Investment Mathematics Redux

Suppose you bought XYZ shares at £100 three years ago. You sold the shares today for £250 and received £10 in dividends. Calculate the HPR.

HPR = (250 − 100 + 10) / 100

HPR = 160 / 100

HPR = 1.60

HPR = 160%

Calculate the expected return using the following data:

Scenario | Probability | Return
Boom | 30% | 20%
Normal | 50% | 10%
Recession | 20% | -5%

A:

E(r) = 0.30(20) + 0.50(10) + 0.20(-5)

E(r) = 6 + 5 - 1

E(r) = 10%

Calculate the risk premium.

Expected return = 10%

Risk-free rate = 3%

Risk Premium = 10% − 3%

Risk Premium = 7%

Meaning investors require 7% additional expected return for accepting risk.

Calculate the Sharpe Ratio.

Expected Return = 12%

Risk-Free Rate = 3%

Standard Deviation = 15%

Sharpe Ratio = (0.12 − 0.03) / 0.15

Sharpe Ratio = 0.09 / 0.15

Sharpe Ratio = 0.60

Which portfolio is better according to the Sharpe Ratio?

Fund A:
Return = 15%
Risk = 30%

Fund B:
Return = 12%
Risk = 10%

Assuming a risk-free rate of 0%:

Sharpe A = 15/30 = 0.50

Sharpe B = 12/10 = 1.20

Fund B is preferred because it generates more return per unit of risk.

An investment has the following possible returns:

Scenario | Probability | Return
Boom | 0.25 | 20%
Normal | 0.50 | 10%
Recession | 0.25 | -4%

Calculate:

1. Expected return

2. Variance

3. Standard deviation

Expected Return:

E(r) = 0.25(20) + 0.50(10) + 0.25(-4)

E(r) = 9%

Variance:

Var(r) = 0.25(20 − 9)² + 0.50(10 − 9)² + 0.25(-4 − 9)²

Var(r) = 72

Standard Deviation:

σ = √72

σ = 8.49%

A risky portfolio has a standard deviation of 20%.

The weight invested in the risky portfolio is 0.7.

Calculate portfolio risk.

σC = yσP

σC = 0.7 × 20

σC = 14%

What is the equation for the risk of a complete portfolio?

σC = yσP

Where:

σC = Risk of complete portfolio

σP = Risk of risky portfolio

y = Weight invested in risky portfolio

A portfolio has the following attributes:

Risky portfolio return = 10%

Risk-free rate = 2%

70% invested in risky portfolio

30% invested in risk-free asset

Calculate the expected return of the complete portfolio.

E(rC) = 0.7(10) + 0.3(2)

E(rC) = 7 + 0.6

E(rC) = 7.6%

What is the equation for the optimal allocation to risky assets?

y* = (E(rP) − rf) / (Aσ²P)

Where:

y* = Optimal allocation to risky assets

E(rP) = Expected return of risky portfolio

rf = Risk-free rate

A = Risk aversion coefficient

σ²P = Variance of the risky portfolio

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Calculate the optimal allocation to risky assets.

Expected return = 12%

Risk-free rate = 4%

Risk aversion coefficient = 4

Standard deviation = 20%

Variance = (0.20)²

Variance = 0.04

y* = (0.12 − 0.04)/(4 × 0.04)

y* = 0.08/0.16

y* = 0.50

Optimal allocation to risky assets = 50%

Calculate y.

Portfolio value = £300,000

Risky portfolio value = £210,000

A:

y = 210,000 / 300,000

y = 0.70

y = 70%

Calculate the proportion invested in the risk-free asset.

Portfolio value = £300,000

Risky portfolio value = £210,000

y = 0.70

1 − y = 0.30

30% invested in risk-free assets

Calculate the value invested in risk-free assets.

Portfolio value = £300,000

Risky portfolio value = £210,000

Risk-free assets = 300,000 − 210,000

Risk-free assets = £90,000

Calculate the slope of the CAL.

Expected return of risky portfolio = 11%

Risk-free rate = 3%

Standard deviation = 16%

Slope = (11 − 3)/16

Slope = 8/16

Slope = 0.50

This is also the Sharpe Ratio.

An investor is considering investing in a risky portfolio PPP and a risk-free asset.

The risky portfolio has:

• Expected return = 14%

• Standard deviation = 25%

The risk-free rate is:

• 4%

The investor's coefficient of risk aversion is:

• A=5A = 5A=5

The investor has £500,000 available to invest.

Required

(a) Calculate the optimal proportion of wealth that should be invested in the risky portfolio.

(b) Calculate the proportion invested in the risk-free asset.

