Calculate the optimal hedge ratio and capm-beta, Portfolio Management

Question 1

An investor would like to buy a futures contract on the ALCOA share. Today's price of the ALCOA share is $17. The maturity of the futures contract is in 6 months and the risk-free interest rate is 6% per annum.

(a) What is the 6-month futures price of the ALCOA share?

(b) If the actual futures price were higher than the no-arbitrage 6-month futures price, which arbitrage could you implement?

Question 2

An investor would like to buy a futures contract on the ALCOA share. Today's price of the ALCOA share is $17. The maturity of the futures contract is in 1 year and the risk-free interest rate is 6% per annum. ALCOA pays a $2 dividend in 3 months from now and a $1 dividend in 9 months from now.

(a) What is the 1-year futures price of the ALCOA share?

(b) If the actual futures price were lower than the no-arbitrage 1-year futures price, which arbitrage could you implement?

Question 3
We are at t=0. A stock is expected to pay a dividend of $10 per share at t=2 months and t=5 months. The stock price is $80, and the risk-free rate is 3.5% per annum. An investor has just taken a short position in a 6-month forward contract on the stock.

(a) What are the forward price and the initial value of the forward contract?

We are now at t=3 (months), the price of the stock is $68 and the risk-free rate of interest is now 3% per annum.

(b) What is the forward price?

(c) What is the value of the short position in the forward contract at this point in time?

Question 4

On BB you will find the file "Q5HW1.xls". This is a Microsoft Excel file.The file contains daily prices on a commodity ("Und") and on the corresponding futures ("Futures").

(a) Bothfor the underlying asset and the futures, compute the price changes ΔS and ΔF.

(b) Compute means, standard deviations, and correlation of the price changes.

Note: the corresponding Excel commands are: AVERAGE, STDEV, and CORREL.

(c) Using the results in (b), calculate the optimal hedge ratio which minimizes risk.Comment on your results.

(d) Suppose that the price exposure is for 1,000,000 units of "Und" and each futures contract is for 55,000 units of the underlying. How many futures contracts should be used to hedge?

Question 5

On BB you will find the file "Q6HW1.xls". This is a Microsoft Excel file.The file contains two time series. "Index" represents the value of an index, and "Portfolio" represents the value of a portfolio in millionsof dollars.

(a) For both the portfolio and the index compute the rate of return Ri = log(P Pi,t /P Pi,t-1)x100, where Pi,t is either the price at time t of the portfolio (i=Port) or the price at time t of the index (i=index).

(b) Compute the CAPM-β of the portfolio with respect to the market.(Assume that the market is represented by the index.)

(c) On December 27th, 2007, the value of the portfolio is $243.54 million. Assume that you wish to use futures contracts on the index to hedge the portfolio's risk. The index on that day is standing at 130.89, and each contract is for the delivery of $250 times the index. What is the hedge (i.e., number of contracts) that minimizes risk?

Question 6

At t=0 the term structure of zero rates is fitted with the following formula:

(a) Plot the term structure of zero rates for T in [0,30] and comment on it.

(b) Compute the price and yield of a 2-year bond providing a semi-annual coupon of $1,000 and a principal of $18,000.

(c) Compute the forward rate between 1 year and 3 years.

Question 7

The 6-month, 12-month, 18-month, and 24-month zero rates are 4%, 4.5%, 4.75% and 5% with semi-annual compounding.

(a) What are the rates with continuous compounding?

(b) What is the forward rate for the 6-month period beginning in 18 months?

(c) What is the value of an FRA that promises to pay you 6% (compounded semi-annually) on a principal of $1 million for the 6-month period starting in 18 months?

Question 8

The futures price for the June 2011 bond futures contract is 118-23.

(a) Calculate the conversion factor for a bond maturing on January 1, 2027, paying a coupon of 10%.

(b) Calculate the conversion factor for a bond maturing on October 1, 2032, paying a coupon of 7%.

(c) Suppose that the quoted prices of the bonds in (a) and (b) are 169.00 and 136.00, respectively. Which bond is cheaper to deliver?

(d) Assuming that the cheaper to deliver bond is actually delivered on June 25, 2011, what is the cash price received for the bond?

Question 9

The December Eurodollar futures contract is quoted as 98.40 and a company plans to borrow $8 million for three months starting in December at LIBOR plus 0.5%.

(a) What rate can the company lock in by using the Eurodollar futures contract?

(b) What position should the company take in the contract?

(c) If the actual three-month rate turns out to be 1.3%, what is the final settlement price on the futures contract?

Question 10

Companies XYZ and ABC have been offered the following rates per annum on a nominal $100mln 5-year loan:

 

Fixed Rate

Floating Rate

Company XYZ

6.80%

LIBOR + 1.0%

Company ABC

8.20%

LIBOR + 2.0%

(a) Which of the two companies is a better company? Explain.

(b) XYZ would like to borrow floating and ABC would like to borrow fixed. You are in the position of the financial intermediary. Design a swap that will net you 0.2% and appears equally attractive to XYZ and ABC.

(c) How should the set up of this problem change if instead of a loan we were dealing with an investment?

Question 11

Under the terms of an interest rate swap, a financial institution has agreed to pay 10% per annum and receive three-month LIBOR in return on a notional principal of $100 million with payments being exchanged every three months. The swap has a remaining life of 14 months. The average of the bid and offer fixed rates currently being swapped for three-month LIBOR is 12% per annum for all maturities (take this as the current term structure of interest rates). The three-month LIBOR rate one month ago was 11.8% per annum. All rates are compounded quarterly.

What is the value of the swap for the financial institution? Value the swap as the difference between two bonds.

Question 12

The price of a stock is $40. The price of a one-year European put option on the stock with a strike price of $30 is quoted as $7 and the price of a one-year European call option on the stock with a strike price of $50 is quoted as $5. Suppose that an investor buys 100 shares, shorts 100 call options, and buys 100 put options.

(a) Draw a diagram illustrating how the investor's profit or loss varies with the stock price over the next year.

(b) How does your answer change if the investor buys 100 shares, shorts 200 call options, and buys 200 put options?

Question 13

Draw a diagram showing the variation of investor's profits and losses for the terminal value of a portfolio consisting of:

(a) One share and a short position in a call option. Interpret.

(b) Two shares and a short position in a call option. Interpret.

(c) One share and a short position in two call options. Interpret.

Question 14

The price c of a European call with strike K=$40 on a non-dividend paying stock with initial price S=$42 and maturity T=6 months is c=$1. The interest rate in the market is r=3%.

Devise, if it is possible, an arbitrage strategy in this market.

Posted Date: 2/16/2013 2:07:30 AM | Location : United States







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