I am wondering which rate should I use in the question of solving swap rate, should I use the rate of forward rates or should I use the spot rate for the later periods? (for Q1-3)
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UNIVERSITY OF PENNSYLVANIA
The Wharton School
FNCE 235/725:
FIXED INCOME SECURITIES
Prof. Domenico Cuoco
Spring 2017
Assignment 2
Background
In late 1993, Procter & Gamble (P&G) wanted to replace an existing receiver IRS
that was about to expire. The company’s financial managers expected interest rates
to continue to drop and approached Bankers Trust (BT) expressing interest in negotiating a new $200 million swap that would allow P&G to continue to pay a negative
spread over the Commercial Paper (CP) rate.
BT proposed an exotic 5-year swap, which included an initial negative spread of
75 basis points over the CP rate for the first 6 months, but with the provision that
the spread would be revised after 6 months based on a complex formula that linked
the spread to the Treasury term structure prevailing at that date. Figure 1 shows a
condensed term sheet for the swap, which was stipulated on November 4, 1993.
On February 4, 1994, the Fed raised the Fed Funds target rate by 25 basis points.
As a consequence of the resulting increase in Treasury rates and of the high leverage
implicit in the formula used to determine the revised swap spread, P&G faced the
prospect of having to pay a huge spread over the CP rate after May 4, 1994 (the date
on which the spread was to be revised). In March 1994, P&G decided to unwind the
swap at a loss of over $100 million.
In October 1994, P&G sued BT alleging fraud, misrepresentation, breach of fiduciary duty, negligent misrepresentation, and negligence in connection with the swap
transaction. During pre-trial discovery, P&G obtained from BT over 6,500 audio tapes
of internal telephone conversations. In one tape, two BT employees were recorded
having the following conversation regarding the swap:
– Oh, my ever-loving God. Do they [P&G] understand that? What they did?
– No. They understand what they did but they don’t understand the leverage, no.
– Are you letting greed get in the way of business decisions? I hope that these people
don’t get blown up ’cause that’s the end of the gravy train. […] They would never
know. They would never be able to know how much money was taken out of that …
That’s the beauty of Bankers Trust.
In this Assignment, you will be asked to price the P&G swap and to asses its risk.
Questions
For simplicity, assume throughout that:
• the semi-annually compounded rate used to determine the payments in the
floating leg of the P&G swap was determined by the formula
CP
0.75% + Spread,
where CP is the 6-month Commercial Paper rate at the start of each interest
accrual period (instead of the average 30-day Commercial Paper rate over each
interest accrual period);
• the Commercial Paper term structure was the appropriate forwarding term
structure for the P&G swap while the Treasury term structure was the appropriate discounting term structure;
• the spread between the continuously compounded Commercial Paper and Treasury rates was equal to 22 basis points for all maturities.
In addition, assume that the Treasury term structure on November 4, 1993 was
as estimated in the FEDS 2006-28 paper.1
Answer the following questions making sure to clearly explain the procedure you
followed to determine your answers.
1. What was the fair swap rate on November 4, 1993 for a vanilla 5-year swap with
semi-annual payments on both legs that exchanged the 6-month CP rate for a
fixed rate?
2. What was the fair swap rate on November 4, 1993 for a vanilla 5-year swap with
semi-annual payments on both legs that exchanged the 6-month CP rate minus
75 basis points for a fixed rate?
1
The paper and the estimated daily term structures can be downloaded from the Fed’s website:
http://www.federalreserve.gov/pubs/feds/2006/200628/200628abs.html.
2
3. It is easy to see that the P&G swap is equivalent to a portfolio that consists
of the vanilla receiver swap in part 2 plus a short position in a fixed income
derivative that pays the semi-annual Spread, as determined on May 4, 1994,
on a notional of $200 million on each of the nine semi-annual payment dates
starting November 4, 1994 and ending November 4, 1998. What compensation
was P&G receiving under the terms of the swap for selling this derivative to
Bankers Trust?
4. Figure 2 shows the weekly time series of the 5-year TCM rate and of the average
of the bid and ask clean prices of the 6.25% 8/15/2023 Treasury bond between
November 5, 1993 and May 6, 1994. Produce a graph showing what the Spread
of the P&G swap would have been if determined on each of the dates in Figure 2.
