Overview

2. 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

3. 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.

4. 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.

5. 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.

6. 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

7. 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

8. 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

9. Macaulay Duration • Macaulay Duration Where D = Macaulay duration (in years) t = number of periods in the future CFt = cash flow to be delivered in t periods N= time-to-maturity DFt = discount factor

10. 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

11. Semiannual Cash Flows • It is important to see that we must express t in years, and the present values are computed using the appropriate periodicinterest rate. For semiannual cash flows, Macaulay duration, D is equal to:

12. 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

13. 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

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

15. Special Case • Maturity of a consol: M = . • Duration of a consol: D = 1 + 1/R

16. 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

17. 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 × ΔR where MD is modified duration

18. 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

19. Dollar Duration • Dollar duration equals modified duration times price • Dollar duration = MD × Price • Using dollar duration, we can compute the change in price as ΔP = -Dollar duration × ΔR

20. Semi-annual Coupon Payments • With semi-annual coupon payments the percentage change in price is ΔP/P = -D[ΔR/(1+(R/2)]

21. 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

22. 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.

23. Duration as Index of Interest Rate Risk

24. 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

25. 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, but Duration 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

26. 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 • DE = -[DA - DLk]A(DR/(1+R))

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

28. 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.

29. 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

30. 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 captured • Convexity • More complex if nonparallel shift in yield curve

31. 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.

32. *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.

33. *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

34. *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

35. *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

36. *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

37. Pertinent Websites Bank for International Settlements Securities Exchange Commission The Wall Street Journal www.bis.org www.sec.gov www.wsj.com