Week 14: Code along 13
Simulation
Dante Moretti
2023-12-05
# 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
## # ℹ 50 more rows5 Simulating growth of a dollar
# Get mean portfolio return
mean_port_return <- mean(portfolio_returns_tbl$returns)
mean_port_return## [1] 0.005899133# Get standard deviation of portfolio returns
stddev_port_return <- sd(portfolio_returns_tbl$returns)
stddev_port_return## [1] 0.02347487# Construct a normal distribution
simulated_monthly_returns <- rnorm(120, mean_port_return, stddev_port_return)
simulated_monthly_returns## [1] -0.0045146013 0.0540553605 0.0046491889 0.0428795683 -0.0054678538
## [6] 0.0082602118 0.0242737681 0.0006162828 0.0287327894 0.0179195693
## [11] -0.0282027434 -0.0059578135 -0.0179767591 0.0155500859 0.0270598919
## [16] 0.0069583313 0.0213152254 0.0398266964 0.0129902029 0.0010981872
## [21] -0.0057339298 -0.0048686432 0.0263951953 0.0264132568 0.0154903009
## [26] -0.0061029118 0.0228146030 -0.0023544273 0.0144501045 0.0341827839
## [31] 0.0173100648 0.0019651980 -0.0112007201 0.0429683937 0.0062944909
## [36] 0.0252501275 0.0123678415 -0.0046841898 0.0052713773 0.0439400095
## [41] 0.0123938841 0.0126301346 -0.0015222962 0.0032876309 0.0084673809
## [46] 0.0135685011 0.0401150288 0.0202628285 -0.0049677419 0.0639258531
## [51] 0.0034649117 0.0120570169 -0.0015030040 0.0088547241 0.0049184327
## [56] 0.0323986945 0.0103372199 0.0013251106 0.0027364595 -0.0290572139
## [61] 0.0421795491 0.0623585031 -0.0160167441 0.0017785575 0.0163909720
## [66] 0.0253023464 -0.0062997609 0.0068669398 0.0378590406 0.0391907499
## [71] 0.0163275641 0.0269946064 0.0104719810 -0.0072638026 -0.0094029690
## [76] -0.0043363406 0.0445020010 0.0191473522 0.0064787568 0.0185989266
## [81] -0.0288523272 0.0113874217 -0.0150997627 -0.0063609286 -0.0069253906
## [86] 0.0139765322 -0.0120153218 -0.0533734384 -0.0208243359 0.0141690522
## [91] -0.0048947858 0.0304788729 0.0157779891 0.0038915559 0.0017110466
## [96] -0.0130994062 -0.0027899341 0.0002130478 -0.0117271087 0.0197916090
## [101] 0.0164349984 -0.0033079737 0.0546208039 -0.0102428598 0.0290601718
## [106] -0.0336799213 -0.0011406040 0.0080101278 0.0121916021 0.0203727653
## [111] -0.0151288239 -0.0133666087 0.0083067835 0.0108334016 -0.0004724018
## [116] -0.0149859771 0.0061203551 0.0474237116 -0.0043422896 0.0120798431# 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 0.995
## 3 1.05
## 4 1.00
## 5 1.04
## 6 0.995
## 7 1.01
## 8 1.02
## 9 1.00
## 10 1.03
## # 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 0.995
## 3 1.05
## 4 1.05
## 5 1.10
## 6 1.09
## 7 1.10
## 8 1.13
## 9 1.13
## 10 1.16
## # ℹ 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 11.45346 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) ## # A tibble: 241 × 1
## growth
## <dbl>
## 1 100
## 2 99.1
## 3 98.5
## 4 98.7
## 5 99.4
## 6 99.2
## 7 98.7
## 8 98.9
## 9 98.0
## 10 97.6
## # ℹ 231 more rowsdump(list = c("simulate_accumulation"),
file = "../00_scripts/accumulate_accumulation.R")7 Running multiple simulations
# Create a vector of 1's 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
set.seed(1234)
monte_carlo_simulation_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 rows
set_names(c("month", names(starts))) %>%
# Transform to long form
pivot_longer(cols = -month, names_to = "sim", values_to = "growth")
monte_carlo_simulation_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_simulation_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_simulation_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 $1 over 120 months")
Line plot with max, median, min
# Step 1 summarize data into max, median, and min of last value
sim_summary <- monte_carlo_simulation_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
summarise(max = max(growth),
median = median(growth),
min = min(growth))
sim_summary## # A tibble: 1 × 3
## max median min
## <dbl> <dbl> <dbl>
## 1 3.88 1.98 1.17# STep 2 plot
monte_carlo_simulation_51 %>%
# Filter for max, median and min sim
group_by(sim) %>%
filter(last(growth) == sim_summary$max |
last(growth) == sim_summary$min |
last(growth) == sim_summary$median) %>%
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, and Minimum Simulation")