Question

a. Portfolio Man wants to create a portfolio as risky as the market and he has $1,000,000 to invest. Given this information, fill in the three missing values of the following table: Asset Investment Beta $170,000 1.6 $140,000 1.5 $130,000 D $200,000 Risk-free asset b. An asset's reward-to-risk ratio is defined as its risk premium divided by its standard deviation. It is a useful statistic to summarize the asset's risk-return trade-off. Consider the following information: Stock A has a reward-to-risk ratio of 0.4 and stock B has a reward-to-risk ratio of 0.33. Stock A's risk premium is 8%, stock B's risk premium is 10% and the market risk premium is 7%. The correlation between stocks A and B is 0.6. Assume the CAPM holds. A portfolio consisting of stocks A and B has 20% more systematic risk than the market. Calculate the total risk of this portfolio c. Julia invests 20% of her wealth in the risk free asset and the rest in FB shares and BHC shares. The risk premium on FB shares is 10%, the risk premium on BHC is 7% and the market risk premium is 8%. Julia's total portfolio of the three assets is 10% less risky than the market portfolio. Calculate the weight of FB in her portfolio. Assume that the Capital Asset Pricing Model holds.

Answer 1

Answer - a

Total amount available for investment is $1,000,000, hence the Risk-free asset investment amount will be the balance amount after deducting the investments in Asset A, B, C & D

Hence,

Risk-free asset investment = Total investment - Investment in Asset A, B, C & D

Risk-free asset investment = $1,000,000 - ($170,000 + $140,000 + $130,000 + $200,000)

Risk-free asset investment = $1,000,000 - $640,000

Risk-free asset investment = $360,000

Now, let us calculate the weights of the investments

Statement showing weights of investment

Asset	Investment ($)	Weights (w)(Investment / $1,000,000)
A	170,000	0.17
B	140,000	0.14
C	130,000	0.13
D	200,000	0.20
Risk-free asset	360,000	0.36
	1,000,000	1

Now, the PortfolioMan wants to create a portfolio as risky as the market, and the maket beta is equal to 1, hence the portfolio beta shall also be equal to 1. The Risk-free asset beta is 0 since there is no risk involved in such investment as its name is risk-free.

Let us assume the beta of asset D is x, and

Portfolio beta = BetaA * WA + BetaB * WB + BetaC * WC + BetaD * WD + BetaRF * WRF

1 = 1.6 * 0.17 + 1.5 * 0.14 + 1.1 * 0.13 + x * 0.20 + 0 * 0.36

1 = 0.272 + 0.21 + 0.143 + 0.20x + 0

1 = 0.625 + 0.20x

0.20x = 1 - 0.625

0.20x = 0.375

x = 1.875

Hence, beta of asset D is 1.875

Answer - b

Following information is given in the question -

1. Reward-to-risk ratio of stock A is 0.4 and of stock B is 0.33
2. Risk premium of stock A is 8% and of stock B is 10%
3. Market premium is 7%
4. Correlation (r) between stock A and B is 0.6
5. Portfolio systematic risk (Portfolio beta) is 20% more than that of market

We have to calculate the the total risk of portfolio consisting of stock A and B, total risk is calculated using portfolio standard deviation ($\sigma$) since it comprises of both systematic and unsystematic risk, while beta comprises of only systematic risk, hence we have to ignore the calculation of beta.

Reward-to-risk ratio = Risk premium (RP) / Standard deviation ($\sigma$)

where, reward-to-risk ratio and risk premium (RP) is given above, hence standard deviation ($\sigma$) can be calculated

Reward-to-risk ratioA = RPA / $\sigma$ A

0.4 = 8 / $\sigma$ A

$\sigma$A = 20

Reward-to-risk ratioB = RPB / $\sigma$ B

0.33 = 10 / $\sigma$ B

$\sigma$B = 30.30

For calculating portfolio standard deviation ($\sigma$PF) we have to first calculate weights (W) of investments in the portfolio containing of stock A and B which is unavailable, and can be calculated using the below mentioned formula-

WA = [($\sigma$B)2 - CovA&B] / [($\sigma$A)2 + ($\sigma$B)2 - 2CovA&B]

Where, $\sigma$ A = 20, $\sigma$ B = 30.30 and CovA&B is unavailable and can be calculated using below mentioned formula-

CovA&B = r * $\sigma$ A * $\sigma$ B

CovA&B = 0.6 * 20 * 30.30

CovA&B = 363.60

On putting these figures in the above formula, we get -

WA = [($\sigma$B)2 - CovA&B] / [($\sigma$A)2 + ($\sigma$B)2 - 2CovA&B]

WA = [(30.30)2 - 363.60] / [(20)2 + (30.30)2 - 2 * 363.60]

WA = [918.09 - 363.60] / [400 + 918.09 - 727.20]

WA = 554.49 / 590.89

WA = 0.94 (approx)

Hence, WB = 1 - WA

WB = 1 - 0.94

WB = 0.06

Now we have weights of investment in the portfolio, let us calculate the portfolio standard deviation ($\sigma$PF) using the below mentioned formula -

($\sigma$PF)2 = ($\sigma$A)2 * (WA)2 + ($\sigma$B)2 * (WB)2 + 2 * $\sigma$ A * $\sigma$ B * WA * WB * r

($\sigma$PF)2 = (20)2 * (0.94)2 + (30.3)2 * (0.06)2 + 2 * 20 * 30.3 * 0.94 * 0.06 * 0.6

($\sigma$PF)2 = 400 * 0.8836 + 918.09 * 0.0036 + 41.0141

($\sigma$PF)2 = 353.44 + 3.3051 + 41.0141

($\sigma$PF)2 = 397.7592

$\sigma$PF =

V397.7592

$\sigma$PF = 19.94

Hence, the total risk of portfolio consisting of stock A and B is 19.94%

Answer - c

Following information is given in the question -

1. Risk premium on FB shares is 10% and on BHC is 7%
2. Market risk premium is 8%
3. Portfolio risk (Portfolio beta) is 10% less risky than that of market
4. Weight of investment in risk-free asset is 20%

Market risk premium is 8% and market beta is always 1,

If CAPMholds true, then

Risk premium = Beta (Market return - Risk-free return)

8 = 1 (Market return - Risk-free return)

(Market return - Risk-free return) = 8

Now, apply the above formula in FB & BHC shares for calculating their beta

Risk premiumFB = BetaFB (Market return - Risk-free return)

10 = BetaFB * 8

BetaFB = 1.25

Risk premiumBHC = BetaBHC (Market return - Risk-free return)

7 = BetaBHC * 8

BetaBHC = 0.875

Now, let us assume the Weight of FB in portfolio be x, therefore Weight of BHC will be (1 - 0.20 - x)

Portfolio beta = 10% less risky than that of market

Portfolio beta = 1 - (10% of 1)

Portfolio beta = 0.9

Now,

Portfolio beta = BetaFB * Weight of FB + BetaBHC * Weight of BHC + BetaRisk-free * Weight of Risk-free

0.9 = 1.25 * x + 0.875 * ((1 - 0.20 - x) + 0 * 0.20

0.9 = 1.25 x + 0.7 - 0.875 x

0.9 - 0.7 = 0.375 x

x = 0.2 / 0.375

x = 0.5333

Hence the Weight of FB in portfolio of Juliaa will be 53.33%