Correct ones please.

1 An investor measures the utility of her wealth using the utility function U(w) = ln(w)

for w > 0.

(i) Derive the absolute and relative risk aversions for this investor's utility

function, and the first derivative of each. [4]

(ii) Comment on what this tells us about the proportion of her assets that this

investor will invest in risky assets. [2]

The investor has 100 available to invest in two possible assets, Asset A and Asset B.

The future value of Asset A depends on an uncertain future event.Every 1 invested in Asset A will be worth 1.30 with probability 0.75 and 0.40

with probability 0.25.

Asset B is risk-free, so every 1 invested in Asset B will always be worth 1.

The investor does not discount future asset values when making investment decisions.

She decides to invest a proportion a of her wealth in Asset A and the remaining

proportion 1 - a in Asset B.

(iii) Express her expected utility of wealth in terms of a. [2]

(iv) Determine the amount that she should invest in each of Asset A and B to

maximise her expected utility, using your result from part (iii).

2.

Consider the following time series model: Y, = 1 +0.6Y,_1 +0.16Y,-2+ & where &, is a white noise process with variance o-. (i) Determine whether Y, is stationary and identify it as an ARMA(p,q) process. [3] (ii) Calculate E(Y,). [2] (iii) Calculate for the first four lags: the autocorrelation values p1, P2, P3. P, and . the partial autocorrelation values v1, v2, (3, V4- [7](i) State the general expression of the exponential families of distributions and use this to derive the relevant expressions for the mean and the variance of these distributions. [6] (ii) Extend the result in (i) to obtain an expression for the third central moment. [4] (iii) Show that the following density function belongs to the exponential family of distributions: [4] (x) = u"T(a) (iv) Using the results in (i) and (ii) obtain the second and third central moments for this distribution. [4]A company's stock is heavily weighted in a fund. The company is about to issue a statement on its recent financial results. There is uncertainty whether the news will be received positively or negatively and the impact that it will have on the stock price. The fund is considering purchasing a put option and a call option on the stock, each with the same strike price close to the current stock price, and the same expiry date. (i) Draw the payoffs from this strategy showing clearly the individual and combined position. [5] (ii) Comment on the circumstances under which you think this strategy will be successful. [2]The manager of a large equity fund is expecting a significant inflow of cash in the near future. (i) Describe how the manager would protect the fund against a rising market in this situation. [3] (ii) Describe the residual risks that remain with the strategy you have described in part (i). [4] (iii) Suggest alternative reasons why the manager might want to hedge the portfolio. [2]