1. Financial Risk Management, Skövde University 1 Chapter 9 Overview This chapter discusses a market value-based model for assessing and managing interest rate risk: This chapter discusses a market value-based model for assessing and 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

2. Financial Risk Management, Skövde University 2 Price Sensitivity and Maturity In general, the longer the term to maturity, the greater the sensitivity to interest rate changes. 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 $1762.34 in five years. Bond B pays $3105.85 in ten years, and both are currently priced at $1000. 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. Financial Risk Management, Skövde University 3 Example (cont.) Bond A: P = $1000 = $1762.34/(1.12) 5Bond A: P = $1000 = $1762.34/(1.12) 5 Bond B: P = $1000 = $3105.84/(1.12) 10Bond B: P = $1000 = $3105.84/(1.12) 10 Now suppose the interest rate increases by 1%. Now suppose the interest rate increases by 1%. Bond A: P = $1762.34/(1.13) 5 = $956.53Bond A: P = $1762.34/(1.13) 5 = $956.53 Bond B: P = $3105.84/(1.13) 10 = $914.94Bond 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. The longer maturity bond has the greater drop in price because the payment is discounted a greater number of times.

4. Financial Risk Management, Skövde University 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. Consequently, less sensitive to changes in R. 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. Consequently, less sensitive to changes in R.

5. Financial Risk Management, Skövde University 5 Price Sensitivity of 6% Coupon Bond

6. Financial Risk Management, Skövde University 6 Price Sensitivity of 8% Coupon Bond

7. Financial Risk Management, Skövde University 7 Remarks on Preceding Slides In general, longer maturity bonds experience greater price changes in response to any change in the discount rate. 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 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 has greater interest rate risk.The 6% bond shows greater changes in price in response to a 2% change than the 8% bond. The first bond has greater interest rate risk.

8. Financial Risk Management, Skövde University 8 Examples 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 immediately and one penny, ten years hence.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. 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 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.The higher the coupon rate, the more similar the bond is to our hypothetical bond with higher value of cash flows arriving sooner.

9. Financial Risk Management, Skövde University 9 Duration Duration Duration Weighted average time to maturity using the relative present values of the cash flows as weights.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.Combines the effects of differences in coupon rates and differences in maturity. Based on elasticity of bond price with respect to interest rate. Based on elasticity of bond price with respect to interest rate.

10. Financial Risk Management, Skövde University 10 Duration Duration Duration D =  n t=1 [t CF t /(1+R) t ] /  n t=1 [CF t /(1+R) t ] =  n t=1 [t CF t /(1+R) t ] /P =  n t=1 [t CF t /(1+R) t ] /PWhere D = duration t = term-to-maturity of each cash flow CF t = cash flow to be delivered in t periods CF t = cash flow to be delivered in t periods n= number of periods in the future R = yield to maturity. P= price of the bonds P= price of the bonds

11. Financial Risk Management, Skövde University 11 Duration Since the price (P) of the bond must equal the present value of all its cash flows, we can state the duration formula another way: Since the price (P) of the bond must equal the present value of all its cash flows, we can state the duration formula another way: D =  n t=1 [t  (Present Value of CF t /P)] Notice that the weights correspond to the relative present values of the cash flows. Notice that the weights correspond to the relative present values of the cash flows.

12. Financial Risk Management, Skövde University 12 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 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: For all other bonds: duration < maturity duration < maturity

13. Financial Risk Management, Skövde University 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. 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 Therefore, each coupon payment is $40 and the per period YTM is (1/2) × 12% = 6%. (1/2) × 12% = 6%. Present value of each cash flow equals CF t ÷ (1+ 0.06) t where t is the period number. Present value of each cash flow equals CF t ÷ (1+ 0.06) t where t is the period number.

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

15. Financial Risk Management, Skövde University 15 Duration equation In general, the duration equation is written as: In general, the duration equation is written as: where m is the number of times per year interest is paid. where m is the number of times per year interest is paid.

16. Financial Risk Management, Skövde University 16 Special Case  Maturity of a consol bond that pays a fixed coupon each year: M = . Theoretically a consol bond has a infinite time to maturity. Theoretically a consol bond has a infinite time to maturity.  Duration of a consol bond: D = 1 + 1/R where R is required yield to maturity. (see proof on next slide) D = 1 + 1/R where R is required yield to maturity. (see proof on next slide) As interest rate rises, the duration of a consol bond falls. See, textbook p222-225. or http://ihome.cuhk.edu.hk/~b100534/Courses /sau_ch10.pdf

17. Financial Risk Management, Skövde University 17 Consol bond: D = 1 + 1/R See G. Hawawini, 1987.

18. Financial Risk Management, Skövde University 18 Example: Duration Gap (see page 227) 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). 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: M L - M D = 2 -2 = 0 Maturity gap: M L - M D = 2 -2 = 0 Duration Gap: D L - D D = 1.883 - 2.0 = -0.117 Duration Gap: D L - D D = 1.883 - 2.0 = -0.117 Deposit has greater interest rate sensitivity than the loan (longer duration), so DGAP is negative.Deposit has greater interest rate sensitivity than the loan (longer duration), so DGAP is negative. The FI exposed to rising interest rates.The FI exposed to rising interest rates.

