Capital Allocation and Leverage

EXERCISE 01

You have €10,000, borrow another €10,000 at the risk-free rate of 4%, and invest €20,000 in a risky portfolio with expected return 20% and standard deviation 16%.

A.Find your portfolio return and volatility relative to your equity.

A.

E[R_C]=w_PE[R_P]+(1-w_P)r_f,\quad\sigma_C=|w_P|\sigma_P

w_P=20{,}000/10{,}000=2,\quad w_f=-1

E[R_C]=2(0.20)-0.04=0.36

\sigma_C=2(0.16)

Expected return 36%; volatility 32%.

Capital Allocation and Leverage

EXERCISE 02

You own €100, borrow €50 at the risk-free rate and invest €150 in the market portfolio.

A.Calculate the portfolio beta.

A.

\beta_P=\sum_iw_i\beta_i

\beta_P=\frac{150}{100}(1)-\frac{50}{100}(0)

1.50.

Capital Allocation and Leverage

EXERCISE 03

The stock has beta 1.2 and expected return 16%. The risk-free asset returns 5%.

A.Find the return of a 50/50 portfolio.
B.Find weights for portfolio beta 0.75.
C.Find portfolio beta for a target return of 8%.

A.

E[R_P]=0.5(0.16)+0.5(0.05)=10.5\%

Expected return = 10.5%.

B.

w=0.75/1.2=0.625

62.5% in the stock; 37.5% risk-free.

C.

w=\frac{0.08-0.05}{0.16-0.05}=0.272727

\beta_P=\frac{0.08-0.05}{0.16-0.05}(1.2)\approx0.32727

Portfolio beta = 0.32727.

Capital Allocation and Leverage

EXERCISE 04

A €1,000 portfolio contains €400 of A (beta 1.3), an unknown amount of B (beta 0.7), and the balance in a risk-free asset. Target beta is 0.90.

A.Find the investment in B and the risk-free balance.

A.

\beta_P=w_A\beta_A+w_B\beta_B

0.90=0.40(1.3)+\frac{B}{1000}(0.7)

B=1000\frac{0.90-0.40(1.3)}{0.7}

B = €542.86; risk-free balance = €57.14.

Capital Allocation and Leverage

EXERCISE 05

Risk-free rate = 2.500%; market risk premium = 5.000%.

Asset	Weight	Beta
1	0.31818	0.7
2	0.18182	0.85
3	0.16364	1.1
4	0.33636	1.4

A.Calculate portfolio beta and expected return.
B.Mix this portfolio with the risk-free asset to earn 9%.

A.

\beta_P=\sum_iw_i\beta_i,\quad E[R_P]=r_f+\beta_PMRP

\beta_P=(0.318182)(0.7)+(0.181818)(0.85)+(0.163636)(1.1)+(0.336364)(1.4)\approx 1.028182

E[R_P]=0.025+(1.028182)(0.05)\approx 0.076409

Beta = 1.02818; expected return = 7.641%.

B.

w_P=\frac{r_{target}-r_f}{E[R_P]-r_f}

w_P=\frac{0.09-0.025}{0.076409-0.025}\approx 1.264368

w_f=1-w_P

Portfolio weight = 126.437%; risk-free weight = -26.437%.

The negative risk-free position means borrowing.

Capital Allocation and Leverage

EXERCISE 06

Assume CAPM. The risk-free return is 4.00%, and market volatility is 8.00%. Construct a portfolio C of the market and risk-free asset targeting 20.00% expected return. B’s printed beta 1.38 is rounded: at a 12% market return its exact CAPM beta for a 15% expected return is 1.375.

Security	Expected return	Volatility	Beta
A	10.00%	10.00%	0.750
B	15.00%	20.00%	1.380

A.Infer the market return and assess B’s pricing.
B.Find each security’s covariance with the market.
C.Find C’s weights and volatility.

A.

E[R_M]=R_f+\frac{E[R_A]-R_f}{\beta_A}

E[R_M]=0.04+\frac{0.1-0.04}{0.75}=12.0000\%

R_B^*=0.04+1.38(0.12-0.04)=15.0400\%

Compare B’s forecast return with this required return: a lower forecast means overpricing; equality means fair pricing.

B.

Cov(R_i,R_M)=\beta_i\sigma_M^2

Cov(A,M)=0.75(0.08)^2=0.004800

Cov(B,M)=1.38(0.08)^2=0.008832

C.

w_M=\frac{E[R_C]-R_f}{E[R_M]-R_f},\quad w_f=1-w_M

w_M=\frac{0.2-0.04}{0.12-0.04}=2.0000

w_f=-1,\quad \sigma_C=|2|(0.08)=16.0000\%

A negative risk-free weight means borrowing. C strictly dominates B in mean–variance terms because its expected return is higher and its volatility lower.

