Industry: Banking and Financial Services Product: ModelRisk Application: Interest Rate Modeling
A treasury team running a $40B asset-liability book ran the standard regulatory drill: a deterministic +100bp parallel shift, which printed a +$80M one-year NII uplift and was duly filed away. The probabilistic re-run, using a calibrated Vasicek short-rate model, showed something the shock test could not: a roughly symmetric NII distribution with P10 near -$70M, P90 near +$57M, and approximately a 1-in-5 chance of NII falling more than $50M below baseline — driven not by a single shock but by the distribution of plausible rate paths the curve might actually take.
The bank had been treating the market-implied forward curve as the rate forecast. In a falling-vol environment that is defensible. In the post-2022 environment, with short-rate vol at roughly 110bp/year and a Fed reaction function that itself behaves like a regime variable, it is not.
The team replaced the deterministic forward with a calibrated Vasicek one-factor short-rate model:
\[ dr_t = a(\theta - r_t)\,dt + \sigma\,dW_t \]
with current short rate \(r_0 = 4.50\%\), long-run mean \(\theta = 3.50\%\), mean-reversion speed \(a = 0.18\), and short-rate volatility \(\sigma \approx 110\) basis points per year. The three parameters are themselves uncertain — \(\theta\) was modeled as Normal around the macro team's central view, \(a\) as Normal truncated to (0.08, 0.35) to keep mean reversion meaningful, and \(\sigma\) as LogNormal to keep volatility positive and to reflect the right-skewed nature of vol-of-vol surprises.
Five thousand 60-month paths produce a fan chart that does what the forward curve cannot — it shows the width of the cone, not just its centerline.
The deterministic forward sits inside the P10-P90 band but slightly above the median, a structural feature of mean-reverting models that the bank's regulatory submissions had been silently absorbing. By year 5 the short rate sits roughly between 2% (P10) and 6% (P90), with a 1-in-100 path reaching above 7%. That 4-percentage-point P10-P90 width is the quantity a +100bp shock test fundamentally cannot estimate.
The bank's asset-liability book has $40B of rate-sensitive assets and $32B of rate-sensitive liabilities, an $8B repricing gap. A naive 1-year NII delta is just \(\text{Gap} \times \Delta r\), but the stochastic curve forces the calculation iteration-by-iteration:
The deterministic +100bp shock answer (+$80M) is plotted as a dotted black line. The simulated distribution shows the median NII change is mildly negative — close to zero, because the mean of the simulated rate path sits just below the starting rate — but the P10/P90 envelope is roughly -$70M to +$57M, a 127M-dollar swing centred near zero. The chart marks the operationally relevant number: the probability that one-year NII falls more than $50M below baseline, which under the calibrated model is 20% — the threshold at which the ALCO committee has to convene a contingency hedging review. That probability does not exist in the deterministic world.
When the team ranked drivers of the NII uncertainty itself — not of rates, but of the width of the NII distribution — the result reordered priorities.
Short-rate volatility \(\sigma\) is the single largest contributor — a 70-basis-point swing in the calibrated vol moves the P10 NII outcome by roughly $38M. Long-run mean \(\theta\) is nearly as large. Calibration risk dominates the rate-path risk itself: the bank's choice of which Vasicek calibration to trust matters more than the realized rate path does. Deposit beta — the elasticity with which retail deposits reprice — sits surprisingly low in the ranking, because the existing book is already well-matched on that bucket.
The most uncomfortable finding came from comparing the single-factor HW1F output against a two-factor extension that adds an independent slope-shock state variable.
The single-factor model produces a tighter cone than the two-factor model — by construction, because it admits only parallel shifts of the curve. The two-factor model widens the 5y rate distribution by roughly 15bp at each tail. For an ALM book whose vulnerability is concentrated in twists (short funding, long lending), the single-factor model systematically under-states the risk that matters. The team now runs both: HW1F for the regulatory submission, the two-factor extension for the internal economic-capital view.
The probabilistic curve had three operational consequences:
Monte Carlo turns the yield curve from a forecast into a distribution of forecasts, and that is the only frame in which a treasury team can distinguish "the rate moved against us" from "we were calibrated to the wrong curve all along."