(Papers) ACET Paper June 2015 "CT8 Financial Economics"

Q.1 Consider a Zero-coupon corporate bond that promises to pay a return of 8 % next period. Suppose that there is a 10% chance that the company will default on the bond payment, in which case there is an equal chance of receiving a return of 4% or 0%.

a) Calculate values for the following measures of investment risk:

(1) Downside semi variance
(2) Shortfall probability based on a risk free rate of return of 5% pa
(3) The expected shortfall below the risk free return conditional on a short fall occurring

b) Discuss the usefulness of “down side semi variance” as a measure of investment risk.

Q.2

a) Explain “Market Price of Risk” in the context of the Capital Asset Pricing Model.
b) When is it optimal to consider only the mean –variance of return from a portfolio when choosing investments?
c) Show that the CAPM result can be written as a single index model; and hence it is consistent with the Arbitrage Pricing Theory

Q.3

a) Distinguish between the lognormal model of security prices and Wilkie model with respect to consistency with market efficiency.

Q.4

a) If U [w] denotes an investors’ utility function, where w denotes the level of wealth, state the conditions [in terms of derivatives of the utility function] for

(i) Absolute risk aversion
(ii) Relative risk aversion

b) State the assumptions underlying the Mean Variance Portfolio theory.

Determine the minimum risk portfolio. What are the expected return and standard deviation of that portfolio?

Q.5 a) Let Bt [t ³0] be a Standard Brownian Motion process starting with B0 =0

i) What is the probability that B2 takes a positive value?
ii) What is the probability that B2 takes a value in the interval [-1, 1]?
iii) Prove that the probability that B1 and B2 both take positive values is 3/8.
iv) Show that if B50 is 3.04, the probability that B100 is negative is approximately 1/3.

b) Use Ito’s lemma to show that the continuous- time lognormal model of security prices can be derived from the assumption that the security prices follow a geometric Brownian motion process.

Q.6

a) What is meant by saying that the market is “arbitrage -free”?
b) The increase in the price of a share over the next year is believed to have a mean of 10% and a S.D of 10%

i) Determine the values of u and d for a one - step binomial tree model that are consistent with the mean and standard deviation of the return on the underlying share, assuming that the share price is twice as likely to go up than to go down.

ii) Hence calculate the value of each of the following options:

- a one year European Call Option with a strike price of 275.
- a one year European Put Option with a strike price of 300
For this purpose, assume a current share price of 250 and the continuously compounded risk free rate of interest to be 7.5% pa. Also assume that dividends can be ignored.