5. Compute the semi-annually compounded 5-year Treasury rate on November 4,
1993 based on the FEDS term structure. On November 4, 1993, the 5-year
TCM rate was 5.03%. What was the spread between the 5-year Treasury rate
and the 5-year TCM rate? What was the reason for this spread?
6. Compute the fair clean price on November 4, 1993 of the 6.25% 8/15/2023
Treasury bond.
7. Compute the fair clean prices on November 4, 1993 of two 6.25% Treasury
bonds, with maturities 5/4/2023 and 11/4/2023, respectively. What is the
error of approximating the fair clean price of the 8/15/2023 bond using linear
interpolation and the fair clean prices of the 5/4/2023 and 11/4/2023 bonds?
8. Use the weekly time series of the 6-month TCM rate between January 8, 1982
and October 29, 1993 posted on Canvas to estimate the annualized standard
deviation of the changes in the 6-month TCM rate and the annualized volatility
of the 6-month TCM rate.
9. Calibrate the Ho-Lee model to the 6-month Treasury rate and construct the
tree for this rate out to 30 years (assume that the standard deviation of the
changes in the 6-month Treasury rate is the same as the standard deviation of
the changes in the 6-month TCM rate estimated in part 8).
10. Compute the tree for the price of a Treasury STRIP maturing on May 4, 1999.
11. Compute the trees for the prices of the two Treasury bonds in part 7.
12. Compute the tree for the price of a deferred annuity that makes semi-annual
payments of $1, with the first payment occurring on November 4, 1994 and the
last payment on November 4, 1998.
3
1. General Terms
Contract date
Maturity date
Notional amount
Fixed rate payer
Floating rate payer
November 4, 1993
November 4, 1998
$200 million
Bankers Trust
Procter & Gamble
2. Fixed Payments
Dates
May 4 and November 4, starting May 4, 1994
Rate
5.30% semi-annually compounded
3. Floating Payments
Dates
May 4 and November 4, starting May 4, 1994
Rate
The semi-annually compounded floating rate is determined by the following formula:
CP
Spread
0.75% + Spread,
where CP is the average of the 30-day Commercial
Paper rate over each interest accrual period.
The Spread is 0% for the first floating payment.
The spread for each floating payment after the first
will be determined on May 4, 1994 by the following
formula:
”
#
98.5
P30
5.78% ⇥ TCM5
max 0,
,
100
where TCM5 is the 5-year Treasury Constant Maturity yield, and P30 the average of the bid and
ask clean prices of the 6.25% 8/15/2023 Treasury
bond (for example, the Spread will be set to 0% if
TCM5 = 5% and P30 = 100 and to 12.2491% if
TCM5 = 6% and P30 = 90).
Figure 1: Condensed term sheet for the P&G swap
5
Date
TCM5
P30
11/05/1993
11/12/1993
11/19/1993
11/26/1993
12/03/1993
12/10/1993
12/17/1993
12/24/1993
12/31/1993
01/07/1994
01/14/1994
01/21/1994
01/28/1994
02/04/1994
02/11/1994
02/18/1994
02/25/1994
03/04/1994
03/11/1994
03/18/1994
03/25/1994
04/01/1994
04/08/1994
04/15/1994
04/22/1994
04/29/1994
05/06/1994
5.03%
5.04%
5.04%
5.13%
5.14%
5.10%
5.18%
5.16%
5.14%
5.21%
5.03%
5.06%
5.05%
5.14%
5.36%
5.40%
5.60%
5.74%
5.85%
5.91%
6.00%
6.19%
6.47%
6.47%
6.60%
6.56%
6.76%
100.56
101.44
98.81
100.09
100.06
100.66
99.59
100.50
98.75
100.28
99.41
99.56
100.44
98.69
97.97
95.19
94.13
92.59
91.81
91.72
90.44
87.66
87.84
87.53
88.16
87.34
84.81
Figure 2: Weekly values of the 5-year TCM rate and of the clean mid price of the
6.25% 8/15/2023 Treasury bond
6
…
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