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 five interpolation dimensions or "sensitivities" available in ProVal:
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valuation interest rate assumption, which is the rate(s) used to discount future payments to determine liabilities;
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experience rate of inflation, which is linked to experience salary inflation and increase rates;
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experience lump sum benchmark yield (e.g., the yield on 30-year government bonds), which is linked to the experience interest rates underlying (pension mode) lump sum factor Benefit Formula Components and lump sum Optional forms of payment.
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an alternate benchmark, which optionally impacts cash balance accrual definitions, universal mode career average with indexation accrual definitions, COLAs, employee contribution refunds, and new entrant asset transfers
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a valuation benchmark, which optionally impacts future changes to valuation assumptions in a forecast.
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 (baseline) and varying the dimension under study between 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, since 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 eleven sets of core projection output to be used for interpolation during a forecast.
In a core projection, ProVal defaults to running all sensitivities for each interpolation dimension. If this selection is changed to run only the baseline lump sum experience interest rate sensitivities, for example, then ProVal will generate no variation among the low, medium/baseline, and high lump sum experience benchmark yield environments. There are situations when ProVal recognizes that low and high sensitivity runs are not needed and will not run those scenarios. 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, career average components with indexation, COLAs, employee contributions with refunds, or new entrant asset transfers and an alternate benchmark is entered in Projection Assumptions and selected for interpolation, ProVal does not run the alternate benchmark scenarios. Similarly, if the projection assumptions do not vary any assumptions with the valuation benchmark, ProVal does not run the valuation benchmark scenarios.
The following discussion presumes that all inflation & alternate benchmark, interest rate & valuation benchmark and lump sum sensitivities have been selected in the Plan; thus the core projection produces eleven 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; a low, medium, or high alternate benchmark; and a low, medium, or high valuation 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 alternate benchmark environment the alternate benchmark is 7%, in a medium yield environment the lump sum benchmark yield is 6.5%, and in a medium valuation benchmark environment, the alternate benchmark is 4%.
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Sensitivity Dimension |
|||||
|
Order of performance of scenarios ↓ |
Valuation Interest Rates |
Experience Inflation Environment |
Lump Sum Experience Interest Rate |
Experience Alternate Benchmark |
Valuation Benchmark |
|
Baseline all |
7.5% |
4% |
6.5% |
7% |
4% |
|
Low interest rates |
5.5% |
4% |
6.5% |
7% |
4% |
|
High interest rates |
9.5% |
4% |
6.5% |
7% |
4% |
|
Low inflation |
7.5% |
2% |
6.5% |
7% |
4% |
|
High inflation |
7.5% |
6% |
6.5% |
7% |
4% |
|
Low lump sum experience interest |
7.5% |
4% |
6.5% |
7% |
4% |
|
High lump sum experience interest |
7.5% |
4% |
6.5% |
7% |
4% |
|
Low alternate benchmark |
7.5% |
4% |
6.5% |
5% |
4% |
|
High alternate benchmark |
7.5% |
4% |
6.5% |
9% |
4% |
|
Low valuation benchmark |
7.5% |
4% |
6.5% |
7% |
2% |
|
High valuation benchmark |
7.5% |
4% |
6.5% |
7% |
6% |
Under the scenarios for the low and high valuation interest rates and valuation benchmark, 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 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 alternate 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:
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Interpolation among the three scenarios that vary valuation interest rates, using medium/baseline for the other sensitivities. This returns liabilities at the target valuation interest rates, holding all other sensitivities constant.
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Interpolation among the three scenarios that vary the experienced inflation environment, holding all other sensitivities constant. (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.
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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 accrual definitions, universal mode career average with indexation accrual definitions, COLAs, employee contribution refunds, or new entrant asset transfers that vary with the alternate benchmark, 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 alternate benchmark and returns liabilities based on the experience crediting rates in the target cash balance accrual definitions, universal mode career average with indexation accrual definitions, COLAs, employee contribution refunds, or new entrant asset transfers benchmark environment, holding all other sensitivities at the baseline. (Please note that ProVal’s interpolation is based on the annualized average compound alternate benchmark experienced for all years prior to the forecast valuation date.) 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 alternate benchmark is defined as the ratio of the liability at the target alternate benchmark under the baseline interest & inflation to the liability at the medium alternate 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.
When the valuation benchmark sensitivity is applicable, similar steps are followed to create a factor reflecting the change in valuation benchmark compared to the baseline (holding all other sensitivities constant at the baseline) and that factor is multiplied (or added in the special case of interpolating effective interest rates) to the results obtained from the aforementioned interpolation steps. If an interpolation is being performed on both the valuation benchmark and interest rate sensitivity dimensions, there is a final adjustment made to reflect the fact that the forecasted valuation benchmark can impact the shape of the projected benefit payment curve, and thus the effective duration of the liabilities, and therefore the interest sensitivity (as compared to what the interest sensitivity would have been at the baseline valuation benchmark). This adjustment factor is determined by taking the ratioof the interest sensitivity of the present value of underlying projected benefit payments (or proxy projected benefit payments when none are available) at the baseline valuation benchmark yield and at the forecasted valuation benchmark yield.
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, cumulative experienced alternate benchmark, or valuation 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 developed by Winklevoss Technologies, uses a proprietary transformation algorithm involving logarithms demonstrated to significantly improve accuracy. Because of the superiority of the method, it is used whenever the pattern of underlying liabilities falls within the expected pattern. ProVal performs non-logarithmic interpolation only for occasional instances of irregularly-shaped liabilities. When ProVal detects a rare case of liabilities appearing to be irregularly shaped with respect to one of the sensitivity dimensions (determined using a series of tests or "triggers") two-point linear non-logarithmic interpolation is applied with respect to that dimension.