.472817e-02 2.730568e-02 -2.810319e-03 -3.505774e-03 3.736269e-02
## [41] 8.489400e-02 2.221951e-02 9.875509e-03 -2.581638e-02 2.496742e-02
## [46] 7.736401e-03 3.310179e-02 3.269904e-02 2.717708e-02 1.080437e-02
## [51] -8.298816e-03 1.660676e-02 1.652941e-02 1.211156e-02 4.593890e-02
## [56] 7.148112e-03 1.285800e-02 -1.923936e-02 4.934753e-02 1.993897e-02
## [61] 5.933093e-03 -1.850021e-02 3.340416e-02 4.037769e-02 2.577244e-02
## [66] -3.938341e-02 2.370417e-02 1.024807e-02 2.467560e-02 1.294410e-02
## [71] 2.808241e-02 -6.250369e-03 1.473310e-02 -1.092611e-02 -2.138732e-02
## [76] 5.430369e-03 3.032100e-02 6.048285e-02 -2.992604e-02 -2.803205e-02
## [81] 6.790042e-03 2.686514e-02 -2.827298e-03 1.211727e-02 1.026275e-02
## [86] -2.324359e-03 1.534780e-02 2.861937e-02 -7.310854e-05 4.753847e-03
## [91] -7.610177e-03 6.062740e-03 -3.366456e-02 -9.190583e-03 -1.573936e-02
## [96] 4.092573e-02 2.766869e-03 4.038860e-03 3.786570e-03 -2.141061e-02
## [101] 2.560926e-02 1.176251e-02 9.116095e-03 -8.108084e-02 -3.661285e-02
## [106] 7.673238e-04 -1.033495e-02 2.971205e-02 -5.980100e-03 -6.417144e-03
## [111] -8.264396e-03 1.982093e-02 2.113696e-02 -1.426928e-02 2.143803e-02
## [116] -3.770224e-04 2.665681e-02 -1.621586e-02 -2.665350e-02 -7.989413e-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 1.02
## 3 1.01
## 4 1.01
## 5 1.02
## 6 1.02
## 7 0.954
## 8 1.02
## 9 0.961
## 10 0.983
## # … 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 1.04
## 5 1.06
## 6 1.08
## 7 1.03
## 8 1.05
## 9 1.01
## 10 0.993
## # … with 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 6.2273086 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 = .005,sd_return = .01) %>% tail()## # A tibble: 6 × 1
## growth
## <dbl>
## 1 290.
## 2 292.
## 3 292.
## 4 298.
## 5 304.
## 6 304.dump(list = c("simulate accumulation"),
file = "../00_scripts/simulate accumulation.R")7 Running multiple simulations
# Create a vector a starting function
sims <- 100
starts <- rep(100, sims) %>%
set_names(paste0("sim", 1:sims))
starts## sim1 sim2 sim3 sim4 sim5 sim6 sim7 sim8 sim9 sim10 sim11
## 100 100 100 100 100 100 100 100 100 100 100
## sim12 sim13 sim14 sim15 sim16 sim17 sim18 sim19 sim20 sim21 sim22
## 100 100 100 100 100 100 100 100 100 100 100
## sim23 sim24 sim25 sim26 sim27 sim28 sim29 sim30 sim31 sim32 sim33
## 100 100 100 100 100 100 100 100 100 100 100
## sim34 sim35 sim36 sim37 sim38 sim39 sim40 sim41 sim42 sim43 sim44
## 100 100 100 100 100 100 100 100 100 100 100
## sim45 sim46 sim47 sim48 sim49 sim50 sim51 sim52 sim53 sim54 sim55
## 100 100 100 100 100 100 100 100 100 100 100
## sim56 sim57 sim58 sim59 sim60 sim61 sim62 sim63 sim64 sim65 sim66
## 100 100 100 100 100 100 100 100 100 100 100
## sim67 sim68 sim69 sim70 sim71 sim72 sim73 sim74 sim75 sim76 sim77
## 100 100 100 100 100 100 100 100 100 100 100
## sim78 sim79 sim80 sim81 sim82 sim83 sim84 sim85 sim86 sim87 sim88
## 100 100 100 100 100 100 100 100 100 100 100
## sim89 sim90 sim91 sim92 sim93 sim94 sim95 sim96 sim97 sim98 sim99
## 100 100 100 100 100 100 100 100 100 100 100
## sim100
## 100# 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: 12,100 × 3
## month sim growth
## <int> <chr> <dbl>
## 1 1 sim1 100
## 2 1 sim2 100
## 3 1 sim3 100
## 4 1 sim4 100
## 5 1 sim5 100
## 6 1 sim6 100
## 7 1 sim7 100
## 8 1 sim8 100
## 9 1 sim9 100
## 10 1 sim10 100
## # … with 12,090 more rows# Find Quantiles
monte_carlo_sim_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
pull(growth) %>%
quantile(Probs = c(0, .25, .5, .75, 1)) %>%
round(2)## 0% 25% 50% 75% 100%
## 78.20 165.78 198.88 227.84 336.098 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 = .5))+
labs(title = "Simulating the growth of 1 dollar over 120 months")
New Line plot with max, median, and minimum
# Step one Summarized data into maximum medium and minimum
sim_summary <- monte_carlo_sim_51 %>%
group_by(sim) %>%
summarize(growth = last(growth)) %>%
ungroup() %>%
summarize(max = max(growth),
median = median(growth),
min = min(growth))
sim_summary## # A tibble: 1 × 3
## max median min
## <dbl> <dbl> <dbl>
## 1 336. 199. 78.2# 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 = .5)) +
theme(plot.subtitle = element_text(hjust = .5)) +
labs(title = "Simulating the growth of 1 dollar over 120 months", subtitle = "Max, Medium, and Min Simulation")