Goal

  1. Import stock prices
  2. Convert prices to returns
  3. Assign a weight to each asset
  4. Build a portfolio
  5. Simulate growth of a dollar
  6. Build a simulation function
  7. Run multiple simulations
  8. Visualize simulations with ggplot

We will simulate future portfolio returns using a Monte Carlo framework.

# Load packages

# Core
library(tidyverse)
library(tidyquant)

# time series
library(timetk)

1 Import stock prices

five stocks: “SPY”, “EFA”, “IJS”, “EEM”, “AGG”
market: “SPY”
from 2012-12-31 to 2017-12-31

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 0.10
w_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        0.1

4 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 rows

5 Simulating growth of a dollar

# Get mean portfolio return
mean_port_return <- mean(portfolio_returns_tbl$returns)
mean_port_return
## [1] 0.005899132
# Get standard deviation of portfolio returns
stddev_port_return <- sd(portfolio_returns_tbl$returns)
stddev_port_return
## [1] 0.0234749
# Construct a normal distribution
set.seed(1234)
simulated_monthly_returns <- rnorm(120, mean_port_return, stddev_port_return)
simulated_monthly_returns[1:20]
##  [1] -0.022436617  0.012411756  0.031356281 -0.049165889  0.015972792
##  [6]  0.017778744 -0.007592832 -0.006932997 -0.007351323 -0.014994418
## [11] -0.005302919 -0.017537891 -0.012323351  0.007412296  0.028423160
## [16]  0.003310191 -0.006096765 -0.015491090 -0.013753390  0.062610624
# Add a dollar
simulated_returns_add_1 <- tibble(returns = c(1, 1 + simulated_monthly_returns))
head(simulated_returns_add_1)
## # A tibble: 6 × 1
##   returns
##     <dbl>
## 1   1    
## 2   0.978
## 3   1.01 
## 4   1.03 
## 5   0.951
## 6   1.02
# Calculate the cumulative growth of a dollar
simulated_growth <- simulated_returns_add_1 %>%
    mutate(growth = accumulate(returns, function(x, y) x * y)) %>%
    select(growth)

head(simulated_growth)
## # A tibble: 6 × 1
##   growth
##    <dbl>
## 1  1    
## 2  0.978
## 3  0.990
## 4  1.02 
## 5  0.971
## 6  0.986
# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr
## [1] 2.606633

6 Simulation function

simulate_accumulation <- function(initial_value, N, mean_return, sd_return) {
    
    # Add a dollar / starting value
    simulated_returns_add_1 <- tibble(returns = c(initial_value, 1, 1 + rnorm(N, mean_return, sd_return)))
    
    # Calculate the cumulative growth of the initial value
    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: 242 × 1
##    growth
##     <dbl>
##  1   100 
##  2   100 
##  3   100.
##  4   100.
##  5   103.
##  6   104.
##  7   106.
##  8   107.
##  9   107.
## 10   106.
## # ℹ 232 more rows
# Optionally save the function (may require the directory to exist)
# dump(list = c("simulate_accumulation"),
#      file = "../00_scripts/simulate_accumulation.R")

7 Running multiple simulations

# Create a vector of 1s as 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
# for reproducible research
set.seed(1234)

monte_carlo_sim_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 names
    set_names(c("month", names(starts))) %>%
    
    # transform to long form
    pivot_longer(cols = -month, names_to = "sim", values_to = "growth")
## New names:
## • `growth` -> `growth...1`
## • `growth` -> `growth...2`
## • `growth` -> `growth...3`
## • `growth` -> `growth...4`
## • `growth` -> `growth...5`
## • `growth` -> `growth...6`
## • `growth` -> `growth...7`
## • `growth` -> `growth...8`
## • `growth` -> `growth...9`
## • `growth` -> `growth...10`
## • `growth` -> `growth...11`
## • `growth` -> `growth...12`
## • `growth` -> `growth...13`
## • `growth` -> `growth...14`
## • `growth` -> `growth...15`
## • `growth` -> `growth...16`
## • `growth` -> `growth...17`
## • `growth` -> `growth...18`
## • `growth` -> `growth...19`
## • `growth` -> `growth...20`
## • `growth` -> `growth...21`
## • `growth` -> `growth...22`
## • `growth` -> `growth...23`
## • `growth` -> `growth...24`
## • `growth` -> `growth...25`
## • `growth` -> `growth...26`
## • `growth` -> `growth...27`
## • `growth` -> `growth...28`
## • `growth` -> `growth...29`
## • `growth` -> `growth...30`
## • `growth` -> `growth...31`
## • `growth` -> `growth...32`
## • `growth` -> `growth...33`
## • `growth` -> `growth...34`
## • `growth` -> `growth...35`
## • `growth` -> `growth...36`
## • `growth` -> `growth...37`
## • `growth` -> `growth...38`
## • `growth` -> `growth...39`
## • `growth` -> `growth...40`
## • `growth` -> `growth...41`
## • `growth` -> `growth...42`
## • `growth` -> `growth...43`
## • `growth` -> `growth...44`
## • `growth` -> `growth...45`
## • `growth` -> `growth...46`
## • `growth` -> `growth...47`
## • `growth` -> `growth...48`
## • `growth` -> `growth...49`
## • `growth` -> `growth...50`
## • `growth` -> `growth...51`
monte_carlo_sim_51
## # A tibble: 6,222 × 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 6,212 more rows
# find quantiles of ending values
monte_carlo_sim_51 %>%
    group_by(sim) %>%
    summarise(growth = dplyr::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 3.88

8 Visualizing simulations with ggplot

monte_carlo_sim_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 and min

# summarize data into max, median, and min of last value
sim_summary <- monte_carlo_sim_51 %>%
    group_by(sim) %>%
    summarise(growth = dplyr::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
# Plot
monte_carlo_sim_51 %>%
    # filter for max, median and min sim
    group_by(sim) %>%
    filter(dplyr::last(growth) == sim_summary$max |
           dplyr::last(growth) == sim_summary$median |
           dplyr::last(growth) == sim_summary$min) %>%
    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 = "Maximum, Median, and Minimum Simulation")