Financial Risk Management, Skövde University 1 Chapter 9 Overview This chapter discusses a market...

Financial Risk Management, Skövde UniveFinancial Risk Management, Skövde Universityrsity

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Chapter 9 OverviewChapter 9 Overview

This chapter discusses a market This chapter discusses a market value-based model for assessing and value-based model for assessing and managing interest rate risk: managing interest rate risk: • DurationDuration• Computation of durationComputation of duration• Economic interpretationEconomic interpretation• Immunization using durationImmunization using duration• * Problems in applying duration* Problems in applying duration


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Price Sensitivity and MaturityPrice Sensitivity and Maturity

In general, the longer the term to In general, the longer the term to maturity, the greater the sensitivity maturity, the greater the sensitivity to interest rate changes. to interest rate changes.

Example: Suppose the zero coupon Example: Suppose the zero coupon yield curve is flat at 12%. Bond A yield curve is flat at 12%. Bond A pays $1762.34 in five years. Bond B pays $1762.34 in five years. Bond B pays $3105.85 in ten years, and both pays $3105.85 in ten years, and both are currently priced at $1000.are currently priced at $1000.


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Example (cont.)Example (cont.)

• Bond A: P = $1000 = $1762.34/(1.12)Bond A: P = $1000 = $1762.34/(1.12)55 • Bond B: P = $1000 = $3105.84/(1.12)Bond B: P = $1000 = $3105.84/(1.12)1010

Now suppose the interest rate Now suppose the interest rate increases by 1%. increases by 1%. • Bond A: P = $1762.34/(1.13)Bond A: P = $1762.34/(1.13)55 = $956.53 = $956.53• Bond B: P = $3105.84/(1.13)Bond B: P = $3105.84/(1.13)10 10 = $914.94= $914.94

The longer maturity bond has the greater The longer maturity bond has the greater drop in price drop in price because the payment is because the payment is discounted a greater number of timesdiscounted a greater number of times..


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Coupon EffectCoupon Effect Bonds with identical maturities will respond Bonds with identical maturities will respond

differently to interest rate changes when differently to interest rate changes when the coupons differ. This is more readily the coupons differ. This is more readily understood by recognizing that coupon understood by recognizing that coupon bonds consist of a bundle of “zero-coupon” bonds consist of a bundle of “zero-coupon” bonds. With higher coupons, more of the bonds. With higher coupons, more of the bond’s value is generated by cash flows bond’s value is generated by cash flows which take place sooner in time. which take place sooner in time. Consequently, less sensitive to changes in Consequently, less sensitive to changes in R.R.

Financial Risk Management, Skövde University

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Price Sensitivity of 6% Coupon Bond

r 8% 6% 4% Range

n

40 $802 $1,000 $1,273 $471

20 $864 $1,000 $1,163 $299

10 $919 $1,000 $1,089 $170

2 $981 $1,000 $1,019 $37


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Price Sensitivity of 8% Coupon Bond

r 10% 8% 6% Range

n

40 $828 $1,000 $1,231 $403

20 $875 $1,000 $1,149 $274

10 $923 $1,000 $1,085 $162

2 $981 $1,000 $1,019 $38


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Remarks on Preceding SlidesRemarks on Preceding Slides In general, longer maturity bonds In general, longer maturity bonds

experience greater price changes in experience greater price changes in response to any change in the discount response to any change in the discount rate.rate.

The range of prices is greater when the The range of prices is greater when the coupon is lower. coupon is lower. • The 6% bond shows greater changes in The 6% bond shows greater changes in

price in response to a 2% change than the price in response to a 2% change than the 8% bond. The first bond has greater 8% bond. The first bond has greater interest rate risk.interest rate risk.


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Examples with equal maturitiesExamples with equal maturities

Consider two ten-year maturity instruments:Consider two ten-year maturity instruments:• A ten-year zero coupon bondA ten-year zero coupon bond• A two-cash flow “bond” that pays $999.99 almost A two-cash flow “bond” that pays $999.99 almost

immediately and one penny, ten years hence.immediately and one penny, ten years hence. Small changes in yield will have a large effect Small changes in yield will have a large effect

on the value of the zero but essentially no on the value of the zero but essentially no impact on the hypothetical bond.impact on the hypothetical bond.

Most bonds are between these extremesMost bonds are between these extremes• The higher the coupon rate, the more similar the The higher the coupon rate, the more similar the

bond is to our hypothetical bond with higher value of bond is to our hypothetical bond with higher value of cash flows arriving sooner.cash flows arriving sooner.


