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Chapter 21

Removing Interest Rate Risk

Portfolio Construction, Management, & Protection, 4e, Robert A. StrongCopyright ©2006 by South-Western, a division of Thomson Business & Economics. All rights reserved.

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The first mistake is usually the cheapest mistake.

A trader adage

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Outline Introduction Interest Rate Futures Contracts Concept of Immunization

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Introduction A portfolio has high interest rate

sensitivity if its value declines in response to interest rate increases• Especially pronounced:

– For portfolios with income as their primary objective

– For corporate and government bonds

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Interest Rate Futures Contracts

Categories of Interest Rate Futures Contracts

U.S. Treasury Bills and Their Futures Contracts

Treasury Bonds and Their Futures Contracts

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Categories of Interest Rate Futures Contracts

Short-Term Contracts Intermediate- and Long-Term Contracts

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Short-Term Contracts The two principal short-term futures

contracts are:• Eurodollars

– U.S. dollars on deposit in a bank outside the U.S.

– The most popular form of short-term futures

– Not subject to reserve requirements

– Carry more risk than a domestic deposit

• U.S. Treasury bills

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Intermediate- and Long-Term Contracts

Futures contract on U.S. Treasury notes is the only intermediate-term contract

The principal long-term contract is the contract on U.S. Treasury bonds

Special-purpose contracts:• Municipal bonds• U.S. dollar index

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U.S. Treasury Bills and Their Futures Contracts

Characteristics of U.S. Treasury Bills Treasury Bill Futures Contracts

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Characteristics of U.S. Treasury Bills

U.S. Treasury bills:• Are sold at a discount from par value

• Are sold with 91-day and 182-day maturities at a weekly auction

• Are calculated following a standard convention and on a bond equivalent basis

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Characteristics of U.S. Treasury Bills (cont’d)

Standard convention:

T-bill price = Face value - Discount amount

Days to maturityDiscount amount = Face value ( ) Ask discount

360

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Characteristics of U.S. Treasury Bills (cont’d)

The T-bill yield on a bond equivalent basis adjusts for:• The fact that there are 365 days in a year

• The fact that the discount price is the required investment, not the face value

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Characteristics of U.S. Treasury Bills (cont’d)

The T-bill yield on a bond equivalent basis:

Discount amount 365Bond equivalent yield

Discount price Days to maturity

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Characteristics of U.S. Treasury Bills (cont’d)

Example

A 182-day T-bill has an ask discount of 5.30 percent. The par value is $10,000.

What is the price of the T-bill? What is the yield of this T-bill on a bond equivalent basis?

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Characteristics of U.S. Treasury Bills (cont’d)

Example (cont’d)

Solution: We must first compute the discount amount to determine the price of the T-bill:

Days to maturityDiscount amount = Face value ( ) Ask discount

360182

$10,000 ( ) 0.053360

$267.94

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Characteristics of U.S. Treasury Bills (cont’d)

Example (cont’d)

Solution (cont’d): With a discount of $267.94, the price of this T-bill is:

T-bill price = Face value - Discount amount

$10,000 $267.94

$9,732.06

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Characteristics of U.S. Treasury Bills (cont’d)

Example (cont’d)

Solution (cont’d): The bond equivalent yield is 5.52%:

Discount amount 365Bond equivalent yield

Discount price Days to maturity

$267.94 365

$9,732.06 182

5.52%

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Treasury Bill Futures Contracts

T-bill futures contracts:• Call for the delivery of $1 million par value of

90-day T-bills on the delivery date of the futures contract

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Treasury Bill Futures Contracts (cont’d)

Example

Listed below is information regarding a T-bill futures contract. What would you pay for this futures contract today?

