Estimating Mortgage Duration with Prepayment Risk
Summary
The document explains how effective duration can be used to estimate a mortgage portfolio’s sensitivity to interest-rate changes for duration gap analysis. It uses a central finite-difference calculation: reprice the mortgage at the current rate and after upward and downward rate shocks, then relate the price difference to the base value and shock size.
Mortgage prepayments complicate this calculation because borrowers’ prepayment options can produce negative convexity. The repricing scenarios therefore need to incorporate assumptions about how prepayment changes with rates; the answer describes adjusting prepayment inputs numerically rather than relying on a closed-form duration formula. Effective duration approximates the local price response with a slope, but the result depends on the chosen rate shocks and prepayment model. The document gives a conceptual method, not a worked mortgage valuation or evidence comparing alternative models.
Key ideas
- Effective duration estimates price sensitivity by repricing at rates above and below the current rate.
- Mortgage prepayment behavior should be reflected in each scenario valuation.
- Embedded prepayment options can give mortgages negative convexity.
- The estimate is a local, first-order approximation and depends on modeling assumptions.
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Full text
# Can duration gap analysis be applied to mortgages?
# 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?
## Answer by David Harper (score 4, accepted)
https://quant.stackexchange.com/a/2797
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 FRMShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.