Topic 1: Risk and Return




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  • Answer ALL
  • This is group coursework. You can find your group in the KEATS announcement sent 11 May.
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  • The coursework will be graded out of 100. The number of points that each question is worth is written in brackets.























Topic 1: Risk and Return


Question 1 [15 marks]

Suppose you hold an asset that delivers a return 𝑅 . You wish to hedge against a decline in the price of this asset using a risk-free asset with return 𝑟 and a market index with return 𝑅 . Using the CAPM, explain how you would allocate money between the risk-free asset and the market index so as to minimize your portfolio variance.


Question 2 [6 marks]

Determine whether the following statements are true/false. Justify your answers in no more than 100 words for each statement.

  1. IBM’s stock has expected return 10% and standard deviation 15%. Coca-Cola’s stock has expected return 12% and standard deviation 13%. No investor should ever buy IBM’s stock.
  2. Diversification can reduce risk only when asset returns are negatively correlated.
  3. As more securities are added to a portfolio, the total risk of the portfolio is expected to fall at a decreasing rate.



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Question 3 [14 marks]

Consider stocks of Firms A and B. Their expected returns are 12% and 11%, respectively, and the volatilities of their returns are 8% and 10%, respectively. Firm A has a market beta of 1.5 and Firm B has a market beta of 1.0, and the correlation between the two stocks is 0.5.


Your risk-averse client wants to invest $5 million with an expected return of 15%. You can put together a portfolio with the market index, Firm A, Firm B, and a risk-free asset. Which of these four securities would you include in the portfolio and how much would you invest in each security?




Topic 2: Forwards and Futures


Question 1 [20 marks]

A 30-day Federal Fund (FF) futures contract is an instrument that enables banks to lock in a Federal funds interest rate for the 30-day period ending on the expiration of the contract.


FF futures are quoted as prices rather than as rates. The price quoted is 100 minus the 30-day Fed funds rate, with the rate being expressed as a percentage (not in decimal form). For example, if you buy/sell 1

FF futures at a price of $96.82, you lock in a Fed funds rate of 100 − 96.82 = 3.18% on your future lending/borrowing. The security underlying the FF futures is a $5,000,000 30-day deposit or loan.

Consider a bank that has surplus cash of $10 million. The bank plans to lend this $10 million in the Fed funds market over June but to hedge its risk, it decides to lock in a return by buying FF futures that expire on June 30.


  1. Calculate the change in the value of one FF futures contract for a 1 basis point change in the Fed funds rate.
  2. Suppose that the bank bought the futures today for $99.780, and their price at expiration (June 30) was $99.824. Explain how the bank will meet its objective of locking in a return on its Fed funds lending over June.


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Question 2 [10 marks]

A pension fund currently has $50 million in the S&P 500 index and $50 million in Treasury bills (risk-free asset). The current 6-month S&P 500 futures price is 1390.5, and one contract is on $250 times the index.

  1. Suppose that the manager is concerned about the performance of the index over the next 6 months. Explain what position in the S&P 500 futures he should take to eliminate all exposure to the market over the next six months.
  2. If the manager decides to switch to a portfolio that invests 30% in the index and 70% in T-bills for a period of 6 months, explain how he could achieve that using S&P 500 futures.





Topic 3: Options and Greeks



Question 1 [10 marks]

A portfolio manager wants to hedge his stock portfolio against changes in stock prices. Construct a strategy using options so that changes in the portfolio value are limited in both directions. What is the cost of this hedge to the manager?


Question 2 [10 marks]

You are long a call option on stock ABC. You have delta-hedged your position. You hear on the radio that the CEO of ABC has been arrested for running a massive Ponzi scheme. The stock price of ABC drops. Explain (qualitatively) how you would adjust your hedge.


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Question 3 [15 marks]

Consider the following options on stock X:

Option Delta Gamma Vega
A 0.5 2.2 1.8
B 0.8 0.6 0.2
C -0.40 1.3 0.7
D 0.70 1.8 1.4


A bank is long 1000 units of Option A, short 200 units of Option B, short 2000 units of Option C, and short 500 units of Option D.

The following two options are also available in the market:

Option Delta Gamma Vega
E 0.6 1.5 0.8
F 0.1 0.5 0.6


  1. Design one strategy that would make the bank’s portfolio delta neutral.
  2. Design one strategy that would make the bank’s portfolio delta and gamma neutral. Explain why this strategy provides a better hedge with respect to changes in the price of stock X than the strategy you constructed in part (a).
  3. Design one strategy that would make the bank’s portfolio delta, gamma, and vega neutral.