bond duration and duration model-student

8/10/2019 Bond Duration and Duration Model-student

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4. As maturity increases, pricesensitivity increases at adecreasing rate.

5. Interest rate risk is inverselyrelated to the bonds coupon rate.

6. Price sensitivity is inversely related

to the yield to maturity at whichthe bond is selling.


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A measure of the effective maturityof a bondThe weighted average of the times

until each payment is received,with the weights proportional tothe present value of the paymentDuration is shorter than maturityfor all bonds except zero couponbonds.Duration is equal to maturity forzero coupon bonds.


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CFt=cash flow at time t

Price)1( yCF w t

t t

t wt DT

t 1


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Price change is proportional toduration and not to maturity

D * = modified duration

(1 )1

P y Dx P y

* P

D y P


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Calculate the price and duration of a 2 yearmaturity, 8% coupon bond makingsemiannual coupon payments when themarket interest rate is 9%.


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Timeuntil

Payment(years)

Cash Flow (Discount rate= 5%) PV of CF (Discount

rate = 5%) Weight

Column(1) x

Column(5)

1 $40.00 0.9524 38.095 0.039 0.039

2 $40.00 0.907 36.281 0.0376 0.0752

3 $40.00 0.8638 34.554 0.0358 0.1074

4 $1040.00 0.8227 855.611 0.8871 3.5484

Column Sums $964.540 1 3.77

D = 3.77 (semiannual) = 1.885

years


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Two bonds have duration of 1.8852 years.One is a 2-year, 8% coupon bond withYTM=10%. The other bond is a zero coupon

bond with maturity of 1.8852 years.Duration of both bonds is 1.8852 x 2 =3.7704 semiannual periods.Modified D = 3.7704/1+0.05 = 3.591

periods


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Suppose the semiannual interest rateincreases by 0.01%. Bond prices fall by:

=-3.591 x 0.01% = -0.03591%Bonds with equal D have the same interestrate sensitivity.

y D P P *


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Coupon ond

The coupon bond,which initially sellsat $964.540, fallsto $964.1942 whenits yield increasesto 5.01%percentage declineof 0.0359%.

ZeroThe zero-couponbond initially sells

for $1,000/1.053.7704 = $831.9704.At the higher yield,it sells for

$1,000/1.053.7704

=$831.6717. Thisprice also falls by0.0359%.


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Rule 1 The duration of a zero-couponbond equals its time to maturity

Rule 2 Holding maturity constant, abonds duration is higher when thecoupon rate is lower

Rule 3 Holding the coupon rateconstant, a bonds durationgenerally increases with its time tomaturity


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Rule 4 Holding other factorsconstant, the duration of a coupon

bond is higher when the bondsyield to maturity is lower

Rules 5 The duration of a levelperpetuity is equal to: (1+y) / y


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The relationship between bondprices and yields is not linear.

Duration rule is a goodapproximation for only smallchanges in bond yields.

Bonds with greater convexity havemore curvature in the price-yieldrelationship.


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n

t t

t t t y

CF y P

Convexity1

22 )()1()1(

1

Correction for Convexity:

21 [ ( ) ]2

P D y Convexity y

P


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Bonds with greater curvature gain more inprice when yields fall than they lose whenyields rise.

The more volatile interest rates, the moreattractive this asymmetry.Bonds with greater convexity tend to havehigher prices and/or lower yields, all else

equal.


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As rates fall, there is a ceiling on thebonds market price, which cannot riseabove the call price.

Negative convexityUse effective duration:

/Effective Duration =

P P r


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The number of outstanding callablecorporate bonds has declined, but theMBS market has grown rapidly.

MBS are based on a portfolio of callableamortizing loans. Homeowners have the right to

repay their loans at any time. MBS have negative convexity.


