The liabilities that are calculated at any given year in a Deterministic Forecast or a Stochastic Forecast are interpolated from the alternate scenarios generated in a Core Projection.

There are four interpolation dimensions available in ProVal:

1. valuation assumption interest rates (i.e., the funding valuation interest rates, accounting valuation discount rate, current liability and PBGC variable premium interest rates in the U.S. qualified mode, solvency liability interest rate in the Canadian registered mode);

2. the rate of experience inflation, which impacts experienced salary inflation and increase rates;

3. an experience lump sum benchmark yield (e.g., the yield on 30-year government bonds), which impacts the experience interest rates underlying (pension mode) lump sum factor Benefit Formula Components and lump sum Optional forms of payment. The evaluation of optional form conversion factors and lump sum factor Benefit Formula Components associated with benefits paid to, and liabilities for, active participants assumed to decrement during the experience period (between the initial valuation date and some future valuation date) will be based on the experience lump sum benchmark yield and related parameters defined in the Projection Assumptions. In a deterministic or stochastic forecast, the emerging inactive liabilities and experience benefit payments will be interpolated along the lump sum yield sensitivity dimension, based on the assumed year of decrement.

4. an asset benchmark, which optionally impacts cash balance accrual definitions, universal mode career average with indexation accrual definitions and employee contribution refunds.

ProVal performs a sensitivity analysis separately for each interpolation dimension selected. For each dimension, three interpolation anchor points are obtained by holding the other interpolation dimensions constant, at the medium values for the inflation environment, the lump sum experience interest rate, and the asset benchmark, and at the “baseline” values (i.e., current values as of the baseline valuation date) for the valuation interest rate, and varying the dimension under study among low, medium/baseline and high to determine the sensitivity to fixed changes (to a lower or higher value, respectively) in that dimension. Therefore, for the first interpolation dimension, ProVal runs the core projection three times to get results under low, medium/baseline and high scenarios for that interpolation dimension. Selecting each additional interpolation dimension adds another two scenarios (low and high, baseline or medium having already been run as one of the scenarios for the first dimension selected). If all sensitivities have been selected, ProVal produces nine sets of core projection output for interpolation during a forecast.

If the Plan Definition used by the Core Projection contains a lump sum factor component but only the baseline lump sum experience interest rate sensitivities has been selected, then ProVal will generate no variation among the low, medium and high lump sum experience interest rate environments, and thus no variation of a lump sum factor component’s experience interest rate from its baseline interest rate (medium value). Conversely, if the Plan Definition does not contain a lump sum factor component but all lump sum experience interest rate sensitivities have been selected, then ProVal recognizes the absence of lump sum factors a priori and does not actually run low and high scenarios. Similarly, if the Plan Definition does not contain any cash balance accrual definitions (or career average components with indexation or employee contributions with refunds) and an asset benchmark is entered in Projection Assumptions and selected for interpolation, ProVal recognizes the absence of applicable components and does not run the asset benchmark scenarios.

The following discussion presumes that all inflation & asset benchmark, interest rate and lump sum sensitivities have been selected in the Plan; thus the core projection produces nine anchor points in the forecast.

Each of the scenarios is based on some combination of: assumed low, baseline or high valuation assumption interest rates; a low, medium or high inflation experience environment; a low, medium, or high lump sum experience interest rate; and a low, medium, or high asset benchmark, as illustrated in the chart below for a sample core projection. The sample is based on a sensitivity change to interest rates of 2% in both directions, with all of the sensitivity change applied to valuation assumption salary inflation / increase rates / crediting rates and to valuation assumption lump sum & optional payment forms; furthermore, the Valuation Assumptions contain an interest rate of 7.5%, and the Projection Assumptions (experience) indicate that in the medium inflation environment the rate of inflation is 4%, in a medium asset benchmark environment the asset benchmark is 7%, and in a medium yield environment the lump sum benchmark yield is 6.5%.

	Sensitivity Dimension
Order of performance of scenarios	Valuation Interest Rates	Experience Inflation Environment	Lump Sum  Experience  Interest Rate	Experience  Asset Benchmark
Baseline all	7.5%	4%	6.5%	7%
Low interest rates	5.5%	4%	6.5%	7%
High interest rates	9.5%	4%	6.5%	7%
Low inflation	7.5%	2%	6.5%	7%
High inflation	7.5%	6%	6.5%	7%
Low lump sum experience interest	7.5%	4%	6.5%	7%
High lump sum experience interest	7.5%	4%	6.5%	7%
Low asset benchmark	7.5%	4%	6.5%	5%
High asset benchmark	7.5%	4%	6.5%	9%

Under the scenarios for the low and high valuation interest rates, the valuation assumptions with respect to the salary inflation rate, the various increase rates / crediting rates and any lump sum factor & optional forms interest rates also move to lower and higher values, respectively, unless the fraction of the assumed interest rate sensitivity change applied to them is zero.

