Interest Rate Risk II Chapter 9

1. Interest Rate Risk IIChapter 9 Financial Institutions Management, 3/e By Anthony Saunders

2. Price Sensitivity and Maturity • 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 $1762.34 in five years. Bond B pays $3105.85 in ten years, and both are currently priced at $1000.

3. Example continued... • 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.

4. Coupon Effect • Bonds with identical maturities will respond differently to interest rate changes when the coupons differ. This is more readily 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.

5. Price Sensitivity of 6% Coupon Bond

6. Price Sensitivity of 8% Coupon Bond

7. Remarks on Preceding Slides • The 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 shows greater changes in price in response to a 2% change than the 8% bond. The first bond is riskier.

8. Duration • Duration • Combines the effects of differences in coupon rates and differences in maturity. • Based on elasticity of bond price with respect to interest rate.

9. Duration • Duration D = Snt=1[Ct• t/(1+r)t]/ Snt=1 [Ct/(1+r)t] Where D = duration t = number of periods in the future Ct = cash flow to be delivered in t periods n= term-to-maturity & r = yield to maturity.

10. Duration • Duration • Weighted sum of the number of periods in the future of each cash flow, (weighted by respective fraction of the PV of the bond as a whole). • 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.

11. Advantages to Duration Measure: • 1. Simplicity • 2. Can be used to determine elasticity between price and YTM: (DP/P)/(Dr/r) = -D[r/(1+r)] • We can rewrite this as: DP = -D[P/(1+r)] Dr • Note the direct relationship between DP and -D.

12. Duration as Index of Interest Rate Risk: • The greater the duration, the greater the price sensitivity and the greater the risk. Higher duration indicates that it takes a longer time to recover the PV of the bond. This agrees with intuition once we realize that ONLY a zero-coupon bond has duration equal to maturity. ALL other bonds will have duration LESS than maturity.

13. 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 an installment loan with two equal payments of $522.61. Loan #2 is a discount loan, which has a single payment of $1,060.90. Loan #3 is structured as a 3% annual coupon bond.

14. Duration as Index of Interest Rate Risk

15. Limits to Duration Measure • Duration relationship may not hold if the bond has a call or prepayment provision. • Convexity. • Negative Convexity.

16. Special Case and an Adjustment • Maturity of a consol: M = . • Duration of a consol: D= 1 + 1/R • Adjusting for semi-annual payments dP/P = -D[dR/(1+ (1/2)R]

17. Immunizing 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 • DE = -[DA - DLk]A(DR/(1+R))

18. Duration and Immunizing • The formula shows 3 effects: • Leverage adjusted D-Gap • The size of the FI • The size of the interest rate shock

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

20. *Limitations of Duration • Only works with parallel shifts in yield curve. • 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.

21. *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. 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).

22. *Modified duration • 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.

23. *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.