Discount

Open High Low Settle Change Settle Change Open Interest

92.43 92.49 92.41 92.48 –.01 7.52 +.01 250

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Treasury Bill Futures Contracts (cont’d)

Example (cont’d)

Solution: First, determine the yield for the life of the T-bill:

7.52% × 90/360 = 1.88%

Next, discount the contract value by the yield:

$1,000,000/(1.0188) = $981,546.92

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Treasury Bonds and Their Futures Contracts

Characteristics of U.S. Treasury Bonds Treasury Bond Futures Contracts

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Characteristics of U.S. Treasury Bonds

U.S. Treasury bonds:• Pay semiannual interest

• Have a maturity of up to 30 years

• Trade readily in the capital markets

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Characteristics of U.S. Treasury Bonds (cont’d)

U.S. Treasury bonds differ from U.S. Treasury notes:• T-notes have a life of less than ten years

• T-bonds are callable fifteen years after they are issued

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Treasury Bond Futures Contracts

U.S. Treasury bond futures:• Call for the delivery of $100,000 face value of

U.S. T-bonds that have a minimum of fifteen years until maturity (fifteen years of call protection for callable bonds)

Bonds that meet these criteria are deliverable bonds

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Treasury Bond Futures Contracts (cont’d)

A conversion factor is used to standardize deliverable bonds:• The conversion is to bonds yielding 6 percent

• Published by the Chicago Board of Trade

• Is used to determine the invoice price

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Sample Conversion Factors

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Treasury Bond Futures Contracts (cont’d)

The invoice price is the amount that the deliverer of the bond receives when a particular bond is delivered against a futures contract:

Invoice price = (Settlement price on position day Conversion factor)

+ Accrued interest

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Treasury Bond Futures Contracts (cont’d)

Position day is the day the bondholder notifies the clearinghouse of an intent to delivery bonds against a futures position• Two business days prior to the delivery date

• Delivery occurs by wire transfer between accounts

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Treasury Bond Futures Contracts (cont’d)

At any given time, several bonds may be eligible for delivery• Only one bond is cheapest to delivery

– Normally the eligible bond with the longest duration

– The bond with the lowest ratio of the bond’s market price to the conversion factor is the cheapest to deliver

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Cheapest to Deliver Calculation

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Concept of Immunization Definition Duration Matching Immunizing with Interest Rate Futures Immunizing with Interest Rate Swaps Disadvantages of Immunizing

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Definition Immunization means protecting a bond

portfolio from damage due to fluctuations in market interest rates

It is rarely possible to eliminate interest rate risk completely

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Duration Matching An Independent Portfolio Bullet Immunization Example Expectation of Changing Interest Rates An Asset Portfolio with a Corresponding

Liability Portfolio

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An Independent Portfolio Bullet immunization is one method of

reducing interest rate risk associated with an independent portfolio• Seeks to ensure that a set sum of money will be

available at a specific point in the future

• The effects of interest rate risk and reinvestment rate risk cancel each other out

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Bullet Immunization Example Assume:

• You are required to invest $936• You are to ensure that the investment will grow

at a 10 percent compound rate over the next 6 years

– $936 × (1.10)6 = $1,658.18

• The funds are withdrawn after 6 years

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Bullet Immunization Example (cont’d)

If interest rates increase over the next 6 years:• Reinvested coupons will earn more interest

• The value of any bonds we buy will decrease– Our portfolio may end up below the target value

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Bullet Immunization Example (cont’d)

Reduce the interest rate risk by investing in a bond with a duration of 6 years

One possibility is the 8.8 percent coupon bond shown on the next two slides:• Interest is paid annually• Market interest rates change only once, at the

end of the third year

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Expectation of Changing Interest Rates

The higher the duration, the higher the interest rate risk

To reduce interest rate risk, reduce the duration of the portfolio when interest rates are expected to increase• Duration declines with shorter maturities and

higher coupons

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An Asset Portfolio with a Liability Portfolio

A bank immunization case occurs when there are simultaneously interest-sensitive assets and interest-sensitive liabilities

A bank’s funds gap is its rate-sensitive assets (RSA) minus its rate-sensitive liabilities (RSL)

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An Asset Portfolio with a Liability Portfolio (cont’d)

A bank can immunize itself from interest rate fluctuations by restructuring its balance sheet so that:

,

,

$ $

where $ dollar value of rate-sensitive

assets and liabilities

dollar-weighted average duration

of assets and liabilities

A A L L

A L

A L

D D

D

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An Asset Portfolio with a Liability Portfolio (cont’d)

