Bonus Slide Set - Duration

8/3/2019 Bonus Slide Set - Duration

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Prepared byKen Hartviksen

INTRODUCTION TO

CORPORATE FINANCELaurence Booth W. Sean Cleary

Duration


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Duration


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Duration G - 3

Lecture Agenda

Learning Objectives

Important Terms

What is Duration? Summary

Conclusions


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Duration G - 4

Important Terms

Consol bond

Duration

Floating Rate Note (FRN)

Immunization

Stripped Bond

Zero Coupon Bond


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Duration G - 5

Duration

An alternative measure of bond price sensitivity is the bonds duration.

Duration measures the life of the bond on a present value basis.

Duration can also be thought of as the average time to receipt of thebonds cash flows.

The longer the bonds duration, the greater is its sensitivity to interestrate changes.


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Duration G - 6

Duration Rules-of-Thumb

Duration of zero-coupon bond (strip bond) = the term left until maturity.

Duration of a consol bond (ie. a perpetual bond) = 1 + (1/k)

where: k = required yield to maturity

Duration of an FRN (floating rate note) = 1/2 year


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Duration G - 7

Other Duration Rules-of-Thumb

Duration and Maturity

Duration increaseswith maturity of a fixed-income asset, but at a decreasingrate.

Duration and Yield

Duration decreasesas yield increases.

Duration and Coupon Interest

The higher the coupon or promised interest payment on the security, the lowerits duration.


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Duration G - 8

Economic Meaning of Duration

duration is a direct measure of the interest rate sensitivity orelasticityof an asset or liability. (ie. what impact will a change inYTM have on the price of the particular fixed-income security?)

interest rate sensitivity is equal to:

dP = - D[ dk/(1+k)]

P

Where: P = Price of bond

C = Coupon (annual)k = YTM

N = Number of periods

F = Face value of bond


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Duration G - 9

Interest Rate Elasticity

the percent change in the bonds price caused by agiven change in interest rates (change in YTM)

(The following slide illustrates how bond price sensitivity can be graphed against

changing discount rates)


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Duration G - 10

Price Elasticity of Stripped Bonds

$0

$5,000

$10,000

$15,000

$20,000

0.0% 5.0% 10.0% 15.0% 20.0%

30 year stripped bond price given different YTM.


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Duration G - 11

Price Sensitivity of a Stripped Bond

Take our previous example where a $20,000 30-year stripped bond has arequired rate of return of 12%:

P0 = $20,000(PVIFn=30, k = 12%)

= $20,000 (.0334)

= $668.00

Assume now that interest rates fall by 16.7% from 12% to 10%. What isthe percentage change in price of the bond?

P0 = $20,000(PVIFn=30, k = 10%)

= $20,000 (.0573)

= $1,146.00

Percentage change in price = ($1,146 - $668) / $668=71.6%

This stripped bond had a 71.6% increase in price with a 2% decrease(200 bp) decrease in required rate of return.


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Duration G - 12

Duration and Coupon Rates

A bonds duration is affected by the size of the coupon rate offered bythe bond.

The duration of a zero coupon bond is equal to the bonds term tomaturity. Therefore, the longest durations are found in stripped bondsor zero coupon bonds. These are bonds with the greatest interest rateelasticity.

The higher the coupon rate, the shorter the bonds duration. Hence thegreater the coupon rate, the shorter the duration, and the lower theinterest rate elasticity of the bond price.


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Duration G - 13

Duration

The numerator of the duration formula represents the present value of futurepayments, weighted by the time interval until the payments occur. The longerthe intervals until payments are made, the larger will be the numerator, and thelarger will be the duration. The denominator represents the discounted futurecash flows resulting from the bond, which is the bonds present value.

maturitytoyieldsbondthek

providedarepaymentsthewhichattimethet

bondthebygeneratedpaymentprincipalorcoupontheCwhere

k

C

k

tC

DUR

t

n

tt

t

n

tt

t

'

:

)1(

)1(

)(

1

1


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Duration G - 14

Duration ExampleA Formula-based Duration Calculation for a Three Year, 7% Coupon Bond