The deterministic or stochastic forecast then interpolates, at each forecast valuation date, among these scenarios, or anchor points, to:

• return liabilities at the target valuation assumption interest rates and at the values determined according to the applied fraction of the assumed interest rate sensitivity change for valuation assumptions with respect to salary increases / increase rates / crediting rates and lump sum factor & optional forms interest rates when these interest rates are at their target values;
• reflect, in the calculation of the various liabilities, the salary inflation / increase rates / crediting rates experienced up to the forecast valuation date (i.e., reflecting inflation experienced during the prior years) under the target (experience) inflation and asset benchmark environments; and
• reflect the benefit payments of all years prior to the forecast valuation date, deriving benefit payments defined in terms of lump sum factors at the (experience) lump sum interest rates resulting from the target (experience) lump sum benchmark yield. Note that it is experience benefit payments that are affected by the lump sum benchmark yield environment, not expected benefit payments.

The following three steps are performed, using the first five scenarios listed in the chart:

1. Interpolation among the three scenarios that vary valuation interest rates, using medium experienced inflation. This returns liabilities at the target valuation interest rates as if the experienced inflation rate were the medium rate for the inflation environment.

2. Interpolation among the three scenarios that vary the experienced inflation environment, using the baseline interest rate. (Please note that ProVal’s interpolation is based on the annualized average compound rate of inflation experienced for all years prior to the forecast valuation date.) This returns liabilities based on the experienced salary inflation rates / increase rates / crediting rates in the target inflation environment as if assumed valuation interest rates were the baseline rates.

3. Combine the two interpolations using the following formula:

Target Liability = (Liability at target interest rate) (Increase or decrease due to inflation)

where

In other words, ProVal adjusts the results of the first step by the percentage change between the target and medium inflation liabilities derived in the second step.

If there are cash balance components in the plan, then ProVal performs additional steps similar to these three, using the last two scenarios of the chart. That is, ProVal adds a fourth step, to interpolate among the three scenarios that vary the experience asset benchmark and returns liabilities based on the experienced crediting rates in the target asset benchmark environment as if assumed valuation interest and inflation rates were the baseline rates. ProVal then combines this interpolated result with that of step (3) using a formula comparable to that used in step (3):  the increase or decrease due to the asset benchmark is defined as the ratio of the liability at the target asset benchmark under the baseline interest & inflation to the liability at the medium asset benchmark under the baseline interest & inflation.

If there are lump sum factors in the plan, then ProVal performs additional steps using the 6th and 7th scenarios of the chart to interpolate among the three scenarios that vary the experience lump sum experience interest rate and returns (actual, not expected) benefit payments at the lump sum experience interest rates, as if the assumed valuation interest rate were the baseline rate and the assumed inflation rate were the medium rate. ProVal then combines this interpolated result with that of the previous step.

In forecasts executed in ProVal prior to 2008, the interpolation techniques applied in steps (1) and (2) are based upon the Lagrange interpolating polynomial in most cases, depending on the user’s input.

Methodology of Logarithmic Versus Non-Logarithmic (LaGrange) Interpolation

Each interpolation anchor point consists of an ordered pair: the value of an independent variable (assumed valuation interest rate, cumulative experienced inflation, lump sum experience interest rate, or asset benchmark) and the corresponding forecast result (the dependent value). Given three interpolation anchor points (low, medium/baseline and high), three points can be drawn on a graph and then, given a fourth independent value (the “target assumption”), the corresponding dependent value (forecast result) can be determined along the curve connecting the three known points.

Logarithmic interpolation, a mathematical technique invented by Winklevoss Technologies, uses a proprietary transformation involving logarithms demonstrated to significantly improve accuracy. Because of the superiority of the method, it has been used almost exclusively in ProVal since 2008. ProVal supports non-logarithmic interpolation only for upward compatibility of ProVal releases and occasional instances of irregularly-shaped liabilities. If ProVal detects that liabilities appear to be irregularly shaped with respect to one of these dimensions (interest rate, experience inflation rate, or lump sum experience interest rate), it will apply non-logarithmic interpolation with respect to that dimension. For these rare cases of non-logarithmic interpolation, ProVal will use two-point linear interpolation.