Investment Science AMA532, S2, 2014/15, Assignment Two

Hand in your solutions to questions 2,5,7,9,11,15,18 and 19 by 6:30pm, 14 April. This isworth 4% towards your final course mark. Late submissions may not be marked, and ifmarked will receive reduced or zero credit.

1. You need to download an excel file under /Evaluation Tools/Assignments at

http://webct2.polyu.edu.hk/webct/public/home.pl

In the excel file, there are two sheets ‘dataset1’ and ‘dataset2’. The sheet ‘dataset1’provides some examples how to use excel. However you may find the ‘help’ in theexcel software will be much more useful, where you can learn how to calculate, such as‘covariance’, ‘transpose’ and ‘inverse’ and ‘matrix multiplication’.

The sheet ‘dataset2’ contains the data of daily closing prices (pij) of 15 assets from1/2/2005 to 30/6/2005, which are needed to do this question.

You need to use the data in ‘dataset2’ and ‘excel’ to do this question.

Assume that the investment period T = 22. Normalize the data in the sheet ‘dataset2’to obtain the rates of return by the formula

Oij =pi,j+T − pi,j

pi,j, i = 1, · · · , 15, j = 1, 2, · · · , K.

For example, when i = 1, j = 1, p1,1 = 71 of the date 1/2/2005 and p1,23 = 71.25 of thedate 23/2/2005, O11 = 0.0035211. The rates of return for the period from 1/2/2005 to23/2/2005, a 15 × 86 matrix (here K = 86), are obtained. Thus the expected rates ofreturn and the covariance matrix of 15 assets can be calculated.

(a) Find the minimum-variance portfolio, what is µ in this case? and

(b) Find the optimal portfolio with r̄ = 0.06. and what are λ and µ ?

Hand in the computer printing out including the original data, means, covariancematrix etc. Present your answers upto 4 decimal points.

2. (Bounds on returns) Consider a universe of just three securities. They have expectedrates of return of 20%, 30%, and 40%, respectively. Two portfolios are known as to lieon the minimum-variance set. They are defined by the portfolio weights

w =

0.30.40.3

,v =

0.50.8−0.3

.

It is known that the market portfolio is efficient.

(a) Given this information, what are the minimum and maximum possible values forthe expected rate of return on the market portfolio?

(b) Now suppose you are told that v represents the minimum-variance portfolio. Doesthis change your answers to part (a)?

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3. (Capital market line) Assume that the expected rate of return on the market portfoliois 15% and the rate of return on T -bills (risk-free rate) is 4%. The standard deviationof the market is 20%. Assume that the market portfolio is efficient.

(a) What is the equation of the capital market line?

(b) (i) If an expected return of 20% is desired, what is the standard deviation of thisposition? (ii) If you have $1,000 to invest, how should you allocate it to achieve theabove position?

(c) If you invest $400 in the risk-free asset and $600 in the market portfolio, how muchmoney should you expect to have at the end of the year?

4. Assume that the following assets are correctly priced according to the security marketline. Derive the security market line. What is the expected return on an asset with aBeta of 3?

r̄1 = 6%, β1 = 0.5,

r̄2 = 12%, β2 = 1.5.

5. In Simpleland there are only two risk stocks A and B, whose details are listed below.

Number of shares Price Expected Standard deviationoutstanding per share rate of return of return

Stock A 500 $10.00 50% 10%Stock B 500 $5.00 30% 8%

Furthermore, the correlation coefficient between the returns of stocks A and B is ρAB =1

3. There is also a risk-free asset, and Simpleland satisfies the CAPM exactly.

(a) What is the expected rate of return of the market portfolio?

(b) What is the standard deviation of the market portfolio?

(c) What is the beta of stock A?

(d) What is the risk-free rate in Simpleland?

6. Election Wizards, Inc. (EWI) has a new idea for producing TV sets, and it is planningto enter the development stage. Once the product is developed (which will be at the endof 1 year), the company expects to sell its new process for a price p, with expected valuep̄ = $24M . However, this sale price will depend on the market for TV sets at the time.By examing the stock histories of various TV companies, it is determined that the finalsales price p is correlated with the market return as E[(p − p̄)(rM − r̄M)] = $20Mσ2

M .

To develop the process, EWI must invest in a research and development project. Thecost c of this project will be known shortly after the project is begun (when a technicaluncertainty will be resolved). The current estimate is that the cost will be eitherc = $20M or c = $16M , and each of these is equally likely. (This uncertainty is

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uncorrelated with the final price and is also uncorrelated with the market.) Assumethat the risk-free rate is rf = 9% and the expected return on the market is r̄M = 33%.

(a) What is the expected rate of return of this project?

(b) What is the beta of this project?

[

Hint: In this case, note that E[(

p − p̄

c

)

(rM − r̄M)]

= E(

1

c

)

E[(p − p̄)(rM − r̄M )].]

(c) Is this an acceptable project based on a CAPM criterion? In particular, what isthe excess rate of return (+ or −) above the return predicted by the CAPM?

7. Someone who believes that the collection of all stocks satisfies a single-factor modelwith the market portfolio serving as the factor. The table below gives you informationon three stocks which make up a portfolio.

Standard deviationBeta of random error term Weight in portfolio

Stock A 2.50 8.0% 20%Stock B 0.90 2.0% 50%Stock C 1.20 3.0% 30%

In addition, you know that the market portfolio has an expected rate of return of 15%and a standard deviation of 20%. The risk-free rate is 6%.

(a) What is the portfolio’s expected rate of return?

(b) Assuming the factor model is accurate, what is the standard deviation of this rateof return?

