Week 15: Code along 13
CHAPTER 11 Monte Carlo Simulation
Gannon Stewart
2023-02-05
# 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$returns)
mean_port_return## [1] 0.005899134# Get standard deviation of portfolio returns
stddev_port_return <- sd(portfolio_returns_tbl$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] -0.0263325105 -0.0101925437 0.0007808795 0.0267470262 0.0192070095
## [6] -0.0007411276 0.0303007441 0.0282540198 -0.0001882714 0.0335469021
## [11] -0.0242303971 -0.0020089451 0.0002886061 -0.0330817618 0.0068202512
## [16] -0.0073463607 -0.0040800702 0.0402455528 -0.0265405719 0.0182654856
## [21] 0.0073773462 0.0190044227 0.0224337195 0.0248599868 0.0499269117
## [26] -0.0374926666 0.0059555535 -0.0051379522 0.0212551712 -0.0271514491
## [31] 0.0211241792 0.0351345462 -0.0077556363 -0.0109085032 -0.0146314284
## [36] 0.0302068051 0.0659358417 -0.0023529265 -0.0388035790 0.0167103564
## [41] 0.0430144902 0.0555177118 -0.0025351014 -0.0143654589 -0.0118101731
## [46] 0.0328256835 -0.0001985289 0.0206519659 0.0399644265 -0.0467601210
## [51] 0.0034171399 0.0516101094 -0.0059667312 0.0070154338 -0.0074312747
## [56] -0.0539795712 0.0155951154 -0.0210226768 0.0236966269 0.0253963791
## [61] 0.0068277849 -0.0234112583 -0.0020511513 0.0425124529 0.0791391808
## [66] 0.0293516877 0.0058161384 -0.0070234642 -0.0002941011 0.0367998977
## [71] 0.0305426819 0.0382362509 -0.0206062115 -0.0195116437 -0.0091178434
## [76] 0.0400755676 -0.0092102990 -0.0208121231 0.0079575797 0.0178657968
## [81] 0.0069745942 0.0099728606 -0.0134053656 0.0082834770 -0.0011024827
## [86] -0.0362617904 0.0363104569 0.0016977468 -0.0102332152 0.0014191946
## [91] 0.0224392183 0.0120381096 0.0117885482 -0.0267559362 0.0076739323
## [96] -0.0098081139 0.0222719132 0.0222421859 0.0166259328 0.0054992977
## [101] 0.0427771810 -0.0260431540 0.0377945140 -0.0176778677 0.0150822877
## [106] 0.0001765890 0.0136192410 0.0087440186 0.0453018825 0.0304316353
## [111] 0.0361190369 0.0003996892 0.0328494977 0.0123987830 0.0688183192
## [116] 0.0113545605 -0.0136305917 -0.0063843012 0.0209541153 -0.0484301314# 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 0.974
## 3 0.990
## 4 1.00
## 5 1.03
## 6 1.02
## 7 0.999
## 8 1.03
## 9 1.03
## 10 1.00
## # … 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 0.974
## 3 0.964
## 4 0.964
## 5 0.990
## 6 1.01
## 7 1.01
## 8 1.04
## 9 1.07
## 10 1.07
## # … with 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 9.7902626 Simulation function
simulate_accumulation <- function(init_value, N, mean, stdev) {
tibble(returns = c(init_value, 1 + rnorm(N, mean, stdev))) %>%
mutate(growth = accumulate(returns, function(x, y) x*y)) %>%
select(growth)
}
simulate_accumulation(1, 120, mean_port_return, stddev_port_return)## # A tibble: 121 × 1
## growth
## <dbl>
## 1 1
## 2 0.960
## 3 1.01
## 4 0.990
## 5 1.02
## 6 1.06
## 7 1.13
## 8 1.12
## 9 1.16
## 10 1.17
## # … with 111 more rows# Save the function
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(paste("sim", 1:sims, sep = ""))
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
monte_carlo_sim_51 <- starts %>%
# Simulate
map_dfc(simulate_accumulation,
N = 120,
mean = mean_port_return,
stdev = 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
## # … with 6,161 more rows8 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 $1 over 120 months")
# Simplify the plot
sim_summary <- monte_carlo_sim_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
summarise(max = max(growth),
median = median(growth),
min = min(growth))
sim_summary## # A tibble: 1 × 3
## max median min
## <dbl> <dbl> <dbl>
## 1 3.67 1.98 1.04monte_carlo_sim_51 %>%
group_by(sim) %>%
filter(last(growth) == sim_summary$max |
last(growth) == sim_summary$median |
last(growth) == sim_summary$min) %>%
# Plot
ggplot(aes(month, 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",
subtitle = "Maximum, Median, and Minimum Simulation")