# Can duration gap analysis be applied to mortgages?

Can a mortgage loan be treated like a bond and its duration calculated using the bond duration formula? More precisely, can I calculate the loan portfolio duration for duration gap analysis, with coupon payment instead of annuity payments, loan value instead of bond value, loan rate in place of yield, and using the bond duration formula?

• Are you referring to individual loans or to mortgage backed securities? Is this from the perspective of the borrower or the lender? For MBS, calculating duration is a common but rather complex task. – Tal Fishman Jan 18 '12 at 20:56
• It is commonly done but usually with extra terms for prepayments risk. – Brian B Jan 18 '12 at 21:27
• say we want to do duration gap analysis for mortgage bank which lends mortgage loans.. so themortgage bankwill havemortgage loan portfolio as one of its interestdependant assets in its balance sheet.my question is thathow should the duration ofthis portfolio be calculated? – Pasha Jan 19 '12 at 21:27

You don't say which duration, but it's generally okay to use effective duration:

$$duration (eff) = \frac{-1}{P(r)}*\frac{Price(r+b) - Price(r-b)}{2*b}$$

where $r$ = rate and $b$ = yield shock.

Although, to address Brian's point, the mortgage contains an embedded call option that creates negative convexity, so the three re-pricings, $P(r)$, $P(r+b)$, $P(r-b)$, need to reflect prepayment assumptions. This cannot be done analytically, to my knowledge (I am aware of no analytical duration with "extra terms" for the prepayment risk). Rather, the effects of prepayment are numerically figured into the re-pricings (typically you increase the PSA a bit at the lower yield). So, you do have numerical inputs into the "analytical" effective duration.

Technically, the effective duration is okay (and has the same role as modified duration: to give a first-order/linear approximation of the price change) because, if you re-price the bond to include varying prepayment rates, then you are just computing the slope (rise/run) of the tangent to the P/Y curve. It is helpful to see how the above formula is merely a slope formula, such that in the case of effective duration applied to negative convexity, it's the slope of a secant line that approximates the tangent.

Reference: Veronesi, Fixed Income Securities, in FRM

• ... I should say, the duration formula is calculating -1/P(r)*slope (slope is the dollar duration, not duration) as this is rise/run: [Price(r+b) - Price(r-b)]/(2*b) – David Harper Jan 19 '12 at 3:56
• Whatabout maculay duration(weighted average of a payment times) can the same methdology applied as forbonds? – Pasha Jan 19 '12 at 21:24
• @Pasha - To my knowledge, Macaulay duration cannot be applied with mortgages/MBS: the prepayment (an embedded option) implies that maturity varies with yield. I suppose you could try Mac duration = effective * (1+yield/k) but I am unclear to what end and don't see how it can have the traditional "weighted average maturity" definition that works for vanilla bonds (i.e., in a vanilla bond, you actually do weight maturities, but not here). – David Harper Jan 19 '12 at 22:20
• Thank you for your detailled reply, any recommendations-books, articles- where I can go deeper with this topic. – Pasha Jan 20 '12 at 19:03
• @Pasha, my recommendation is the text i cited above (Fixed Income Securities by Pietro Veronesi). I learned what i know from Bruce Tuckman's Fixed Income Securities, which is EXCELLENT still after ~10 years. You cannot go wrong with either, both are assigned in FRM. Of course, Fabozzi also has a couple of popular books. – David Harper Jan 29 '12 at 21:24