(c) Calculate the expected return of the complete portfolio.

(d) Calculate the standard deviation of the complete portfolio.

(e) Calculate the monetary amount invested in the risky portfolio and the risk-free asset.

(f) Explain how your answer would change if the investor's risk aversion coefficient increased from 5 to 10.

You will have to GPT this question

Suppose a portfolio invests 50% in a bond fund returning 8% and 50% in an equity fund returning 13%. Calculate expected return.

E(rp) = w1E(r1) + w2E(r2)

E(rp) = 0.5(8%) + 0.5(13%)

E(rp) = 4% + 6.5%

E(rp) = 10.5%

What is the equation for expected return of a two-asset portfolio?

E(rp) = w1E(r1) + w2E(r2)

w1 = weight in asset 1

w2 = weight in asset 2

E(r1) = expected return of asset 1

E(r2) = expected return of asset 2

What is the equation for portfolio variance?

σp² = w1²σ1² + w2²σ2² + 2w1w2σ1σ2ρ12

Where:

σp² = portfolio variance

w1 = weight in asset 1

w2 = weight in asset 2

σ1 = standard deviation of asset 1

σ2 = standard deviation of asset 2

ρ12 = correlation coefficient between assets 1 and 2

Given the following data, calculate portfolio return and risk:

Debt return = 8%

Debt SD = 12%

Equity return = 13%

Equity SD = 20%

Correlation = 0.30

Weights = 50% debt and 50% equity

Expected return:

E(rp) = 0.5(8%) + 0.5(13%)

E(rp) = 10.5%

Portfolio variance:

σp² = w1²σ1² + w2²σ2² + 2w1w2σ1σ2ρ12

σp² = (0.5²)(12²) + (0.5²)(20²) + 2(0.5)(0.5)(12)(20)(0.30)

σp² = 36 + 100 + 36

σp² = 172

Portfolio standard deviation:

σp = √172

σp = 13.11%

Note: depending on rounding, this may appear as approximately 13.1% or 13.23%.

Asset A has a standard deviation of 10%, Asset B has a standard deviation of 20%, the correlation is 0.25, and the weights are 40% in A and 60% in B. Calculate portfolio variance and standard deviation.

σp² = w1²σ1² + w2²σ2² + 2w1w2σ1σ2ρ12

σp² = (0.4²)(10²) + (0.6²)(20²) + 2(0.4)(0.6)(10)(20)(0.25)

σp² = 16 + 144 + 24

σp² = 184

σp = √184

σp = 13.56%

A portfolio has a standard deviation of 15%. Asset A has SD = 10%, Asset B has SD = 20%, and the weights are 50% in each asset. Calculate the correlation coefficient.

σp² = w1²σ1² + w2²σ2² + 2w1w2σ1σ2ρ12

15² = (0.5²)(10²) + (0.5²)(20²) + 2(0.5)(0.5)(10)(20)ρ12

225 = 25 + 100 + 100ρ12

225 = 125 + 100ρ12

100 = 100ρ12

ρ12 = 1

A portfolio has variance of 100. Asset A has SD = 8%, Asset B has SD = 12%, and the weights are 50% in each asset. Calculate the covariance between the two assets.

Using covariance form:

σp² = w1²σ1² + w2²σ2² + 2w1w2Cov12

100 = (0.5²)(8²) + (0.5²)(12²) + 2(0.5)(0.5)Cov12

100 = 16 + 36 + 0.5Cov12

100 = 52 + 0.5Cov12

48 = 0.5Cov12

Cov12 = 96

Asset A has expected return of 6%, Asset B has expected return of 14%. What weight in Asset B is required to achieve a portfolio expected return of 10%?

E(rp) = wAEA + wBEB

Let wB = x, so wA = 1 − x

10 = (1 − x)(6) + x(14)

10 = 6 − 6x + 14x

10 = 6 + 8x

4 = 8x

x = 0.5

Weight in Asset B = 50%

Weight in Asset A = 50%

Asset A has expected return of 5%, Asset B has expected return of 15%. What weight in Asset B is required to achieve a portfolio expected return of 12%?

Let wB = x, so wA = 1 − x

12 = (1 − x)(5) + x(15)

12 = 5 − 5x + 15x

12 = 5 + 10x

7 = 10x

x = 0.7

Weight in Asset B = 70%

Weight in Asset A = 30%

Q: A risky portfolio has:

Expected Return = 14%

Standard Deviation = 20%

Risk-Free Rate = 4%

An investor has a risk aversion coefficient of A = 5 and £400,000 available to invest.

Required:

(a) Calculate the Sharpe Ratio.

