# Quantitative Techniques OPEN BOOK EXAMINATION

 ECOM037 Quantitative Techniques Duration: 3 hours

# Answer ALL questions

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.

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:

• 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.
• For mathematical operators: add +, subtract -, multiply *, and divide /.
• Where appropriate, use an underscore to indicate a subscript, Eg r_f for rf.
• Use the ^ character for power, eg x^2 for x2, x^0.5 for √x.
• 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 returns of a mutual fund via a linear regression model using Carhart’s (1997) factors as explanatory variables. The sample size is 100 observations, the residual sum of squares is 5, and the explained sum of squares is 10. The level of significance is 5%.

1. Write down the regression model which uses Carhart’s factors as explanatory variables and define the variables.

## [5 marks]

1. What is the formula, in matrix notation, which provides the ordinary least squares estimators for the model? Define the dimensions of the vectors and matrices appearing in the formula.

## [5 marks]

1. How would you test the statistical significance of each one of the regressors on a stand-alone basis? In which case would you reject the null hypothesis of no significance?

## [5 marks]

1. Calculate the F-statistic to assess the null hypothesis of no significance of the regressors. In which case would you reject the null hypothesis?

## [5 marks]

1. Define Carhart’s (1997) alpha and explain how you would test whether a mutual fund delivers a positive Carhart’s (1997) alpha.

[5 marks]

## Question 2

1. 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]

1. Explain in detail how the implied volatility of an option can be computed.

[5 marks]

1. Prove that the duration of a zero-coupon bond equals its maturity.

[5 marks]

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]

1. Consider two bonds A and B with payments , where . 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]

## Question 3

1. Suppose we form a portfolio invested in Mercedes and BMW stocks. The two stocks are equally weighted in the portfolio. The spreadsheet 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 Mercedes BMW Mean return 1.78% -0.48% Variance 0.0456 0.0276 Standard deviation 21.35% 16.61% Covariance 0.0020

[5 marks]

1. 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]

1. Explain in detail all five propositions on envelope portfolios.

[5 marks]

1. Consider two portfolios, x and y, whose convex combinations compose the envelope 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]

1. Let us now compute an envelope portfolio with constant where the vector z

solves the system of simultaneous linear equations . 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]

## Question 4

1. Why is it desirable to conduct Monte Carlo simulations using as many replications of the experiment as possible?
• marks]

1. Explain in detail how pseudo-random numbers are generated in Excel.
• marks]

1. In which range of numbers do we expect the random numbers drawn from a standard normal distribution to fall? Explain why.

• marks]

1. 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 her money in this investment strategy. The Excel spreadsheet below reports the end-ofyear 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:

Page 9                                                                                                     ECOM037 (2021)

[10 marks] End of Examination/  George Skiadopoulos

End of Paper