When the forecast yield curve's shape differs from the baseline valuation interest rate assumption, the liability is determined by discounting the relevant benefit payments. Discounting benefit payments is possible when payments don’t depend on the interest rate, or there’s another benefit with matching duration where payments don’t depend on the interest rate. If benefit payments aren’t available, ProVal uses duration-based interpolation instead.

Discounting benefit payments 

When projected benefit payments for a particular benefit do not vary significantly with the valuation interest assumption, ProVal determines the liability by:

1. Calculating an average payment timing factor (e.g., 0.45) based on the projected benefit payments and baseline liability in the core projection
2. Discounting the projected benefit payments at the specified forecast interest rate yield curve and average payment timing to determine the forecasted liability.

When projected benefit payments for a particular benefit do vary significantly with the valuation interest assumptions because, for example, the benefit is based on lump sum factors or optional payment form conversion factors that vary with the valuation interest rate, then ProVal attempts to find another benefit (from the same liability) that has a similar effective duration but that is interest rate invariant.  If ProVal finds such a benefit, then that benefit's projected benefit payments, scaled to match the baseline liability of the benefit in question, are used as described above.

Duration-based interpolation 

When it isn't possible to discount projected benefit payments, ProVal determines the liability by:

1. Calculating the effective duration of the liability for the benefit in question. This essentially identifies what part of the yield curve is important.
2. Selecting prototypical benefit payments with that same duration.
3. Calculating present values for the prototypical benefit payments, using the low, baseline and high interest rate assumptions from the Core Projection as well as the forecast interest rate yield curve.
4. Performing a Lagrange interpolation (with logarithmic transformation) using present values in step 3 as independent variables and the low, baseline, and high liabilities from the Core Projection as dependent variables to estimate the forecasted liability.

In step 1, ProVal calculates the effective duration of a liability by determining its rate of change with respect to the change in the interest rate, where these elements are determined from the interest rate sensitivity scenarios of the Core Projection.  If the effective duration turns out to be 13.5 years, for example, then the prototypical benefit payments in step 2 are typically largest in the range surrounding 13.5, say years 10-17, with smaller values in other years. 

Restrictions

The methodology above works best when the change from the baseline liabilities to the high/low interest rate liabilities in the Core Projection is attributable solely to the change in the interest rate.  If part of the change in liability is due to the change in interest rate and part of it is due to the fact that salary inflation (or some other factor irrelevant to interest rate changes) follows the change in the interest rate and therefore changed the benefit payments, then less information is available to make an accurate forecast.  Accordingly, ProVal requires that the applicable interest rate sensitivity fractions (other than sensitivity fractions for lump sum factors and optional payment forms) be set to zero in the Projection Assumptions when forecasting to a yield curve library entry.