The University of Nottingham

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1 The University of Nottingham BUSINESS SCHOOL A LEVEL 2 MODULE, SPRING SEMESTER COMPUTATIONAL FINANCE Time allowed TWO hours Candidates may complete the front cover of their answer book and sign their desk card but must NOT write anything else until the start of the examination period is announced Answer THREE Questions only All questions carry equal marks Only silent, self contained calculators with a Single-Line Display or Dual-Line Display are permitted in this examination. Dictionaries are not allowed with one exception. Those whose first language is not English may use a standard translation dictionary to translate between that language and English provided that neither language is the subject of this examination. Subject specific translation dictionaries are not permitted. No electronic devices capable of storing and retrieving text, including electronic dictionaries, may be used. DO NOT turn examination paper over until instructed to do so ADDITIONAL MATERIAL: Formula sheets are on pages 5 6.

2 2 1. You are considering of investing in three assets: A, B, and C. If investing 1000 each in the three assets for a year, your financial analyst provides the following forecasts based on different states of the economy: Economic state Probability Year-end value forecasts ( ) Asset A Asset B Asset C Recession Moderate growth Strong growth Based on the analyst s forecasts, (a) find the expected return per pound for each of the three assets over the coming year. [35 marks] (b) calculate the return variance for each of the three assets. (c) work out the return correlation coefficients between A and B, between A and C, and between B and C. [35 marks] 2. Today, you observe that the three-month (91 days) spot LIBOR is 2.5% p.a. and the nine-month (275 days) spot LIBOR is 3% p.a. You also observe that the three-month forward six-month LIBOR is quoted at 2.8% p.a. (a) Based on your observations, is there an arbitrage opportunity? (answer the question with mathematical evidence) (b) If your answer to (a) is yes, use a forward rate agreement (FRA) with a notional principal of 10 million to make an arbitrage profit. i. Design an arbitrage strategy today. [35 marks] ii. Illustrate how much your strategy will produce in nine months. [35 marks] (c) What assumption(s) have you made in your answers to (b)? [10 marks] 3. (a) Explain why dividend reinvestment is usually assumed in calculating the holdingperiod return of a security. (b) A UK investor purchased 1000 shares of an American company on 1 June 2010 at $60 per share, when the exchange rate was $1.5/ and the UK CPI level was The company paid $1.5 annual dividend per share on 20/10/2010, when the after dividend price was $50 per share. The company declared a 5-for-1 stock split effective on 25/3/2011, when the after split price was $12 per share. The investor sold all his shares of the company at $12.6 per share on 31/5/2011, when the exchange rate was $1.6/ and the UK CPI level was i. What is the nominal return per dollar over the one-year holding period? [35 marks] ii. What is the nominal return per pound over the one-year holding period? [25 marks] iii. What is the real return per pound over the one-year holding period?

3 3 4. (a) Describe what is meant by systematic and unsystematic risk. What is meant by systematic risk principle? [25 marks] (b) An analyst provides the following estimates for the coming year (return means rate of return): Expected return (%) CAPM Beta Stock X Stock Y Market portfolio 10 1 Risk-free rate 5 0 i. Based on the analyst s estimates, draw and label a graph showing the security market line and position stocks X and Y relative to it. [25 marks] ii. Based on the analyst s estimates, compute the Jensen alphas (i.e., the CAPM alphas) both for Stock X and for Stock Y. Show your work. iii. You agree with the analyst s estimates except for the estimated risk-free rate, which you believe will be 7%. Based on your belief, select the stock providing the higher expected CAPM risk-adjusted return and justify your selection (i.e., show your calculations). 5. You are thinking to invest in two risky assets and a risk-free asset. The risk-free asset provides an annual rate of return of 3.5%. Based on your careful investigation, you select the risky assets A and B and work out their annual rates of return in the past four years: Year Asset A Asset B 1 5% 15% 2 12% 25% 3 5% 12% 4 8% 20% The probability of return realised in each year is 1/4. To allocate your investment, you follow Markowitz s mean-variance portfolio analysis. You estimate the basic inputs of the mean-variance portfolio analysis such as expected return, standard deviation, and correlation coefficient using the sample information over the past four years. (a) For the two risky assets A and B, what are your estimated expected annual rates of return, return standard deviations, and their return correlation coefficient? Show your calculations. (b) Based on your estimates in (a), find the mean-variance-efficient portfolio (MVE) consisting of Assets A and B.

