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Slide 25.17 Interest Rate Futures Contracts
.A Pricing of Treasury Bonds
Slide 25.18 –
Slide 25.19 Pricing of Treasury Bonds
Recall the general expression for the value of a bond:
Bond value = present value of coupons + present value of par
Bond value = [C[1 – 1/(1+R)T] / R] + [FV / (1+R)T]
This formula assumes a flat yield curve. If this is not the case, then
each cash flow must be discounted at the rate specific to the timing
of the cash flow.
.B Pricing of Forward Contracts
Slide 25.20 Pricing of Forward Contracts
Slide 25.21 –
Slide 25.22 Pricing of Forward Contracts: Example
With the forward contract, you are agreeing to purchase the bond
at a specified point in the future. The value of the bond at that time
is just the present value of the subsequent cash flows. This price
can then be discounted to the present to find the value at time 0.
.C Futures Contracts
Slide 25.23 Pricing of Futures Contracts
Futures contracts are priced similarly to forwards, with the
exception of the daily resettlement.
.D Hedging in Interest Rate Futures
Slide 25.24 Hedging in Interest Rate Futures
25.2. Duration Hedging
Slide 25.25 –
Slide 25.26 Duration Hedging
Duration is a measure of interest rate (i.e., price) risk.
.A The Case of Zero Coupon Bonds
Long term bonds are more sensitive to changes in interest rates.
.B The Case of Two Bonds with the Same Maturity but with Different
Coupons
Low coupon bonds are more sensitive to changes in interest rates.
.C Duration
Slide 25.27 Duration Formula
Slide 25.28 –
Slide 25.29 Calculating Duration: Example
Slide 25.30 Duration
In 1938, Frederick MaCaulay defined the duration of an asset with
cash payments (C1, ... ,CT) as the weighted average maturity of
an asset stated in terms of present values:
T
tT
t
t
t
t
t
t
t
R
C
R
C
tDuration
1
11
1
Hicks (see also Hopewell and Kaufman [1974]) independently
developed the same measure in 1939 and noted that it represented
the price elasticity of an asset with respect to a change in the level
of interest rates. The relationship between changes in interest rate,
changes in bond prices, and duration is:
0
01
0
1
1
1R
RR
Duration
P
P
Lecture Tip: You may want to show students that Excel has a built
in function for finding duration. It is =duration(…).
.D Matching Liabilities with Assets
By matching the duration of financial assets and liabilities, a
change in interest rates has the same impact on the value of the
assets and liabilities, leaving the value of equity unchanged.
Duration as a measure of price elasticity is important in many
portfolio management applications. In banking, matching the
duration of financial assets and liabilities is referred to as "asset-
liability management." In insurance, duration hedging is used in
"insulating" a portfolio of assets and liabilities against changes in
interest rates. The basic concept has applications in corporate
finance as well. Following is an example of hedging interest rate
risk using futures contracts.
Example of an Interest-rate Futures Hedge
Your firm has just leased a downtown hotel to a national hotel
chain. The lessee has agreed to pay your firm $10M per year for
the next 20 years. You can hedge the risk of a rise in inflation (and
hence a fall in the value of the lease contract) over this period by
forming a short hedge in the T-bond futures market.
To keep the analysis simple, assume that a 10% discount rate
applies to both the lease receipts and the T-bonds. The present
value of the 20-year lease contract is $85.136M. The duration of
the lease contract is calculated as:
Year CF PV (r=10%) PV Weight Year × Wt
1 10M 10M/1.1 = 9.0909M 9.0909/85.1356 = .1068 1 × .1068 = .1068
2 10M 10M/1.12 = 8.2645M 8.2645/85.1356 = .0971 2 × .0971 = .1941
. . .
. . .
20 10M 10/1.1 20
=1.4864M 1.4864/85.1356 = .0175 20 × .0175 = .3492
Total PV = 85.1356M Duration = 7.508
years
A Short Futures Hedge
Suppose there is a futures contract trading on an exchange based
on 12-year T-Bonds with 10% coupons selling at par. The duration
of the T-Bond is 7.495 years.
An offsetting short-futures position in the T-Bond can hedge the
interest rate risk of the expected future lease receipts. The firm can
sell $85.1M worth of T-Bond (85.1 contracts @$100,000 each)
using futures contracts. This short position has a duration of 7.495
years. The amount and duration of the financial asset (the lease
receipts) are very close to those of the financial liability (the
futures obligation). Consequently, the hedged position should be
relatively insensitive to changes in interest rates.
Suppose the interest rate increases from 10% to 12%. The effects
on the long (lease) and short (T-Bond futures) positions are as
follows:
Lease Receipts T-Bond Futures Asset – Liabilities
Value at r = 10% $85.136M $85.136M $0.000M
Value at r = 12% $74.694M $74.588M $0.106M
Change in value $10.442M $10.548M $0.106M
When interest rate increases to 12%, your firm loses on the lease
agreement, which falls in value by $10.442M to $74.694M. The
value of the futures contracts obligation also decreases (by
$10.548M to $74.588M). The value of your firm's net position
actually increases by $0.106M. In terms of reducing risk, the
change in value on the net position ($0.106M) is substantially
smaller than the change in value of the unhedged position
($10.442M).
This example was not a perfect hedge for two reasons. First, the
duration of the lease and T-bond futures contracts were not
identical. Second, matching duration only provides a perfect hedge
if the change in interest rate is infinitesimal and the yield curve is
flat.