Capital Allocation and Leverage

EXERCISE 07

Assume CAPM. The risk-free return is 3.00%, and market volatility is 8.00%. Construct a portfolio C of the market and risk-free asset targeting 21.00% expected return.

Security	Expected return	Volatility	Beta
A	12.00%	8.00%	0.800
B	18.00%	16.00%	1.600

A.Infer the market return and assess B’s pricing.
B.Find each security’s covariance with the market.
C.Find C’s weights and volatility.

A.

E[R_M]=R_f+\frac{E[R_A]-R_f}{\beta_A}

E[R_M]=0.03+\frac{0.12-0.03}{0.8}=14.2500\%

R_B^*=0.03+1.6(0.1425-0.03)=21.0000\%

Compare B’s forecast return with this required return: a lower forecast means overpricing; equality means fair pricing.

B.

Cov(R_i,R_M)=\beta_i\sigma_M^2

Cov(A,M)=0.8(0.08)^2=0.005120

Cov(B,M)=1.6(0.08)^2=0.010240

C.

w_M=\frac{E[R_C]-R_f}{E[R_M]-R_f},\quad w_f=1-w_M

w_M=\frac{0.21-0.03}{0.1425-0.03}=1.6000

w_f=-0.6,\quad \sigma_C=|1.6|(0.08)=12.8000\%

A negative risk-free weight means borrowing. C strictly dominates B in mean–variance terms because its expected return is higher and its volatility lower.

Capital Allocation and Leverage

EXERCISE 08

The market return is 15%, volatility 20%, and risk-free return 5%. Complete the missing entries and assess pricing. Then form a portfolio of 55% Scherzer, 25% market and 20% risk-free.

Security	Forecast return	Volatility	Market correlation	Beta
Scherzer	13%	38%	?	0.90
Kershaw	16%	?	0.40	1.10
Price	25%	65%	0.35	?

A.Complete the risk data, including market and risk-free entries.
B.Assess each share under CAPM.
C.Calculate the portfolio return and volatility.

A.

\beta_i=\rho_{iM}\frac{\sigma_i}{\sigma_M}

\rho_{SM}=\frac{0.9(0.20)}{0.38}=0.473684

\sigma_K=\frac{1.1(0.20)}{0.40}=55\%,\quad\beta_P=\frac{0.35(0.65)}{0.20}=1.1375

Market beta and self-correlation equal 1. Risk-free volatility and beta equal zero. Its correlation is mathematically undefined because its variance is zero; its covariance is zero.

B.

R_i^*=5\%+\beta_i(10\%)

Security	Required return	Alpha	Assessment
Scherzer	14%	−1.00 pp	Overpriced
Kershaw	16%	0.00 pp	Fair
Price	16.375%	+8.625 pp	Underpriced

In the theoretical CAPM exercise, prefer positive-alpha exposure and avoid negative-alpha exposure, subject to portfolio constraints.

C.

E[R_P]=0.55(0.13)+0.25(0.15)+0.20(0.05)=11.9\%

\sigma_P=\sqrt{0.55^2(0.38)^2+0.25^2(0.20)^2+2(0.55)(0.25)(0.9)(0.20)^2}

23.6814%

Capital Allocation and Leverage

EXERCISE 09

Risk-free return is 2%. Market return is 8.5% and volatility 16%. You have 100,000 of equity and borrow 20,000 at the risk-free rate. Compare investing the full 120,000 in the market (portfolio A) or in Alfa (portfolio B).

Security	Forecast return	Volatility	Market correlation	Beta
Alfa	8.5%	20%	?	1
Gamma	4.4%	12.8%	?	0.4
Lambda	9.8%	20%	0.96	?

A.Complete the table and assess pricing.
B.Calculate both leveraged portfolios and select one.

A.

\rho_{AM}=\frac{1(0.16)}{0.20}=0.80,\quad\rho_{GM}=\frac{0.4(0.16)}{0.128}=0.50

\beta_L=0.96\frac{0.20}{0.16}=1.20

R_i^*=2\%+\beta_i(8.5\%-2\%)

Alfa requires 8.5% and Lambda 9.8%: both are fairly priced. Gamma requires 4.6% but forecasts 4.4%: it lies below the SML and is overpriced. Market beta and correlation are 1; risk-free beta and volatility are zero, and risk-free correlation is undefined.

B.

w_{risky}=120000/100000=1.20,\quad w_f=-0.20

E[R_A]=E[R_B]=1.20(8.5\%)-0.20(2\%)=9.8\%

\sigma_A=1.20(16\%)=19.2\%,\quad\sigma_B=1.20(20\%)=24\%

Prefer A in mean–variance terms: same expected return, lower volatility.