# 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$returns)
mean_port_return
## [1] 0.005899133
# Get standard deviation of portfolio returns
stddev_port_return <- sd(portfolio_returns_tbl$returns)
stddev_port_return
## [1] 0.02347491
# Construct a normal distribution
simulated_monthly_returns <- rnorm(120, mean_port_return, stddev_port_return)
simulated_monthly_returns
##   [1]  1.523903e-02 -1.121119e-02  1.126377e-03  3.151855e-02 -1.398831e-05
##   [6] -2.168851e-03  3.325471e-02  1.575873e-02  2.315761e-03 -8.302125e-03
##  [11] -3.784811e-03  5.909666e-03  8.101011e-03 -1.556215e-02  1.063507e-02
##  [16] -1.896348e-03  1.735573e-02 -3.662087e-04  1.242812e-03 -3.443536e-02
##  [21]  3.044973e-02  2.738864e-02  1.575422e-02  1.845068e-03  4.912482e-02
##  [26]  2.270662e-02  4.923382e-03  3.187287e-02  2.634639e-03  3.085189e-02
##  [31] -3.797788e-02 -4.173710e-03  4.644489e-02 -2.768056e-02 -2.254615e-03
##  [36] -1.204463e-02 -1.833817e-03  1.492728e-02 -1.636004e-02 -1.314652e-02
##  [41]  5.722075e-02 -2.629796e-02  5.140360e-02 -8.761450e-03  9.740050e-03
##  [46] -4.126162e-03  2.166142e-02  3.538110e-02  1.925323e-03 -1.001516e-02
##  [51]  2.243303e-02  1.453870e-02  1.689194e-02  4.209543e-02 -9.611177e-03
##  [56] -3.294738e-03  1.542448e-02  2.374771e-02  1.691410e-02  2.161759e-02
##  [61] -1.041703e-03  2.663340e-02 -2.300530e-02 -2.541510e-02  3.726631e-02
##  [66]  6.444940e-02  7.600792e-03  1.060148e-02  3.976333e-02  2.108132e-02
##  [71]  2.514381e-02  1.834943e-03  1.084595e-02  1.104839e-02 -4.371305e-03
##  [76] -2.405674e-02  3.522498e-02  3.093984e-02 -1.051277e-02  9.356066e-03
##  [81]  6.521062e-02  1.976706e-02  1.384553e-02 -2.057527e-03  1.380292e-02
##  [86]  1.373500e-02 -4.321342e-02  2.440610e-02 -3.653189e-02  3.894090e-02
##  [91]  1.079303e-02 -3.569212e-02  9.197518e-03  3.092815e-02  6.072742e-03
##  [96]  2.497103e-02  4.705703e-04  6.459918e-02 -1.237251e-02  3.651501e-04
## [101] -1.244790e-02 -3.805402e-02  2.491048e-02  5.574009e-02  2.989650e-02
## [106]  2.047505e-03 -9.038807e-03 -1.826936e-02 -3.619000e-02  4.296368e-02
## [111] -2.848151e-02 -1.227635e-02 -1.032712e-03  8.960338e-03  1.136101e-02
## [116]  3.278614e-02 -9.159713e-04  3.219859e-02 -2.032080e-02  2.121264e-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   1.02 
##  3   0.989
##  4   1.00 
##  5   1.03 
##  6   1.00 
##  7   0.998
##  8   1.03 
##  9   1.02 
## 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   1.02
##  3   1.00
##  4   1.00
##  5   1.04
##  6   1.04
##  7   1.03
##  8   1.07
##  9   1.09
## 10   1.09
## # … with 111 more rows
# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr
## [1] 10.69112

6 Simulation function

simulate_accumulation <- function(initial_value, N, mean_return, sd_return) {
   
  # Add a dollar
simulated_returns_add_1 <- tibble(returns = c(initial_value, 1, 1 + rnorm(N, mean_return, sd_return)))
 
# Calculate the cumulative growth of a dollar
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) %>% tail()
## # A tibble: 6 × 1
##   growth
##    <dbl>
## 1   242.
## 2   242.
## 3   239.
## 4   241.
## 5   238.
## 6   237.
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(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
# Simulate
# 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") 

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
## # … with 6,212 more rows
# Find quantiles 
monte_carlo_sim_51 %>% 
    
    group_by(sim) %>%
    summarise(growth = 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

# Line plot of simulations
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 maximum, median & minimun

# Step 1: Summarize data into max, median, and min of last value
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  3.88   1.98  1.17
# Step 2: Plot
monte_carlo_sim_51 %>%
    
    # Filter for max, median, min sim
    group_by(sim) %>%
    filter(last(growth) == sim_summary$max |
               last(growth) == sim_summary$median |
               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")