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# Overview - PowerPoint PPT Presentation

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

• 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.

• 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.

• 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.

• 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.

• 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]

Where

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.

• 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

• 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%

• Maturity of a consol: M = .

• Duration of a consol: D = 1 + 1/R

• 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.

• 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

• 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.

• 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.

• With semi-annual coupon payments:

(ΔP/P)/(ΔR/R) = -D[ΔR/(1+(R/2)]

• 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 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))

• The formula shows 3 effects:

• The size of the FI

• The size of the interest rate shock

• 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.

• 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

• 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).

• 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.

• 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.

• 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.

• 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.

Bank for International Settlements www.bis.org

Securities Exchange Commission www.sec.gov

The Wall Street Journal

www.wsj.com