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BUFN 722

BUFN 722. ch-9 Interest Rate Risk – part 2 Duration, convexity, etc. 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

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BUFN 722

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  1. BUFN 722 ch-9 Interest Rate Risk – part 2 Duration, convexity, etc. BUFN722- Financial Institutions

  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 BUFN722- Financial Institutions

  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 $1762.34 in five years. Bond B pays $3105.85 in ten years, and both are currently priced at $1000. BUFN722- Financial Institutions

  4. 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. BUFN722- Financial Institutions

  5. 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. BUFN722- Financial Institutions

  6. Price Sensitivity of 6% Coupon Bond BUFN722- Financial Institutions

  7. Price Sensitivity of 8% Coupon Bond BUFN722- Financial Institutions

  8. 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 has greater interest rate risk. BUFN722- Financial Institutions

  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. BUFN722- Financial Institutions

  10. 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 (per period basis). BUFN722- Financial Institutions

  11. Duration • Since the price 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 Ct/Price)] • Notice that the weights correspond to the relative present values of the cash flows. BUFN722- Financial Institutions

  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 BUFN722- Financial Institutions

  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. BUFN722- Financial Institutions

  14. Duration of 2-year, 8% bond: Face value = $1,000, YTM = 12% BUFN722- Financial Institutions

  15. Special Case • Maturity of a consol: M = . • Duration of a consol: D = 1 + 1/R BUFN722- Financial Institutions

  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. BUFN722- Financial Institutions

  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 BUFN722- Financial Institutions

  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. BUFN722- Financial Institutions

  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. BUFN722- Financial Institutions

  20. Semi-annual Coupon Payments • With semi-annual coupon payments: (dP/P)/(dR/R) = -D[dR/(1+(R/2)] BUFN722- Financial Institutions

  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 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. BUFN722- Financial Institutions

  22. Duration as Index of Interest Rate Risk BUFN722- Financial Institutions

  23. Immunizing theBalance 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)) BUFN722- Financial Institutions

  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 BUFN722- Financial Institutions

  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. BUFN722- Financial Institutions

  26. Immunization and Regulatory Concerns • Regulators set target ratios for a bank’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 BUFN722- Financial Institutions

  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. BUFN722- Financial Institutions

  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. BUFN722- Financial Institutions

  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). BUFN722- Financial Institutions

  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. BUFN722- Financial Institutions

  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. BUFN722- Financial Institutions

  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. BUFN722- Financial Institutions

  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. BUFN722- Financial Institutions

  34. BUFN722- Financial Institutions

  35. Duration Gap Analysis %ΔP - DUR x Δi/(1+i) i 5%, from 10% to 15%  ΔAsset Value = %ΔP x Assets = -2.7 x .05/(1+.10) x $100m = -$12.3m ΔLiability Value = %ΔP x Liabilities = -1.03 x .05/(1+.10) x $95m = -$4.5m ΔNW = -$12.3m - (-$4.5m) = -$7.8m DURgap = DURa - [L/A x DURl] = 2.7 - [(95/100) x 1.03] = 1.72 %ΔNW = - DURgap x Δi/(1+i) = - 1.72 x .05/(1+.10) = -.078 = -7.8% ΔNW = -.078 x $100m = -$7.8m BUFN722- Financial Institutions

  36. Example of Finance Company Friendly Finance Company Assets Liabilities ---------------------------------------------------------------------------------------------- Cash and Deposits $ 3 m | Commercial Paper $ 40 m | Securities | Bank Loans less than 1 year $ 5 m | less than 1 year $ 3 m 1 to 2 year $ 1 m | 1 to 2 year $ 2 m greater than 2 year $ 1 m | greater than 2 year $ 5 m | Consumer Loans | Long-Term Bonds less than 1 year $ 50 m | and other long-term 1 to 2 year $ 20 m | debt $ 40 m greater than 2 year $ 15 m | | Capital $ 10 m Physical capital $ 5 m | BUFN722- Financial Institutions

  37. Duration of Finance Company's Assets and Liabilities BUFN722- Financial Institutions

  38. Gap and Duration Analysis • If i 5% • Gap Analysis • GAP = RSA - RSL = $55 m - $43 m = $12 million • ΔIncome = GAP x Δi = $12 m x 5% = $0.6 million • Duration Gap Analysis • DURgap = DURa - [L/A x DURl] • = 1.16 - [90/100 x 2.77] = -1.33 years • %ΔNW = - DURgap X Δi /(1+i) • = -(-1.33) x .05/(1+.10) • = .061 = 6.1% BUFN722- Financial Institutions

  39. Managing Interest-Rate Risk • Problems with GAP Analysis • 1. Assumes slope of yield curve unchanged and flat • 2. Manager estimates % of fixed rate assets and liabilities that are rate sensitive BUFN722- Financial Institutions

