Empirical Duration vs Analytical Duration

Take a mortgage-backed security's stated analytical duration — computed by discounting a model's projected cash flows — and compare it to how the security's price has actually moved against Treasury yields over the past year. The two numbers routinely disagree, sometimes by half. That gap isn't a modeling bug to be fixed; it reflects two genuinely different questions being asked of the same bond.

Two different measurements

Analytical duration perturbs a pricing model's input yield up and down by a small amount, recomputes the model price both times using its cash-flow assumptions, and reads off the sensitivity:

$$D_{analytical} = -\frac{1}{P} \times \frac{P_{+\Delta y} - P_{-\Delta y}}{2\Delta y}$$

In words: nudge the model's yield input up and down a little, see how much the model's price changes, and scale that change back into a duration figure — the answer depends entirely on how well the model's cash-flow assumptions (like a prepayment model for a mortgage bond) reflect reality.

Empirical duration instead runs a regression of the bond's actual historical price returns against actual historical changes in a benchmark yield:

$$D_{empirical} = -\frac{\text{Cov}(\Delta P/P,\ \Delta y)}{\text{Var}(\Delta y)}$$

In words: this is just the slope coefficient from regressing percentage price changes on yield changes — it measures how the bond has behaved, independent of any model of why.

Worked example

A mortgage-backed security's prepayment model implies an analytical (effective) duration of 5.0. Over the past year, regressing its weekly price returns against changes in the 10-year Treasury yield produces a slope of 3.2.

1. Analytical duration: 5.0, based on the model's prepayment assumptions.
2. Empirical duration: 3.2, based on how the bond has actually traded.
3. Interpretation: the security has behaved as though it were noticeably shorter-duration than the model predicts — likely because dealers and investors have been pricing in faster prepayments (homeowners refinancing sooner) than the model's baseline assumption captures, or because MBS prices are also influenced by supply-demand and liquidity factors the model doesn't include.
4. Practical use: a hedger using the model's 5.0 to size a Treasury hedge would over-hedge the position relative to how it has actually moved; using 3.2 would track realized behavior more closely, at the cost of relying on a backward-looking, sample-dependent estimate.

What this means in practice

Empirical duration is used most heavily on mortgage-backed securities and other bonds with prepayment or call optionality, where investor behavior and market technicals routinely diverge from a model's clean assumptions. Risk desks often quote both figures side by side and hedge somewhere between them, or reconcile the gap by adjusting the model's own assumptions (like the prepayment speed) until analytical duration better matches the empirical estimate.