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Overview

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. Price Sensitivity and Maturity.

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Overview

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  1. 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

  2. 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 $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 because the payment is discounted a greater number of times.

  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.

  5. Price Sensitivity of 6% Coupon Bond

  6. Price Sensitivity of 8% Coupon Bond

  7. 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 shows greater changes in price in response to a 2% change than the 8% bond. The first bond has greater interest rate risk.

  8. 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.

  9. 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.

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

  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: D = Snt=1[t  (Present Value of CFt/P)] • Notice that the weights correspond to the relative present values of the cash flows.

  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 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. 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 = 1.885 - 2.0 = -0.115 • Deposit has greater interest rate sensitivity than the loan, so DGAP is negative. • FI exposed to rising interest rates.

  17. 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

  18. Economic Interpretation • Duration is a measure of interest rate sensitivity or elasticity of a liability or asset: [dP/P]  [dR/(1+R)] = -D Or equivalently, dP/P = -D[dR/(1+R)] = -MD × dR where MD is modified duration.

  19. Economic Interpretation • To estimate the change in price, we can rewrite this as: dP = -D[dR/(1+R)]P = -(MD) × (dR) × (P) • Note the direct linear relationship between dP and -D.

  20. Semi-annual Coupon Payments • With semi-annual coupon payments: (dP/P)/(dR/R) = -D[dR/(1+(R/2)]

  21. 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.

  22. Duration as Index of Interest Rate Risk

  23. 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))

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

  25. 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.

  26. 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

  27. *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.

  28. *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.

  29. *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).

  30. *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.

  31. *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.

  32. *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.

  33. *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.

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

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