代做FN3023 Investment management 2023帮做Python语言

FN3023 Investment management

Question 1

(a)    Explain the priority rules applied in executing orders in limit order markets. (7 marks)

(b)    How many variance terms and how many different covariance terms does an investor need to calculate the risk of a 100-asset portfolio? Suppose all assets have a standard deviation of 30% and a correlation with each other of 0.4. What is the standard deviation of a fully diversified portfolio of such assets? (9 marks)

(c)    Chris wants to form. an options portfolio that is immunised against small changes   in the underlying stock price while protecting the value of the stock from increases in volatility. The stock price (S) currently is £900. The Black-Scholes call option value is given by

c = SN(d1) − Xe − r(T −t)N(d2),

where X is the exercise price, r is the risk-free interest rate, and t is the time to maturity. Cumulative normal probability function N(.) for parameters d1  and d2 are 0.5471 and 0.4745 respectively. Derive the proportion in which to invest in the call and put options. How many put options should Chris buy for 20 call options? (9 marks)

Question 2

(a)    Explain why shareholders of a company are normally seen as the ‘owners’ of the company, distinct from debt holders and employees who also receive a share of  the company’s cash flow. What is the purpose of having different categories of share ownership? (7 marks)

(b)    A variance-averse investor has a utility function U(μ, σ2) = μ − 0.25σ2 , where μ is the portfolio expected return and σ is portfolio standard deviation. The risk-free     rate of return is 3%, the average return on the market index is 16%, and thestandard deviation of the market index is 68%. Derive a formula for the optimal weight to be placed on the market and solve for it in this case. What risk-aversion coefficient would justify an optimal investment consisting of only 1% of funds being placed in the market index? (9 marks)

(c)    A stock is usually traded at an average price of $50. The covariance of successive returns on the stock, i.e., trade-by-trade changes in the price, is about -0.25. Derive Roll’s model and estimate the bid-ask spread of the stock expressed as a percentage of the stock’s average price? (9 marks)

Question 3

(a)    A futures contract is not well correlated with your equity portfolio. Show how you  would determine the size of the position taken in the futures contract, if you were to minimise risk to the portfolio. Let P represent the value of your current portfolio, F represent the current futures value, h represent the size of the position taken in the futures contract,  and Δ represent changes in portfolio value. (7 marks)

(b)    Investors are variance-averse and hold the market portfolio. They predict that E(ri) − rF  = Constant ∗ Cov(ri, rM), where E(ri) is the expected return of stock i;    rF  is the risk-free rate; and Cov(ri, rM) is the covariance between the return of the stock and the return on the market index. Suppose the market consists of only 2 stocks, A and B, and of a risk-free asset.   Let E(rA ) = 9%; E(rB) = 7%; rF   = 2%; the variance of stock A, Var(rA ), is 0.35;     the variance of stock B, Var(rB), is 0.18; and the correlation between stock A and stock B, ρAB , is 0.4. What is the composition of the market index? (9 marks)

(c)    In a risk-neutral world where the market is perfectly competitive, Mark, a market maker, clears buy and sell orders. Mark believes that there is equal probability    that the share is worth £120 or £80. Uninformed traders buy with a probability of 0.5 and sell with a probability of 0.5. Informed traders buy if the share is worth    £120 and sell if the share is worth £80. Informed trades constitute 10% of the trades, and the remaining ones are uninformed.

Find the initial bid and ask prices. (9 marks)

Question 4

(a)    A company has a pension liability for the next year of £1,000. This is expected to grow at a rate of 2% each year indefinitely.

Derive formulas for the value and duration of this liability. At a discount rate of 4%, what is the duration?

Hint: for 0 < x < 1,

x + x2  + x3  + ⋯ =

x + 2x2  + 3x3  + ⋯ = (7 marks)

(b)    Suppose you invest in an equally weighted portfolio consisting of ‘n’ stocks, each with a beta equal to 1.1 and an idiosyncratic variance of 10%. Idiosyncratic risk is independent across the stocks and the market index has a variance of 15%.

How many such stocks would be needed to keep the proportion of idiosyncratic risk below 5% of total portfolio risk?

If the correlation between idiosyncratic risks is in fact 0.5, what will the total portfolio risk be with 20 stocks? (9 marks)

(c)    The value of a portfolio follows a geometric Brownian motion with the instantaneous return μ = 5%, and the instantaneous standard deviation σ = 25%.

The log return and standard deviation over 30 days are given by

(μ − σ2) and   σ ,  respectively.

What is the 30-day, 1% VaR of the portfolio? You should give your answer in terms of log returns. The probability that z is less than or equal to -2.326 is approximately 1% for a standard normal random variable with zero mean and unit variance. (9 marks)

Question 5

(a)    The security market line shows the combinations of risky and risk-free

investments. With the use of a diagram, explain what happens to the tangency portfolio if investors have different interest rates for savings and borrowing. (7 marks)

(b)    The variance and the return of the market portfolio are 20% and 10%,

respectively. The risk-free return is 2%. A portfolio has a beta of 0.8 and

idiosyncratic risk with a variance of 15%. What is the required return on the

portfolio in order that it matches the market portfolio in terms of the Sharpe ratio? Demonstrate what this implies for the M2  ratio. (9 marks)

(c)    A risk-neutral competitive market maker clears the market for trading in a stock    after observing the incoming orders from a noise trader and an informed trader     (who perfectly knows the true value of the stock). The noise trader buys and sells 1 share of the stock with equal probability, whereas the informed trader buys the  stock if the value is $14 (high) and sells the stock if the value is $5 (low).

The market maker believes initially that there is a 45% probability of high value and a 55% probability of low value. What profits can the informed trader expect to make in this market? (9 marks)

Question 6

(a)    Explain two key assumptions of Markowitz’s portfolio theory. Discuss one empirical implication for each of the assumptions. (7 marks)

(b)    Considering the positions specified in the Table 1, fill in the required intrinsic

values and provide a rough sketch of the combination for a range of expiry prices spanning the given exercise prices.

Table 1

Positions	Maximum profit	Maximum loss	Break- even price(s)	Profit/loss at an expiry price of 160
Buy 1 ABC June £160 Call at 32p
Buy 1 ABC June £180 Put at 38p
Combination of the above positions

(9 marks)

(c)    The price of Wand stock is currently £150. Over the next year the stock price will increase by 8% or decrease by 8%. Risk-neutral probability for an upward price   movement is 0.75. Risk free interest rate is 4% per year. Angie would like to buy

a put on the stock to lock in a guaranteed minimum value of £150 at year-end.

(i)     Suppose the desired put option with exercise price of £150 is traded. What   is the profit/loss on the combination of stock and put option if the stock price at year-end is £100?

(ii)    Angie does not find put options on Wand stock. What combination of stock and risk-free loan positions ensures Angie a payoff equal to the payoff that would be provided by a put option with exercise price of £150? (9 marks)