# 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 
## # ℹ 50 more rows

5 Simulating growth of a dollar

# Get mean portfolio return
mean_port_return <- mean(portfolio_returns_tbl$returns)
mean_port_return
## [1] 0.005899133
# Get standard deviation of portfolio returns
stddev_port_return <- sd(portfolio_returns_tbl$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] -4.439347e-03 -1.599810e-02  5.465433e-02 -7.024307e-03 -3.141452e-04
##   [6] -2.912793e-02 -2.351206e-02 -2.931805e-02 -2.127070e-02 -3.445639e-02
##  [11] -8.565149e-03  3.486428e-02  4.027911e-02  1.511694e-02  2.304597e-02
##  [16] -1.086235e-02  2.900091e-02  7.747554e-05  3.698890e-02 -3.589995e-02
##  [21]  3.865170e-02  1.192054e-03  8.748463e-03 -3.009789e-02  4.670723e-02
##  [26]  7.752175e-03  3.080668e-02 -1.571742e-02 -4.984604e-03 -5.379276e-03
##  [31] -2.243152e-02  7.745622e-03 -4.960711e-02 -1.441349e-02 -5.288841e-03
##  [36]  2.467825e-02 -6.199894e-03  2.694201e-02  3.021866e-02 -2.614379e-02
##  [41]  3.618225e-02  1.953456e-02  9.930489e-03  3.453903e-02  2.085452e-02
##  [46]  6.400006e-02  2.808741e-03 -2.119638e-02 -1.013681e-02 -2.760347e-03
##  [51]  8.894716e-03  1.601992e-02  1.657469e-02  2.238959e-02  1.371741e-02
##  [56] -1.687561e-03 -1.088536e-02  3.349364e-02 -4.158089e-02 -7.877091e-03
##  [61]  3.991034e-02  1.729740e-03  8.200688e-03  5.197183e-03  1.585171e-02
##  [66] -1.281548e-03 -1.986416e-02 -4.292079e-04 -1.770394e-03 -1.773743e-02
##  [71] -5.605619e-02  2.749308e-02 -1.005188e-02 -2.775557e-03 -5.806642e-03
##  [76]  5.902644e-02 -4.118275e-04 -7.419323e-03  1.898437e-04  4.861186e-02
##  [81] -3.164660e-02 -1.346279e-02 -1.597053e-02 -1.654437e-02  6.182225e-03
##  [86]  2.870388e-03 -2.581739e-02  2.864692e-02 -1.141175e-02 -5.048957e-02
##  [91]  3.211397e-02  5.160401e-02 -8.583927e-03  6.544886e-03  2.691446e-02
##  [96] -5.848777e-03  2.553322e-02  1.452096e-02 -5.268857e-02 -1.509824e-02
## [101]  4.236817e-02  2.980236e-02  7.373999e-03  4.343143e-02 -1.217333e-02
## [106] -3.295748e-02 -3.077165e-04  4.476276e-02  2.342963e-04 -4.901750e-02
## [111]  1.899412e-02  5.458371e-02 -1.611390e-02 -8.474613e-03  1.733468e-02
## [116]  4.718663e-03  1.320179e-02 -3.645986e-02 -3.780402e-02  2.992457e-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   0.996
##  3   0.984
##  4   1.05 
##  5   0.993
##  6   1.00 
##  7   0.971
##  8   0.976
##  9   0.971
## 10   0.979
## # ℹ 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  0.996
##  3  0.980
##  4  1.03 
##  5  1.03 
##  6  1.03 
##  7  0.996
##  8  0.972
##  9  0.944
## 10  0.924
## # ℹ 111 more rows
# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr
## [1] 3.674131

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 + 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   335.
## 2   340.
## 3   347.
## 4   349.
## 5   345.
## 6   342.
dump(list = c("simulate_accumulation"), file = "../00_scripts/simulate_accumulation.R")

7 Running multiple simulations

# Create a Vector of 1's 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 
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: 6,171 × 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
## # ℹ 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% 
## 0.88 1.73 2.10 2.38 3.58

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")
## $title
## [1] "Simulating Growth of $1 Over 120 Months"
## 
## attr(,"class")
## [1] "labels"

Line Plot with Max, Median, and Min

# 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  3.58   2.10 0.880
# Step 2 Plot 
monte_carlo_sim_51 %>%
   
    # Filter for max, median, and 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, Minimum")
## $title
## [1] "Simulating Growth of $1 Over 120 Months"
## 
## $subtitle
## [1] "Maximum, Median, Minimum"
## 
## attr(,"class")
## [1] "labels"