Problem Set 2

Modules 3 and 4

• A one-year long forward contract on a non-dividend-paying stock is entered into when the stock price is $38 and the risk-free rate of interest is 8% per annum with continuous compounding.
• What are the forward price and the initial value of the forward contract?
• Six months later, the price of the stock is $46 and the risk-free interest rate is still 8%.? What are the forward price and the value of the forward contract?

• The spot price of oil is $110 per barrel and the cost of storing a barrel of oil for one year is $3.25, payable at the end of the year. The risk-free interest rate is 6% per annum, continuously compounded. What is an upper bound for the one-year futures price of oil?

What are the forward price and the initial value of the forward contract?

Three months later, the price of the stock is $48 and the risk-free rate of interest is still 8% per annum. What are the forward price and the value of the short position in the forward contract?

Show that both portfolios have the same duration.

Show that the percentage changes in the values of the two portfolios for a 0.1% per annum increase in yields are the same.

What are the percentage changes in the values of the two portfolios for a 5% per annum increase in yields?

• A financial institution has entered into a 10-year currency swap with company Y. Under the terms of the swap, the financial institution receives interest at 3% per annum in Swiss francs and pays interest at 8% per annum in U.S. dollars. Interest payments are exchanged once a year. The principal amounts are 7 million dollars and 10 million francs. Suppose that company Y declares bankruptcy at the end of year 6, when the exchange rate is $0.80 per franc. What is the cost to the financial institution? Assume that, at the end of year 6, the interest rate is 3% per annum in Swiss francs and 8% per annum in U.S. dollars for all maturities. All interest rates are quoted with annual compounding.

• Assume that the default probability for a company in a year, conditional on no earlier defaults is and the recovery rate is. The risk-free interest rate is 5% per annum. Default always occur half way through a year. The spread for a five-year plain vanilla CDS where payments are made annually is 120 basis points and the spread for a five-year binary CDS where payments are made annually is 160 basis points. Estimate ? R?and?. ?

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Problem Set 2
Modules 3 and 4

1. A one-year long forward contract on a non-dividend-paying stock is entered into
when the stock price is $38 and the risk-free rate of interest is 8% per annum with
continuous compounding.
a) What are the forward price and the initial value of the forward contract?
b) Six months later, the price of the stock is $46 and the risk-free interest rate is
still 8%. What are the forward price and the value of the forward contract?

2. The spot price of oil is $110 per barrel and the cost of storing a barrel of oil for one
year is $3.25, payable at the end of the year. The risk-free interest rate is 6% per
annum, continuously compounded. What is an upper bound for the one-year futures
price of oil?

3. A stock is expected to pay a dividend of $1 per share in two months and in five
months. The stock price is $50, and the risk-free rate of interest is 8% per annum
with continuous compounding for all maturities. An investor has just taken a short
position in a six-month forward contract on the stock.
a) What are the forward price and the initial value of the forward contract?
b) Three months later, the price of the stock is $48 and the risk-free rate of interest
is still 8% per annum. What are the forward price and the value of the short
position in the forward contract?

4. It is May 5, 2013. The quoted price of a government bond with a 12% coupon that
matures on July 27, 2024, is 110-24. What is the cash price?

5. It is July 30, 2015. The cheapest-to-deliver bond in a September 2015 Treasury

bond futures contract is a 13% coupon bond, and delivery is expected to be made
on September 30, 2015. Coupon payments on the bond are made on February 4
and August 4 each year. The term structure is flat, and the rate of interest with
semiannual compounding is 12% per annum. The conversion factor for the bond is
1.5. The current quoted bond price is $110. Calculate the quoted futures price for the
contract.

6. Portfolio A consists of a one-year zero-coupon bond with a face value of $2,000 and
a 10-year zero-coupon bond with a face value of $6,000. Portfolio B consists of a
5.95-year zero-coupon bond with a face value of $5,000. The current yield on all
bonds is 10% per annum.
(a) Show that both portfolios have the same duration.
(b) Show that the percentage changes in the values of the two portfolios for a
0.1% per annum increase in yields are the same.
(c) What are the percentage changes in the values of the two portfolios for a 5%
per annum increase in yields?

7. A financial institution has entered into a 10-year currency swap with company Y.
Under the terms of the swap, the financial institution receives interest at 3% per
annum in Swiss francs and pays interest at 8% per annum in U.S. dollars. Interest
payments are exchanged once a year. The principal amounts are 7 million dollars
and 10 million francs. Suppose that company Y declares bankruptcy at the end of
year 6, when the exchange rate is $0.80 per franc. What is the cost to the financial
institution? Assume that, at the end of year 6, the interest rate is 3% per annum in
Swiss francs and 8% per annum in U.S. dollars for all maturities. All interest rates
are quoted with annual compounding.

8. The LIBOR zero curve is flat at 5% (continuously compounded) out to 1.5 years.
Swap rates for 2- and 3-year semiannual pay swaps are 5.4% and 5.6%,
respectively. Estimate the LIBOR zero rates for maturities of 2.0, 2.5, and 3.0 years.
(Assume that the 2.5-year swap rate is the average of the 2- and 3-year swap rates
and use LIBOR discounting.)

9. A company enters into a total return swap where it receives the return on a corporate
bond paying a coupon of 5% and pays LIBOR. Explain the difference between this
and a regular swap where 5% is exchanged for LIBOR.

10. Assume that the default probability for a company in a year, conditional on no earlier
defaults is and the recovery rate is . The risk-free interest rate is 5% per annum.
R

Default always occur half way through a year. The spread for a five-year plain vanilla
CDS where payments are made annually is 120 basis points and the spread for a
five-year binary CDS where payments are made annually is 160 basis points.
Estimate
and .
R