19. Financial Risk Management, Skövde University 19 Features of Duration Duration and maturity: Duration and maturity: D increases with M, but at a decreasing rate.D increases with M, but at a decreasing 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.

20. Financial Risk Management, Skövde University 20 Economic Interpretation Duration is a measure of interest rate sensitivity or elasticity of a liability or asset: Duration is a measure of interest rate sensitivity or elasticity of a liability or asset: [dP/P] /[dR/(1+R)] = -D See proof p225. Or equivalently, dP/P = -D[dR/(1+R)] = -MD × dR where MD=D/(1+R) is modified duration.

21. Financial Risk Management, Skövde University 21 Economic Interpretation To estimate the change in price (p), we can rewrite this as: To estimate the change in price (p), we can rewrite this as: dP = -D[dR/(1+R)]P = -(MD) × (dR) × (P) Modified duration is duration divided by one plus the interest rate. Modified duration is duration divided by one plus the interest rate. *Note the direct linear relationship between dP and -D.

22. Financial Risk Management, Skövde University 22 Semi-annual Coupon Payments With semi-annual coupon payments: With semi-annual coupon payments: (dP/P)/(dR/R) = -D[dR/(1+(R/2)] The derivation is similar to See textbook page 225

23. Financial Risk Management, Skövde University 23 EX: Consider three loan plans, all of which have maturity of 2 years. The loan amount is $1,000 and the current interest rate is 3%. EX: Consider three loan plans, all of which have maturity 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.

24. Financial Risk Management, Skövde University 24 Duration as measure of Interest Rate Risk

25. Financial Risk Management, Skövde University 25 Immunizing the Balance Sheet of an FI Duration Gap: calculation of D A and D L Duration Gap: calculation of D A and D L From the balance sheet, E=A-L. Therefore, E=A-L.

26. Financial Risk Management, Skövde University 26 Immunizing the Balance Sheet of an FI ThusApply we have

27. Financial Risk Management, Skövde University 27 Duration and Immunizing The formula shows 3 effects: The formula shows 3 effects: 1.Leverage adjusted Duration Gap: first item 2.The size of the FI: second item 3.The size of the interest rate shock: third item

28. Financial Risk Management, Skövde University 28 An example: p234 Suppose D A = 5 years, D L = 3 years and rates are expected to rise from 10% to 11%. Also, A = $100 M, L = $90 M and E = $10 M. Find change in E, i.e. the net worth of the FI. Suppose D A = 5 years, D L = 3 years and rates are expected to rise from 10% to 11%. Also, A = $100 M, L = $90 M and E = $10 M. Find change in E, i.e. the net worth of the FI. D A - D L k]A[R/(1+R)] = -[5 - 3(90/100)]100[.01/1.1] = - $2.09 M. Methods of immunizing balance sheet. Methods of immunizing balance sheet. Adjust D A, D L or k.Adjust D A, D L or k.

29. Financial Risk Management, Skövde University 29 Immunization and Regulatory Concerns Regulators set target ratios for an FI’s capital (net worth): Regulators set target ratios for an FI’s capital (net worth): Capital (Net worth) ratio = E/ACapital (Net worth) ratio = E/A If target is to set (E/A) = 0: If target is to set (E/A) = 0: D A = D LD A = D L To have E = 0 (immune to rate changes): To have E = 0 (immune to rate changes): We must have D A = kD L

30. Financial Risk Management, Skövde University 30 Some Limitations of Duration model 1.Immunizing the entire balance sheet can be costly. Duration model can be used in combination with hedge positions to immunize. 2.Immunization is a dynamic process since duration depends on instantaneous R. 3.Large interest rate change effects not accurately captured. Convexity 4.More complex if nonparallel shift in yield curve.

31. Financial Risk Management, Skövde University 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. 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. Financial Risk Management, Skövde University 32 *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). 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).

33. Financial Risk Management, Skövde University 33 *Modified duration P/P = -D[R/(1+R)] + (1/2) CX (R) 2 or P/P = -MD R + (1/2) CX (R) 2 Where MD=D/(1+R) 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 10 8.

34. Financial Risk Management, Skövde University 34 *Calculation of CX Example: convexity of 8% coupon, 8% yield, six-year maturity Eurobond priced at $1,000. Example: convexity of 8% coupon, 8% yield, six-year maturity Eurobond priced at $1,000. CX = 10 8 [P - /P + P + /P] = 10 8 [(999.53785-1,000)/1,000 + (1,000.46243-1,000)/1,000)] = 10 8 [(999.53785-1,000)/1,000 + (1,000.46243-1,000)/1,000)] = 28.

35. Financial Risk Management, Skövde University 35 *Duration Measure: Other Issues Default risk Default risk Floating-rate loans and bonds Floating-rate loans and bonds Duration of demand deposits and passbook savings Duration of demand deposits and passbook savings Mortgage-backed securities and mortgages Mortgage-backed securities and mortgages Duration relationship affected by call or prepayment provisions.Duration relationship affected by call or prepayment provisions.

36. Financial Risk Management, Skövde University 36 *Contingent Claims Interest rate changes also affect value of off-balance sheet 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.Duration gap hedging strategy must include the effects on off-balance sheet items such as futures, options, swaps, caps, and other contingent claims.