Week 15: Code along 13
CHAPTER 11 Monte Carlo Simulation
Daniel Lee
2022-09-19
# 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] -0.0074036290 -0.0392647701 -0.0245822701 0.0233605155 0.0129797489
## [6] 0.0105201875 0.0031873046 -0.0104116835 0.0147742889 -0.0252898046
## [11] 0.0415514005 0.0115759023 0.0333351674 -0.0249587302 -0.0012459779
## [16] 0.0062158512 0.0106608033 0.0212370392 -0.0066497357 0.0300463216
## [21] 0.0403141740 -0.0049745591 -0.0274650253 -0.0056551154 0.0773448237
## [26] 0.0232182251 0.0061103945 0.0487800860 0.0274412633 0.0241793724
## [31] 0.0108143508 -0.0111513335 -0.0050390445 0.0009220917 0.0170469625
## [36] -0.0288226806 -0.0363270229 0.0497469796 0.0144851209 0.0080070328
## [41] -0.0218755840 0.0387438427 -0.0045946208 0.0307514023 0.0094730040
## [46] -0.0106596245 0.0071316580 -0.0072363474 -0.0287518528 0.0175662238
## [51] 0.0230188410 -0.0340756779 0.0074220440 0.0506415801 0.0606863302
## [56] 0.0258373058 0.0280661019 0.0031390401 0.0321917949 -0.0049335972
## [61] 0.0140750047 -0.0418618242 0.0419053843 0.0437511450 -0.0167560962
## [66] 0.0135880507 -0.0031547881 0.0241039403 -0.0250406265 0.0082389651
## [71] 0.0141753425 0.0143918418 0.0271777670 0.0261127272 0.0167509081
## [76] 0.0224148632 -0.0153801444 0.0202163303 -0.0181837712 -0.0158399811
## [81] 0.0098546407 0.0413090251 0.0086686548 -0.0289821293 0.0366679444
## [86] -0.0142046607 0.0068093929 -0.0154395585 -0.0198971682 -0.0189047312
## [91] -0.0376102018 0.0167024062 0.0095828738 -0.0113422813 -0.0094381911
## [96] 0.0254827560 -0.0053706960 0.0323724023 0.0033947184 -0.0288522857
## [101] 0.0517912209 -0.0310799482 0.0592058232 -0.0347703412 0.0242387876
## [106] 0.0201767772 0.0130577119 -0.0349938980 0.0170026479 -0.0503195903
## [111] -0.0051118895 0.0594034295 0.0052906368 0.0112452450 -0.0125150792
## [116] 0.0377109426 -0.0034064284 -0.0040746566 0.0078980695 0.0355433724# 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 0.993
## 3 0.961
## 4 0.975
## 5 1.02
## 6 1.01
## 7 1.01
## 8 1.00
## 9 0.990
## 10 1.01
## # … 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 0.993
## 3 0.954
## 4 0.930
## 5 0.952
## 6 0.964
## 7 0.974
## 8 0.978
## 9 0.967
## 10 0.982
## # … with 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 8.4128026 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 0.998
## 3 0.997
## 4 1.02
## 5 1.02
## 6 1.04
## 7 0.991
## 8 1.05
## 9 1.04
## 10 1.05
## # … 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 column month
mutate(month = seq(1:nrow(.))) %>%
# Rearrange 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 rows# Find quantiles
monte_carlo_sim_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
pull(growth) %>%
quantile(probs = c(0, .25, .5, .75, 1)) %>%
round(2)## 0% 25% 50% 75% 100%
## 1.13 1.56 1.84 2.16 3.048 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 3.04 1.84 1.13monte_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)) +
theme(plot.subtitle = element_text(hjust = 0.5)) +
labs(title = "simulating Growth of $1 over 120 months",
subtitle = "Maximum, Median, and Minimum Simulation")