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DurationDuration

DurationDuration• Weighted average time to maturity Weighted average time to maturity

using the relative present values of the using the relative present values of the cash flows as weights.cash flows as weights.

• Combines the effects of differences in Combines the effects of differences in coupon rates and differences in coupon rates and differences in maturity. maturity.

• Based on elasticity of bond price with Based on elasticity of bond price with respect to interest rate.respect to interest rate.


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DurationDuration Duration Duration

D = D = nnt=1t=1[t[t CFCFt t /(1+R)/(1+R)tt] / ] / nn

t=1t=1[CF[CFtt/(1+R)/(1+R)tt]] = = nn

t=1t=1[t[t CFCFt t /(1+R)/(1+R)tt] /P] /PWhereWhere

D = durationD = durationt = term-to-maturity of each cash flowt = term-to-maturity of each cash flow

CFCFtt = cash flow to be delivered in t periods = cash flow to be delivered in t periodsn= number of periods in the futuren= number of periods in the futureR = yield to maturity.R = yield to maturity.

P= price of the bondsP= price of the bonds


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DurationDuration Since the price (P) of the bond must Since the price (P) of the bond must

equal the present value of all its cash equal the present value of all its cash flows, we can state the duration flows, we can state the duration formula another way:formula another way:

D = D = nnt=1t=1[t [t (Present Value of CF (Present Value of CFtt/P)]/P)]

Notice that the weights Notice that the weights correspond to the correspond to the relativerelative present present values of the cash flows.values of the cash flows.


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Duration of Zero-coupon BondDuration of Zero-coupon Bond

For a zero coupon bond, duration For a zero coupon bond, duration equals maturity since 100% of its equals maturity since 100% of its present value is generated by the present value is generated by the payment of the face value, at payment of the face value, at maturity.maturity.

For all other bonds:For all other bonds: duration < maturityduration < maturity


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Computing durationComputing duration Consider a 2-year, 8% coupon bond, with Consider a 2-year, 8% coupon bond, with

a face value of $1,000 and yield-to-a face value of $1,000 and yield-to-maturity of 12%. Coupons are paid semi-maturity of 12%. Coupons are paid semi-annually.annually.

Therefore, each coupon payment is $40 Therefore, each coupon payment is $40 and the per period YTM is and the per period YTM is

(1/2) × 12% = 6%.(1/2) × 12% = 6%. Present value of each cash flow equals CFPresent value of each cash flow equals CFtt

÷ (1+ 0.06)÷ (1+ 0.06)tt where where tt is the period number. is the period number.


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Duration of 2-year, 8% bond: Face value = $1,000, YTM = 12%

t years CFt PV(CFt) Weight (W)

W × years

1 0.5 40 37.736 0.041 0.020

2 1.0 40 35.600 0.038 0.038

3 1.5 40 33.585 0.036 0.054

4 2.0 1,040 823.777 0.885 1.770

P = 930.698 1.000 D=1.883 (years)


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Duration equation Duration equation In general, the In general, the

duration duration equation is equation is written as: written as:

where m is the where m is the number of number of times per year times per year interest is paid.interest is paid.

N

tmt

t

N

tmt

t

m

m

mR

CFmR

tCF

D

1

1

1

1


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Special CaseSpecial Case Maturity of a consol bond that pays a fixed Maturity of a consol bond that pays a fixed

coupon each year: M = coupon each year: M = .. Theoretically a consol bond has a infinite Theoretically a consol bond has a infinite

time to maturity.time to maturity. Duration of a consol bond: Duration of a consol bond: D = 1 + 1/R where R is required yield to D = 1 + 1/R where R is required yield to

maturity. (see proof on next slide)maturity. (see proof on next slide)As interest rate rises, the duration of a consol As interest rate rises, the duration of a consol

bond falls. bond falls. See, textbook p222-225.See, textbook p222-225.ororhttp://ihome.cuhk.edu.hk/~b100534/http://ihome.cuhk.edu.hk/~b100534/

Courses/sau_ch10.pdfCourses/sau_ch10.pdf


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Consol bond: D = 1 + 1/RConsol bond: D = 1 + 1/RSee G. Hawawini, 1987.See G. Hawawini, 1987.


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Example: Duration Gap Example: Duration Gap (see page 227)(see page 227)

Suppose the bond in the previous Suppose the bond in the previous example is the only loan asset (L) of an example is the only loan asset (L) of an FI, funded by a 2-year certificate of FI, funded by a 2-year certificate of deposit (D). deposit (D).

