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# PROBLEM SET 3 Fixed Income Markets BKMME138 VIOREL ROSCOVAN Due April 28, 2010 at noon in my mailbox (T9-25) Problem Set 3.A: Floaters, Inverse

1. Today, time 0, a newly issued (zero cost) 1-year semi-annual pay plain vanilla interest rate swap has a
swap rate of 8%. The time 0.5 cash flow to the counterparty who is short \$100 notional amount of the
swap is \$1.00. Assume there are no swap spreads.
(a) What is the 1-year par rate?
(b) What is the 0.5-year zero rate?
(c) What is the price of \$1 par of a 1-year zero?
(d) What is the value of \$100 par of a 1-year semi-annual pay inverse floating rate note that pays
16% minus floating?
2. (a) Suppose the 0.5-year zero rate is 6% and the 1-year zero rate is 8%. Consider a 1-year, plain
vanilla, semi-annual pay, fixed-for-floating interest rate swap.
i. What is the swap rate that will make this swap worth zero?
ii. What is the dollar duration of \$100 notional amount of this zero-cost swap?
(b) Your liabilities have a market value of \$100,000 and a duration of 3. Your assets have a market
value of \$100,000 and a duration of 5. Determine the notional amount of a position in the swap
from part (a) that you require to immunize your net position against parallel shifts in interest
rates. Ignore convexity
1. At time 0.5, the price of \$1 par of a zero maturing at time 1 will be either \$0.96 or \$0.98. The current
price of the zero maturing at time 1 is \$0.94 and the current price of the zero maturing at time 0.5 is
\$0.97. Consider also a claim that pays off \$1 at time 0.5 if the zero maturing at time 1 is worth \$0.96,
and 0 otherwise. This information is summarized in the payoff diagrams below.
(a) Determine a portfolio of the 0.5- and 1-year zeroes that has the same payoff as the claim at time
0.5.
(b) What is the value of the claim today in the absence of arbitrage?
(c) What are the risk-neutral probabilities of the two possible time 0.5 values of the zero maturing
at time 1?
2. At time 0.5, the price of \$1 par of a zero maturing at time 1 will be either \$0.96 or \$0.98. The riskneutral
probability of the each outcome is 50%. The current price of \$1 par of a zero maturing at time
0.5 is 0.97.
(a) What is the price at time 0 of the zero maturing at time 1 in the absence of arbitrage?
(b) Multiple choice question: Which of the two zeroes above has the higher true expected return from
time 0 to time 0.5? Pick one answer.
i. The 0.5-year zero.
ii. The 1-year zero.
iii. They have the same true expected return.
iv. There is not enough information provided to tell.
3. At time 0.5, the price of \$1 par of a zero maturing at time 1 will be either \$0.97 or \$0.99. Consider
two other assets. Asset #1 pays off \$1 at time 0.5 if the price of the zero maturing at time 1 is \$0.97
and pays off 0, otherwise. Asset #2 pays off \$1 at time 0.5 if the price of the zero maturing at time
1 is \$0.99 and pays off 0, otherwise. The prices today of Assets #1 and #2 are both \$0.49. This
information is summarized below.
(a) What is the price today of the zero maturing at time 1 in the absence of arbitrage?
(b) What is the price today of the zero maturing at time 0.5 in the absence of arbitrage?
4. At time 0, the zero maturing at time 0.5 has a price of 0.97 per \$1 par, and the zero maturing at time
1 has a price of 0.9409 per \$1 par. At time 0.5, there will be one of two possible states: in the “up”
state, the price of \$1 par of a zero maturing at time 1 will be \$0.96; in the “down” state, the price of
this zero will be \$0.98. The risk-neutral probability of the each outcome is 50%. This information is
summarized below:
Consider a 0.5-year floating rate note with rate set in arrears. This note has a single payoff at time 0.5
equal to par plus a coupon based on the 0.5-year rate observed at time 0.5. In terms of the notation
from class, for each \$1 par, the cash flow at time 0.5 is 1 + 0:5r1=2
(a) What is the payoff of \$100,000 par of this note at time 0.5, in the up state?
(b) What is the payoff of \$100,000 par of this note at time 0.5, in the down state?
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(c) Use the risk-neutral probabilities to compute the value of this note at time 0.
(d) Write down, but do not solve, equations that determine the par amounts N0:5 and N1 of the
zeroes maturing at time 0.5 and time 1 to hold in a portfolio that replicates the time 0.5 payoff
of the rate-set-in-arrears floater.
