ECOM053 Quantitative Methods in Finance 【QMUL】

ECOM053 Quantitative Methods in Finance

Duration: 3 hours THIS IS AN OPEN BOOK EXAMINATION TO BE CONDUCTED ONLINE. YOU MAY REFER TO ANY OF THE COURSE MATERIALS, OR ANY OTHER SOURCE OF INFORMATION. YOU MAY ALSO USE A SPREADSHEET OR CALCULATOR. YOU CANNOT SUBMIT HANDWRITTEN ANSWERS ANSWERS ARE TO BE TYPED AND SUBMITTED TO BOTH QMPLUS & EMAILED TO: ecom053-exam@qmul.ac.uk PLEASE ENSURE THAT YOUR WORKING IS CLEARLY SHOWN WITH ALL STEPS OF YOUR CALCULATION INCLUDED IN YOUR ANSWER DOCUMENT, INCLUDING ANY FORMULA USED. When writing formulas, please note the following: • Answer ALL questions in the sheets following the exam paper It is acceptable to use the standard alphabet rather than greek letters. The following are recommended: m for μ, s for σ, w for ω, r for ρ, d for Δ, b for β. • • Where appropriate, use an underscore to indicate a subscript, Eg r_f for rf. For mathematical operators: add +, subtract -, multiply *, and divide /. • Use the ^ character for power, eg x^2 for x2, x^0.5 for √x. • • Examiner: Dr Panagiotis Koutroumpis As an alternative to x^.5 you may type sqrt(x). Use brackets as necessary. To make your answer clearer use different brackets where appropriate, eg [] {} (). © Queen Mary University of London, 2021

Question 1

An investor models the excess return for Amazon via the market model. Suppose that the ̂̂ sample size is 102 observations, the value of 𝛽 for Amazon is 𝛽 = 1.107, the standard error ̂̂ associated with the coefficient 𝛽 is SE(𝛽) = 0.0348. She tests the null hypotheses that the value of the stock’s beta is one, against a one-sided alternative that the stock’s beta is greater than the market’s beta. The level of significance is 5%. a. Write down the regression model which uses the market model’s factors as explanatory variables and define the variables. b. Writedownthenullandthealternativehypothesis. [5 marks] [5 marks] c. Conduct the test. What do you conclude? Are the analyst’s claims empirically verified? You can find the t-test’s critical values in Table A1 in the Appendix at the end of the exam paper. [5 marks]

1. The analyst also tells you that shares in Chris Mining PLC have no systematic risk, in the sense that the returns on its shares are completely unrelated to movements in the market. The value of beta and its standard error are calculated to be 0.5 and 0.25, respectively, and the model is estimated using 52 observations. Write down the null and alternative hypothesis. Form and interpret a 95% and a 99% confidence interval for beta. You can find the t-test’s critical values in Table A1 in the Appendix at the end of the exam paper. [5 marks]

2. Dothetestedhypothesesconcerntheactualvaluesofthecoefficients(i.e.β),ortheir

̂ estimated values (i.e. 𝛽), and why? © Queen Mary University of London 2021 Turn over [5 marks]

Page 3 ECOM053 (2021)

Question 2

a. Let the stocks of IBM, Google and Amazon. You have a time series of observations for each stock. The sample size is T. Explain in detail, all steps which are necessary to compute the sample variance-covariance matrix of the three stocks. [5 marks] [5 marks] [5 marks] b. Explainindetailhowtheimpliedvolatilityofanoptioncanbecomputed. c. Prove that the duration of a zero-coupon bond equals its maturity.

1. Do you agree with the statement “Let a fixed income portfolio which has a duration equal to zero. This portfolio is subject to no interest rate risk.” [5 marks]

2. Consider two bonds A and B with payments 𝐶𝑡 , where 𝑡 = 1,2, ... ,10. Both bonds have $1,000 face value. Bond A has just been issued, it bears coupon rate of 7%, and it will mature in 10 years. Bond B was issued 5 years ago, when interest rates were higher. This bond has $1,000 face value and bears a 13% coupon rate. When issued, this bond had a 15-year maturity, so its remaining maturity is 10 years. The yield to maturity is 7% (see Cell B2). Using the Excel spreadsheet below, estimate the duration of each of the two bonds A (Cell B20) and B (Cell E20), using the mathematical formula of the Macaulay duration measure. Which bond has the longest duration? Show your calculations and interpret your results. [5 marks]

© Queen Mary University of London 2021 Turn over

Page 4 ECOM053 (2021)

Question 3

a. SupposeweformaportfolioinvestedinMercedesandBMWstocks.Thetwostocks are equally weighted in the portfolio. The table below displays each stock’s mean return, variance, and standard deviation, as well as their covariance. Estimate the mean and variance of this portfolio, as well as its standard deviation. Show in detail your calculations. Asset returns Mean return Variance Standard deviation Covariance Mercedes 1.78% 0.0456 21.35% BMW -0.48% 0.0276 16.61% 0.0020 b. The graph below shows the envelope frontier. Based on the graph below do you agree with the statement “Portfolio P(1) and P(2) are two efficient portfolios”? Explain your answer. [5 marks] c. Explain in detail all five propositions on envelope portfolios. [5 marks] [5 marks] © Queen Mary University of London 2021 Turn over

Page 5 ECOM053 (2021) d. Considertwoportfolios,xandy,whoseconvexcombinationscomposetheenvelope frontier in the graph below (curve ABC). Also marked, are other portfolios, some of which contain short positions of either x or y. Do you agree with the statement “Every convex combination of any two efficient portfolios is efficient”? Explain your answer. [5 marks] © Queen Mary University of London 2021 Turn over

Page 6 ECOM053 (2021) e. Let us now compute an envelope portfolio with constant 𝑐 = 4%, where the vector z solves the system of simultaneous linear equations 𝐸(𝑟) − 𝑐 = 𝑆𝑧. Then, this solution produces a portfolio x and y on the envelope of the feasible set. Write down and explain the Excel formula used to calculate the values in Cells A20, F20, F24 (portfolio weights and sum of portfolio weights), F26:F28 (mean, variance and standard deviation), and B30:B31 (covariance, correlation), in the Excel spreadsheet below: [5 marks] © Queen Mary University of London 2021 Turn over

Page 7 ECOM053 (2021)

Question 4

a. Why is it desirable to conduct Monte Carlo simulations using as many replications of the experiment as possible? b. Explain in detail how pseudo-random numbers are generated in Excel. [5 marks] [5 marks] c. In which range of numbers do we expect the random numbers drawn from a standard normal distribution to fall? Explain why. [5 marks] d. Marcus is 45-year old. He has a new job and intends to save £10,000 today and in each of the next 14 years (15 deposits altogether). He is considering to invest in an investment policy in which he would invest 30% of his assets in a risk-free bond with 3% continuously compounded annual interest and the remaining 70% in a risky asset that has lognormally distributed returns with mean μ = 12% and standard deviation σ = 35%. Marcus applied Monte Carlo simulation to decide whether he should invest his money in this investment strategy. The Excel spreadsheet below reports the end-of-year wealth based on one simulation that he conducted. Write down and explain the Excel formula used to calculate the yellowed values in cells E11 and F11, in the Excel spreadsheet below: © Queen Mary University of London 2021