Chapter 9

Interest Rate Risk II

Chapter 9

Copyright © 2010 McGraw-Hill Ryerson Ltd., All Rights Reserved..

Prepared by Lois Tullo, Schulich School of Business, York University

FINA 481 Fall 2015 A.Addas 2

Chapter Outline

• This chapter presents the duration model and duration gap as measures of an FI’s interest rate risk: – BasiC arithmetic to calculate Duration– Economic meaning of Duration– Immunization using Duration– Problems in applying duration


Duration: A Simple Introduction

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

• Example: Suppose a $100 loan with a 15% interest rate, with half repayment after ½ a year, and the remaining at the end of the year.Weights


Duration: A Simple Introduction

• => Duration = 0.5349 x 0.5 + 0.4651 x 1 = 0.7326 years– This is the Weighted Average Time to Maturity of

this loan• Think of it this way: on a TVM basis, the bank’s

initial investment in the loan is recovered after 0.7326 years

• After that it earns a profit


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.


Example

– Bond A: P = $1000 = $1762.34/(1.12)5 – Bond B: P = $1000 = $3105.84/(1.12)10

• Now suppose the interest rate increases by 1%. – Bond A: P = $1762.34/(1.13)5 = $956.53– Bond B: P = $3105.84/(1.13)10 = $914.94

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


Duration: Definition and Features


Duration Formula

N

tt

N

tt

N

ttt

N

ttt

PV

tPV

DFCF

tDFCFD

1

1

1

1

D = Duration in YearsCFt = Cash Flow at the end of Period tN = Bond MaturityDFt = Discount Factor = R = Annual Yield to Maturity

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


Coupon Effect• Bonds with identical maturities will respond

differently to interest rate changes when the coupons differ.

• Think of coupon bonds as 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.

Þ Consequently, less sensitive to changes in R.

Duration of a 6yr bond with an 8% Coupon

2yr Bond, 8% coupon, 12% Yield





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. – The 6% bond will show a greater change in price in

response to a 2% change than the 8% bond. The 6% bond has greater interest rate risk.


Balance Sheet Example

• Consider FI held bond with 8% YTM, 10% coupon, 12 years to maturity, FV=$1000

• Financed by issuing bond at 10% YTM, 10% coupon, 12 years to maturity, FV=$1000

• PV of each bond =?


Balance Sheet Example

Market Value Balance Sheet:


Impact of Maturity


Impact of Maturity


Impact of Coupon Rate


Impact of Coupon Rate


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 essentially 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.


Duration of Zero-coupon Bond

• For a zero coupon bond, 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


Duration of a Consol Bond

• Consol bonds are perpetual (never mature)– Not available in Canada

• Maturity of a consol: M = .• Duration of a consol: D = 1 + 1/R• If R = 5% => D = 1 + 1/0.05 = 21 years– Based on TVM, investors would recover their

investment in 21 years. Subsequent cash flows are pure profits.


Duration Gap

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

• Maturity gap: ML - MD = 2 -2 = 0

• Duration Gap: DL - DD = 1.883 - 2.0 = -0.117– Deposit has greater interest rate sensitivity than

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


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.


Features of Duration• Duration and maturity:– D increases with M, but at a decreasing rate.


Economic Meaning of Duration

• Duration is a measure of interest rate sensitivity or elasticity of a liability or asset:

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


Economic Meaning of Duration

• 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.


Semi-annual Coupon Payments

• With semi-annual coupon payments:

(ΔP/P)/(ΔR/R) = -D[ΔR/(1+(R/2)]


Immunization example:• In 2010 an insurer guarantees a lump sum

payment in 2015 of $1,469 equal to an annually compounded rate of 8% over 5 years. Consider 2 immunization strategies:

• Discounted Bond –P = 680.58 = 1000 / (1.08) 5

– Buy 1.469 of these bonds• Six Year Coupon Bond (Table 9.2)– P5 = 1080 / 1.08 = $1000– Duration = 4.993 years (same as liability)


Immunizing the Balance Sheet of an FI

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

DE=DA-DL. In the same manner used to determine the change in bond prices, we can find the change in value of equity using duration.

– DE = [-DAA + DLL] DR/(1+R) or

– Or: DE = -[DA - DLk]A(DR/(1+R))

where k = L/A


Duration and Immunizing

• The formula shows 3 effects:– Leverage adjusted D-Gap (in years)• Measure of exposure to interest rate risk

– The size of the FI• Measured by Assets

– The size of the interest rate shock

– E = [Leverage adjusted Duration Gap] x Assets x Interest Rate Shock


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.• = -[DE DA - DLk]A[DR/(1+R)]

= -[5 - 3(90/100)]100[.01/1.1] = - $2.09 million• Methods of immunizing balance sheet.– Adjust DA , DL or k.


An example:• If rates rise by 1%:– ∆A/A = -5 x (0.01)/1.1 = -0.04545– A = 100 + (-0.04545 x 100) = 95.45– ∆L/L = -3 x (0.01)/1.1 = -0.02727– L = 90 + (-0.02727 x 90) = 87.54– E = A – L = 7.91


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



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

– E = -(5-(0.9) x 5) x $100m x (0.01/1.1) = -$0.45m



• Set E = 0:– DA = kDL

• Ex: Set DL=5.55– E = -(5-(0.9) x 5.55) x $100m x (0.01/1.1) = 0


Difficulties in Applying 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.


Difficulties in Applying Duration

– Large interest rate change effects not accurately captured.• Convexity

– More complex if nonparallel shift in yield curve.


Convexity

• The duration measure is a linear approximation of a non-linear function. Þ 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.


Convexity

• Recall 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).


Modified duration & Convexity

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


Calculation of CX

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

CX = 108[DP-/P + DP+/P] = 108[(999.53785-1,000)/1,000 +

(1,000.46243-1,000)/1,000)]= 28.


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.


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.


Chapter Summary

• This chapter presented the duration model and duration gap as measures of an FI’s interest rate risk: – Basis arithmetic to calculate Duration– Economic meaning of Duration– Immunization using Duration– Problems in applying duration


Pertinent Websites

Bank for International Settlements www.bis.org Securities Exchange Commission www.sec.govThe Wall Street Journal www.wsj.com

http://www.bis.org/

http://www.sec.gov/

http://www.wsj.com/

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Chapter 9

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