Correlation effect

Assume a market with only two risky assets (\(X\) and \(Y\)). Their expected returns and standard deviation are given below.

Use the slider to see the effect of changing the correlation between those asset returns (\(\rho =\) ) on the envelope of feasible portfolios.

What happens as you decrease the correlation between the returns of \(X\) and \(Y\)?

Are all feasible portfolios always efficient portfolios?

Assume that:

Asset	Expected Return (\(E[r]\))	Standard Deviation (\(\sigma\))
X	0.12	0.2
Y	0.17	0.25

viewof rho = Inputs.range([-1, 1], {step: .01, label: "𝜌", value:0.5})

function seq(start, end, step) {
  const length = Math.floor((end - start) / step) + 1;
  return Array(length).fill().map((_, i) => start + (i * step));
}

// Calculate portfolios data
portfolios = {
  const er_x = 0.12;
  const er_y = 0.17;
  const sd_x = 0.2;
  const sd_y = 0.25;
  const cov = rho * sd_x * sd_y;
  
  const w_x = seq(-1.4, 1.4, 0.01);
  
  return w_x.map(wx => {
    const wy = 1 - wx;
    return {
      w_x: wx,
      w_y: wy,
      er_p: wx * er_x + wy * er_y,
      sd_p: Math.sqrt(
        Math.pow(wx,2) * Math.pow(sd_x,2) + 
        Math.pow(wy,2) * Math.pow(sd_y,2) + 
        2 * wx * wy * cov
      )
    };
  });
}

// Individual assets data
asset_x = [{
  er: 0.12,
  sd: 0.2,
  label: "X"
}]

asset_y = [{
  er: 0.17,
  sd: 0.25,
  label: "Y"
}]

Plot.plot({
  caption: `Investment opportunity set`,
  x: {label: "Standard deviation", zero: true, domain: [0,0.6]},
  y: {label:"Expected return", domain: [0, 0.3]},
  marks: [
    Plot.ruleY([0]),
    Plot.lineY(portfolios, {x: "sd_p", y: "er_p", stroke:"blue", clip: true}),
    Plot.dot(asset_x, {x: "sd", y: "er", r:5, fill: "red"}),
    Plot.dot(asset_y, {x: "sd", y: "er", r:5, fill: "brown"}),
    Plot.text(asset_x, {x: "sd", y: "er", text: "label", dy: -7, lineAnchor: "bottom", fontSize: 12}),
    Plot.text(asset_y, {x: "sd", y: "er", text: "label", dy: -7, lineAnchor: "bottom", fontSize: 12}),
  ]
})

Plot.plot({
  caption: `Portfolio expected return as a function of the weight of X in the portfolio`,
  x: {label: "Weight of asset X in the portfolio", zero: true},
  y: {label:"Portfolio expected return", domain: [0, 0.3]},
  marks: [
    Plot.ruleY([0]),
    Plot.lineY(portfolios, {x: "w_x", y: "er_p", stroke:"orange"}),
  ]
})

Plot.plot({
  caption: `Portfolio standard deviation as a function of the weight of X in the portfolio`,
  x: {label: "Weight of asset X in the portfolio", zero: true},
  y: {label:"Portfolio standard deviation", domain: [0,0.6]},
  marks: [
    Plot.ruleY([0]),
    Plot.lineY(portfolios, {x: "w_x", y: "sd_p", stroke:"green", clip: true}),
  ]
})