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Presentation Transcript

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... Now suppose the interest rate increases by 1%. The longer maturity bond has the greater drop in price because the payment is discounted a greater number of times.

- Bond A: P = $1000 = $1762.34/(1.12)5
- Bond B: P = $1000 = $3105.84/(1.12)10

- Bond A: P = $1762.34/(1.13)5 = $956.53
- Bond B: P = $3105.84/(1.13)10 = $914.94

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

- 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

- 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

- 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 = 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.

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.

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 CFt ÷ (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: 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

- 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:
[ΔP/P] [ΔR/(1+R)] = -D

Or equivalently,

ΔP/P = -D[ΔR/(1+R)] = -MD × ΔR

where MD is modified duration.

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

- With semi-annual coupon payments:
(ΔP/P)/(ΔR/R) = -D[ΔR/(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.

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

- The formula shows 3 effects:
- Leverage adjusted D-Gap
- The size of the FI
- The size of the interest rate shock

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

- 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

- 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.

*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.

*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).

*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

- 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

- 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

- 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

Bank for International Settlements www.bis.org

Securities Exchange Commission www.sec.gov

The Wall Street Journal

www.wsj.com

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