Hedging

INTEREST RATE RISK



THE DURATION GAP MODEL

• Comprehensive measure of interest rate risk

• Considers Market Values (

Example: Bonds



• Considers the MATURITY STRUCTURE of all assets and liabilities.





• Considers the degree of LEVERAGE on the Balance Sheet.







• Considers the TIMING of every Cash Flow ( payments as well as cashinflows.



← DURATION = COMPREHENSIVE MEASURE of INTEREST RATE RISK, i.e., measures the interest sensitivity (elasticity) of asset and liability values.




Consider a “Straight Bond”:



(
















Multiplying both sides by [pic], we obtain:

[pic]



Duration = ELASTICITY Measure ( a relative ∆ (%) in P due to a relative ∆ (%) in R or i.Also:

[pic]



In general when cash flows are paid/received m times per year:



[pic] Normally m = 1 or 2









MODIFIED DURATION (MD) is defined as follows:



[pic] or, for discreet ∆ in i or R.

[pic][pic]

∆ in VALUE directly proportional to the Magnitude of Modified Duration.

IOW:

[pic]



D ( Varies INVERSELY with y ( y↑ (distant CF´s do not contribute much to the value of an asset/liability.

↔ MOST VALUE comes from early cash flows.

D ( Varies INVERSELY with C (coupon): To see this, it is important to understand that



There are 2 opposite effects of ∆i: 1. Capital Loss/Gain



2. Reinvestment Effect.

[pic]



Time (t) until Maturity determines which effectdominates:

SELL NEXT DAY ( ONLY CAPITAL EFFECT MATTERS.

MAINTAIN UNTILL MATURITY ( ONLY REINVESTMENT EFFECT MATTERS.



Between these 2 extremes [pic] point at which both effects cancel out

NEUTRALIZE exactly.



INTERPRETATION D IMMUNIZE USE of DURATION

This point = DURATION

D ( Varies DIRECTLY with M(aturity)

However, D = finite,even when M ( [pic]



Let us look at 2 extreme situations:

1. Duration of a 0-Coupon Bond

No intermediate CF’s ( Reinvestment effect = 0

All value comes from the last and only CF at M ( D = M

Proof:

[pic]

[pic]

Now, [pic]

[pic]

← 0-coupon Bond has the highest INTEREST RATE SENSITIVITY: If ∆+i → IntermediateCF’s cannot be reinvested because they do not exist!



2. Perpetual Bond (“consol bond” -> issued to finance the BOER wars in South-Africa)

M → [pic]

All value comes from reinvestment!

¿D = [pic]? ¡NO! = [pic]

Proof:















































( Even when M →[pic], D = finite = [pic]

NB: Up till now,only small changes in i (or yield), were considered.

For large movements, i.e. substantial ∆i → we need to take into account the CONVEXITY of the PRICE – YIELD CURVE:


















IMMUNIZATION OF THE BALANCE SHEET OF A FINANCIAL INSTITUTION (FI).




Duration GAP Model



[pic]

Asset Duration = Weighted average of individual asset durationsof the FI. Likewise,

[pic]

where [pic] market weights of the assets and liabilities of the FI.



IOW, “The duration of a PORTFOLIO of ASSETS /LIABILITIES = A market-value weighted average of the individual durations of the assets/liabilities on the FI´s Balance Sheet”.

[pic]

[pic]





Now, we know that:

[pic]

where [pic] is a LEVERAGE GACTOR of the FI.← The EFFECT of a ∆i on a FI´s EQUITY, may be separated in 3 multiplicative effects:



1. Leverage adjusted DURATION GAP = [pic]

= ”Duration Mismatch” between Assets and Liabilities.

The LARGER the GAP → The HIGHER the SENSITIVITY to a ∆i (+ exposure!)



2. SIZE of the FI → A = Total Asset Size → A larger size → In dollar terms ($), the loss will be bigger....