CHAPTER 9Interest Rate Risk II

Copyright © 2011 by The McGraw-Hill Companies, Inc. All Rights Reserved.McGraw-Hill/Irwin

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Overview

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

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Price Sensitivity and Maturity

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

Example: Suppose the zero coupon yield curve is flat at 12%. Bond A pays $1790.85 in five years. Bond B pays $3207.14 in ten years, and both are currently priced at $1000.

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Example continued...– Bond A: P = $1000 = $1790.85/(1.06)10

– Bond B: P = $1000 = $3207.14/(1.06)20

Now suppose the annual interest rate increases by 1% (0.5 % semiannually). – Bond A: P = $1762.34/(1.065)10 = $954.03– Bond B: P = $3105.84/(1.065)20 = $910.18

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

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Coupon Effect Bonds with identical maturities will

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

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Remarks on Preceding Slides

In general, longer maturity bonds experience greater price changes in response to any change in the discount rate

The range of prices is greater when the coupon is lower– A 6% bond will have a larger change in

price in response to a 2% change than an 8% bond

– The 6% bond has greater interest rate risk

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Extreme Examples With Equal Maturities

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

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

effect on the value of the zero but almost no impact on the hypothetical bond

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

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

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Duration Duration

– Weighted average time to maturity using the relative present values of the cash flows as weights

– Combines the effects of differences in coupon rates and differences in maturity

– Based on elasticity of bond price with respect to interest rate

– The units of duration are years

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Macaulay Duration Macaulay Duration

WhereD = Macaulay duration (in years)t = number of periods in the future

CFt = cash flow to be delivered in t periodsN= time-to-maturity DFt = discount factor

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Duration Since the price (P) of the bond equals the

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

Notice the weights correspond to the relative present values of the cash flows

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Semiannual Cash Flows It is important to see that we must

express t in years, and the present values are computed using the appropriate periodic interest rate. For semiannual cash flows, Macaulay duration, D is equal to:

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Duration of Zero-coupon Bond For a zero-coupon bond, Macaulay

duration equals maturity since 100% of its present value is generated by the payment of the face value, at maturity

For all other bonds, duration < maturity

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

face value of $1,000 and yield-to-maturity of 12%

Coupons are paid semi-annually Therefore, each coupon payment is $40

and the per period YTM is (1/2) × 12% = 6%

Present value of each cash flow equals CFt ÷ (1+ 0.06)t where t 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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Special Case Maturity of a consol: M = . Duration of a consol: D = 1 + 1/R

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Features of Duration Duration and maturity

– D increases with M, but at a decreasing rate

Duration and yield-to-maturity– D decreases as yield increases

Duration and coupon interest– D decreases as coupon increases

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Economic Interpretation Duration is a measure of interest rate

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

Or equivalently,ΔP/P = -D[ΔR/(1+R)] = -MD × ΔRwhere MD is modified duration

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Economic Interpretation To estimate the change in price, we

can rewrite this as:ΔP = -D[ΔR/(1+R)]P = -(MD) × (ΔR) × (P)

Note the direct linear relationship between ΔP and -D

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Dollar Duration Dollar duration equals modified

duration times price Dollar duration = MD × Price Using dollar duration, we can

compute the change in price asΔP = -Dollar duration × ΔR

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Semi-annual Coupon Payments

With semi-annual coupon payments the percentage change in price isΔP/P = -D[ΔR/(1+(R/2)]

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Immunization Matching the maturity of an asset

investment with a future payout responsibility does not necessarily eliminate interest rate risk

Matching durations will immunize against changes in interest rates

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An Example Consider three loan plans, all of which have

maturities of 2 years. The loan amount is $1,000 and the current interest rate is 3%.

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

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

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

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Duration as Index of Interest Rate Risk

Yield Loan Value 2% 3% ΔP 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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Balance Sheet Immunization Duration is a measure of the interest

rate risk exposure for an FI If the durations of liabilities and

assets are not matched, then there is a risk that adverse changes in the interest rate will increase the present value of the liabilities more than the present value of assets is increased

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Duration Gap Suppose that a 2-year coupon bond is the

only loan asset (A) of an FI. A 2-year certificate of deposit is the only liability (L). If the duration of the coupon bond is 1.8 years, then:Maturity gap: MA - ML = 2 -2 = 0, butDuration Gap: DA - DL = 1.8 - 2.0 = -0.2– Deposit has greater interest rate sensitivity than

the bond, so DGAP is negative– FI exposed to rising interest rates

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Immunizing the Balance Sheet of an FI

Duration Gap: – From the balance sheet, E=A-L.

Therefore, E=A-L. In the same manner used to determine the change in bond prices, we can find the change in value of equity using duration.

– E = [-DAA + DLL] R/(1+R) or– EDA - DLk]A(R/(1+R))

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Duration and Immunizing The formula shows 3 effects:

– Leverage adjusted D-Gap– The size of the FI– The size of the interest rate shock

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An Example Suppose DA = 5 years, DL = 3 years and

rates are expected to rise from 10% to 11%. (Rates change by 1%). Also, A = 100, L = 90 and E = 10. Find change in E.

DA - DLk]A[R/(1+R)] = -[5 - 3(90/100)]100[.01/1.1] = - $2.09.

Methods of immunizing balance sheet.– Adjust DA , DL or k.

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Immunization and Regulatory Concerns

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

If target is to set (E/A) = 0:– DA = DL

But, to set E = 0:– DA = kDL

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Limitations of Duration– Immunizing the entire balance sheet need not

be costly– Duration can be employed in combination with

hedge positions to immunize– Immunization is a dynamic process since

duration depends on instantaneous R– Large interest rate change effects not

accurately capturedConvexity

– More complex if nonparallel shift in yield curve

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Convexity The duration measure is a linear

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

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*Convexity Those who are familiar with calculus

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

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*Modified Duration & Convexity

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

– Where MD implies modified duration and CX is a measure of the curvature effect.

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

– Commonly used scaling factor is 108

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*Calculation of CX Example: convexity of 8% coupon,

8% yield, six-year maturity Eurobond priced at $1,000CX = 108[P-/P + P+/P]

= 108[(999.53785-1,000)/1,000 + (1,000.46243-1,000)/1,000)]

= 28

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*Duration Measure: Other Issues

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

passbook savings Mortgage-backed securities and

mortgages– Duration relationship affected by call or

prepayment provisions

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*Contingent Claims Interest rate changes also affect

value of off-balance sheet claims– Duration gap hedging strategy must

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

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Pertinent WebsitesBank for International

SettlementsSecurities Exchange

Commission The Wall Street Journal

www.bis.org

www.sec.gov

www.wsj.com