Efficient frontier

Assume a market with only two risky assets (\(X\) and \(Y\)). Their expected returns and standard deviation are given below. The correlation between the returns of these assets is \(\rho_{x,y} = 0.5\)

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

Use the slider to identify the portfolio created combining the two assets, by changing the weight of asset \(X\) (\(w_x\)) in the portfolio (\(p\)).

What happens when you change the weight of asset \(X\) in the portfolio?

Can we create a portfolio with a negative asset weight? How?

\(w_x + w_y = 1\)

\(w_y = 1 - w_x =\)

\(E[r_p] = w_x E[r_x] + (1 - w_x) E[r_y]\) =

\(\sigma_p = \sqrt{w_x^2 \sigma^2_x + w_y^2 \sigma^2_x + 2 w_x w_y cov(x,y)}\) =

viewof w_x = Inputs.range([-1.4, 1.4], {step: .01, label: "w_x", value:0.5})

er_x = 0.12
er_y = 0.17
sd_x = 0.2
sd_y = 0.25
rho = 0.5
cov = rho * sd_x * sd_y

// Create portfolios data
portfolios = {
  const data = [];
  for (let w_x = -1.4; w_x <= 1.4; w_x += 0.01) {
    const w_y = 1 - w_x;
    const er_p = w_x * er_x + w_y * er_y;
    const sd_p = Math.sqrt(w_x**2 * sd_x**2 + w_y**2 * sd_y**2 + 2 * w_x * w_y * cov);
    data.push({w_x, w_y, er_p, sd_p});
  }
  return data;
}

// Calculate minimum variance portfolio weight
w_x_min_var_port = (sd_y**2 - cov)/(sd_x**2 + sd_y**2 - 2*cov)

// Create efficient portfolios data
efficient_portfolios = portfolios.filter(d => d.w_x <= w_x_min_var_port)

// Create asset data points
asset_x = [{er: er_x, sd: sd_x, label: "X"}]
asset_y = [{er: er_y, sd: sd_y, label: "Y"}]

// Create portfolio P data based on selected w_x
portfolio_p = {
  const w_y = 1 - w_x;
  const er_p = w_x * er_x + w_y * er_y;
  const sd_p = Math.sqrt(w_x**2 * sd_x**2 + w_y**2 * sd_y**2 + 2 * w_x * w_y * cov);
  return [{w_x, w_y, er_p, sd_p, label: "P"}];
}

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:"gray"}),
    Plot.lineY(efficient_portfolios, {x: "sd_p", y: "er_p", stroke:"blue", strokeWidth: 3}),
    Plot.dot(portfolio_p, {x: "sd_p", y: "er_p", r:5, fill: "green"}),
    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(portfolio_p, {x: "sd_p", y: "er_p", text: "label", dy: -7, lineAnchor: "bottom", fontSize: 12}),
    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.dot(portfolio_p, {x: "w_x", y: "er_p", r:5, fill: "green"}),
    Plot.text(portfolio_p, {x: "w_x", y: "er_p", text: "label", dy: -7, lineAnchor: "bottom", fontSize: 12}),
  ]
})

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"}),
    Plot.dot(portfolio_p, {x: "w_x", y: "sd_p", r:5, fill: "green"}),
    Plot.text(portfolio_p, {x: "w_x", y: "sd_p", text: "label", dy: -7, lineAnchor: "bottom", fontSize: 12}),
  ]
})