Question

Homework 7

VOLATILITY

1. Find the mean and standard deviation of returns for stock ABC.

Year

% Return

(using logs)

2003

3.45

2004

7.81

2005

8.34

2006

6.21

2007

-8.81

2008

-5.30

2009

2.01

2. Find the expected return and standard deviation of a portfolio which holds $300 in asset A and $700 in asset B. There is a correlation of -0.6 between asset A and B.

Asset

Expected

Return

Expected

Standard Deviation

A

8.3%

12.1%

B

4.6%

7.3%

3. Find the expected return and standard deviation of a portfolio which holds $2,000 in General Electric and $3,500 in Walmart. There is a correlation of -0.4 between asset General Electric and Walmart. For General Electric the expected return is 2.5% and the standard deviation is 4.1%. For Walmart the expected return is 1.4% and the standard deviation is 1.3%.

4. Calculate the Sharpe Ratio for the individual assets and the portfolios in questions 2 and 3, assuming the risk-free rate is 1.2%.

Answer 1

Stock ABC	Returns
2003	3.45
2004	7.81
2005	8.34
2006	6.21
2007	-8.81
2008	-5.3	Mean	1.96
2009	2.01	Std dev	6.14

We will use the below formulas to find Expected return and standard deviation of portfolios:

E(Rp) = w1*E(Ra) + w2* E(Rb)

Std dev(p) = (w1*w1) * (std dev(a) * std dev(a)) + (w2*w2) * (std dev(b) * std dev(b)) + 2*(w1)*(w2)*(Covariance(a,b))

correlation(a,b) = cov(a,b) / (std dev(a) * std dev(b))

where, w1 and w2 are weights of assets A and B

E(Rp) = expected return of portfolio

E(Ra) = expected return on asset A

E(Rb) = expected return on asset B

Std dev(p) = std dev of portfolio

for part 2:

w1 = $300 / ($300+$700) = 30%

w2 = $700 / ($300+$700) = 70%

E(Ra) = 8.3%

E(Rb) = 4.6%

std dev(a) = 12.1%

std dev(b) = 7.3%

Covariance(a,b) = - 0.0053

Using the aforementioned formulas:

Expected return of portfolio of asset A and B : 5.71%

Standard deviation of portfolio of asset A and B : 1.7%

For part 3:

w1(GE) = $2000 / ($2000+$3500) = 36.36%

w2(walmart) = $3500 / ($2000+$3500) = 63.63%

E(Ra)(GE) = 2.5%

E(Rb)(walmart) = 1.4%

std dev(a)(GE) = 4.1%

std dev(b)(walmart) = 1.3%

covariance(GE,Walmart) = - 0.0002

Using the aforementioned formulas:

Expected return of portfolio with investments GE and Walmart : 1.81%

Standard deviation of portfolio with investments GE and Walmart : 1.22%

For part 4 :

Sharpe Ratio = (Return on portfolio - risk free rate) / std dev of portfolio

Sharpe Ratio for part 2 = ( 5.71% - 1.2%) / 1.7% = 2.653

Sharpe Ratio for part 3 = ( 1.81% - 1.2%) / 1.22% = 0.5