1

Ch. 11 Outline

Interest rate futures – yield curve Discount yield vs. Investment Rate %”

(bond equivalent yield):

Pricing interest rate futures contractsPricing interest rate futures contracts Spreading with interest rate futuresSpreading with interest rate futures

Speculation Example

The ED futures contract has a face value of $1 million. Suppose the discount yield at the time of purchase was 2.74%. In the middle of March 2005, interest rates have risen to 7.00%.

What is the speculator’s dollar gain or loss?

2

3

Speculating With Eurodollar Futures – Initial price

Face value - $1 million; Disc yield-2.74% (price of 97.26; or 100-97.26)

150,993$360

900274.1000,000,1$Price

360

90YieldDiscount -1Value FacePrice

4

Speculating With Eurodollar Futures (cont’d)

The price with the new interest rate of 7.00% is:

00.500,982$360

900700.1000,000,1$Price

360

90YieldDiscount -1Value FacePrice

5

Speculating With Eurodollar Futures (cont’d)

Speculation Example (cont’d)

The speculator’s dollar loss is therefore:

00.650,10$00.150,993$00.500,982$

6

Hedging With Eurodollar Futures

Hedging ExampleFace value 10 mil, Disc Yield – 1.24

(at 98.76; or 100- 98.76)

.

7

Hedging With Eurodollar Futures (cont’d)

Hedging Example (cont’d)When you receive the $10 million in three months, assume interest rate have fallen to 1.00%. $10 million in T-bills would then cost:

This is $6,000 more than the price at the time you established the hedge.

00.000,975,9$360

9001.1000,000,10$Price

8

Hedging With Eurodollar Futures (cont’d)

Hedging Example (cont’d)

In the futures market, you have a gain that will offset the increased purchase price. When you close out the futures positions, you will sell your contracts for $6,000 more than you paid for them.

9

Pricing Interest Rate Futures Contracts

Interest rate futures prices come from the implications of cost of carry:

tC

S

tF

CSF

t

t

tt

time tozero timefromcarry ofcost

pricecommodity spot

at timedelivery for price futures

where

)1(

,0

,0

10

Computation

Cost of carry is the net cost of carrying the commodity forward in time (the carry return minus the carry charges)– If you can borrow money at the same rate that a

Treasury bond pays, your cost of carry is zero

Solving for C in the futures pricing equation yields the implied repo rate (implied financing rate)

11

Implied Repo or Financing rate

12

Arbitrage With T-Bill Futures

If an arbitrageur can discover a disparity between the implied financing rate and the available repo rate, there is an opportunity for riskless profit– If the implied financing rate is greater than the

borrowing rate, then he/she could borrow, buy T-bills, and sell futures

13

NOB spread (trading the yield curve)

slope increases (long term R increases more than short term or short term even decreases) buy notes sell bonds

14

The NOB Spread

The NOB spread is “notes over bonds” Traders who use NOB spreads are

speculating on shifts in the yield curve– If you feel the gap between long-term rates and

short-term rates is going to narrow ( yield curve slope decreases or flattens), you could sell T-note futures contracts and buy T-bond futures

15

Trading Spreads

16

TED spread (different yield curves)

The TED spread is the difference between the price of the U.S. T-bill futures contract and the eurodollar futures contract, where both futures contracts have the same delivery month (T-bill yield<ED yield)– If you think the spread will widen, buy the

spread (buy T-bill, sell ED)

© 2004 South-Western Publishing17

Chapter 12

Futures Contracts and Portfolio Management

18

Outline

The concept of immunization Altering portfolio duration with futures Duration as a convex function as opposed

to market risk measure beta

19

Introduction

An immunized bond portfolio is largely protected from fluctuations in market interest rates– Seldom possible to eliminate interest rate risk

completely – A portfolio’s immunization can wear out, requiring

managerial action to reinstate the portfolio– Continually immunizing a fixed-income portfolio can

be time-consuming and technical

20

Bond Risks

A fixed income investor faces three primary sources of risk:– Credit risk– Interest rate risk– Reinvestment rate risk

21

Bond Risks (cont’d)

Credit risk is the likelihood that a borrower will be unable or unwilling to repay a loan as agreed– Rating agencies measure this risk with

bond ratings– Lower bond ratings mean higher expected

returns but with more risk of default– Investors choose the level of credit risk

that they wish to assume

22

Bond Risks (cont’d)

Interest rate risk is a consequence of the inverse relationship between bond prices and interest rates– Duration is the most widely used measure of a

bond’s interest rate risk

23

Bond Risks (cont’d)

Reinvestment rate risk is the uncertainty associated with not knowing at what rate money can be put back to work after the receipt of an interest check– The reinvestment rate will be the

prevailing interest rate at the time of reinvestment, not some rate determined in the past

24

Duration Matching

Bullet immunization Change of portfolio duration with interest

rate futures

25

Introduction

Duration matching selects a level of duration that minimizes the combined effects of reinvestment rate and interest rate risk

Two versions of duration matching:– Bullet immunization– Bank immunization

26

Bullet Immunization

Seeks to ensure that a predetermined sum of money is available at a specific time in the future regardless of interest rate movements

27

Bullet Immunization (cont’d)

Objective is to get the effects of interest rate and reinvestment rate risk to offset– If interest rates rise, coupon proceeds can be

reinvested at a higher rate– If interest rates fall, proceeds can be reinvested

at a lower rate

28

Bullet Immunization (cont’d)

Bullet Immunization Example

A portfolio managers receives $93,600 to invest in bonds and needs to ensure that the money will grow at a 10% compound rate over the next 6 years (it should be worth $165,818 in 6 years).

