# 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] 0.0234749
# Construct a normal distribution
simulated_monthly_returns <- rnorm(120, mean_port_return, stddev_port_return)
simulated_monthly_returns
##   [1]  0.0259174146 -0.0368490344  0.0119352382  0.0108517577 -0.0301590534
##   [6]  0.0083040637  0.0212925817 -0.0007544321 -0.0034891591  0.0680913286
##  [11] -0.0102537486  0.0009733599  0.0224508505  0.0175006628 -0.0062418089
##  [16]  0.0053362895  0.0403674272 -0.0070585139  0.0389783108  0.0124892351
##  [21] -0.0383116458  0.0233869395  0.0118329785  0.0086511462 -0.0053666018
##  [26]  0.0106128702 -0.0114469744  0.0031763933  0.0149133421  0.0237335581
##  [31]  0.0228282154 -0.0010857455  0.0540071683  0.0132128471 -0.0009771709
##  [36] -0.0491072965 -0.0014698466  0.0048012497  0.0174313470 -0.0120651273
##  [41] -0.0125270465  0.0096121464  0.0230801869  0.0239777384  0.0034155217
##  [46]  0.0103766110 -0.0434485742  0.0643203434  0.0481965989  0.0215070793
##  [51]  0.0330703385 -0.0133855195  0.0009125400  0.0262661726 -0.0167542110
##  [56]  0.0030823970  0.0131527157  0.0206187430 -0.0117624426 -0.0275000457
##  [61] -0.0084854138  0.0017045908 -0.0188865125  0.0274072743 -0.0085364312
##  [66] -0.0077686583 -0.0020333339  0.0373195391  0.0014195721  0.0212091183
##  [71]  0.0014420771  0.0599745057 -0.0235587520  0.0163382272  0.0230763675
##  [76]  0.0126332424  0.0132645032 -0.0160414391  0.0166984262  0.0265198529
##  [81]  0.0138987910  0.0242776883 -0.0052693761 -0.0041975511  0.0035962081
##  [86]  0.0317587567 -0.0079172198  0.0528821523  0.0426074975  0.0249461735
##  [91]  0.0104000266 -0.0097451677 -0.0055280795 -0.0001097425  0.0547056103
##  [96]  0.0253682638  0.0090137223  0.0192915144 -0.0072160430  0.0415620713
## [101]  0.0265120386  0.0166904362  0.0096594873  0.0149936467  0.0098064810
## [106]  0.0339558890  0.0179296249  0.0320419491  0.0008301255  0.0287742139
## [111]  0.0234955038 -0.0097825735 -0.0260446750  0.0106076573 -0.0008077891
## [116]  0.0166035789  0.0343196306  0.0154212802 -0.0048558609 -0.0168776875
# 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   1.03 
##  3   0.963
##  4   1.01 
##  5   1.01 
##  6   0.970
##  7   1.01 
##  8   1.02 
##  9   0.999
## 10   0.997
## # 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  1.03 
##  3  0.988
##  4  1.00 
##  5  1.01 
##  6  0.980
##  7  0.988
##  8  1.01 
##  9  1.01 
## 10  1.01 
## # ℹ 111 more rows
# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr
## [1] 12.10556

6 Simulation function

simulated_accumulation <- function(initial_value, N, mean_return, sd_return) {
    
    # Construct a normal distribution
simulated_monthly_returns <- rnorm(120, mean_port_return, stddev_port_return)

# Add a dollar
simulated_returns_add_1 <- tibble(returns = c(initial_value, 1 + rnorm(120, 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)
}

simulated_accumulation(initial_value = 100, N = 240, mean_return = 0.005, sd_return = 0.01) %>%
tail()
## # A tibble: 6 × 1
##   growth
##    <dbl>
## 1   171.
## 2   173.
## 3   177.
## 4   178.
## 5   176.
## 6   174.
dump(list = c("simulate_accumulation"), 
     file = "../00scripts/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 = ~simulated_accumulation(initial_value = .x,
                                        N             = 120, 
                                        mean_return   = mean_port_return,
                                        sd_return     = stddev_port_return)) %>%
    
    # Add column
    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 quantities
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 1.61 1.99 2.35 3.68

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 for $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) %>%
    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  3.68   1.99  1.17
# 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 for $1 over 120 months", 
         subtitle = "Maximum, Median, and Minimum Simulation")
## $title
## [1] "simulating growth for $1 over 120 months"
## 
## $subtitle
## [1] "Maximum, Median, and Minimum Simulation"
## 
## attr(,"class")
## [1] "labels"