Week 15: Code along 13
CHAPTER 11 Monte Carlo Simulation
Sondre Asheim
2023-12-03
# 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$returns)
mean_port_return## [1] 0.005899135# Get standard deviation of portfolio returns
stddev_port_return <- sd(portfolio_returns_tbl$returns)
stddev_port_return## [1] 0.0234749# Construct a normal distribution
simulated_monthly_returns <- rnorm(120, mean_port_return, stddev_port_return)
simulated_monthly_returns## [1] 0.0305056514 0.0020841599 -0.0050212575 0.0151410566 -0.0011475373
## [6] -0.0198173710 0.0289455824 -0.0252900806 -0.0280111639 0.0165116575
## [11] -0.0340551993 -0.0093109714 -0.0047088325 -0.0101858098 -0.0287233788
## [16] 0.0040765149 -0.0054314817 0.0271858201 -0.0062452677 0.0189030843
## [21] 0.0196314895 -0.0236959503 0.0403675099 0.0105876569 0.0004419260
## [26] 0.0330823665 0.0198623498 0.0266265674 0.0260906702 0.0110313403
## [31] 0.0188993772 -0.0046343707 0.0296058250 0.0286271635 -0.0266243718
## [36] -0.0048312965 -0.0073125622 0.0067490982 0.0141932907 0.0730718310
## [41] -0.0292839508 0.0247904040 0.0083377258 0.0198747104 0.0348471820
## [46] -0.0193024979 -0.0110898764 0.0356837656 0.0140785852 0.0257377796
## [51] -0.0142439410 0.0256907366 0.0037589843 0.0505667199 0.0120780671
## [56] -0.0396284543 -0.0158111472 0.0848784716 0.0061364940 0.0285833962
## [61] 0.0086019940 -0.0387544273 -0.0010345007 -0.0089757897 0.0038289479
## [66] -0.0218448995 0.0284432109 0.0368634374 0.0079245260 0.0158398925
## [71] -0.0034228778 0.0293240462 0.0490845258 0.0037503230 0.0194526318
## [76] 0.0364522009 0.0239434289 -0.0052169923 -0.0089376677 0.0018147909
## [81] 0.0351727260 -0.0001730556 0.0156532498 0.0345308513 -0.0186251605
## [86] 0.0209610202 0.0350271664 0.0126421372 0.0255660258 0.0002574457
## [91] 0.0022992086 0.0157978982 0.0073265045 0.0260194645 0.0202242003
## [96] 0.0469478193 -0.0451328971 0.0027989719 -0.0022611291 -0.0046464851
## [101] 0.0323418224 0.0096732835 0.0005373015 -0.0088805332 -0.0078518319
## [106] 0.0221637185 0.0008665170 -0.0029116920 0.0004292359 0.0059710091
## [111] -0.0049779536 -0.0128220075 0.0143181801 0.0033461425 -0.0294349541
## [116] 0.0541090262 0.0040788272 0.0202232082 0.0131054520 -0.0101602514# 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 1.03
## 3 1.00
## 4 0.995
## 5 1.02
## 6 0.999
## 7 0.980
## 8 1.03
## 9 0.975
## 10 0.972
## # 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 1.03
## 3 1.03
## 4 1.03
## 5 1.04
## 6 1.04
## 7 1.02
## 8 1.05
## 9 1.02
## 10 0.995
## # ℹ 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 10.302746 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 = 120, mean_return = 0.005, sd_return = 0.01) %>%
tail()## # A tibble: 6 × 1
## growth
## <dbl>
## 1 165.
## 2 166.
## 3 169.
## 4 173.
## 5 174.
## 6 173.7 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(simulate_accumulation,
N = 120,
mean = mean_port_return,
sd_return = stddev_port_return) %>%
# Add the column, month
mutate(month = seq(1:nrow(.))) %>%
# Arrange column names
select(month, everything()) %>%
set_names(c("month", names(starts))) %>%
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# Calculate the quantiles for simulated values
probs <- c(0, 0.25, 0.5, 0.75, 1)
monte_carlo_sim_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
pull(growth) %>%
# Find the quantiles
quantile(probs = probs) %>%
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, col = sim)) +
geom_line() +
theme(legend.position = "none") +
theme(plot.title = element_text(hjust = 0.5)) +
labs(title = "Simulating Growth of $1 over 120 months")
Line plot with max, median, and min
# Step 1 Summarize data into max, median, and min of last value
sim_summary <- monte_carlo_sim_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
summarise(max = max(growth),
median = median(growth),
min = min(growth))
sim_summary## # A tibble: 1 × 3
## max median min
## <dbl> <dbl> <dbl>
## 1 3.88 1.98 1.17monte_carlo_sim_51 %>%
group_by(sim) %>%
filter(last(growth) == sim_summary$max |
last(growth) == sim_summary$median |
last(growth) == sim_summary$min) %>%
ungroup() %>%
# Plot
ggplot(aes(x = month, y = growth, col = 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")