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 ($\rho =$ 0.5) on the envelope of feasable portfolios.

What happens as you decrease the correlation between the returns of $X$ and $Y$?

Are all feasable 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})

𝜌

rho = 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"
}]

seq = ƒ(start, end, step)

portfolios = Array(281) [Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, …]

asset_x = Array(1) [Object]

asset_y = Array(1) [Object]

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"}),
    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}),
  ]
})

Investment opportunity set

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"}),
  ]
})

Portfolio expected return as a function of the weight of X in the portfolio

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"}),
  ]
})

Portfolio standard deviation as a function of the weight of X in the portfolio