Effective Duration of a Prepayment-Sensitive MBS

Worked Solution

How to Think About It: The whole point of effective duration is that it handles bonds whose cash flows change when yields move. A plain vanilla bond has fixed coupons -- push yields up or down and only the discount factors change. An MBS is fundamentally different: changing yields changes prepayment behavior, which changes the cash flows themselves. Macaulay and modified duration assume fixed cash flows, so they miss this entirely. Before writing any formulas, the key economic picture is: when rates drop, prepayments spike, you get your principal back early (just when you least want it -- reinvestment rates are low), and the bond shortens. When rates rise, prepayments slow, and the bond extends (locking you into below-market coupons longer). This asymmetry is negative convexity, and effective duration is the tool that captures it.

Quick Sanity Checks: - For a plain fixed-coupon bond with no embedded options, effective duration should converge to modified duration as $\Delta y \to 0$. - MBS effective duration should be shorter than modified duration when rates are low (fast prepayments shorten the bond). - MBS effective duration should be longer than modified duration when rates are high (slow prepayments extend the bond). - The sign is always positive for a long bond position (price falls when yields rise).

Derivation:

The finite-difference formula for effective duration is:

$D_{\text{eff}} = \frac{P(y - \Delta y) - P(y + \Delta y)}{2 \cdot P(y) \cdot \Delta y}$

where: - $P(y)$ is the price at the current yield $y$, - $P(y \pm \Delta y)$ are prices computed after a parallel shift of $\pm \Delta y$ (typically 25-50 bps), - Crucially, each price $P(y \pm \Delta y)$ is computed by regenerating the full cash flow schedule at the shifted yield.

The computation procedure for each shifted price is:

1. Shift the yield curve by $\pm \Delta y$ in parallel. 2. Recompute $\text{CPR}(y \pm \Delta y)$ using a prepayment model (e.g., PSA model, or a more sophisticated logistic/econometric model that maps mortgage rates to prepayment speeds). 3. Convert CPR to SMM (single monthly mortality): $\text{SMM} = 1 - (1 - \text{CPR})^{1/12}$. 4. Generate the monthly cash flows $\text{CF}_t(y \pm \Delta y)$ for each month $t$. Each month's cash flow is the sum of: - Scheduled principal payment - Interest on the remaining balance - Prepaid principal $= \text{SMM} \times (\text{remaining balance after scheduled payment})$ 5. Discount the regenerated cash flows at the shifted yield to get $P(y \pm \Delta y)$: $P(y \pm \Delta y) = \sum_{t=1}^{T} \frac{\text{CF}_t(y \pm \Delta y)}{(1 + (y \pm \Delta y)/12)^t}$ 6. Plug into the finite-difference formula above.

Why Effective Duration Differs from Macaulay/Modified Duration:

Macaulay duration is the weighted-average time to cash flows (weights = PV of each cash flow / price). Modified duration equals Macaulay duration divided by $(1 + y/k)$ and measures price sensitivity under the assumption that cash flows are fixed. Both treat the bond's cash flow schedule as a constant.

For an MBS, this assumption is badly wrong. The key differences:

• Cash flow feedback loop: When $y$ drops, $\text{CPR}$ rises, principal returns faster, and the effective maturity shortens. Modified duration does not see this -- it would overstate the price gain from a rate decline because it assumes the long-dated cash flows remain intact.
• Negative convexity: A plain bond has positive convexity (price gains from rate drops exceed losses from equal rate rises). An MBS can have negative convexity: rate drops trigger prepayments that cap price appreciation. Effective duration computed at $y - \Delta y$ and $y + \Delta y$ captures this asymmetry because it re-prices with the actual shifted cash flows.
• Magnitude: In a low-rate environment, MBS effective duration can be 2-4 years while modified duration of the underlying collateral might suggest 6-8 years. The difference is entirely due to the prepayment option embedded in the mortgage.

Practical Interpretation: On a trading desk, you would never hedge an MBS position using modified duration -- you would massively over-hedge. Effective duration (often called option-adjusted duration, OAD) is the right risk measure because it reflects how the position actually behaves when the curve shifts. The choice of $\Delta y$ matters: too small and numerical noise dominates; too large and you miss the local slope. Most practitioners use 25-50 bps.

Answer: The effective duration formula is $D_{\text{eff}} = [P(y - \Delta y) - P(y + \Delta y)] / [2 \cdot P(y) \cdot \Delta y]$, where each shifted price is computed by regenerating the cash flow schedule with the updated $\text{CPR}(y \pm \Delta y)$. It differs from Macaulay/modified duration because those measures assume fixed cash flows, while effective duration accounts for the prepayment option that causes the bond's cash flows -- and effective maturity -- to change with yields. This is why MBS exhibit negative convexity and require option-adjusted duration for proper hedging.