The duration of the lease and the T-bond also change when interest
rate changes. The new duration for the lease is 7.020 years and for
the T-bond it is 7.185 years when interest rate is 12%. Notice that
the difference in duration widens at the new interest rate. To
minimize the exposure of the net position, you can close the
existing futures contract and sell $74.694M of new futures
contracts of 12% T-Bond with duration = 6.938 years. To hedge
interest rate risk effectively you must rebalance your positions
when the interest rate changes.
Lecture Tip: Duration and Hedging
1. Duration and the Yield Curve
Ideally, the yield curve is based on pure discount bonds. The term
structure of interest rates refers to the current spot rates of interest
on pure discount bonds that differ only in their maturity. In
practice, the difficulty lies in inferring the term structure of pure
discount bonds from the available selection of coupon-bearing
bonds. Duration provides a surrogate for the maturity of a pure
discount bond because the duration of a pure discount bond is its
maturity.
2. The Effectiveness of Hedging Risk using Duration Matching
While duration has a wide following in the practitioners'
literature, it has often been criticized in academic circles.
Duration as a measure of price elasticity suggests a linear
relationship between duration and return. Several studies (e.g.,
Gultekin and Rogalski [1976]) have demonstrated that duration
and return are not linearly related for discrete bond price changes.
Gultekin and Rogalski [1984] and Ingersol, Skelton, and Well
[1977] found that Macaulay's duration performed no better than
simple maturity in explaining bond returns. Both of these latter
studies suggest multiple factor models to explain the price
volatility of bonds in response to changes in interest rates (see
Richard [1978] and Schaefer and Schwartz [1978]). Cox,
Ingersol, and Ross [1979] derive a general measure of price
volatility based on a model of the term structure which assumes
that interest rates change through a stochastic process.
25.3. Swaps Contracts
Slide 25.31 Swaps Contracts
Swaps are arrangements between two counterparties to exchange
cash flows over time. Thus, swaps are essentially a series of
forward contracts.
Slide 25.32 The Swap Bank
The swap bank acts as either a broker (matching counterparties) or
dealer (serving as one of the counterparties).
.A Interest Rate Swaps
Slide 25.33 –
Slide 25.41 An Example of an Interest Rate Swap
Just as two companies can agree to exchange currencies at specific
future dates, they can also agree to exchange the cash flows
associated with respective loan agreements.
Interest rate swaps are generally used to convert a fixed rate obligation
to a floating rate obligation, or vice versa, depending on the needs
of the company.
Only the net interest payment is exchanged since we are dealing with
one currency.
.B Currency Swaps
Slide 25.42 –
Slide 25.48 An Example of a Currency Swap
Two firms agree to exchange a specific amount of one currency for a
specific amount of another currency at specific future dates.
Lecture Tip: The following example illustrates that a currency swap is
essentially a “parallel loan.”
Example: Two multinational companies with foreign projects
need to obtain financing. Company A is based in England and has
a U.S. project. Company B is based in the U.S. and has a British
project.
1. Both firms want to avoid exchange rate fluctuations.
2. Both firms receive currency for investment at time zero and repay
loans as funds are generated by the foreign project.
3. Both firms could avoid exchange rate fluctuations if they arrange
loans in the country of the project. Funds generated in England for
the U.S. company (B) would be in pounds and repayment would be
in pounds. The opposite would be true for the British company.
4. Both firms can borrow cheaper in their home countries than they
can in the country where the project originates.
The firms arrange loans for the initial investment in their home
currency and then use the proceeds from the project, converted to
the home currency through a swap agreement, to repay the loan.
Cash flows for the swap: Assume a fixed exchange rate of $2
per £1, a fixed interest rate of 10% for both firms, and a four-year
loan.
Year 0 1 2 3 4
Company
A
-£100,000
+
$200,000
-$20,000
+
£10,000
-$20,000
+
£10,000
-$20,000
+
£10,000
-$20,000
+
£10,000
Company
B
-$200,000
+
£100,000
-£10,000
+
$20,000
-£10,000
+
$20,000
-£10,000
+
$20,000
-£10,000
+
$20,000
These cash flows are the same as those for a parallel loan. The
firms have effectively fixed the exchange rate for the $200,000
(£100,000) loan and enough of the cash flows to repay the loans in
the home currencies.
.C Credit Default Swaps
Slide 25.49 Credit Default Swaps
A Credit Default Swap (CDS) effectively serves as insurance
against the default of a bond. The first counterparty (protection
buyer) pays a periodic CDS spread to the second counterparty
(protection seller) in exchange for covering the full bond value
should a default occur.
The terms and structure seem clear; however, the lack of an
organized exchange creates significant counterparty risk.
Lecture Tip: Obviously, a discussion of Lehmann Brothers and
AIG would be particularly relevant at this point.
.D Exotics
Slide 25.50 Variations of Basic Swaps
Slide 25.51 –
Slide 25.52 Risks of Interest Rate and Currency Swaps
Slide 25.53 Pricing a Swap
25.4. Actual Use of Derivatives
Slide 25.54 Actual Use of Derivatives
The use of derivatives, particularly interest rate and currency, is
widespread, but may be primarily concentrated in large, publicly-
held corporations.
Slide 25.55 Quick Quiz