As an example, the duration of a bond with $1,000 par value and a 7 percentcoupon rate, three years remaining to maturity, and a 9 percent yield to maturityis:

years

DUR

80.2

)09.1(

1070$

)09.1(

70$

)09.1(

70$

)09.1(

)3(1070$

)09.1(

)2(70$

)09.1(

70$

321

321


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Duration G - 15

Duration ExampleA Formula-based Duration Calculation for a Zero Coupon Bond

As an example, the duration of a zero-coupon bond with $1,000 par value andthree years remaining to maturity, and a 9 percent yield to maturity is:

0.3

)09.1(

1000$

)09.1(

)3(1000$

3

3

years

DUR


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Example of a Duration CalculationUsing a Spreadsheet Model

Example

Assume a 10% coupon bond with three years left to maturity and a required return of 8%.

Coupon Rate = 10.00%

Required Return = 8.00%

Time Cashflow PVIF Present Value Weight

Time

Weighted

CFs

0

0.5 50 0.96225 $48.11 4.55% 0.022767679

1 50 0.925926 $46.30 4.38% 0.043816419

1.5 50 0.890973 $44.55 4.22% 0.063243554

2 50 0.857339 $42.87 4.06% 0.0811415172.5 50 0.824975 $41.25 3.90% 0.097598077

3 1050 0.793832 $833.52 78.89% 2.36662759

Bond Price = $1,056.60 100.00% 2.675194837 =Duration


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Duration of a Portfolio

Bond portfolio mangers commonly attempt to immunize their portfolio,or insulate their portfolio from the effects of interest rate movements.

This is a common challenge when the investment portfolio is dedicated

to funding a future liability.


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Duration G - 18

Duration of a Portfolio

Insurance Company Example

A life insurance company knows that they need $100 million 30 years from now coveractuarially-determined claims against a group of life insurance policies just no sold toa group of 30 year olds.

The insurance company has invested the premiums into 30-year government bonds.Therefore there is no default risk to worry about. The company expects that if therealized rate of return on this bond portfolio equals the yield-to-maturity of the bondportfolio, there wont be a problem growing that portfolio to $100 million. The problemis, that the coupon interest payments must be reinvested and there is a chance thatrates will fall over the life of the portfolio.

If this happens the portfolios terminal value will be less than the liability the insurancecompany needs to finance. This shortfall in investment returns will have to be borneat the expense of the Insurance companys shareholders.


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Duration G - 19

Duration of a Portfolio ...Interest Rate Risk

The life insurance company example illustrates a key risk in fixed-income portfolio management - interest rate risk.

The portfolio manager, before-the-fact calculates the bond portfoliosyield-to-maturity. This is an ex antecalculation.

As such, a nave assumption assumption is made that the couponinterest received each year is reinvested at the yield-to-maturity for theremaining years until the bond matures.

Over time, however, interest rates will vary and reinvestmentopportunities will vary from that which was forecast.


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Duration G - 20

Duration of a PortfolioImmunization

The insurance company will want to IMMUNIZE their portfolio from thisreinvestment risk.

The simplest way to do this is to convert the entire bond portfolio tozero-coupon/stripped bonds. Then the ex anteyield-to-maturity willequal ex post(realized) rate of return. (ie. the ex anteYTM is locked insince there are no intermediate cash flows the require reinvestment).

If the bond portfolio manager matches the duration of the bond portfoliowith the expected time when they will require the $100 m, then interestrate risk will be largely eliminated.


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Summary

In this slide set you have:

Learned how to calculate duration

Learned the meaning of duration Learned the factors that influence the duration ofa bond

Learned some uses of duration


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Copyright

Copyright 2007 John Wiley & SonsCanada, Ltd. All rights reserved.Reproduction or translation of this workbeyond that permitted by AccessCopyright (the Canadian copyrightlicensing agency) is unlawful. Requestsfor further information should beaddressed to the PermissionsDepartment, John Wiley & Sons Canada,Ltd. The purchaser may make back-upcopies for his or her own use only andnot for distribution or resale. The authorand the publisher assume noresponsibility for errors, omissions, ordamages caused by the use of these filesor programs or from the use of theinformation contained herein.

Bonus Slide Set - Duration

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