Maturity gap: MMaturity gap: MLL - M - MDD = 2 -2 = 0 = 2 -2 = 0 Duration Gap: Duration Gap: DDLL - D - DDD = 1.883 - 2.0 = -0.117 = 1.883 - 2.0 = -0.117

• Deposit has greater interest rate sensitivity Deposit has greater interest rate sensitivity than the loan (longer duration), so DGAP is than the loan (longer duration), so DGAP is negative. negative.

• The FI exposed to rising interest rates.The FI exposed to rising interest rates.


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Features of DurationFeatures of Duration

Duration and maturity:Duration and maturity:• D increases with M, but at a decreasing D increases with M, but at a decreasing

rate.rate.

Duration and yield-to-maturity:Duration and yield-to-maturity:• D decreases as yield increases.D decreases as yield increases.

Duration and coupon interest:Duration and coupon interest:• D decreases as coupon increases.D decreases as coupon increases.


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Economic Interpretation Economic Interpretation Duration is a measure of interest rate Duration is a measure of interest rate

sensitivity or sensitivity or elasticity of a liabilityelasticity of a liability or or asset:asset:[dP/P] /[dR/(1+R)] = -D [dP/P] /[dR/(1+R)] = -D

See proof p225.See proof p225.Or equivalently,Or equivalently,dP/P = -D[dR/(1+R)] = -MD × dRdP/P = -D[dR/(1+R)] = -MD × dRwhere MD=D/(1+R) is where MD=D/(1+R) is modified modified

durationduration..


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Economic InterpretationEconomic Interpretation

To estimate the change in price (p), To estimate the change in price (p), we can rewrite this as:we can rewrite this as:

dP = -D[dR/(1+R)]P = -(MD) × (dR) × (P)dP = -D[dR/(1+R)]P = -(MD) × (dR) × (P)

Modified duration is duration divided by Modified duration is duration divided by one plus the interest rate.one plus the interest rate.

*Note the direct *Note the direct linearlinear relationship between relationship between dP and -D.dP and -D.


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Semi-annual Coupon PaymentsSemi-annual Coupon Payments

With semi-annual coupon payments:With semi-annual coupon payments:

(dP/P)/(dR/R) = -D[dR/(1+(R/2)](dP/P)/(dR/R) = -D[dR/(1+(R/2)]

The derivation is similar to The derivation is similar to

See textbook page 225See textbook page 225


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EX: Consider three loan plans, all of EX: Consider three loan plans, all of which have which have maturity of 2 yearsmaturity of 2 years. The . The loan amount is loan amount is $1,000$1,000 and the current and the current interest rate is interest rate is 3%.3%.

Loan #1, is a two-payment loan with Loan #1, is a two-payment loan with two equal payments of $522.61 each. two equal payments of $522.61 each.

Loan #2 is structured as a 3% annual Loan #2 is structured as a 3% annual coupon bond. coupon bond.

Loan # 3 is a discount loan, which has a Loan # 3 is a discount loan, which has a single payment of $1,060.90. single payment of $1,060.90.


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Duration as measure of Interest Duration as measure of Interest Rate RiskRate Risk

Yield Loan Value 2% 3% dP n D

Equal Payment

$1014.68 $1000 $14.68 2 1.493

3% Coupon $1019.42 $1000 $19.42 2 1.971

Discount $1019.70 $1000 $19.70 2 2.000


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Immunizing the Balance Sheet of an FIImmunizing the Balance Sheet of an FI Duration Gap: calculation of DDuration Gap: calculation of DAA and and

DDLL

From the balance sheet, E=A-L. From the balance sheet, E=A-L. Therefore, Therefore, E=E=A-A-L. L.


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Immunizing the Balance Sheet of an FIImmunizing the Balance Sheet of an FI

Thus Thus

ApplyApply

we havewe have


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Duration and ImmunizingDuration and Immunizing

The formula shows 3 effects:The formula shows 3 effects:

1.1. Leverage adjusted Duration Gap: first Leverage adjusted Duration Gap: first itemitem

2.2. The size of the FI: second itemThe size of the FI: second item3.3. The size of the interest rate shock: The size of the interest rate shock:

third itemthird item


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An example: p234An example: p234 Suppose DSuppose DAA = 5 years, D = 5 years, DLL = 3 years = 3 years

and rates are expected to rise from and rates are expected to rise from 10% to 11%. Also, A = $100 M, L = 10% to 11%. Also, A = $100 M, L = $90 M and E = $10 M. Find change in $90 M and E = $10 M. Find change in E, i.e. the net worth of the FI.E, i.e. the net worth of the FI.

