Stockholm School of Economics in Riga Financial Economics, Spring 2010 Jevgenijs Babaicevs

Problem Set II: The Law of One Price and Optimal Portfolio Choice Problem 1: Arbitrage and the Law of One Price a) Consider two securities that pay risk-free cash flows over the next two years and that have the current market prices shown in the following table:

Security

Market Price Today ($)

Cash Flow in One Year ($)

Cash Flow in Two Years ($)

B1

94

100

0

B2

85

0

100

(i) What is the no-arbitrage price of a security that pays cash flows of $100 in one year and $100 in two years? (ii) What is the no-arbitrage price of a security that pays cash flows of $100 in one year and $500 in two years? (iii) Suppose a security with cash flows of $50 in one year and $100 in two years is trading for a price of $130. What arbitrage opportunity is available? b) There is an arbitrage opportunity in each of the following two cases. For each case, explain the source of the arbitrage opportunity and how you would trade to exploit it. Payoffs State 1 State 2

Case 1

Asset Price

Asset 1

0.5

1

Asset 2

3

5

Payoffs State 1 State 2

Case 2

Asset Price

-0.5

Asset 1

0.5

1

2

-2.5

Asset 2

2.5

3

10

Problem 2: The Price of Risk The following table shows the no-arbitrage prices of securities A and B. Cash Flows in One Year ($) Weak Economy Strong Economy

Security

Market Price Today ($)

A

231

0

600

B

346

600

0

1

a) What are the payoffs of a portfolio of one share of security A and one share of security B? b) What is the market price of this portfolio? What expected return will you earn from holding this portfolio? c) Suppose security C has a payoff of $600 when the economy is weak and $1800 when the economy is strong. (i) Security C has the same payoffs as what portfolio of the securities A and B? What is the no-arbitrage price of security C? (ii) What is the expected return of security C if both states are equally likely? What is its risk premium? What is the difference between the return of security C when the economy is strong and when it is weak? (iii) If security C had a risk premium of 10%, what arbitrage opportunity would be available? Problem 3: Don't Put All Your Eggs in One Basket: Diversify! Assume A and B are the only securities traded in the market. Expected returns, standard deviations, and the correlation coefficient between the returns of these securities are shown in the following table: Security

Expected Return

Standard Deviation

Stock A

20%

20%

Stock B

15%

25%

Correlation coefficient -0.4

a) Given the expected return and standard deviation of stock B, would anyone be interested in investing in it? Explain! b) Toms, a prominent Latvian investor, invests 60% of his money in stock A and the rest in stock B. What is the expected return and standard deviation of his portfolio? c) Toms is not satisfied: He wants to form a portfolio (from stock A and B) with the lowest risk. He asks you to solve for the portfolio weights analytically and calculate the expected return and standard deviation of his rebalanced portfolio! d) Additionally two more stocks (stock C and stock D) have been just introduced. Notice that volatilities of stock A and stock B have changed accordingly and the variance-covariance matrix is as follows:

 44 

A B   A 10 10    B  15  C 5   D 12 5

C D   10   20 0   12 

(i) Fill in the missing values and interpret numbers on the main diagonal! (ii) Madara suggest Toms constructing a portfolio consisting of 25% invested in stock A, 40% in stock B, 20% in stock C, and the rest in stock D. Calculate the variance of his new portfolio! (iii) What are the betas of all four stocks relative the portfolio?

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Problem 4: A Simple One... Key characteristics of the two securities are summarized in the following table: Security

Expected Return

Standard Deviation

Security 1

10%

5%

Security 2

16%

8%

a) Which security should an investor choose if she wants to (i) maximize expected returns, (ii) minimize risk (assume the investor cannot form a portfolio)? b) Suppose the correlation of returns on the two securities is +1.0. What is the optimal combination of Security1 and Security 2 that should be held by the investor whose objective is to minimize risk (assume short-selling is not allowed)? c) Suppose the correlation of returns on the two securities is -1.0. What fraction of the investor's net worth should be held in Security 1 and in Security 2 in order to produce a zero risk portfolio? d) What is the expected return on the portfolio in c)? How does this compare with the risk-free return on Treasury Bills of 10%? Would the investor want to invest in Treasury Bills?

Problem 5: How Well Diversified is Your Portfolio? a) How many variance terms and how many covariance terms do you need to calculate the risk of a 100-share portfolio? b) Suppose all stocks have a standard deviation of 30% and a correlation coefficient of 0.4 with each other. What is the standard deviation of the returns on a portfolio that has equal holdings in 100 stocks? c) What is the standard deviation of a fully diversified portfolio of such stocks?

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Additional Problems for Your own Study Problem 6: Dell vs. Microsoft Historical data on the key risk characteristics of Dell and Microsoft stocks are shown in the following table. Stock

Beta

Standard Deviation

Dell

2.21

62.7%

Microsoft

1.81

50.7%

Correlation coefficient 0.66

Assume the standard deviation of the return on the market portfolio to be 15%. a) What is the standard deviation of a portfolio invested half in Dell and half in Microsoft? b) What is the standard deviation of a portfolio invested one-third in Dell, one-third in Microsoft and one-third in Treasury Bills? c) What is the standard deviation if the portfolio is split evenly between Dell and Microsoft and is financed with 50% margin? d) What is the approximate standard deviation of a portfolio composed of 100 stocks with betas of 2.21 like Dell? How about 100 stocks like Microsoft? Answer:

a)  p  51.7%

b)  p  34.5%

c)  p  103.4%

d)  pDell  33.15%  pMicrosoft  27.15%

Problem 7: Diversification Principle Once Again Assume there are three states of the world and two different financial assets. Assets’ returns are described in the following discrete probability distribution function.

State 1

Returns State 2

State 3

Asset 1

8%

-2%

12%

Asset 2

-5%

14%

9%

Probability

0.5

0.3

0.2

Assets

a) What is the mean return on Asset 1 and Asset 2? b) What is the variance of the return on Asset 1 and Asset 2? c) What is the covariance of the returns of two assets? What is the correlation coefficient between the two returns? d) Consider an equally-weighted portfolio of Asset 1 and Asset 2. Compute the mean return and standard deviation of this portfolio! Answer:

a) E  r1   5.8% E  r2   3.5%

b)  r21  28.36  r22  75.25

d) E  rp   4.65%  p  12.35%

c)  r1 , r2  27.1  r1 , r2  0.587

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