# 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$returns)
mean_port_return
## [1] 0.005899137
# Get standard deviation of portfolio returns
stddev_port_return <- sd(portfolio_returns_tbl$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] -0.0028084775  0.0238513821  0.0303860142  0.0138368401  0.0307924405
##   [6]  0.0120980408  0.0203271984 -0.0188221203  0.0235329720 -0.0271449152
##  [11]  0.0053272914 -0.0197314704  0.0334464849  0.0150221450 -0.0182574718
##  [16]  0.0458013994  0.0208890765  0.0235622185  0.0075048505 -0.0059958063
##  [21]  0.0100716297  0.0084071427  0.0329257671  0.0049769362  0.0045641110
##  [26]  0.0225872153  0.0163357296  0.0226192820 -0.0207665409  0.0087663163
##  [31] -0.0110513947 -0.0228693824  0.0256201194  0.0226694956  0.0037713277
##  [36] -0.0231379127 -0.0035004119 -0.0406344684  0.0181929634  0.0094447549
##  [41]  0.0260989783 -0.0092156787 -0.0184188691  0.0181872549  0.0259340063
##  [46]  0.0131870997 -0.0031466852  0.0202565966 -0.0133735263  0.0133645348
##  [51]  0.0183905026  0.0030139061 -0.0166402677  0.0334401098  0.0527753171
##  [56]  0.0145714706  0.0043342158  0.0321304649  0.0110416840  0.0098613151
##  [61]  0.0159789995 -0.0371479390  0.0326574523 -0.0281474577  0.0226371474
##  [66]  0.0102156716 -0.0407914305  0.0174723394  0.0001499351 -0.0231970364
##  [71]  0.0022002636  0.0446152305  0.0286882816 -0.0214980207 -0.0207335378
##  [76]  0.0302179376  0.0031459497  0.0133451177 -0.0195739208  0.0276984558
##  [81]  0.0509500819 -0.0038275685  0.0224287447 -0.0259254274  0.0077405192
##  [86] -0.0257536752 -0.0021994572  0.0248566826 -0.0090023715  0.0410821809
##  [91] -0.0057906944 -0.0260293298 -0.0171586601 -0.0327572211  0.0217883961
##  [96] -0.0005271104  0.0164770439  0.0046565740 -0.0168676900 -0.0062702987
## [101]  0.0431362307  0.0169723276 -0.0205842520  0.0110928934 -0.0310123407
## [106] -0.0021764726  0.0169165517 -0.0282542833 -0.0082999039 -0.0056479163
## [111] -0.0083246784 -0.0084822650  0.0176479211  0.0165913350  0.0024199947
## [116] -0.0043517636  0.0014200438 -0.0048752624  0.0334352925  0.0580649499
# 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   1.02 
##  4   1.03 
##  5   1.01 
##  6   1.03 
##  7   1.01 
##  8   1.02 
##  9   0.981
## 10   1.02 
## # … 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  1.02 
##  4  1.05 
##  5  1.07 
##  6  1.10 
##  7  1.11 
##  8  1.14 
##  9  1.11 
## 10  1.14 
## # … with 111 more rows
# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr
## [1] 7.04123

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  1.01 
##  3  1.01 
##  4  1.00 
##  5  0.961
##  6  0.945
##  7  0.953
##  8  0.935
##  9  0.939
## 10  0.972
## # … 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% 
##  1.25  1.31  1.59  1.85  2.28  2.67  3.12

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  3.27   1.85  1.24
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()