9-1

Chapter 9 part B

Portfolio Immunization Using

Duration

9-2

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)

9-3

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.

9-4

Immunizing the Balance Sheet of an FI in $

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

Assets $1000, MD = 4%Liabilities $800, MD = 5%Equity $200, MD = NA (treat as zero)

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

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

9-5

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 liabilities for R

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

9-6

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

Note that k = Liabilities/Assets

9-7

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

9-8

Duration and Immunizing

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

9-9

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.

9-10

*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.

9-11

*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.

9-12

*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.

9-13

Residential Mortgages

Typical term is 360 months Rate can be fixed or floating

Classic ARM adjusts every year Hybrid ARMs now very popular

Fixed first 3, 5 7 or 10 years

Fully amortizing loans No prepayment penalty (call price =100) Prepayments accelerate cash flows, so they reduce

duration Duration can move dramatically due to rate

changes (loan rate – new loan rate)