Risk and number of stocks

A variation on a very typical plot, this one shows the relationship between the number of stocks in a portfolio and the portfolio’s volatility. The twist is that it also shows the “elbow point” for each level of correlation. The elbow point is the number of stocks at which the marginal benefit of adding more stocks to the portfolio diminishes. In other words, it’s the point at which the portfolio’s volatility stops decreasing as quickly as it did with fewer stocks.

The method I used to identify the elbow point is the so-called Kneedle algorithm, which is a simple way to find the “knee” or “elbow” in a curve based on perpendicular distance. The algorithm works by fitting a line between the first and last points of the curve and then finding the point farthest from that line. This point is the elbow point.

library(ggplot2)
library(dplyr)
library(tidyr)

# Function to calculate portfolio volatility
calc_portfolio_vol <- function(n_assets, correlation, sigma = 1) {
  sqrt((1/n_assets) + ((n_assets-1)/n_assets) * correlation) * sigma
}

# Function to find elbow point using Kneedle algorithm
find_elbow <- function(x, y) {
  # Normalize the data
  x_norm <- (x - min(x)) / (max(x) - min(x))
  y_norm <- (y - min(y)) / (max(y) - min(y))
  
  # Calculate the line between first and last point
  slope <- (y_norm[length(y_norm)] - y_norm[1]) / (x_norm[length(x_norm)] - x_norm[1])
  intercept <- y_norm[1] - slope * x_norm[1]
  
  # Calculate distance from points to line
  distances <- abs(slope * x_norm + intercept - y_norm) / sqrt(slope^2 + 1)
  
  # Return original x value at maximum distance
  return(x[which.max(distances)])
}

# Create plot with elbow points
portfolio_vol_elbow_plot <- function() {
  # Parameters
  n_assets_seq <- 1:50
  correlations <- seq(0, 0.95, 0.05)
  sigma <- 1
  
  # Create data frame
  plot_data <- expand.grid(
    n_assets = n_assets_seq,
    correlation = correlations
  ) %>%
    mutate(
      volatility = mapply(calc_portfolio_vol, n_assets, correlation),
      correlation = factor(correlation, levels = sort(unique(correlation), decreasing = TRUE))
    )
  
  # Find elbow points for each correlation
  elbow_points <- plot_data %>%
    group_by(correlation) %>%
    summarize(
      elbow_n = find_elbow(n_assets, volatility),
      elbow_vol = calc_portfolio_vol(elbow_n, first(as.numeric(as.character(correlation))))
    )
  
  # Create plot
  ggplot(plot_data, aes(x = n_assets, y = volatility, color = correlation)) +
    geom_line(linewidth = 0.8) +
    geom_point(data = elbow_points, 
               aes(x = elbow_n, y = elbow_vol),
               size = 3, shape = 16) +
    scale_color_viridis_d(
      name = "Correlation",
      option = "magma",
      begin = 0.2,
      end = 0.8,
      direction = -1,
      guide = guide_legend(
        keyheight = 0.5,    # Reduce key size
        keywidth = 1.5,     # Make keys wider than tall
        label.theme = element_text(size = 8),  # Smaller text
        title.theme = element_text(size = 9),  # Smaller title
        ncol = 1,           # Force single column
        spacing = 0.1,      # Reduce spacing between keys
        override.aes = list(linewidth = 1)  # Thinner lines in legend
      )
    ) +
    labs(
      title = "Portfolio Volatility vs Number of Assets",
      subtitle = "With Detected Elbow Points",
      x = "Number of Assets",
      y = "Portfolio Volatility"
    ) +
    theme_minimal() +
    theme(
      plot.title = element_text(size = 14, face = "bold"),
      plot.subtitle = element_text(size = 12),
      legend.position = "right",
      legend.margin = margin(1, 1, 1, 1),  # Reduce legend margin
      legend.box.spacing = unit(0.5, "lines"),  # Reduce spacing around legend box
      panel.grid.minor = element_line(color = "gray90"),
      panel.grid.major = element_line(color = "gray85")
    ) +
    scale_x_continuous(breaks = seq(0, 50, 10)) +
    scale_y_continuous(breaks = seq(0, 1, 0.1))
}

# Execute the plotting function
portfolio_vol_elbow_plot()