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Duration model Duration model Presentation Transcript

    • Interest Rate Risk
    • -Duration model
  • 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
    • 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 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 .
  • 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.
  • Price Sensitivity of 6% Coupon Bond
  • Price Sensitivity of 8% Coupon Bond
  • 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.
  • 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.
  • Duration
    • Duration
      • D =  N t=1 [CF t • t/(1+R) t ]/  N t=1 [CF t /(1+R) t ]
    • Where
    • D = duration
    • t = number of periods in the future
    • CF t = cash flow to be delivered in t periods
    • N= time-to-maturity
    • R = yield to maturity.
  • 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 =  N t=1 [t  (Present Value of CF t /P)]
    • Notice that the weights correspond to the relative present values of the cash flows.
  • 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
  • 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 CF t ÷ (1+ 0.06) t where t is the period number.
  • Duration of 2-year, 8% bond: Face value = $1,000, YTM = 12%
  • Special Case
    • Maturity of a consol: M =  .
    • Duration of a consol: D = 1 + 1/R
  • 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: M L - M D = 2 -2 = 0
    • Duration Gap: D L - D D = 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.
  • 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
  • Economic Interpretation
    • Duration is a measure of interest rate sensitivity or elasticity of a liability or asset:
    • [ Δ P/P]  [ Δ R/(1+R)] = -D
    • Or equivalently,
    • Δ P/P = -D[ Δ R/(1+R)] = -MD × Δ R
    • where MD is modified duration.
  • Economic Interpretation
    • To estimate the change in price, we can rewrite this as:
    • Δ P = -D[ Δ R/(1+R)]P = -(MD) × ( Δ R) × (P)
    • Note the direct linear relationship between Δ P and -D.
  • Semi-annual Coupon Payments
    • With semi-annual coupon payments:
    • ( Δ P/P)/( Δ R/R) = -D[ Δ R/(1+(R/2)]
  • 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.
  • Duration as Index of Interest Rate Risk
  • Immunizing the Balance Sheet of an FI
    • Duration Gap:
      • From the balance sheet, E=A-L. Therefore,  E=  A-  L. In the same manner used to determine the change in bond prices, we can find the change in value of equity using duration.
      •  E = [-D A A + D L L]  R/(1+R) or
      •  E  D A - D L k]A(  R/(1+R))
  • Duration and Immunizing
    • The formula shows 3 effects:
      • Leverage adjusted D-Gap
      • The size of the FI
      • The size of the interest rate shock
  • An example:
    • Suppose D A = 5 years, D L = 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.
    •  D A - D L k]A[  R/(1+R)]
      • = -[5 - 3(90/100)]100[.01/1.1] = - $2.09.
    • Methods of immunizing balance sheet.
      • Adjust D A , D L or k.
  • 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:
      • D A = D L
    • But, to set  E = 0:
      • D A = kD L
  • *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.
  • *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.