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# Problem Set 2

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University of Toronto Scarborough

Finance

MGFC30H3

Ata

Fall

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UNIVERSITY OF TORONTO SCARBOROUGH
DEPARTMENT OF MANAGEMENT
MGTC71: Introduction to Derivatives Markets
Problem Set – 2 (Supplemental Questions)
_______________________________________________________________________________
Interest Rate & Interest Rate Futures: (Required for Test-1)
Problem-1
A firm entered into a forward rate agreement with a bank. The firm agreed to pay the bank a semi-
annually compounded interest rate of 4% per annum on a notional deposit of $6 million for a 6-month
period starting in two months. The current two- month and eight-month USD LIBOR rates are 2% p.a.
and 2.5% p.a., respectively. What is the value of this forward rate agreement to the firm?
Problem-2
a) You manage a portfolio of 10,000 5 years zero bonds with a face value of 1,000 and YTM of 5%
continuously compounded. You think his portfolio is too risky and intend to bring its duration down to 0
by selling the original portfolio and reinvesting the proceeds into 3 months treasury bills. What is his new
portfolio allocation (how much long (or short) in 5 years bond, how much in treasury bills).
b) As a bond portfolio manager you have just sold 10,000 unit of 10 year zero coupon bonds with a face
value of 1,000 and YTM of 8%. This manager wants to hedge his position and thinks one of three T-
bonds will be cheapest to deliver; some data for these three bonds are as follows:
Bond Quoted Price ($) Conversion Factor Duration inMarch
A 99.5 1.0382 12.4 years
B 143.5 1.5188 13.2 years
C 119.75 1.2615 15.7
b1) If the March futures bond price is quoted at 93-08 which bond is cheapest to deliver? Explain why by
showing calculations.
b2) How should the portfolio manager use March T-bond futures to produce a new portfolio having an
overall duration of zero? How many contracts? Long or short?
Problem-3
A portfolio manager manages a bond portfolio of 5,000 5 years zero bonds with a face value of 1,000 and
YTM of 3% continuously compounded. The standard deviation of the change in the yield is assumed to
be 2%. You think his portfolio is too risky and intend to bring its duration down to 0 by selling the
original portfolio and reinvesting the proceeds into 6 months treasury bills. What is his new portfolio
allocation (how much long (or short) in 5 years bond, how much in treasury bills).
1 b) The futures price for March T-bonds is 96-27. The manager of a $100M bond portfolio, the duration of
which will be 6.7 years in March, wishes to use these futures for hedging this portfolio. This manager
thinks one of two T-bonds will be cheapest to deliver; some data for these two bonds are as follows:
Bond Quoted Price ($) Conversion Factor Duration inMarch
A 111-22 1.1462 9.3 years
B 106-26 1.0987 10.2 years
b1) Which bond is cheapest to deliver? Explain why by showing calculations.
b ) How should the portfolio manager use March T-bond futures to produce a new portfolio having an
2
overall duration of zero? How many contracts? Long or short? Verify your solution i.e. show that your
new combined portfolio is perfectly hedged.
Swap: (Required for Test-2)
Problem-1
Suppose company A and B encounter the following borrowing terms:
Fixed % Floating %
Company A 3.6 LIBOR + 1
Company B 4.0 LIBOR + 2
A wants a fixed rate loan while B wants a floating rate loan.
a) If the swap rate is 2.5%, show that both A and B would benefit from the swap. What is the gain?
b) What is he maximum gain from swap? Find a swap rate that splits this gain evenly between A and B.
Problem-2
Companies A and B face the following borrowing costs:
Fixed rate Floating rate
A 7% 6-m LIBOR
B 9% 6-m LIBOR + 1%
a) Explain any comparative advantage A and/or B mayhave in fixed rate and/or floating borrowing. If the
two sides decide to engage in a swap what would be is maximum gain possible?
b) Suppose B wants a fixed rate loan while A wants a floating rate loan.
If the swap rate is 7.5%, show that both A and B would gain from the swap. What is the gain?
Ignore Parts (a) and (b): Suppose the one-year zero rate is 4% and the 2-year zero rate is 6% both
continuously compounded. Consider a 2-year, annual pay, fixed-for-floating interest rate swap between
risk-free borrowers.
a) Find the equilibrium swap rate as of today?
2 b) If after three months the zero curve flatten to 5%, what would be the value of the swap contract. Who
gained and who lost?
Problem 3:
A Corp. wishes to be a floating-rate dollar borrower, which it can be at LIBOR + 1%. B Corp. strongly
prefers fixed-rate debt, but it must pay 1.5% more than the 6.25% coupon that A’s notes would carry.
a) If B Corp, can borrow LIBOR +1.5%. What is the maximum possible cost savings to A. from
engaging in a currency swap with B.? What should be Swap rate for A. to enjoy this maximum gain?
a) If B Corp, can borrow LIBOR +.5%. What is maximum gain from the swap trade? Suppose a bank
charges 0.8% to arrange the swap and A and B split the resulting cost savings. How much A will pay for its
floating-rate money and B will pay for its fixed-rate money
Ignore Parts (a) and (b): Suppose the LIBOR spot rate are given by the table below. Consider a 3-year
plan vanilla interest swap whose float payments are the LIBOR rate.
Maturity LIBOR
1.0 1.0
2.0 1.5
3.0 2.0
a) Find the par swap rate?
b) If after 6 months the yield curve flattens to 2%, what would be the value of the Swap contract. Who
will lose and who will gain by how much?
3 Solutions
Interest Rate & Interest Rate Futures
Problem-1:
We need the six-month forward rate but since we only learned the continuously compound version of the forward
rate formula I first convert the 2 months and 8 months discrete zero rates to CC version, find the forward rate then
convert the forward rate back to discrete version.
0.02
r2c= 6 ln(1 + ) 1.997%
6
0.025
r8c= 1.5 ln(1 + ) = 2.479%
1.5
f = 0.6667×2.479% −0.1667×1.997% = 2.64
c

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