Hedging Risk with Forwards
and Futures
Derivatives contracts, including forwards and futures have
become a huge market with over $100 trillion in value.
Futures (forward) contracts are agreements between
two agents where one agrees to purchase and the
other to sell (deliver) a given amount of a specific
commodity at a specific price at a future date.
Like an order for a house, car, etc. at a fixed price.
Futures- standardized, traded on exchanges and settled daily.
Forwards - non-standard, traded over the counter and settled
on the maturity date.
Micro-Hedging - Hedging
One Asset or Liability
Situation 1: A futures or forward contract is available for the
asset or liability you wish to hedge.
Example: Suppose you own some 30-year U.S. Treasury
Bonds with a 7% yield and a market value of
$1,000,000. There are one-year futures contracts on
30-year U.S. Treasuries priced at 93-08 (93 8/32)
which is $93.25 per $100 face value of bonds. Each
contract represents $100,000 face value. You believe
interest rates will rise, how do you hedge your bonds?
Answer: Sell $1,000,000 of futures. How many contracts?
Each contract’s market value = $93.25(1000) = $93,250
# of contracts = $1,000,000/$93,250 = 10.72 or 10 contracts.
(Round down because convexity offsets the remainder.)
Cross-hedging Interest Rate
Risk with Futures
Situation 2: A futures or forward contract is not available for
the asset or liability you wish to hedge.
In this situation, we need to cross-hedge. This mean to hedge
with a futures contract on a similar asset.
In general, for an effective hedge, we want the change in the
value of the spot commodity (our asset) to be equal to minus
the change in the value of the futures. The amount of futures
needed per unit of spot is;
h = hedge ratio = -DSpot price / DFutures price
- this means if the spot and futures price move together
(opposite), sell (buy) futures to hedge.
Hedging Your Corporate Bond
with Treasury Futures
You can estimate your bond’s price change as
(1) %DPC   DurC x[(YnC  Yo C ) /(1  Yo C )]
and the price change of the Treasury futures bond as
(2)
% DPT   DurT x[(YnT  YoT ) / (1  YoT )]
Pc, (PT)
Durc, (DurT)
Yoc, (YoT)
Ync , (YoT)
= Price of Corporate (Treasury) bond
= Duration of Corporate (Treasury) bond
= old yield of Corporate (Treasury) bond
= new yield of Corporate (Treasury) bond
Note: We ignore convexity here but it could be added.
Hedge Ratio
To get the hedge ratio, divide equation (1) by (2) to get
%DP Dur DY (1  Y )
h

x
x
%DP Dur DY (1  Y )
C
C
C
T
o
T
T
where
T
DY  Y  Y
C
C
n
o
C
C
o
and DY T  YnT  YoT
h = the units (dollars) of futures to be sold per unit (dollar) of
spot. It assumes a parallel shift in yield curve.
NOTE: This hedges only interest rate risk - default risk is
ignored. If default risk and interest rate risk are
negatively correlated (because GNP increase -> r increase
but default decreases), then less hedging may be required.
Basis Risk
• Whenever we cross-hedge, we are subject to basis risk. The
basis is the difference between the futures price and the spot
price. Basis risk arises when the futures price does not
change one-for-one with the hedged-asset’s spot price.
• Basis risk for hedging corporate bonds using Treasury
futures comes from the middle term in the previous equation.
DYC / DYT
We know the durations and the yields of the corporate bond
and the Treasury bond. But when Treasury yields change by
1%, corporate yields often change by more than 1%. If we
run a regression of past corporate yield changes on Treasury
yield changes we can estimate the relationship between them.
Example
Suppose you own $1,000,000 of a 30-year B-rated corporate
bond with a duration of 14 and a yield of 12%. You plan to
hedge its interest rate risk with the one-year futures contracts
on 30-year U.S. Treasuries priced at 93-08. The bond
deliverable on the Treasury futures has a duration of 16 and a
6% yield. Assume that we run the following regression:
(B-Rated Corporate Yield) = 6.5 + 1.45(Treasury Yield)
This result means that for every 1% change in Treasury
yields, B-rated corporate yields change by 1.45%. We use
this result to get our hedge ratio:
h = (14/16) (1.45) (1.06/1.12) = 1.2
To get the number of futures contracts to sell, first get the
value of Treasury futures required for the hedge:
Value of futures = $1,000,000 (1.20) = $1,200,000.
Value of each futures contract = 93.25 ($100,000) = $93,250
# of contracts to sell = $1,200,000/$93,250 =12.86 or 12.
This same approach can be used to “macro-hedge” a
portfolio of bonds once we know the market-value weighted
average duration of the portfolio. If we are short some bonds,
their weight is a negative and the net value of the portfolio is
reduced by the market value of the shorted bonds.
Macro-Hedging - Hedging a
Duration Gap
Macro-hedging involves hedging the net risk of a portfolio or
balance sheet. Recall that the change in a firm’s equity is:
DEquity = -[DA - kDL]A(Yn - Yo)/(1 + Yo)
A typical function financial firms, especially banks, is to
provide liquidity by taking short-term deposits and investing
in long-term assets such as loans. The firm’s equity serves as
a cushion to absorb interest rate risk. For example, a small
business deposits money in a bank and requires a long-term
fixed-rate loan for new plant. The bank is exposed if interest
rates rise after the loan is made. The bank could require its
customers to make deposits in long-term fixed rate CDs or it
can lay off is duration gap risk in the futures market.
