Week 15: Code along 13
CHAPTER 11 Monte Carlo Simulation
Damian Senesac
2022-12-05
# 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$returns)
mean_port_return## [1] 0.005899135# Get standard deviation of portfolio returns
stddev_port_return <- sd(portfolio_returns_tbl$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] 0.0088710537 0.0170672059 -0.0004548328 -0.0140356281 0.0210063301
## [6] -0.0059013736 -0.0435855400 0.0053841200 -0.0024706302 0.0696154300
## [11] 0.0175616378 0.0083256783 0.0271655800 -0.0276928130 0.0249499773
## [16] 0.0108461777 0.0195012780 0.0638649531 -0.0445499038 0.0231739708
## [21] 0.0361700753 -0.0425469005 0.0166360546 0.0093973975 0.0648637750
## [26] 0.0141909180 -0.0014697181 0.0175257048 -0.0731629039 -0.0035686183
## [31] 0.0329499543 0.0019131318 -0.0010975345 0.0465140525 -0.0089545079
## [36] 0.0298840807 0.0242530786 0.0316577135 0.0070920438 -0.0115832484
## [41] 0.0516704220 0.0034136741 0.0102909188 0.0136497861 -0.0037870653
## [46] 0.0518976911 -0.0196637202 0.0060863114 -0.0249436598 0.0248619504
## [51] -0.0213938275 -0.0085720068 -0.0457776448 -0.0011556409 0.0061809123
## [56] 0.0123099958 -0.0087335524 0.0436117937 0.0056594502 -0.0218654251
## [61] 0.0157953292 0.0072071045 0.0104007637 -0.0026197654 -0.0079073016
## [66] -0.0226060578 -0.0314943267 0.0488004579 -0.0033462225 -0.0307928758
## [71] -0.0095604447 -0.0063433009 0.0148220991 0.0241331374 0.0102014963
## [76] -0.0097325306 0.0109617027 0.0307683414 0.0172645745 0.0015128868
## [81] 0.0090606042 0.0695612330 -0.0166043247 -0.0059767555 -0.0282303928
## [86] 0.0115112558 -0.0001327350 0.0210776981 -0.0027495826 -0.0349658147
## [91] 0.0227724373 -0.0207762852 0.0223110316 0.0196293975 0.0049464522
## [96] 0.0008012361 0.0207015923 0.0386370880 0.0188604117 -0.0009489028
## [101] -0.0087144044 -0.0116988421 0.0258086131 0.0283806137 -0.0214784242
## [106] 0.0090264227 0.0299189366 0.0244584783 -0.0268641985 0.0101658561
## [111] 0.0404513585 0.0172319310 0.0007481837 0.0255751537 0.0124331534
## [116] -0.0250963825 0.0144170416 0.0045894729 -0.0291550012 -0.0228863904# 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 0.986
## 6 1.02
## 7 0.994
## 8 0.956
## 9 1.01
## 10 0.998
## # … 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 1.01
## 3 1.03
## 4 1.03
## 5 1.01
## 6 1.03
## 7 1.03
## 8 0.982
## 9 0.987
## 10 0.984
## # … with 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 7.4465886 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>
## 1 355.
## 2 360.
## 3 354.
## 4 355.
## 5 357.
## 6 353.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("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 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 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 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
## # … 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 $1 over 120 months")
Line plot with max, median, and min
# Step 1: Summarize data into max, median, and min of the last value
sim_summary <- monte_carlo_sim_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
summarise(max = max(growth),
median = median(growth),
min = min(growth))
sim_summary## # A tibble: 1 × 3
## max median min
## <dbl> <dbl> <dbl>
## 1 3.88 1.98 1.17# Step 2 Plot
monte_carlo_sim_51 %>%
# Filter for max, median, and min sim
group_by(sim) %>%
filter(last(growth) == sim_summary$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")