8. Write the CAPM shown below in price form

r̄i = 0.03 + 0.20βi.

9. Stock X has an expected return of 5.5% and a risk of β = 0.7. Stock Y has an expectedreturn of 16% and a risk of β = 2. The market’s expected return is 8%, and rf = 3%.

(a) What are the Jensen values of stocks X and Y? Draw these values on the (β, r̄)-space.

(b) According to the CAPM, which stock is a better buy?

10. Two stocks are believed to satisfy the two-factor model

r1 = a1 + 2f1 + f2, r2 = a2 + 3f1 + 4f2.

In addition, there is a risk-free asset with a rate of return of 5%. It is known thatr̄1 = 12% and r̄2 = 16%. What are the values of λ0, λ1 and λ2 for this model?

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11. (a) Three widely diversified portfolios are shown in the following table.

Portfolio Expected return bi1 bi2

A 18 1.0 0.8B 14 2.5 1.0C 10 0.8 0.2

Find the APT model.

(b) Based on the data specified from (a), there is a portfolio named D constructed byplacing 1/3 of the funds in portfolio A, 1/3 of the funds in portfolio B, and 1/3 of thefunds in portfolio C. Another portfolio is given by

Portfolio Expected return bi1 bi2

E 15 1.4333 0.6667

Compare the portfolios D and E.

(c) To make $3 profit from arbitraging D and E, how much amount of funds one needto buy E and sell D short? Present the solution also in table form.

12. A record of annual percentage rates of return of the stock S is shown below.

Month Percent rate of return Month Percent rate of return1 2.0 13 −3.22 −1.5 14 1.53 .5 15 3.54 3.2 16 3.15 −3.7 17 −2.76 3 18 2.77 −2.1 19 4.28 3.1 20 1.49 −2.7 21 1.710 .4 22 −2.911 1.4 23 −2.912 2.2 24 2.1

(a) Estimate the arithmetic mean rate of return, expressed in percent per year.

(b) Estimate the arithmetic standard deviation of these returns, again as percent peryear.

(c) Estimate the accuracy of the estimates found in parts (a) and (b).

(d) How do you think the answers to (c) would change if you had 2 years of weeklydata instead of monthly data?

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13. (HSI) Historical data of closing prices for three stocks and one index in 2005 are shownin table below.

Year HSBC HSB Cheung Kong HSIJul 120 107 80 15Aug 122 109 78 16Sept 119 106 76 15Oct 123 109 79 16Nov 124 110 81 17Dec 118 107 80 14

Consider the single factor model: ri = ai + bif + εi. Calculate the following quantities:

A. the average return on each stock and the index.

B. the sample variance of each stock and the index.

C. the correlation between the stock and the index.

D. bi for each stock.

E. ai for each stock.

F. the variance of the error for each stock.

Check if the single factor model is accurate.

14. An investor has utility function U(x) = x1/3 + 1 for salary. He has a new job whichpays $100,000 with a bonus. The bonus will be $0, $10,000, $20,000, $30,000, $40,000,or $50,000, each with equal probability. What is the certainty equivalent of this joboffer?

15. Let U(x) be a utility function. The Arrow-Pratt relative risk aversion coefficient isdefined by

µ(x) = −xU ′′(x)

U ′(x).

Show that the utility functions U(x) = lnx and U(x) = γxγ have constant relative riskaversion coefficients.

16. Let U(x) = ax− 1

2bx2, x ≤ a/b, where a > 0 and b ≥ 0 be a concave utility function.

Show that the mean-variance optimization problem is equivalent to the maximizationof the expected utility of quadratic concave function U(x).

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17. Consider the following two investments:

Investment A Investment B

Outcome Probability Outcome Probability4 2/5 5 1/212 1/5 10 1/414 2/5 12 1/4

(a) Which is preferred if the utility function is U(x) = 2x − .04x2?

(b) In investment B, the probability of a $5 return is 1/2 and the probability of $10return is 1/4. What values would these probabilities have to change to so that theinvestor is indifferent between investments A and B?

18. Suppose the utility function is U(x) =√

x.

(a) What is the utility at wealth levels $50,000 and $150,000?

(b) What is the expected utility for a simple prospect with two wealth levels in (a)and half-half opportunity?

(c) What is the certainty equivalent of the risky prospect?

(d) Does this utility function also display risk aversion?

(e) Does this utility function display more or less risk aversion than the log utilityfunction?

19. An investor is considering to organize a forum to exhibit an education product on thecoming Sunday. The success of the event depends on the weather condition on thatday which could be one of the following three possibilities: Good (G), Moderate (M),and Raining (R). The returns and probabilities together with a risk-free opportunity(e.g. deposit money in a bank) are given below

Return ProbabilityG 4.0 .4M 1.0 .4R −2.0 .2

Risk free 1.2 1.0

The investor’s utility function is U(x) = lnx. Find the optimal portfolio.

20. Repeat Question ?? with the following data.

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Return ProbabilityG 5.0 .3M 3.0 .3R 1.0 .4

Risk free 1.3 1.0

21. (a) There are three investment opportunities in the film venture shown below.

Returns of Returns of Returns ofthe the risk-free the residual

Event film venture alternative rightsHigh success 3.0 1.2 6.0

Moderate success 1.0 1.2 0Failure 0.0 1.2 0

Find the state prices.

(b) In addition to the three investment in (a), the promoter of the film venture offers anew investment designed to attract reluctant investors. One unit of this new investmenthas a payoff of $3,000 if the venture is highly successful, and it refunds the originalinvestment otherwise. What is the price of this money-back guaranteed investmentbetween the film venture and the risk-free asset?

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