Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-1 Chapter 16 Interest Rate Risk Measurements and Immunization Using Duration

Copyright © 2000 by Harcourt, Inc. All rights reserved.

16-1

Chapter 16Interest Rate Risk Measurements and

Immunization Using Duration


16-2

Interest Rate Risk Defined

The potential variation in returns due to UNEXPECTED changes in interest rates.


16-3

Fundamental Principles of Financial Asset Pricing

A single investor is unable to influence the price of a financial asset.

The supply and demand for a financial asset also influences price.

All else equal, the price of a riskier asset will be lower than that of a less risky one because financial market participants are risk averse.• Risk aversion causes investors to demand higher

expected rates of return for riskier investments.


16-4

Financial Asset Pricing

n

t

t

y

CP

10 1

where:

y= discount rate that makes the sum of the present value of assets’ cash flows (Ct) equal to its price (P0)


16-5

All else equal, prices and yields change simultaneously in the opposite direction.

• If y rises, the price of the security falls.

• If y falls, the price of the security rises.


16-6

Calculating Ex Ante Effective Annual Yields (y) for Bonds

The ex ante effective annual yield (y) that an investor expects to receive over the life of the bond is the discount rate that makes the present value of the bond’s future cash flows equal to its current price.


16-7

)Value(PVIFMaturity

)IFAPayment(PVCoupon

,y

,y

0n

nP

where:

P0 = price (market value) of the bond

PVIFA= present value of the annuity factor

[1-(1/(1+ y)n)/ y)]

PVIF= present value factor [1/(1+ y)n]


16-8

In order to earn the ex ante effective annual yield, an investor would have to:

• hold the bond until maturity; and

• invest each coupon payment at the y rate over the life of the bond.


16-9

Calculate y for a bond with 15 years to maturity, a coupon rate of 7.5% and a price of 116% of $1,000 par

value.

P0 = 1.160 × $1,000 = $1,160

Coupon Payment = .075 × $1,000 = $75

$1,160 = $75(PVIFA y, 15) + $1,000(PVIF y,15)

y = 5.87%

An investor will only receive an annual effective yield of 5.87% if the bond is held to maturity and coupons can be reinvested at a 5.87% rate each year.

)Value(PVIFMaturity )IFAPayment(PVCoupon ,y,y 0 nnP


16-10

Calculating Ex Post (Actual) Effective Annual Yield

1P

FV EAY

1

0

n

where:

FV = future value of the cash flows received at the end of the life of the bond


16-11

ValueMaturity )IFAPayment(FVCoupon FV ,y n

where:

FVIFA= future value of the annuity factor

[ ((1+ y)n - 1)/ y)]


16-12

What is the EAY for a bond with 15 years to maturity, a coupon rate of 7.5% , bought at a price of $1,160 and for which coupons where reinvested at a 5.87% rate?

$75(FVIFA 5.87%,15) + $1000

$75(23.047) + $1000 = $2,728.50

ValueMaturity )IFAPayment(FVCoupon FV ,y n

1 EAY1

0 n

PFV

($2,728.50/$1,160)(1/15) - 1

(2.35185)0.0067 - 1 = 0.0587 or 5.87%


16-13

If all coupon payments are reinvested at the y rate and the bond is held until maturity, the EAY will be equal to the ex ante effective annual rate, y


16-14

The Two Sides of Interest Rate Risk

Reinvestment Rate Risk

• The risk of interest rates falling and having to reinvest coupon payment at a lower rate than y, resulting in a lower ex-post yield.

Price or Market Value Risk

• The risk of rates rising and the market price of the bond falling if the bond must be sold prior to maturity, also resulting in a lower ex-post yield.


16-15

Example of Reinvestment Rate RiskUse the previous example to calculate EAY, assuming rates have dropped to 5% over the

life of the bond after the bond was purchased.

FV of cash flows

$75(FVIFA 5%,15) + $1000

$75(21.579) + $1000 = $2,618.39

EAY = ($2,618.39/$1,160)(1/15) - 1

= (2.2573)0.0067 - 1 = 0.0558 or 5.58%

Because of reinvestment rate risk, the investor has a lower ex post yield than the expected y of 5.87%.