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PROBLEM SET 3 Fixed Income Markets BKMME138 VIOREL ROS ¸COVAN Due April 28, 2010 at noon in my mailbox (T9-25) Problem Set 3.A: Floaters, Inverse Floaters, Swaps 1. Today, time 0, a newly issued (zero cost) 1-year semi-annual pay plain vanilla interest rate swap has a swap rate of 8%. The time 0.5 cash ﬂow to the counterparty who is short \$100 notional amount of the swap is \$1.00. Assume there are no swap spreads . (a) What is the 1-year par rate? (b) What is the 0.5-year zero rate? (c) What is the price of \$1 par of a 1-year zero? (d) What is the value of \$100 par of a 1-year semi-annual pay inverse ﬂoating rate note that pays 16% minus ﬂoating? 2. (a) Suppose the 0.5-year zero rate is 6% and the 1-year zero rate is 8%. Consider a 1-year, plain vanilla, semi-annual pay, ﬁxed-for-ﬂoating interest rate swap. i. What is the swap rate that will make this swap worth zero? ii. What is the dollar duration of \$100 notional amount of this zero-cost swap? (b) Your liabilities have a market value of \$100,000 and a duration of 3. Your assets have a market value of \$100,000 and a duration of 5. Determine the notional amount of a position in the swap from part (a) that you require to immunize your net position against parallel shifts in interest rates. Ignore convexity. Problem Set 3.B: No Arbitrage Pricing and Risk-Neutral Proba- bilities 1. At time 0.5, the price of \$1 par of a zero maturing at time 1 will be either \$0.96 or \$0.98. The current price of the zero maturing at time 1 is \$0.94 and the current price of the zero maturing at time 0.5 is \$0.97. Consider also a claim that pays oﬀ \$1 at time 0.5 if the zero maturing at time 1 is worth \$0.96, and 0 otherwise. This information is summarized in the payoﬀ diagrams below. (a) Determine a portfolio of the 0.5- and 1-year zeroes that has the same payoﬀ as the claim at time 0.5. (b) What is the value of the claim today in the absence of arbitrage?
(c) What are the risk-neutral probabilities of the two possible time 0.5 values of the zero maturing at time 1? 2. At time 0.5, the price of \$1 par of a zero maturing at time 1 will be either \$0.96 or \$0.98. The risk- neutral probability of the each outcome is 50%. The current price of \$1 par of a zero maturing at time 0.5 is 0.97. (a) What is the price at time 0 of the zero maturing at time 1 in the absence of arbitrage? (b) Multiple choice question: Which of the two zeroes above has the higher true expected return from time 0 to time 0.5? Pick one answer. i. The 0.5-year zero. ii. The 1-year zero. iii. They have the same true expected return. iv. There is not enough information provided to tell. 3. At time 0.5, the price of \$1 par of a zero maturing at time 1 will be either \$0.97 or \$0.99. Consider two other assets. Asset #1 pays oﬀ \$1 at time 0.5 if the price of the zero maturing at time 1 is \$0.97 and pays oﬀ 0, otherwise. Asset #2 pays oﬀ \$1 at time 0.5 if the price of the zero maturing at time 1 is \$0.99 and pays oﬀ 0, otherwise. The prices today of Assets #1 and #2 are both \$0.49. This information is summarized below. (a) What is the price today of the zero maturing at time 1 in the absence of arbitrage? (b) What is the price today of the zero maturing at time 0.5 in the absence of arbitrage? 4. At time 0, the zero maturing at time 0.5 has a price of 0.97 per \$1 par, and the zero maturing at time 1 has a price of 0.9409 per \$1 par. At time 0.5, there will be one of two possible states: in the “up” state, the price of \$1 par of a zero maturing at time 1 will be \$0.96; in the “down” state, the price of this zero will be \$0.98. The risk-neutral probability of the each outcome is 50%. This information is summarized below: Consider a 0.5-year ﬂoating rate note with rate set in arrears . This note has a single payoﬀ at time 0.5 equal to par plus a coupon based on the 0.5-year rate observed at time 0.5. In terms of the notation from class, for each \$1 par, the cash ﬂow at time 0.5 is 1 + 0 . 5 r 1 / 2 (a) What is the payoﬀ of \$100,000 par of this note at time 0.5, in the up state? (b) What is the payoﬀ of \$100,000 par of this note at time 0.5, in the down state? 2
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