# 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$returns)
mean_port_return
## [1] 0.005899132
# Get standard deviation of portfolio returns
stddev_port_return <- sd(portfolio_returns_tbl$returns)
stddev_port_return
## [1] 0.02347489
# Construct a normal distribution
simulated_monthly_returns <- rnorm(120, mean_port_return, stddev_port_return)
simulated_monthly_returns
##   [1] -0.0176876343  0.0263267809 -0.0251221723 -0.0029447707  0.0270746667
##   [6]  0.0170174363  0.0176333254  0.0471894844  0.0435182327  0.0182114018
##  [11]  0.0310709516  0.0062692098  0.0189507277  0.0448018997  0.0176912393
##  [16]  0.0253755395  0.0104504621  0.0417037518 -0.0027198350 -0.0394351580
##  [21]  0.0225068594  0.0233432150  0.0087798228  0.0160762443  0.0007009085
##  [26] -0.0058032122  0.0337615825  0.0238858685 -0.0008701007 -0.0289764307
##  [31]  0.0462759767 -0.0115089304 -0.0137250166 -0.0043613716 -0.0065715599
##  [36]  0.0559889208 -0.0307669792  0.0434300028 -0.0069762109  0.0503680622
##  [41]  0.0214169080 -0.0050448120  0.0193497380  0.0475846432  0.0357244200
##  [46] -0.0447146766  0.0084363270 -0.0144880861 -0.0019736688  0.0128770173
##  [51]  0.0140578075 -0.0286434052 -0.0071276603 -0.0075598609  0.0312931837
##  [56]  0.0060062294 -0.0104352328 -0.0571062787  0.0095529081  0.0245380118
##  [61] -0.0429075244 -0.0286514826 -0.0037187203  0.0341031867 -0.0035324253
##  [66]  0.0077593729 -0.0232850464  0.0138663066  0.0454882201  0.0176538158
##  [71]  0.0203340613 -0.0140637590  0.0263396588 -0.0364437617  0.0062053995
##  [76] -0.0163410920  0.0092574979 -0.0026424693 -0.0259169570  0.0334840194
##  [81]  0.0070305766  0.0192456705  0.0327428430  0.0505771131  0.0075899002
##  [86] -0.0035677537  0.0263968537  0.0124627581  0.0662605920  0.0291154348
##  [91] -0.0305986207 -0.0078928183 -0.0562646367  0.0334443729  0.0104679142
##  [96]  0.0324720942  0.0598023725  0.0101280859  0.0020111017  0.0472551662
## [101]  0.0102173066  0.0160286311  0.0121637569  0.0092776604  0.0070463688
## [106] -0.0088695899  0.0113663685 -0.0145216653  0.0054822994 -0.0148734808
## [111] -0.0046355238 -0.0108539927 -0.0084171777  0.0362975229  0.0243946714
## [116] -0.0042345000  0.0034352339  0.0214656443  0.0057230930 -0.0242013564
# 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   0.982
##  3   1.03 
##  4   0.975
##  5   0.997
##  6   1.03 
##  7   1.02 
##  8   1.02 
##  9   1.05 
## 10   1.04 
## # 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  0.982
##  3  1.01 
##  4  0.983
##  5  0.980
##  6  1.01 
##  7  1.02 
##  8  1.04 
##  9  1.09 
## 10  1.14 
## # ℹ 111 more rows
# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr
## [1] 10.2081

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   338.
## 3   343.
## 4   347.
## 5   350.
## 6   350.
dump(list = c("simulate_accumulation"),
     file = "../00_scripts/simulate_accumulation.R")

7 Running multiple simulations

simulate_accumulation(initial_value = 100, N = 240, mean_return = 0.005, sd_return = 0.01)
## # A tibble: 241 × 1
##    growth
##     <dbl>
##  1   100 
##  2   100.
##  3   101.
##  4   100.
##  5   103.
##  6   105.
##  7   106.
##  8   107.
##  9   107.
## 10   109.
## # ℹ 231 more rows
simulate_accumulation(initial_value = 100, N = 240, mean_return = 0.005, sd_return = 0.01) 
## # A tibble: 241 × 1
##    growth
##     <dbl>
##  1   100 
##  2   101.
##  3   104.
##  4   104.
##  5   104.
##  6   105.
##  7   106.
##  8   105.
##  9   106.
## 10   106.
## # ℹ 231 more rows
simulate_accumulation(initial_value = 100, N = 240, mean_return = 0.005, sd_return = 0.01) 
## # A tibble: 241 × 1
##    growth
##     <dbl>
##  1  100  
##  2   98.0
##  3   98.3
##  4   98.5
##  5   98.0
##  6   99.0
##  7   99.4
##  8  102. 
##  9  102. 
## 10  103. 
## # ℹ 231 more rows
# 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
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, 0.25, 0.5, 0.75, 1)) %>%
    round(2)
##   0%  25%  50%  75% 100% 
## 1.00 1.76 2.05 2.38 3.53

8 Visualizing simulations with ggplot

monte_carlo_sim_51 %>%
    
    ggplot(aes(x = month, y = growth, color = sim)) +
    geom_line() +
    theme(legend.position = "none") + 
    
    labs(title = "Simulating growth of $1 over 120 months")

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) %>%
    summarize(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  3.53   2.05  1.00
# 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 Simulation")