16-16

Example of Market Price RiskUsing the previous example, calculate the EAY for

the bond if rates after 5 years rise to 6.5% and remain at that level for the remainder of the life of

the bond. Assume that the investor will sell the bond after 5 years.

Price of the bond in 5 years $75(PVIFA 6.5,10) + $1,000(PVIF 6.5,10 )

$75(7.188) +1000(0.532) = $1,071.89FV of cash flows

$75(FVIFA 5.87,5) + $1,071.89 $75(5.6225) + $1,071.89 = $1,493.58


16-17

EAY

($1,493.58/$1,160)(1/5) - 1

(1.2876).2 - 1 =.0519 or 5.19%

Because of market price risk, the annual effective yield received ex post is 5.19% over 5 years, 68 basis points lower than the expected effective yield of 5.87%.


16-18

Bond Theorems

Theorem I. Bond prices must move inversely to bond yields.

Theorem II. Holding the coupon rate constant, percentage changes in bond prices are greater the longer term to maturity of a bond.

Theorem III. The percentage price change for a bond will increase at a decreasing rate as N increases.


16-19

Theorem IV. (Convexity Theorem) Starting with a given market yield y, holding other factors constant, the rise in price with a fall in y will be greater than the fall in price with the same absolute value rise in y.

Theorem V. Holding N constant and starting from the same y, the higher the coupon rate, the smaller the percentage change in price for a given change in yield.


16-20

SummaryA Bond’s Interest Rate Risk Depends On

The current market yield (y) The coupon rate of the bond The maturity of the bond


16-21

Duration Defined

It is the weighted average time over which the cash flows from an investment are expected.

Duration is a measure of the effective maturity.

• For most securities, the cash benefits are received before the maturity date.

Duration is also a measure of interest rate risk.

• Securities with higher durations will have a greater change in market value with a change in market rates.


16-22

N

t

t

N

t

t

y

Cy

tC

Dur

1

1

)1(

)1(

where:

Ct = cash flow in time period t

t = the time each cash flow comes in

y = current market yield


16-23

Find the duration for the following three bonds:Bond 1 has an 8% coupon rate, $1000 par, n=6

Bond 2 is a zero-coupon bond with a $1,000 par, n=6Bond 3 has a 6% coupon rate, $1,000 par, n=6

BOND 1: 8% COUPON RATE, $1,000 MATURITY VALUE, 6 YEARS TO MATURITY

Year Cash Flow PVIF 8% PV Cash Flows t × PV Cash Flow

1 $80 .9259 $74.07 $74.072 80 .8573 68.58 137.163 80 .7938 63.50 190.504 80 .7350 58.80 235.205 80 .6806 54.45 272.256 1080 .6302 680.62 4,083.72

Sum $1000.00 4,992.90

Duration = $4,992.90/$1,000 = 4.993 years


16-24

BOND 2: ZERO COUPON RATE, $1,000 MATURITY VALUE, 6 YEARS TO MATURITY


1 $0 .9259 $0 $02 0 .8573 0 03 0 .7938 0 04 0 .7350 0 05 0 .6806 0 06 1000 .6302 630.20 3,781.20

Sum $630.20 $3,781.20

Duration = $3,781.20/$630.20 = 6 years


16-25

BOND 3: 6% COUPON RATE, $1,000 MATURITY VALUE, 6 YEARS TO MATURITY


1 $60 .9259 $55.55 $55.552 60 .8573 51.44 102.883 60 .7938 47.63 142.894 60 .7350 44.10 176.405 60 .6806 40.84 272.256 1060 .6302 668.01 4,008.07

Sum $907.57 4,689.97

Duration = $4,689.97/$907.57 = 5.17 years

Note that duration is a negative function of a bond’s coupon payment. The duration for zero coupon will always be equal to its maturity.


16-26

Alternative Formula for Duration to Use with Long-term Securities

NyyNy

NDur ,PVIFA1)(P

PaymentCoupon

0

y

yn

1/(11PVIFA

where:


16-27

For Bond 1, with N = 6 years, Coupon Payment = $80, y = 8%, P0 = $1,000 and

PVIFA factor for 8% and 6 years = 4.6229

Dur = 6 - [($80/($1000×.08)]×[6-(1.08)(4.6229)]

= 6 - (1) × (1.0073) = 4.993 years

The same as was calculated using the regular formula.

