Risky asset + risk free asset

Consider a simple economy with only two assets: a risk-free asset ( $r_{f}$ ) and a risky asset ( $r_{p}$ ). You can invest in a portfolio ( $c$ ) that is comprised of the risk-free asset, or the risky asset, or a combination of the two assets. The proportion you choose to invest in the risky asset is here represented by $y$ . $A$ is the investor risk aversion coefficient, and we assume that the expected utility function of the investor is given by:

$E [U (r_{c})] = E [r_{c}] - \frac{A}{2} σ_{r_{c}}^{2}$

The blue line in the plot below represents all possible investment opportunities, and is designated as the capital alocation line (CAL).

The orange dot in the plot represents the portfolio that results by combining the risk-free and the risky asset together, where the proportion invested in the risk-free asset is $(1 - y) =$ 0.50, and the proportion invested in the risky asset is $y =$ 0.50.

Assume the following information about the assets:

$r_{f} =$ 0.02

$E [r_{p}] =$ 0.1

$σ_{r_{p}} =$ 0.22

Use the slider below to see how changing the proportion you invest in the risky asset change the risk and return of portfolio $c$ .

viewof y = Inputs.range([0, 1.7], {step: .01, label: "y", value:0.5})
viewof A = Inputs.range([1, 4], {step: .1, label: "A", value:4.0})

y = 0.5

A = 4

e_r_p = 0.1;
sd_r_p = 0.22;
r_f = 0.02;
  
// Calculate portfolio values based on y and A
function calculateValues(y, A) {
  // Expected return and variance of portfolio c
  let e_r_c = y * e_r_p + (1 - y) * r_f;
  let var_r_c = Math.pow(y, 2) * Math.pow(sd_r_p, 2);
  
  // Utility
  let u = e_r_c - (A/2) * var_r_c;
  
  // Generate indifference curve data
  const ic = Array.from({length: 40}, (_, i) => {
    let sd = i * 0.01;
    return {
      sd: sd,
      y_ic: u + (A/2) * Math.pow(sd, 2)
    };
  });
  
  // Label for indifference curve
  let label_ic = {
    sd: ic[ic.length-15].sd,
    y_ic: ic[ic.length-15].y_ic,
    label: `A: ${A}; E[U(r)] = ${u.toFixed(5)}`
  };
  
  // Generate CAL data
  let cal = Array.from({length: 200}, (_, i) => {
    let y_val = i * 0.01;
    return {
      y: y_val,
      e_r_c: y_val * e_r_p + (1 - y_val) * r_f,
      sd_r_c: y_val * sd_r_p
    };
  });
  
  // Point p (risky asset)
  let p = {
    e_r: e_r_p,
    sd: sd_r_p,
    label: "p"
  };
  
  // Point c (portfolio)
  let c = {
    e_r: e_r_c,
    sd: Math.sqrt(var_r_c),
    label: "c"
  };
  
  return {
    ic: ic,
    label_ic: label_ic,
    cal: cal,
    p: p,
    c: c
  };
}

// Create reactive values that update when y or A changes
results = calculateValues(y, A)

ic = results.ic;
label_ic = [results.label_ic];
cal = results.cal;
p = [results.p];
c = [results.c];

e_r_p = 0.1

sd_r_p = 0.22

r_f = 0.02

calculateValues = ƒ(y, A)

results = Object {ic: Array(40), label_ic: Object, cal: Array(200), p: Object, c: Object}

ic = Array(40) [Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, …]

label_ic = Array(1) [Object]

cal = Array(200) [Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, …]

p = Array(1) [Object]

c = Array(1) [Object]

Plot.plot({
  caption: `Investment opportunity set`,
  x: {label: "Standard deviation", domain:[0, .40]},
  y: {label:"Expected return", domain:[0, 0.16]},
  marks: [
    Plot.ruleY([0]),
    Plot.lineY(cal, {x: "sd_r_c", y: "e_r_c", stroke:"blue"}),
    Plot.dot(p, {x: "sd", y: "e_r", r:5, fill: "red"}),
    Plot.text(p, {x: "sd", y: "e_r", text: "label", dy: -7, lineAnchor: "bottom", fontSize: 12}),
    Plot.dot(c, {x: "sd", y: "e_r", r:5, fill: "orange"}),
    Plot.text(c, {x: "sd", y: "e_r", text: "label", dy: -7, lineAnchor: "bottom", fontSize: 12}),
    Plot.lineY(ic, {x: "sd", y: "y_ic", stroke:"black"}),
    Plot.text(label_ic, {x: "sd", y: "y_ic", text: "label", dx: -22, dy: -7, lineAnchor: "bottom", fontSize: 12}),
  ]
})

Investment opportunity set

This investor maximizes its utility by investing $y = \frac{E [r_{c}] - r_{f}}{A σ_{r_{p}}^{2}}$ = 0.41

We are here assuming that investors can borrow at the risk-free asset to invest in the risky asset. How do we know that? What changes when that is not the case?