DDAA - D - DLLk]A[k]A[R/(1+R)]R/(1+R)]

= -[5 - 3(90/100)]100[.01/1.1] = - $2.09 M.= -[5 - 3(90/100)]100[.01/1.1] = - $2.09 M.

Methods of immunizing balance sheet.Methods of immunizing balance sheet.• Adjust DAdjust DAA , D , DLL or k. or k.


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Immunization and Regulatory Concerns Immunization and Regulatory Concerns

Regulators set target ratios for an Regulators set target ratios for an FI’s capital (net worth): FI’s capital (net worth): • Capital (Net worth) ratio = E/ACapital (Net worth) ratio = E/A

If target is to set If target is to set (E/A) = 0:(E/A) = 0:• DDAA = D = DLL

To have To have E = 0 (immune to rate E = 0 (immune to rate changes):changes):We must have DWe must have DAA = kD = kDLL


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Some Limitations of Duration modelSome Limitations of Duration model

1.1. Immunizing the entire balance sheet can be Immunizing the entire balance sheet can be costly. Duration model can be used in costly. Duration model can be used in combination with hedge positions to immunize.combination with hedge positions to immunize.

2.2. Immunization is a dynamic process since Immunization is a dynamic process since duration depends on instantaneous R.duration depends on instantaneous R.

3.3. Large interest rate change effects not Large interest rate change effects not accurately captured.accurately captured.ConvexityConvexity

4.4. More complex if nonparallel shift in yield curve.More complex if nonparallel shift in yield curve.


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*Convexity*Convexity The duration measure is a linear The duration measure is a linear

approximation of a non-linear approximation of a non-linear function. If there are large changes in function. If there are large changes in R, the approximation is much less R, the approximation is much less accurate. All fixed-income securities accurate. All fixed-income securities are convex. Convexity is desirable, are convex. Convexity is desirable, but greater convexity causes larger but greater convexity causes larger errors in the duration-based estimate errors in the duration-based estimate of price changes. of price changes.


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*Convexity*Convexity

Recall that duration involves only the Recall that duration involves only the first derivative of the price function. first derivative of the price function. We can improve on the estimate We can improve on the estimate using a Taylor expansion. In practice, using a Taylor expansion. In practice, the expansion rarely goes beyond the expansion rarely goes beyond second order (using the second second order (using the second derivative).derivative).


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*Modified duration*Modified duration

P/P = -D[P/P = -D[R/(1+R)] + (1/2) CX (R/(1+R)] + (1/2) CX (R)R)22 or or P/P = -MD P/P = -MD R + (1/2) CX (R + (1/2) CX (R)R)22

Where MD=D/(1+R) implies modified Where MD=D/(1+R) implies modified duration and CX is a measure of the duration and CX is a measure of the curvature effect. curvature effect.

CX = Scaling factor × [capital loss from CX = Scaling factor × [capital loss from 1bp rise in yield + capital gain from 1bp 1bp rise in yield + capital gain from 1bp fall in yield]fall in yield]

Commonly used scaling factor is 10Commonly used scaling factor is 1088..


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*Calculation of CX*Calculation of CX

Example: convexity of 8% coupon, Example: convexity of 8% coupon, 8% yield, six-year maturity Eurobond 8% yield, six-year maturity Eurobond priced at $1,000.priced at $1,000.

CX = 10CX = 1088[[PP--/P + /P + PP++/P]/P]

= 10= 1088[(999.53785-1,000)/1,000 + [(999.53785-1,000)/1,000 + (1,000.46243-1,000)/1,000)](1,000.46243-1,000)/1,000)]

= 28.= 28.


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*Duration Measure: Other Issues*Duration Measure: Other Issues

Default riskDefault risk Floating-rate loans and bondsFloating-rate loans and bonds Duration of demand deposits and Duration of demand deposits and

passbook savingspassbook savings Mortgage-backed securities and Mortgage-backed securities and

mortgagesmortgages• Duration relationship affected by call or Duration relationship affected by call or

prepayment provisions. prepayment provisions.


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*Contingent Claims*Contingent Claims

Interest rate changes also affect Interest rate changes also affect value of off-balance sheet claims.value of off-balance sheet claims.• Duration gap hedging strategy must Duration gap hedging strategy must

include the effects on off-balance sheet include the effects on off-balance sheet items such as futures, options, swaps, items such as futures, options, swaps, caps, and other contingent claims.caps, and other contingent claims.

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