NyyNy

NDur ,PVIFA1)(P

PaymentCoupon

0


16-28

Why is duration important for assessing an institution’s interest rate risk?

Because for a given change in market yields, percentage changes in asset prices are proportional to the assets’ duration.

)1(%%

0

00 y

yDurPPP

where:%P0 = percentage change in price%y = percentage change in market yield y = the original market rate prior to the change


16-29

What is the percentage change in price for Bond 1 with a duration of 4.993, a P0 = $1,000, and y

= 8% if rates increase to 8.5%?

)1(%%

0

00 y

yDurPPP

- 4.99[0.50%/(1.08)] = - 2.31%

Multiplying (0.0231 × $1,000), the bond’s price would fall by approximately $23.10.


16-30

Duration assumes a linear relationship between price changes and changes in market yields.

• Bond price changes follow a convex pattern with changes in market yields.

Duration will overestimate the percent fall in price with a rise in rates.

Duration wil1 underestimate the percent rise in price with a fall in rates.


16-31

Estimating Interest Rate Elasticity

y 1yDur - E

Interest rate elasticity is the percentage price change expected for a 1% change in market yields.


16-32

Calculate the interest rate elasticity for Bonds 1, 2, and 3.

y 1yDur - E

Bond 1: E = -4.993(0.08/1.08) = -0.3699

Bond 2: E = -6.000(0.08/1.08) = -0.4444

Bond 3: E = -5.170(0.08/1.08) = -0.3830


16-33

For Bond 1, for every 1% change in the bond’s yield of 8% , the bondholder could have expected a price change of 0.3699%.

Bonds 2 and 3 have higher elasticity than Bond 1, which implies greater interest rate risk.

In this case, elasticities describe the same risk ranking as their relative duration, because each bond has the same market yield of 8%.

If market yields are different, calculating elasticities provides a better comparison of different securities’ relative interest rate risk.


16-34

Modified Duration

y 1Dur (MD)Duration Modified

Multiplying MD by any y gives the approximate change in value of that fixed-rate asset or liability with a change in y.


16-35

Limitations of Duration-Based Measures of Interest Rate Risk

Duration assumes that there will be a parallel shift in the yield curve so that for a given level of default risk, yields across the entire term structure change equally.

Duration assumes a flat yield curve because the same y discount rate is used for early and late cash flows.

As market rate (y ) changes, so will a security’s duration.


16-36

Portfolio Immunization

It is a duration based strategy that makes a portfolio immune to interest rate risk over a given holding period.

Used to achieve a realized annual rate of return at the end of the holding period that is no less than the expected annual yield at the beginning of the period.


16-37

Suppose an insurance company issues a $1 million guaranteed investment contract that guarantees an annual effective yield of 8%. The company is considering two alternatives:

1. Bond 1: six-year bonds with an 8% coupon rate at $1,000 par value which sell for $1,000 per bond, with y = 8%.

2. Bond 2: five-year bonds with an 8% coupon rate and $1,000 par value, which sell for $1,000 per bond, with y = 8%.


16-38

Duration of Bond 1 = 4.993 (from earlier calculations)

Duration of Bond 2

NyyNy

NDur ,PVIFA1)(P

PaymentCoupon

0

5 - [(80/80)] - [5 - (1.08)(3.9927)]

5 - 0.68788 = 4.312


16-39

ILLUSTRATION OF DURATION AND PORTFOLIO IMMUNIZATION.

