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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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  • 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
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
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
    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.

    Remarks on preceding slides
    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.

    Extreme examples with equal maturities
    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.


    • 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

      D = SNt=1[CFt• t/(1+R)t]/ SNt=1 [CFt/(1+R)t]


      D = duration

      t = number of periods in the future

      CFt = cash flow to be delivered in t periods

      N= time-to-maturity

      R = yield to maturity.


    • 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.

    Duration of zero coupon bond
    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
    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.

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

    Special case
    Special Case

    • Maturity of a consol: M = .

    • Duration of a consol: D = 1 + 1/R

    Duration gap
    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.

    Features of duration
    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
    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.

    Economic interpretation1
    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.

    Semi annual coupon payments
    Semi-annual Coupon Payments

    • With semi-annual coupon payments:

      (ΔP/P)/(ΔR/R) = -D[ΔR/(1+(R/2)]

    An example
    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.

    Immunizing the balance sheet of an fi
    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))

    Duration and immunizing
    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 example1
    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.

    Immunization and regulatory concerns
    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

    Limitations of duration
    *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.


    • 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.


    • 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).

    Modified duration convexity
    *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.

    Calculation of cx
    *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.

    Duration measure other issues
    *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.

    Contingent claims
    *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.

    Pertinent websites
    Pertinent Websites

    Bank for International Settlements www.bis.org

    Securities Exchange Commission www.sec.gov

    The Wall Street Journal