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Often sell for more than their principalbalance.Homeowners do not refinance as soon as

rates drop, so implicit call price is not a firmceiling on MBS value.Tranches the underlying mortgage pool isdivided into a set of derivative securities


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Duration measures the interest rate sensitivity ofan asset or liabilitys value to small changes ininterest rates

The duration gap is a measure of overall interestrate risk exposure for an FITo find the duration of the total portfolio of assets(DA) (or liabilities (D L)) for an FI

First determine the duration of each asset (orliability) in the portfolio Then calculate the market value weighted

average of the duration of the assets (orliabilities) in the portfolio

)1/(

securityaof uemarket valin the%

R R D


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The change in the market value of the assetportfolio for a change in interest rates is:

Similarly, the change in the market value of theliability portfolio for a change in interest rates is:

)1()( R R

D A A A

)1()(

R

R D L L L


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Finally, the change in the market value of equity ofan FI given a change in interest rates is determinedfrom the basic balance sheet equation:

By substituting and rearranging, the change in networth is given as:

where k is L / A = a measure of the FIs leverage

E L A E L A

)1()(

R

R AkD D E L A


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The effect of interest rate changes on the marketvalue of equity or net worth of an FI breaks downto three effects:

The leverage adjusted duration gap = ( D A kD L )measured in yearsreflects the duration mismatch on an FIs balancesheetthe larger the gap, the more exposed the FI tointerest rate risk

The size of the FI The size of the interest rate shock


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Duration Gap Example

Assets $ Amount Weight Duration Weight x DurationT-bills 90 2.96% 0.500 0.015T-notes 55 1.81% 0.900 0.016T-bonds 176 5.78% 4.393 0.254Loans 2,724 89.46% 3.000 2.684

Liabilities & Equity $ Amount Weight Duration Weight x DurationDeposits 2,092 68.70% 0.500 0.344Federal funds 102 3.35% 0.010 0.000Borrowings 536 17.60% 5.500 0.968Equity 315 10.34% 0.000

3,045 100.00% DurL 1.312k 72.05%Duration Gap DurA - kDurL 2.023R 12%

R 0.005E -$27.51

1.120.005

3,0452.023

R)(1

R AkDur Dur E

LA


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Suppose a bank with $500 million in assetshas an average asset duration of 3 years, andan average liability duration of 1 year. Thebank also has a total debt ratio of 90%. R maybe thought of as the required return onequity or perhaps as the average interest ratelevel. If r is 12% and the bank is expecting a50 basis point increase in interest rate byhow much will the equity value change?


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(a) Calculate the leverage-adjusted duration gapof an FI that has assets of $1million invested30-year, 10 percent semiannual couponTreasury bonds selling at par and whose

duration has been estimated at 9.94 years. Ithas liabilities of $900,000 financed througha two years. 7.25 percent semiannualcoupon note selling at par?

(b) What is the impact in equity values if allinterest rates falls 20 basis points that is= 0.0020 R)(1

R


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The leverage adjusted duration gap = ( D A kD L )

= [9.94 (900,000/1,000,000)(1.8975)] = 8.23 years

b. Change in net worth using leveraged adjustedduration gap is given by

R)(1

R AkDur Dur E LA

50.464,16$)002.0(1mill8.23E


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Difficulties emerge when applying the durationmodel to real-world FI balance sheets Duration matching (immunization) can be costly as

restructuring the balance sheet is time consuming, costly,

and generally not desirable Immunization is a dynamic problemDuration of assets and liabilities change as theyapproach maturityThe rate at which the duration of assets and liabilitieschange may not be the same

Duration is not accurate for large interest rate changesunless convexity is modeled into the measure

Convexity is the degree of curvature of the price-yieldcurve around some interest rate level


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Substitution swapIntermarket swap

Rate anticipation swapPure yield pickupTax swap


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Select a particular holding periodand predict the yield curve at end ofperiod.Given a bonds time to maturity atthe end of the holding period, its yield can be read from the

predicted yield curve and the end-

of-period price can be calculated.


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Two passive bond portfolio strategies:

1. Indexing2. Immunization

Both strategies see market prices asbeing correct, but the strategies have

very different risks.


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Bond indexes contain thousands of issues,many of which are infrequently traded.Bond indexes turn over more than stockindexes as the bonds mature.Therefore, bond index funds hold only arepresentative sample of the bonds in theactual index.


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Immunization is a way to control interest raterisk.

Widely used by pension funds, insurancecompanies, and banks.


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Immunize a portfolio by matching the interestrate exposure of assets and liabilities. This means: Match the duration of the assets and

liabilities. Price risk and reinvestment rate risk exactly cancel

out.Result: Value of assets will track the value ofliabilities whether rates rise or fall.


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Cash flow matching = automaticimmunization.Cash flow matching is a dedicationstrategy.

Not widely used because of constraintsassociated with bond choices.

bond duration and duration model-student

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