If the dollar-duration value of the asset side exceeds the dollar-duration of the liability side:• The value of RSA will fall to a greater extent

than the value of RSL

• The net worth of the bank will decline

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An Asset Portfolio with a Liability Portfolio (cont’d)

To immunize if RSA are more sensitive than RSL:• Get rid of some RSA• Reduce the duration of the RSA• Issue more RSL• Raise the duration of the RSL

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Immunizing with Interest Rate Futures

Financial institutions use futures to hedge interest rate risk

If interest rate are expected to rise, go short T-bond futures contracts

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Immunizing with Interest Rate Futures (cont’d)

To hedge, first calculate the hedge ratio:

where price of bond portfolio as a percentage of par

duration of bond portfolio

price of futures contract as a percentage

duration of cheapest-to-deliver bond eligible

b bctd

f f

b

b

f

f

P DHR CF

P D

P

D

P

D

for delivery

conversion factor for the cheapest-to-deliver bondctdCF

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Immunizing with Interest Rate Futures (cont’d)

Next, calculate the number of contracts necessary given the hedge ratio:

Portfolio valueNumber of contracts

$100,000HR

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Immunizing with Interest Rate Futures (cont’d)

Example

A bank portfolio manager holds $20 million par value in government bonds that have a current market price of $18.9 million. The weighted average duration of this portfolio is 7 years. Cheapest-to-deliver bonds are 8.125s28 T-bonds with a duration of 10.92 years and a conversion factor of 1.2786.

What is the hedge ratio? How many futures contracts does the bank manager have to short to immunize the bond portfolio, assuming the last settlement price of the futures contract was 94 15/32?

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Immunizing with Interest Rate Futures (cont’d)

Example

Solution: First calculate the hedge ratio:

0.945 71.2786

0.9446875 10.920.8199

b bctd

f f

P DHR CF

P D

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Immunizing with Interest Rate Futures (cont’d)

Example

Solution: Based on the hedge ratio, the bank manager needs to short 155 contracts to immunize the portfolio:

$18,900,000Number of contracts 0.8199

$100,000

154.96

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Immunizing with Interest Rate Swaps

Interest rate swaps are popular tools for managers who need to manage interest rate risk

A swap enables a manager to alter the level of risk without disrupting the underlying portfolio

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Immunizing with Interest Rate Swaps (cont’d)

A basic interest rate swap involves:• A party receiving variable-rate payments

– Believes interest rates will decrease

• A party receiving fixed-rate payments– Believes interest rates will rise

The two parties swap fixed-for-variable payments

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Immunizing with Interest Rate Swaps (cont’d)

The size of the swap is the notional amount• The reference point for determining how much

interest is paid

The price of the swap is the fixed rate to which the two parties agree

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Immunizing with Interest Rate Swaps (cont’d)

Interest rate swaps introduce counterparty risk:• No institution guarantees the trade

• One party to the swap may not honor its agreement

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Disadvantages of Immunizing Opportunity Cost of Being Wrong Lower Yield Transaction Costs Immunization Is Instantaneous Only

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Opportunity Cost of Being Wrong

With an incorrect forecast of interest rate movements, immunized portfolios can suffer an opportunity loss

For example, if a bank has more RSA than RSL, it would benefit from a decline in interest rates• Immunizing would have reduced the benefit

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Lower Yield The yield curve is usually upward sloping

Immunizing may reduce the duration of a portfolio and shift fund characteristics to the left on the yield curve

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Transaction Costs Buying and selling bonds requires

brokerage commissions• Sales may also result in tax liabilities

Commissions with the futures market are lower• The futures market is the method of choice for

immunization strategies

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Immunization Is Instantaneous Only

A portfolio is theoretically only immunized for an instant• With each day that passes, durations, yields to maturity,

and market interest rates change It is not practical for any but the largest portfolios to

make daily adjustments to account for changing immunization needs

Smaller portfolios may be initially immunized and revised only after weeks have passed or when conditions have changed enough to make revision cost effective