Interest Rate Risk Interest Rate Risk IIII

Chapter 9

© 2008 The McGraw-Hill Companies, Inc., All Rights Reserved.McGraw-Hill/Irwin

9-2

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

Note that slides 3-16 refer to Macaulay duration

9-3

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.

9-4

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.

9-5

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.

9-6

Price Sensitivity of 6% Coupon Bond

r 8% 6% 4% Range

n

40 $802 $1,000 $1,273 $471

20 $864 $1,000 $1,163 $299

10 $919 $1,000 $1,089 $170

2 $981 $1,000 $1,019 $38.28

9-7

Price Sensitivity of 8% Coupon Bond

r 10% 8% 6% Range

n

40 $828 $1,000 $1,231 $403

20 $875 $1,000 $1,149 $274

10 $923 $1,000 $1,085 $162

2 $981 $1,000 $1,019 $37.72

9-8

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.

9-9

Macaulay 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.

9-10

Macaulay Duration

Duration

D = Nt=1[CFt• t/(1+R)t]/ N

t=1 [CFt/(1+R)t]

WhereD = 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.

9-11

Macaulay 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 = Nt=1[t (Present Value of CFt/P)]

Notice that the weights correspond to the relative present values of the cash flows.

9-12

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

9-13

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.

Note the discount rate stated as an annual yield is 1.06^2-1 = 12.36%

9-14

Duration of 2-year, 8% bond:

Face value = $1,000, YTM = 12.36%

t years CFt PV(CFt) Weight (W)

W × years

1 0.5 40 37.736 0.041 0.020

2 1.0 40 35.600 0.038 0.038

3 1.5 40 33.585 0.036 0.054

4 2.0 1,040 823.777 0.885 1.770

P = 930.698 1.000 D=1.883 (years)

9-15

Duration Gap

Suppose the bond in the previous example is the only loan asset of an FI, funded by a 2-year certificate of deposit that pays all interest at maturity.

Maturity gap: MA - ML = 2 -2 = 0

Duration Gap: DA - DL = 1.883 - 2.0 = -0.117 Deposit has greater interest rate sensitivity than the

loan, so DGAP is negative. (by this calc) Is FI exposed to rising or falling interest rates?

Note that above numbers do not adjust for equity, that is A>L.

If equity =10%, then DL x k = DL x .90/1.0 = 2.0 x .9 = 1.8

12 &1

9-16

Features of Duration

Duration and maturity: Duration increases with Maturity, but at a

decreasing rate. Duration and yield-to-maturity:

Duration decreases as yield increases. Duration and coupon interest:

Duration decreases as coupon increases

9-17

Economic Interpretation

Duration is a measure of interest rate sensitivity or elasticity of a liability or asset:

ΔP/P = -D[ΔR/(1+R)] = -MD × ΔR

where MD is modified duration.

More simply MD = D/(1+r)

9-18

Economic Interpretation

To estimate the change in price, we can rewrite this as:

ΔP = -D[ΔR/(1+R)]P = -(MD) × (ΔR) × (P)

ΔP = -MD X ΔR X P

Note the direct linear relationship between ΔP and -D.

9-19

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.

9-20Duration as Index of Interest Rate RiskDiscounted at 3%

Price Loan Value 3% ΔP N D MD

Equal Payment

$1000 $14.68 2 1.493 1.449%

3% Coupon $1000 $19.42 2 1.971

1.913%

Discount $1000 $19.70 2 2.000 1.942%

9

9-21

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 = [-DAA + DLL] R/(1+R) or

EDA - DLk]A(R/(1+R)) or, more simply

E$DA - MDLk] x A x R

9-22

Immunizing the Balance Sheet of an FI in $

-DA x A x R/(1+R)=-MDAA x R-MDLL x R/(1+R)=-MDLL x R

Assets $1000, MD = 4%Liabilities $800, MD = 5%Equity $200, MD = NA

.04 x $1000 x 1 = $40 for a 1% rate change

.05 x $800 x 1 = $40 for a 1% rate changeInstitution is matched.

9-23

Immunizing the Balance Sheet of an FI in $

-DAA x R/(1+R)=-MDAA x R

= $ gain/loss on assets for R

-DLL x R/(1+R)=-MDLL x R

= $ gain/loss on assets for R

Equity does not need to be considered since all $ gains/losses on assets & liabilities are accounted for

9-24

Immunizing the Balance Sheet of an FI in %

EDA - MDLk]A x R k=L/A k x MDL = unlevered MDL = L/A x MDL

9-25

Duration and Immunizing

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

9-26

Immunizing the Balance Sheet of an FI in %

EDA - MDLk]A x R k=L/A k x MDL = unlevered MDL = L/A x MDL

Assets $1000, MD = 4%Liabilities $800, MD = 5%Equity $200, MD = NA

MDA = 4%

Unlevered MDL = 5% x 800/1000 =4%Institution is matched.

9-27

*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.

9-28

*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.

9-29

*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.