Week 15: Code along 13
CHAPTER 11 Monte Carlo Simulation
Zach Marvel
2024-12-02
# 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] 0.02347492# Construct a normal distribution
simulated_monthly_returns <- rnorm(120, mean_port_return, stddev_port_return)
simulated_monthly_returns## [1] 9.732627e-03 1.949517e-02 4.770101e-03 1.542031e-03 3.971969e-03
## [6] -6.784557e-04 -9.315755e-03 3.069273e-03 -9.233337e-03 5.996148e-04
## [11] -7.444696e-03 1.084356e-02 2.675974e-02 9.400882e-03 -1.757291e-02
## [16] 1.602427e-02 3.006643e-02 -6.675293e-03 -2.753717e-02 -1.106436e-03
## [21] 2.292201e-02 7.341977e-03 -1.761424e-02 -1.832924e-02 4.786687e-03
## [26] -3.137995e-02 2.776340e-03 5.126525e-03 1.057583e-02 8.524009e-03
## [31] 1.552838e-02 2.789353e-05 2.537530e-02 -3.609716e-02 1.975668e-02
## [36] 3.815478e-02 1.019257e-02 1.843028e-02 2.256028e-02 -1.988669e-02
## [41] -6.309173e-03 -9.668299e-03 -2.393120e-02 2.365401e-02 -7.406336e-03
## [46] 2.479126e-02 -1.709766e-02 3.498953e-02 3.785163e-02 2.578918e-03
## [51] -2.206581e-02 2.107472e-02 2.395062e-02 -5.123318e-03 1.225189e-02
## [56] -1.138134e-02 -9.216749e-03 -8.572055e-03 1.474636e-02 -2.950683e-02
## [61] 2.553296e-02 1.032367e-02 -1.039099e-02 2.558364e-02 -6.251581e-03
## [66] -3.784426e-03 7.233067e-03 9.396172e-04 3.179533e-02 1.615395e-02
## [71] 3.780395e-02 5.459258e-03 -7.696742e-03 1.146076e-02 -6.011421e-03
## [76] -6.513380e-03 2.084349e-04 -1.540996e-04 -2.138738e-03 6.678325e-02
## [81] 2.969732e-03 -5.120168e-02 1.083522e-02 -5.237528e-02 -1.017762e-02
## [86] 8.349316e-03 2.513346e-02 -6.755628e-03 -5.332223e-03 -1.128762e-02
## [91] 3.908798e-02 3.708287e-02 2.289638e-02 -2.781825e-03 4.351217e-03
## [96] 4.579754e-02 -2.348807e-03 1.650387e-02 2.754359e-02 3.529799e-02
## [101] -1.542444e-03 -3.508101e-02 4.133593e-03 -9.198174e-04 4.392539e-02
## [106] 3.583086e-03 1.187009e-03 4.668066e-02 -2.625263e-02 1.577011e-02
## [111] -9.145624e-03 3.865353e-03 -3.803831e-03 2.099808e-02 1.801609e-02
## [116] 8.616470e-03 5.640790e-02 6.930972e-03 5.623174e-03 1.130331e-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 1.01
## 3 1.02
## 4 1.00
## 5 1.00
## 6 1.00
## 7 0.999
## 8 0.991
## 9 1.00
## 10 0.991
## # 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 1.03
## 5 1.04
## 6 1.04
## 7 1.04
## 8 1.03
## 9 1.03
## 10 1.02
## # ℹ 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 6.8728486 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)
(simulated_growth)
}
simulate_accumulation(initial_value = 100, N = 240, mean_return = 0.005, sd_return = 0.01) %>%
tail()## # A tibble: 6 × 1
## growth
## <dbl>
## 1 323.
## 2 318.
## 3 315.
## 4 318.
## 5 315.
## 6 314.dump(list = c("simulate_accumulation"),
file = "../00_scripts/simulate_accumulation.R")7 Running multiple simulations
# Create a vector of its as a starting point
sims <- 51
starts <- rep(1, sims) %>%
set_names(paste0("sims", 1:sims))
starts## sims1 sims2 sims3 sims4 sims5 sims6 sims7 sims8 sims9 sims10 sims11
## 1 1 1 1 1 1 1 1 1 1 1
## sims12 sims13 sims14 sims15 sims16 sims17 sims18 sims19 sims20 sims21 sims22
## 1 1 1 1 1 1 1 1 1 1 1
## sims23 sims24 sims25 sims26 sims27 sims28 sims29 sims30 sims31 sims32 sims33
## 1 1 1 1 1 1 1 1 1 1 1
## sims34 sims35 sims36 sims37 sims38 sims39 sims40 sims41 sims42 sims43 sims44
## 1 1 1 1 1 1 1 1 1 1 1
## sims45 sims46 sims47 sims48 sims49 sims50 sims51
## 1 1 1 1 1 1 1# Simulate
# for reproducable research
set.seed(1234)
monte_carlo_sims_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()) %>%
# Rearrand column names
set_names(c("month", names(starts))) %>%
# Transform to long form
pivot_longer(cols = -month, names_to = "sim", values_to = "growth")
monte_carlo_sims_51## # A tibble: 6,171 × 3
## month sim growth
## <int> <chr> <dbl>
## 1 1 sims1 1
## 2 1 sims2 1
## 3 1 sims3 1
## 4 1 sims4 1
## 5 1 sims5 1
## 6 1 sims6 1
## 7 1 sims7 1
## 8 1 sims8 1
## 9 1 sims9 1
## 10 1 sims10 1
## # ℹ 6,161 more rows# Find quantiles
monte_carlo_sims_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_sims_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 $1 over 120 months")
Line plot with max, median, and min
# Step 1 Summarize data into max, median, and min of last value
sim_summary <- monte_carlo_sims_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
summarise(max = max(growth),
median = median(growth),
min = min(growth))
sim_summary## # A tibble: 1 × 3
## max median min
## <dbl> <dbl> <dbl>
## 1 3.88 1.98 1.17# Step 2 Plot
monte_carlo_sims_51 %>%
# Fitler for max, median, and min sim
group_by(sim) %>%
filter(last(growth) == sim_summary$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")