Week 15: Code along 13
CHAPTER 11 Monte Carlo Simulation
Anton Jellvik
2023-04-30
# 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] 0.0138205843 0.0110005892 -0.0319146534 0.0252262932 -0.0024547563
## [6] -0.0125211618 0.0133143452 0.0134744792 -0.0179736101 0.0115131029
## [11] 0.0034196334 0.0383829521 0.0165065170 -0.0329336663 -0.0177576030
## [16] -0.0287524818 -0.0002520760 0.0388882446 -0.0136402802 0.0197202746
## [21] 0.0173451527 -0.0022404643 -0.0171986539 -0.0185998567 0.0003188318
## [26] -0.0384928765 0.0152315659 0.0393678746 -0.0173889014 0.0198267497
## [31] -0.0077403762 0.0335256176 -0.0011919303 0.0469267892 -0.0037296740
## [36] 0.0150237634 -0.0230867401 -0.0162005537 0.0283317624 0.0335869275
## [41] -0.0036552739 0.0178435442 0.0371522955 0.0231327821 0.0136848664
## [46] 0.0175122405 -0.0135805491 -0.0149338192 0.0095172949 0.0357247028
## [51] 0.0181830908 -0.0080478320 0.0254538564 0.0104550796 -0.0201719541
## [56] 0.0477687814 0.0051657561 -0.0158181516 0.0114427737 0.0375410009
## [61] 0.0071077327 0.0353512156 0.0225019370 -0.0267194028 0.0285622246
## [66] -0.0091146088 -0.0221061921 0.0132290713 0.0231641565 0.0147830791
## [71] -0.0040350029 0.0078307717 0.0030644625 0.0197718726 0.0195323914
## [76] 0.0316239391 0.0434743160 0.0094466256 -0.0022578906 0.0232481028
## [81] -0.0074541411 0.0065176906 -0.0307175502 0.0114724107 -0.0113747454
## [86] 0.0149250394 -0.0119034144 -0.0205853096 0.0160158591 -0.0400840973
## [91] 0.0374423169 -0.0331139770 0.0303034245 -0.0151921892 -0.0089546451
## [96] 0.0092555822 -0.0248822001 0.0392539456 -0.0172101050 -0.0315789651
## [101] 0.0006185035 0.0051372786 0.0357364584 0.0663286169 -0.0182517069
## [106] -0.0218742728 0.0134903770 0.0120379359 -0.0047421081 0.0103636596
## [111] 0.0038659879 0.0222599530 0.0254390627 0.0283233022 -0.0226719803
## [116] -0.0044748621 -0.0080141894 0.0007966172 -0.0491117786 -0.0265170511# 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 1.01
## 4 0.968
## 5 1.03
## 6 0.998
## 7 0.987
## 8 1.01
## 9 1.01
## 10 0.982
## # … 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 1.02
## 4 0.992
## 5 1.02
## 6 1.01
## 7 1.00
## 8 1.02
## 9 1.03
## 10 1.01
## # … with 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 5.8963116 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 1.05
## 3 1.05
## 4 1.03
## 5 1.07
## 6 1.08
## 7 1.08
## 8 1.12
## 9 1.11
## 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 rows# Calculate the quantiles for simulated values
probs <- c(.005, .025, .25, .5, .75, .975, .995)
monte_carlo_sim_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
pull(growth) %>%
# Find the quantiles
quantile(probs = probs) %>%
round(2)## 0.5% 2.5% 25% 50% 75% 97.5% 99.5%
## 1.13 1.28 1.63 1.93 2.37 3.38 3.728 Visualizing simulations with ggplot
monte_carlo_sim_51 %>%
ggplot(aes(x = month, y = growth, col = sim)) +
geom_line() +
theme(legend.position = "none")
# 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 3.82 1.93 1.09monte_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()