STRATEGY 1: DURATION MATCH; BUY BOND 1 WITH APPROXIMATELY A

5-YEAR DUATION Coupon payments reinvested at y and bond sold at end of 5 years.Scenario (1) Market interest rates stay at y = 8% for 5 years

Ex post effective annual yield for Bond 1:FV of cash flows per bond at the end of year 5

80(FVIFA 5,8%) + (1080/1.08) = $1,469.33Ex post effective annual rate is (1469.33/1000)0.20

- 1 = .08 or 8%Scenario (2) Market interest rates stay at y = 9% for 5 years






- 1 = .08 or 8%


16-40

STRATEGY 2: MATURITY MATCH; BUY BOND 2 WITH A MATURITY OF 5 YEARS

Coupon payments reinvested at y and bond sold at end of 5 years.Scenario (1) Market interest rates stay at y = 8% for 5 years


80(FVIFA 5,8%) + 1000 = $1,469.33Ex post effective annual rate is (1469.33/1000)0.20




- 1 = .0814 or 8.14%Scenario (3) Market interest rates stay at y = 7% for 5 years



- 1 = .0.786 or 7.86%


16-41

In this example, holding bonds with a duration of 5 years will lock in an annual return of 8% over the holding period.

Reasoning: With immunization, the expected return on a portfolio is protected from both reinvestment and market value risk.

• Reinvestment risk exactly offsets market value risk.


16-42

Immunization Assumptions and Limitations

Assumes that there is only one parallel shift in the yield curve immediately after the beginning of the period.

• Market shifts may occur in the middle of the investment period and an investor may be less than perfectly immunized.

It is difficult to find an investment with a duration exceeding 10 years.


16-43

Differences in Funding Gap and Duration

A financial institution’s funding gap serves as a measure of reinvestment risk for a given time period.

Funding gap ignores:• time value of money; and• the effects of changes in the market value of assets and

liabilities on a financial institution’s balance sheet. Duration gap provides a better overall measure of

interest rate risk including market price as well as reinvestment risk.


16-44

Duration Gap

AssetssLiabilitieDurL -DurA DGAP

where:

DurA = duration of assets which is the weighted average of the duration of assets on the balance sheet

DurL = duration of liabilities which is the weighted average of the duration of liabilities on the balance sheet


16-45

DGAP > 0 Increase Decrease > Decrease Fall

Decrease Increase > IncreaseRise

DGAP < 0 Increase Decrease <Decrease Rise

Decrease Increase < IncreaseFall

DGAP = 0 Increase Decrease =Decrease Zero

Decrease Decrease =Decrease Zero

Position

y

in Market Value

Assets Liabilities Equity

DURATION GAP AND CHANGES IN THE MARKET VALUE OF EQUITY


16-46

Change in the Value of a Depository Institution’s Equity-to-Asset Ratio

)]y 1/(yDGAP[( Assets) tal(Equity/To Value

where:

y = average rate on assets


16-47

Calculating Excelsior Bank’s DGAP

AssetsCash Dur = 0 $10,0006-month T-bills y = 6% Dur = .5 years $40,00010 year Fixed-Rate Loan Dur = ? $50,000

Total Assets $100,000

Liabilities and Equity

Transactions Deposit Dur = 0 $40,000(rate =2%)

1-year CD, rate = 7% Dur = 1 $50,000Equity $10,000

Total Liabilities and Equity $100,000


16-48

Duration of a Loan

NyyNy

NDur ,PVIFA1)Amount(Loan

PaymentCoupon

Coupon Payment (Loan Payment)

Loan Amount/(PVIFA 10%, 10) = $50,000/6.1446 = $8,137.27 per year

Dur = 10 - [($8,137.27/($50,000×.10)] ×

[10 - (1.10)(6.1446)] = 10 - (1.6275)(3.241) =

4.73 years


16-49

Duration of Assets

($10,000/$100,000)0 + ($40,000/$100,000)0.5 +($50,000/$100,000)4.73 = 2.565 years

Duration of Liabilities

($40,000/$90,000)0 + ($50,000/$90,000)1 = 0.555 years

2.565 yrs - ($90,000/$100,000)0.5556 yrs = 2.065 years

AssetssLiabilitieDurL -DurA DGAP


16-50

Change in the Value of Equity from a 1% Rise in Rates

- 2.065[0.01/(1.074)] = - 0.0192 or -1.92%

)]y 1/(yDGAP[( Assets) tal(Equity/To Value

y = ($10,000/$100,000)0 + ($40,000/$100,000)0.06 + ($50,000/$100,000)0.10 = 0.074 or 7.4%

Documents

Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-1 Chapter 16 Interest Rate Risk Measurements and Immunization Using Duration