Diversification limits

Assume that you are in a market where all assets have the same characteristics: each individual asset’s variance is equal to 50 and the covariance to any other asset is equal to 10. The following graph depicts the variance of a portfolio (p) with an increasing number of these assets, where the weight of all assets in a portfolio is equal (1/N).

Use the slider to change the number of assets in the portfolios. What do you observe?

viewof N = Inputs.range([1, 300], {step: 1, label: "N", value:1})
viewof limit = Inputs.toggle({label: "Add limit", values: [10, 0], value: 0})

portfolios = {
  const var_i = 50;  // individual asset variance
  const cov = 10;    // covariance between assets
  
  return Array.from({length: N}, (_, i) => {
    const n = i + 1;
    return {
      n: n,
      var_p: (1/n) * var_i + ((n-1)/n) * cov
    };
  });
}

\[\begin{align} \sigma^2 &= 50 \\ cov &= 10 \\ \end{align}\]

\(\sigma^2_p = \frac{\sigma^2}{N} + \frac{N-1}{N}\times cov =\)

\[ \lim_{N \to \infty} \sigma^2_p = cov = 10 \]

Plot.plot({
  caption: `Variance of a portfolio with an increasig number of assets`,
  x: {label: "N", zero: true, domain: [0, 300]},
  y: {label:"Variance"},
  marks: [
    Plot.ruleY([0]),
    Plot.ruleY([limit]),
    Plot.dot(portfolios, {x: "n", y: "var_p", r:5, fill:"orange"})
  ]
})