# 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 0.10
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 
## # … with 50 more rows

5 Simulating growth of a dollar

# Get mean portfolio return
mean_port_return <- mean(portfolio_returns_tbl$returns)
mean_port_return
## [1] 0.005899135
# Get standard deviation of portfolio returns
stddev_port_return <- sd(portfolio_returns_tbl$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] -2.975645e-03 -2.344941e-02 -1.844338e-02  1.647597e-02 -9.795139e-04
##   [6]  5.037635e-03  9.252760e-03 -3.575821e-03 -6.832204e-02  2.053834e-02
##  [11] -1.534253e-02  1.298810e-02  2.341726e-02  3.121199e-03 -1.348816e-03
##  [16] -1.457046e-02  7.649371e-04  3.371695e-02  3.782471e-02  1.085971e-02
##  [21]  1.491975e-02 -3.483244e-02 -1.405882e-02  1.717978e-02 -2.702758e-02
##  [26] -1.264655e-02  4.679241e-03  1.095994e-02  3.206934e-02 -3.700855e-02
##  [31]  2.918906e-02  2.698953e-02  1.157323e-03 -2.411801e-03  5.264835e-02
##  [36]  2.791562e-02 -2.168244e-02  1.298480e-04 -1.734290e-02  2.196801e-02
##  [41]  6.821352e-03  5.392110e-03  5.283699e-02 -1.346327e-02  3.676988e-02
##  [46] -8.683608e-05 -1.071384e-02 -1.282907e-02 -2.749273e-03  6.921513e-03
##  [51]  4.495758e-02 -9.525517e-03  1.489063e-02  2.569936e-02 -2.463284e-02
##  [56]  1.895199e-02  2.819906e-03 -8.294100e-04 -1.505825e-02  8.818669e-03
##  [61]  2.366732e-02 -2.922263e-03  2.502148e-02  4.188951e-02 -2.184690e-02
##  [66]  1.119432e-02  1.052948e-02  3.298311e-03 -4.414924e-02  9.076979e-03
##  [71]  2.503441e-02  2.707517e-02  5.648103e-02 -2.121610e-02  4.102979e-02
##  [76] -2.147857e-03  3.722155e-02 -1.488833e-02 -1.812911e-02 -1.352890e-02
##  [81]  1.307771e-02  3.896068e-02 -6.685259e-03  3.936542e-02  4.136924e-03
##  [86]  2.173793e-03  2.334009e-02 -1.859110e-02 -2.374712e-02 -3.787347e-04
##  [91]  1.834299e-02  2.178717e-02 -1.677340e-02  1.551339e-02  1.702882e-02
##  [96]  3.250575e-02  3.080509e-03  2.943817e-02  3.331750e-02 -1.572436e-02
## [101] -2.016747e-02 -9.706746e-03  1.657743e-02  1.423408e-02 -1.873047e-02
## [106]  2.164331e-02  4.726478e-02  2.074560e-03  2.251346e-02 -1.914115e-02
## [111] -2.973694e-03  2.863973e-02  5.964567e-02  5.058268e-02 -1.213754e-02
## [116]  2.350452e-02  3.104873e-02  1.894140e-02  3.046791e-02  1.129218e-02
# 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   0.997
##  3   0.977
##  4   0.982
##  5   1.02 
##  6   0.999
##  7   1.01 
##  8   1.01 
##  9   0.996
## 10   0.932
## # … 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  0.997
##  3  0.974
##  4  0.956
##  5  0.971
##  6  0.970
##  7  0.975
##  8  0.984
##  9  0.981
## 10  0.914
## # … with 111 more rows
# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr
## [1] 9.161192

6 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   1.03
##  3   1.07
##  4   1.05
##  5   1.06
##  6   1.06
##  7   1.08
##  8   1.06
##  9   1.09
## 10   1.08
## # … 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% 
##  1.17  1.36  1.67  2.01  2.32  3.22  3.34

8 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  3.36   2.01  1.11
monte_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()