Money-weighted returns – finedtech by David Harper, CFA, FRM

Claude 3.5 Sonnet wrote much of this for me (the telltale sign is the loops), but it required many iterations.

library(tidyverse)
# library(FinancialMath)

# Set seed for reproducibility
set.seed(123)

# Parameters
initial_account_balance <- 10000
initial_price <- 100
volatility <- 0.2
initial_dividend <- 2.00  # Initial dividend is $2.00
dividend_growth_rate <- 0.05
years <- 5
risk_free_rate <- 0.05

# Generate stock prices
stock_prices <- numeric(years)
stock_prices[1] <- initial_price
for (i in 2:years) {
  stock_prices[i] <- round(stock_prices[i-1] * exp(rnorm(1, mean = 0, sd = volatility)),2)
}

# Calculate dividends
dividends <- numeric(years)
dividends[1] <- initial_dividend
for (i in 2:years) {
  dividends[i] <- dividends[i-1] * (1 + dividend_growth_rate)
}

# Function to generate investment vector
generate_investment_vector <- function() {
  vec <- c(runif(4), 1)  # Generate 4 random numbers between 0 and 1, and add 1 at the end
  vec[5] <- -1  # Set last element to -1 (100% divestment)
  return(vec)
}

# Function to calculate cash flows and portfolio value
calculate_cash_flows <- function(investment_vector) {
  cash_balance <- initial_account_balance
  shares <- 0
  cash_flows <- numeric(years)
  cash_balance_history <- numeric(years)
  
  for (year in 1:years) {
    if (year > 1) {
      # Add dividends from previous year and grow cash balance
      cash_balance <- cash_balance * (1 + risk_free_rate) + shares * dividends[year-1]
    }
    
    cash_balance_history[year] <- cash_balance
    
    if (year < years) {
      # Years 1 to 4: Potential investments
      if (investment_vector[year] > 0) {
        investment <- investment_vector[year] * cash_balance
        new_shares <- investment / stock_prices[year]
        shares <- shares + new_shares
        cash_balance <- cash_balance - investment
        cash_flows[year] <- -investment
      } else {
        cash_flows[year] <- 0
      }
    } else {
      # Final year (year 5): Liquidate all shares and cash balance
      final_value <- (shares * stock_prices[year]) + cash_balance
      cash_flows[year] <- final_value
    }
  }
  
  return(list(cash_flows = cash_flows, cash_balance_history = cash_balance_history, final_shares = shares))
}

# Function to calculate IRR (Money-Weighted Return)
calculate_irr <- function(cash_flows) {
  irr <- function(r) {
    sum(cash_flows / (1 + r)^(0:(length(cash_flows)-1)))
  }

  # Try to find the root within a reasonable range
  tryCatch({
    mwr <- uniroot(irr, c(-0.9, 5))$root
    return(mwr)
  }, error = function(e) {
    # If uniroot fails, return NA
    return(NA)
  })
}

# Function to calculate Time-Weighted Return (TWR)
calculate_twr <- function(stock_prices, dividends) {
  hpr <- numeric(length(stock_prices) - 1)
  
  for (i in 1:(length(stock_prices) - 1)) {
    hpr[i] <- (stock_prices[i+1] + dividends[i] - stock_prices[i]) / stock_prices[i]
  }
  
  twr <- prod(1 + hpr)^(1/(length(hpr))) - 1
  return(twr)
}

# Simulate Money-Weighted Returns (MWR) and Time-Weighted Returns (TWR)
n_simulations <- 10000
mwr_results <- numeric(n_simulations)
twr_results <- numeric(n_simulations)
failed_irr_count <- 0

# Specific scenario
specific_scenario <- c(0.25, 1/3, .5, 1, -1)
specific_result <- calculate_cash_flows(specific_scenario)
specific_mwr <- calculate_irr(specific_result$cash_flows)
specific_twr <- calculate_twr(stock_prices, dividends)

cat("\nSpecific Scenario {25%, 33.3%, 50%, 100%, -100%}:\n")


Specific Scenario {25%, 33.3%, 50%, 100%, -100%}:

cat("Investment Vector:", specific_scenario, "\n")

Investment Vector: 0.25 0.3333333 0.5 1 -1

cat("Cash Flows:", specific_result$cash_flows, "\n")

Cash Flows: -2500 -2641.667 -2831.026 -3165.971 13850.05

cat("Cash Balance History:", specific_result$cash_balance_history, "\n")

Cash Balance History: 10000 7925 5662.053 3165.971 265.9224

cat("Final Shares:", specific_result$final_shares, "\n")

Final Shares: 114.8569

cat("MWR:", specific_mwr, "\n")

MWR: 0.09269345

cat("TWR:", specific_twr, "\n\n")

TWR: 0.06511804

# Run simulations
for (sim in 1:n_simulations) {
  investment_vector <- generate_investment_vector()
  result <- calculate_cash_flows(investment_vector)
  mwr <- calculate_irr(result$cash_flows)
  
  if (is.na(mwr)) {
    failed_irr_count <- failed_irr_count + 1
  } else {
    mwr_results[sim] <- mwr
  }
}

# Calculate TWR (same for all simulations as it doesn't depend on cash flows)
twr <- calculate_twr(stock_prices, dividends)

# Remove non-finite values from mwr_results
mwr_results <- mwr_results[is.finite(mwr_results)]

# Print results
cat("\nStock Prices:", stock_prices, "\n")


Stock Prices: 100 89.4 85.38 116.61 118.27

cat("Dividends:", dividends, "\n")

Dividends: 2 2.1 2.205 2.31525 2.431013

cat("\nMoney-Weighted Returns (MWR) Summary:\n")


Money-Weighted Returns (MWR) Summary:

print(summary(mwr_results))

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
0.04825 0.07064 0.09257 0.10793 0.12391 0.93867

cat("\nTime-Weighted Return (TWR):", twr, "\n")


Time-Weighted Return (TWR): 0.06511804

cat("Number of failed IRR calculations:", failed_irr_count, "\n")

Number of failed IRR calculations: 0

# Plot
mwr_df <- data.frame(MWR = mwr_results)
p <- ggplot(mwr_df, aes(x = MWR)) +
    geom_density(fill = "seagreen2", color = "seagreen4", alpha = 0.7) +
    geom_vline(aes(xintercept = twr), color = "purple2", linetype = "dashed", size = 1) +
    theme_minimal() +
    labs(title = "Distribution of Money-Weighted Returns (10,000 trials)",
         subtitle = "Time-weighted return = 6.51%",
         x = "Return",
         y = "Density") +
    theme(axis.text.x = element_text(size = 12, face = "bold"),
          plot.subtitle = element_text(size = 12, color = "purple4", face = "bold")) +
    annotate("text", x = twr, y = 0, label = "TWR", vjust = -1, hjust = -0.3, color = "purple4") +
    scale_x_continuous(labels = scales::percent_format(accuracy = 0.1)) +
    coord_cartesian(xlim = c(0, 0.3))
print(p)