Hedging a Duration Gap
To find the number of contracts (NF) required to hedge the
duration gap, define the change in the value of the futures as
DF = (-DF) (NF) (PF) (Yn - Yo)/(1 + Yo)
where PF is the price of on futures contract.
DEquity = -[DA - kDL]A(Yn - Yo)/(1 + Yo)
Set these two equal so: DEquity = DF
Assuming that (Yn - Yo)/(1 + Yo) is the same for both gives:
-[DA - kDL]A = (-DF) (NF) (PF)
And solving for NF gives
NF = [DA - kDL]A / (DF)(PF)
Example - Ch. 24 - Prob. 17.
Tree Bank has assets of $150 million with duration 6,
liabilities of $135 million with duration 4 and equity of $15
million. Market interest rates are 10%. How do you hedge
the balance sheet with Tbond futures priced are $95 per $100
face value (each contract $100,000 face value) for the
underlying 20 year, 8% coupon bond?
A. Because DA > kDL we need to sell futures to hedge.
B. To get the number of contracts needed for the hedge, we
need DF . The underlying bond has a yield of
95 = 8[PVA20,?] + 100[PV20,?]
Solving for the yield (?) gives Y = 8.525%
Assuming semi-annual coupons, the semi-annual D is
D
(1  .0426) [(1  .0426)  40(.04  .0426)]

 20.10
.0426
[.04[(1  .0426)  1]  .0426]
40
NF = [6 - (.9)4]150 / (20.10/2)(.095) = 377.06
C. Show you are hedged for rate changes of 1% and -.5%.
For a 1% increase.
DEquity = -[6 - (.9)4]150(.01)/(1.01) = -3,272,272
DFutures Value = -10.05(0.01/1.08525)(95,000)(377)
= -3,316,662
Since you sold futures your net position is
Net = -3,272,272 - (-3,316,662) = 43,935
For a .5% decrease.
DEquity = -[6 - (.9)4]150(-.005)/(1.01) = 1,636,363
DFutures Value = -10.05(-0.005/1.08525)(95,000)(377)
= 1,658,331
Since you sold futures your net position is
Net = 1,636,363 - (-1,658,331) = -21,967
D. If we use $1,000,000 face value Tbills with a price of 98
per 100 face value and .25 duration. How many contracts?
NF = [6 - (.9)4]150 / (.25)(.98) = 1469
E. We may use Tbill futures because they are more liquid.
Problems - more basis risk and transactions costs.
Hedging Credit Risk with
Credit Futures or Forwards
Financial firms that make loans can explicitly or implicitly
suffer losses because a borrower’s credit quality declines
after the loan is made - this is credit risk.
Systematic credit risk (an economy-wide decline in credit
quality) can be hedge with credit spread futures.
Firm-specific credit risk (a decline in credit quality for a
single borrower) requires a specialized forward contract.
Below we consider how to hedge with specialized forwards,
however, the process is the same using futures for systematic
credit risk.
Systematic Changes in Yield
Spread Over Time
Typical Situation of a Credit
Spread Forward in Steps
1. As an example, consider a borrower with a $100 million
loan outstanding who also has a benchmark A-rated bond
outstanding.
2. Suppose the lender wishes to avoid a fall in loan value
from an increase in required yield due to a drop in credit
rating.
3. The lender approaches, for example, an insurance
company willing to buy a credit spread forward covering the
$100 million loan.
4. Assume when the forward is constructed, the yield spread
between the borrower’s bond and the comparable duration
Tbond is 2%. If the spread widens (narrows), the insurance
company (lender) pays the lender (insurance company) to
offset its loss (gain).
Calculating Gains and Losses
on Credit Spread Forwards
The credit spread forward’s value is set equal to the
estimated change in the value of the loan, which is:
(SF - ST)(MD)(A)
where SF is the borrower’s bond yield spread observed when
the forward is constructed, ST is the spread at the maturity
date of the forward, MD is the loan’s modified duration and
A is the value of the loan.
Example: Suppose that the yield spread for the $100 million
loan borrower’s bonds fall to 1% from 2 %. If the loan’s
modified duration is 10, and the lender has sold credit futures
to an insurance company, who pays who and how much?
Credit Futures Value = (.02 - .01)(10)(100) = 10
The lender pays the insurance company $10 million.
Using Catastrophe Futures to
Hedge Casualty Risk
1. Suppose you are an insurance company insuring $10
million in home value.
2. There are catastrophe futures contracts for home insurance
that sell at a price of 0.90 and each contract has a face value
of $25,000. The 0.90 price represents the average loss ratio
expected for all insurance firms over the life of the contract.
The loss ratio is the ratio of country-wide insurance
premiums to payments to homeowners for home damage.
3. You are worried that this year will bring many hurricanes
and the loss ratio will be 1.20.
4. How do you hedge with futures?
Note: Futures may not be available on CBOT but options are.
Forwards may be available from private sources.
5. You need to buy catastrophe futures.
6. How many?
# contracts = 10,000,000/25,000 = 400
7. What happens if the loss ratio ends up at 1.20 at the
maturity date?
Your gain = (1.20 - .90)(25,000)(400) = 3,000,000
8. What happens if the loss ratio ends up at 0.80 at the
maturity date?
Your lose = (.80 - .90)(25,000)(400) = 1,000,000
Question: The CBOT is considering offering medical claims
futures similar to catastrophe futures. What problem may
occur with them that can’t occur with catastrophe futures?