4 4 (c) Work out the expected return, standard deviation, and Sharpe ratio of the MVE found in (b). What is the optimal capital allocation line? (d) You want your investment to yield a return of 7% p.a. How much should you allocate your investment in each of the three chosen assets? What risk (standard deviation) should you prepare to bear?

5 5 Simple rate, i s : B t,t = 1/[1+i s (T t)] Discount rate, i d : B t,t = 1 i d (T t) Discretely compounded rate, i dc : B t,t = Formula Sheet 1 (1+i dc /m) m(t t) Continuously compounded rate, r cc : B t,t = e r cc(t t) No-arbitrage forward-forward rate: f T,m = ( 1+iL d L M 1+i S d S M 1 Payoff of a long FRA: V T (FRA) = A(i T,T+m k) days M days 1+i T,T+m M Return in domestic currency: R t,dc = S t S t 1 (1+R t,fc ) 1 Real return: R real = R nom h 1+h Decomposed buy-and-hold return: R P,τ = Expected value: E[X] = xf(x) x N Variance: V ar(x)=e[(x µ) 2 ]=E[X 2 ] µ 2, ) M d L d S w i τ 1 t=1 (1+R i,t) N j=1 w j τ 1 t=1 (1+R j,t) R i,τ Var ( N c i X i ) Variance of a portfolio consisting of two assets A and B: σ 2 P = w2 A σ2 A +(1 w A) 2 σ 2 B + 2w A(1 w A )ρ AB σ A σ B = N c 2 N 1 i σ2 X i + Skewness: µ 3 = E[(X µ X ) 3 ], where µ X stands for the expected value of X Kurtosis: µ 4 = E[(X µ X ) 4 ], where µ X stands for the expected value of X { } Covariance: Cov(X,Y )=E [X E(X)][Y E(Y )] Correlation coefficient: ρ = Corr(X, Y ) = Cov(X,Y ) σ X σ Y Sample variance: s 2 x = 1 n 1 n Sample covariance: s xy = 1 n 1 (x i X) 2 n (x i X)(y i Y ) N j=i+1 2c i c j σ Xi X j

6 6 Sample correlation coefficient: ˆρ = s xy s x s y OLS estimator: b = (X X) 1 X y, s 2 e = 1 n K t-statistic: T = b j β s bj t n K F -statistic: F = R2 /(K 1) (1 R 2 )/(n K) F K 1,n K Capital market line: E[R P ] = R f + E[R M] R f σ M n Optimal weight of a portfolio consisting two risky assets A and B: E( R A )σb 2 w A = E( R B )Cov(R A,R B ) E( R A )σb 2 +E( R B )σa 2 [E( R A )+E( R B )]Cov(R A,R B ) where R A and R B are excess returns of risky assets A and B relative to the risk-free rate. The capital asset pricing model (CAPM): E(R i R f ) = β i E(R m R f ) The intertemporal capital asset pricing model (ICAPM): E(R R f ) = β m E(R m R f )+β n E(R n R f ) The liquidity-augmented CAPM (LCAPM): E(R R f ) = β m E(R m R f )+β l E(LIQ) Arbitrage pricing theory (APT): E[R i ] = λ 0 +β i1 λ 1 +β i2 λ 2 + +β ik λ K The Fama French three-factor model (FF3FM): E(R R f ) = β m E(R m R f )+β s E(SMB)+β h E(HML) σ P e 2 i END