29

Bullet Immunization (cont’d)

Bullet Immunization Example (cont’d)

The portfolio manager buys $100,000 par value of a bond selling for 93.6% with a coupon of 8.8%, maturing in 8 years, and a yield to maturity of 10.00%.

30

Bullet Immunization Example (cont’d)Panel A: Interest Rates Remain Constant

Bullet Immunization (cont’d)

 Year 1 Year 2 Year 3 Year 4 Year 5 Year 6$8,800 $9,680 $10,648 $11,713 $12,884 $14,172

  $8,800 $9,680 $10,648 $11,713 $12,884    $8,800 $9,680 $10,648 $11,713      $8,800 $9,680 $10,648        $8,800 $9,680         

        Interest $68,805        Bond         Total $165,817

$8,800

$97,920

31

Bullet Immunization (cont’d)

Bullet Immunization Example (cont’d)Panel B: Interest Rates Fall 1 Point in Year 3

 Year 1 Year 2 Year 3 Year 4 Year 5 Year 6$8,800 $9,680 $10,648 $11,606 $12,651 $13,789

  $8,800 $9,680 $10,551 $11,501 $12,536    $8,800 $9,592 $10,455 $11,396      $8,800 $9,592 $10,455        $8,800 $9,592         

        Interest $66,568        Bond        Total $166,218

$8,800

$99,650

32

Bullet Immunization (cont’d)

Bullet Immunization Example (cont’d)Panel C: Interest Rates Rise 1 Point in Year 3

 Year 1 Year 2 Year 3 Year 4 Year 5 Year 6$8,800 $9,680 $10,648 $11,819 $13,119 $14,563

  $8,800 $9,680 $10,745 $11,927 $13,239    $8,800 $9,768 $10,842 $12,035      $8,800 $9,768 $10,842        $8,800 $9,768         

        Interest $69,247        Bond         Total $165,477

$8,800

$96,230

33

Bullet Immunization (cont’d)

Bullet Immunization Example (cont’d)

The compound rates of return in the three scenarios are 10.10%, 10.04%, and 9.96%, respectively.

34

Duration Shifting

The higher the duration, the higher the level of interest rate risk

If interest rates are expected to rise, a bond portfolio manager may choose to bear some interest rate risk (duration shifting)

35

Duration Shifting (cont’d)

The shorter the maturity, the lower the duration

The higher the coupon rate, the lower the duration

A portfolio’s duration can be reduced by including shorter maturity bonds or bonds with a higher coupon rate

36

Duration Shifting (cont’d)

Maturity

Coupon

Lower Higher

Lower Ambiguous Duration Lower

Higher Duration Higher

Ambiguous

37

Hedging With Interest Rate Futures

A financial institution can use futures contracts to hedge interest rate risk

The hedge ratio is:

)1(

)1(

bff

ctdbbctd YTMDP

YTMDPCFHR

38

Hedging With Interest Rate Futures (cont’d)

The number of contracts necessary is given by:

ratio hedge000,100$

par value portfolio contracts #

39

Hedging With Interest Rate Futures (cont’d)

Futures Hedging Example

A bank portfolio holds $10 million face value in government bonds with a market value of $9.7 million, and an average YTM of 7.8%. The weighted average duration of the portfolio is 9.0 years. The cheapest to deliver bond has a duration of 11.14 years, a YTM of 7.1%, and a CBOT correction factor of 1.1529.

An available futures contract has a market price of 90 22/32 of par, or 0.906875. What is the hedge ratio? How many futures contracts are needed to hedge?

40

Hedging With Interest Rate Futures (cont’d)

Futures Hedging Example (cont’d)

The hedge ratio is:

9898.0078.114.11906875.0

071.10.997.01529.1

HR

41

Hedging With Interest Rate Futures (cont’d)

Futures Hedging Example (cont’d)

The number of contracts needed to hedge is:

98.989898.0000,100$

0$10,000,00 contracts #

42

Summary of Immunization and duration hedging

Bullet immunization (bond with target yield and duration = target date)

Duration as a measure of sensitivity to interest rate changes

Duration is a convex function hedge ratio does not change linearly (BPV)

43

Examples for review

Spot rate is $1.33 per 1€. The US 3m T-bill rate is 2.7% and the Forward 3m rate is 1.327011. What is the risk free rate of the European Central Bank if the interest rate parity condition determined this forward rate? (3.6%)

The spot rate is CAD 2.2733 per 1£. If the inflation rate in Canada is 3.4% a year and the inflation rate in UK is 2.3% per year, according to the purchasing power parity the forward exchange rate should be……….? (2.285838)