(b) Calculate the optimal allocation to the risky portfolio.

(c) Calculate the proportion invested in the risk-free asset.

(d) Calculate the expected return of the complete portfolio.

(e) Calculate the standard deviation of the complete portfolio.

(f) Calculate the monetary amount invested in the risky portfolio and the risk-free asset.

(g) Explain how your answer would change if risk aversion increased from 5 to 10.

(a)

Sharpe Ratio

= (14 − 4) / 20

= 10 / 20

= 0.50

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(b)

y* = (E(rP) − rf) / (Aσ²)

y* = (0.14 − 0.04) / [5(0.20²)]

y* = 0.10 / 0.20

y* = 0.50

50% invested in risky portfolio.

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(c)

1 − y

= 1 − 0.50

= 0.50

50% invested in risk-free asset.

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(d)

E(rC)

= yE(rP) + (1 − y)rf

= 0.50(14%) + 0.50(4%)

= 9%

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(e)

σC = yσP

σC = 0.50(20%)

σC = 10%

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(f)

Risky Portfolio

= 0.50 × £400,000

= £200,000

Risk-Free Asset

= £200,000

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(g)

Increasing risk aversion reduces y*.

The investor would allocate less wealth to the risky portfolio and more wealth to the risk-free asset.

Calculate utility for a portfolio with expected return = 10%, standard deviation = 20%, and risk aversion coefficient A = 4.

U = E(r) − ½Aσ²

U = 0.10 − 0.5(4)(0.20²)

U = 0.10 − 2(0.04)

U = 0.10 − 0.08

U = 0.02

Utility = 2%

An investor has A = 4. Which portfolio gives higher utility?

Portfolio A: Expected return = 8%, SD = 10%
Portfolio B: Expected return = 12%, SD = 25%

Portfolio A:

U = 0.08 − 0.5(4)(0.10²)

U = 0.08 − 0.02

U = 0.06

Portfolio B:

U = 0.12 − 0.5(4)(0.25²)

U = 0.12 − 0.125

U = -0.005

Portfolio A is preferred because it has higher utility.

Portfolio A has expected return 10% and SD 15%. Portfolio B has expected return 8% and SD 15%. Which portfolio dominates?

Portfolio A dominates Portfolio B because it has the same level of risk but a higher expected return.

What is the equation for the Market Risk Premium?

Market Risk Premium = E(rM) − rf

Where:

E(rM) = Expected return on the market portfolio

rf = Risk-free rate

Calculate the Market Risk Premium.

Expected Market Return = 10%

Risk-Free Rate = 3%

Market Risk Premium

= 10% − 3%

= 7%

What is the equation for calculating the proportion allocated to the optimal portfolio M?

y = (E(rM) − rf) / (AσM²)

Where:

y = proportion invested in the market portfolio

A = risk aversion coefficient

σM² = variance of the market portfolio

What is the equation for Portfolio Beta?

βP = Σ wiβi

Portfolio beta is the weighted average of the constituent asset betas.

Calculate Portfolio Beta.

Asset | Weight | Beta

A | 60% | 1.2

B | 40% | 0.8

βP

= 0.60(1.2) + 0.40(0.8)

= 1.04

What determines expected return under CAPM?

Expected return is determined only by Beta.

Not:

• Variance

• Standard deviation

• Total risk

What is the CAPM equation?

E(ri) = rf + βi[E(rM) − rf]

Calculate expected return using CAPM.

Risk-Free Rate = 3%

Expected Market Return = 11%

Beta = 1.5

E(ri)

= 3 + 1.5(11 − 3)

= 3 + 12

= 15%

A stock has an expected return of 14%.

Risk-Free Rate = 4%.

Expected Market Return = 9%.

Calculate Beta.

14 = 4 + β(9 − 4)

14 = 4 + 5β

10 = 5β

β = 2

Suppose the Market Risk Premium is 8%.

Toyota Beta = 1.10

Ford Beta = 1.25

A portfolio invests:

25% in Toyota

75% in Ford

Calculate the portfolio risk premium.

Portfolio Beta

= 0.25(1.10) + 0.75(1.25)

= 1.2125

Portfolio Risk Premium

= 1.2125 × 8%

= 9.7%

What is the equation for Alpha?

α = Actual Return − CAPM Return

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Calculate Alpha.

Actual Return = 16%

CAPM Return = 13%

α = 16% − 13%

α = +3%

Positive alpha indicates undervaluation.

A stock has Beta = 1.4.

The market risk premium is 6%.

Calculate the stock's risk premium.

Risk Premium

= Beta × Market Risk Premium

= 1.4 × 6%

= 8.4%