  40. Managing Interest-Rate Risk • Strategies for Managing Interest-Rate Risk • In example above, shorten duration of bank assets or lengthen duration of bank liabilities • To completely immunize net worth from interest-rate risk, set DURgap = 0 1. Reduce DURa = 0.98 DURgap = 0.98 - [(95/100) x 1.03] = 0 2. Raise DURl = 2.80 DURgap = 2.7 - [(95/100) x 2.80] = 0 BUFN722- Financial Institutions

  41. Duration: A Measure of Interest Rate Sensitivity The weighted-average time to maturity on an Investment, using the relative values of the cash flows as weights N N  CFt  tPVt  t t = 1(1 + R)tt = 1 D = N = N CFt PVt t = 1 (1 + R)t t = 1 This measure is known as Macaulay’s Duration BUFN722- Financial Institutions

  42. Key facts about Duration • Everything else equal, • 1. When the maturity of a bond lengthens, the duration rises as well. • 2. When interest rates rise, the duration of a coupon bond falls • 3. The higher is the coupon rate on the bond, the shorter is the duration of the bond. • 4. Duration is additive: the duration of a portfolio of securities is the weighted-average of the durations of the individual securities, with the weights equaling the proportion of the portfolio invested in each. BUFN722- Financial Institutions

  43. Example of Duration Calculation Table 3 1 CFt CFt X t Percent of Initial t CFt (1 + 4%)2t (1 + 4%)2t (1 + 4%)2t Investment Recovered 48.08 46.23 44.45 42.74 41.10 39.52 38.00 767.22 1,067.34 .5 1 1.5 2 2.5 3 3.5 4 50 50 50 50 50 50 50 1,050 0.9615 0.9246 0.8890 0.8548 0.8219 0.7903 0.7599 0.7307 24.04/1,067.34 = 0.02 46.23/1,067.34 = 0.04 66.67/1,067.34 = 0.06 85.48/1,067.34 = 0.08 102.75/1,067.34 = 0.10 118.56/1,067.34 = 0.11 133.00/1,067.34 = 0.13 3,068.88/1,067.34 = 2.88 24.04 46.23 66.67 85.48 102.75 118.56 133.00 3,068.88 3,645.61 3,645.61 1,067.34 D = = 3.42 years BUFN722- Financial Institutions

  44. Calculating Duration, R =10% 10-yr 10% annual Coupon Bond BUFN722- Financial Institutions

  45. Calculating Duration, R = 20% 10-yr 10% annual Coupon Bond BUFN722- Financial Institutions

  46. Duration and Interest-Rate Risk %ΔP - DUR x ΔR/(1+R) Modified Duration = Duration / (1 + R) R 10% to 11% (.10 to .11) => D = + .01: Table 4 -10% coupon bond %ΔP = 6.76 x .01/(1+.10) = -.0615 = -6.15%. Actual decline = 6.23% (need to add correction for convexity) The duration measure is a less accurate measure of price sensitivity the larger the change in interest rates 20% required return on 10% coupon bond, DUR = 5.72 years %ΔP = - 5.72 x .01/(1+.10) = -.0520 = -5.20% • The greater is the duration of a security, the greater is the percentage change in the market value of the security for a given change in interest rates. Therefore, the greater is the duration of a security, the greater is its interest-rate risk.

  47. Features of the Duration Measure • Duration and Coupon Interest • the higher the coupon payment, the lower its duration • Duration and Yield to Maturity • duration decreases as yield to maturity increases • Duration and Maturity • Duration increases with the maturity of a bond but at a decreasing rate BUFN722- Financial Institutions

  48. Economic Meaning of Duration • Measure of the average life of a bond • Measure of a bond’s interest rate sensitivity (elasticity) BUFN722- Financial Institutions

  49. Interest Rate Risk &Value of Cash Flows • I. Interest Rate Risk: Duration Measures • A. Interest Rate Risk of Zero-Coupon Bonds • Value Function — the plot of a bond’s price vs. interest rate • Dollar duration (DD) is a measure of a bond’s absolute price sensitivity to interest rate changes: • Modified duration (MD) measures a bond’s relative (%) change in price due to an interest rate change. BUFN722- Financial Institutions

  50. Changes in price or percentage can be approximated by: • Modified duration is the maturity of the Zero divided by one plus the spot rate. • B. Interest Rate Risk of Coupon Bonds and Other Fixed Cash Flows • Modified duration of a complex instrument is determined by summing the weighted modified durations of each of its cash flows. The weights are based upon the present value of the cash flow divided by the present value of the whole instrument. • . BUFN722- Financial Institutions

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