Two Hours. Mathematical formula books and statistical tables are to be provided THE UNIVERSITY OF MANCHESTER. 22 January :00 16:00

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1 Two Hours MATH38191 Mathematical formula books and statistical tables are to be provided THE UNIVERSITY OF MANCHESTER STATISTICAL MODELLING IN FINANCE 22 January :00 16:00 Answer ALL TWO questions in Section A. Answer TWO of the THREE questions in Section B. If more than TWO questions from section B are attempted then credit will be given to the best TWO answers. Electronic calculators may be used, provided that they cannot store text. 1 of 6 P.T.O.

2 SECTION A MATH38191 Answer ALL of the three questions A1. (a) Define (net) return R t and log return r t at time t in terms of the price P t > 0 at time t and the price P t 1 > 0 at time t 1. Show that r t = log(1 + R t ). (b) Consider a portfolio of two assets with constant weights w 1 = w 2 = 0.5 over time. Express the portfolio (net) return R t in terms of the asset (net) returns R it, and the portfolio log return r t in terms of the asset log returns r it, i = 1, 2. Which portfolio return is easier to work with and why? (c) What is a random walk? What is a geometric random walk? Give mathematical definitions in general terms. (d) What is the random walk model for financial markets? What is unpredictable under this model and in what sense? (e) What evidence has been given in the course against the random walk model? 2 of 6 P.T.O.

3 MATH38191 A2. Suppose that a risk-free asset returns µ 0 = 3%, and that two risky assets have mean returns µ 1 = 4% and µ 2 = 6% with standard deviations σ 1 = 0.1 and σ 2 = 0.2 respectively. The correlation between the two stock returns is ρ = 0.1. (a) Find the tangency portfolio of the two risky assets. You may use the formula γ T = (µ 1 µ 0 )σ 2 2 (µ 2 µ 0 )ρσ 1 σ 2 (µ 1 µ 0 )σ (µ 2 µ 0 )σ 2 1 (µ 1 µ 0 + µ 2 µ 0 )ρσ 1 σ 2 without proof. (b) If 1000 are invested in the tangency portfolio, how much will be spent on each asset? (c) Calculate the mean return µ T and standard deviation σ T of the tangency portfolio. (d) Calculate the Sharpe ratio of the tangency portfolio and comment on it. (e) Sketch the feasible region and efficient frontier of all portfolios consisting of the three assets allowing short selling/borrowing at margin. Indicate the tangency portfolio on the graph. (f) Find a portfolio on the efficient frontier in (e) with mean return 5% and explain how it works. 3 of 6 P.T.O.

4 SECTION B MATH38191 Answer TWO of the three questions B1. (a) Define skewness and (excess) kurtosis. Explain the reason for the term 3 in the definition of (excess) kurtosis. Give an example of a distribution (by name and parameter value only) with skewness 0 and (excess) kurtosis greater than 0. (b) Suppose that U U[0, 1] and X = c/u 1/a, where a > 0 and c > 0 are constants. Show that X has a Pareto distribution with tail index a. (c) Use (b) or otherwise to show that the kth non-central moment of X is E[X k ] = c k a/(a k) provided that a > k. Use this result to find the mean and variance of X. (d) Given that the skewness of X is γ 1 = 2(1 + a) a 2 a 3 a when a > 3, and the (excess) kurtosis of X is γ 2 = 6(a3 + a 2 6a 2) a(a 3)(a 4) when a > 4, what conclusions can be drown from these results about the distribution of X? 4 of 6 P.T.O.

5 MATH38191 B2. (a) In the Capital Asset Pricing Model (CAPM) R i = µ 0 + β i (R M µ 0 ) + ε i, i = 1, 2,..., n, what are the terms R i, R M and ε i? What properties are assumed for ε i? What properties of R i can be deduced? [3+3+2 marks] (b) What is the interpretation of β i in the CAPM and what does β 2 i represent? What is the Security Market Line? (c) The following output from R was obtained. The data used were daily closing prices of GE and the S&P500 index from 1-Nov-93 to 31-Oct-95, and annual percentage T-bill rates. > data <- read.csv("capm.csv", header=t) > data <- data[1:502,] > attach(data) > y <- Close.ge[2:502]/Close.ge[1:501] -1 -Close.tbill[2:502]/100/251 > x <- Close.sp500[2:502]/Close.sp500[1:501] -1 -Close.tbill[2:502]/100/251 > fit <- lm(y ~ x) > summary(fit) Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) x <2e-16 *** (i) What model was fitted in the above output? (ii) Is there significant evidence against the CAPM? Test appropriate hypothesis at the 5% significance level. (iii) Is there significant evidence that β 1? Test appropriate hypothesis at the 5% significance level. 5 of 6 P.T.O.

6 MATH38191 B3. (a) What is the definition of VaR(α), the (1 α) 100% Value at Risk? Is it in the upper or lower tail of the loss distribution? What is the probability for the loss to be greater than VaR(α) when α = 0.05? [2+1+1 marks] (b) Define Expected Shortfall (ES) as a conditional expectation of loss and use the definition to show that ES(α) = 1 α V ar(α) where f(x) is the density of the loss distribution. xf(x)dx, [2+4 marks] (c) Assuming a t ν (µ, λ) distribution for return= loss with location parameter µ and scale parameter λ, write down the formula for VaR(α) and show that ES(α) = µ + λ f [ ] ν(fν 1 (α)) ν + F 1 ν (α) 2, α ν 1 where f ν and F ν are the pdf and cdf of t ν respectively. You may use the fact that without proof. c xf ν (x)dx = f ν (c) ν + c2 ν 1 [2+4 marks] (d) Using 500 daily log returns of the SP500 index, the following results were obtained. > fitdistr(r500, "t") m s df > qt(0.05, df= ) [1] > dt( , df= ) [1] Calculate estimates of VaR(0.05) and ES(0.05) for an investment of $10000 on the SP500 index and interpret the answers. END OF EXAMINATION PAPER 6 of 6