General Description

The Pension Benefit Guaranty Corporation (PBGC) uses the price of annuities to set the interest factors it uses to determine the present value of future benefits in the plans it has trusteed.  This paper provides a description of, and rationale for, the procedures PBGC uses to set these interest factors.

PBGC’s interest factors, when used in conjunction with a specified mortality table,[1] are intended to “match” the resulting present value of future benefits from the plans it has trusteed (PBGC’s benefit obligations) with the price the private sector annuity market would charge to annuitize those same benefit payment obligations.[2]  Each year, PBGC uses two American Council of Life Insurers (ACLI) quarterly surveys of annuity prices to calibrate its interest factor set.  The interest factor set has a “select and ultimate” structure where the select factor applies for the first 20 or 25 years and the ultimate factor applies for all remaining years.  PBGC uses monthly changes in a corporate bond index to adjust the interest factors each month between the yearly calibrations.    

A detailed description of PBGC’s procedures for calculating its interest factor set is presented below.   Separate sections discuss, respectively, the ACLI annuity price survey, general and specific procedures for selecting the best interest factor set, and how PBGC makes monthly adjustments to interest factors. 

ACLI Annuity Price Survey

The ACLI conducts a quarterly survey of annuity prices for the PBGC.[3]  The survey collects information on how private sector annuity providers would price immediate annuities for men and women beginning at ages 50, 55, 60, . . . , and 80 and deferred annuities beginning at age 65 for men and women aged 30, 35, . . . , and 60.  These prices are net of administrative expenses.  In all, the survey collects annuity prices for 28 different points—prices for 7 immediate annuity points for men and 7 for women and prices for 7 deferred annuity points for men and 7 for women. 

When the ACLI receives the surveys, it removes any information that could identify the responding company.  It then forwards the pricing  information to the PBGC.   The ACLI does not inform the PBGC of how many or which annuity providers received the survey or which responded.  Participation in the survey is voluntary, and the number of responses the ACLI receives can vary from quarter to quarter. 

The survey results show annuity prices grouped by company.  That is, the prices for the immediate and deferred annuities for all ages are listed separately for each company.  The companies are not explicitly identified, but are named Company A, Company B, and so on.  The letter codes are randomly generated each quarter, so a given company will have different letter codes applied to it in different quarterly surveys.

Selecting the Interest Factor Set—General Procedure

PBGC uses a “select and ultimate” interest factor structure, where the select factor applies for the first 20 or 25 years (the “select period”), and the ultimate factor applies for all remaining years (to age 120, in the current mortality table).  The two interest factors and the select period constitute an “interest factor set”.  PBGC currently uses a version of the GAM 94 mortality table when selecting the best interest factor set.  The best interest factor set is highly sensitive to the mortality table used when determining it.  In general, if a different mortality table were used, a different “best” interest factor set would be obtained. 

In examining the responses to the survey, PBGC first tests for companies whose prices could be considered “outliers”.  The prices of “outlier” companies are not used because PBGC wants to avoid skewing its interest factor set when the survey prices quoted by one company are unreasonably high or low relative to the survey prices of other responding companies.  There are three tests for these outliers.  All three tests must be met for a company to be considered an outlier.  The tests are:

 

1.         The price quotes from a particular company must be the highest (lowest) for at least all but two of the annuity-price data points used.  PBGC currently uses the 14 annuity-price data points for men when calculating its interest factor set.  Thus, to be considered an outlier, the company’s price quotes must be the highest (lowest) for at least 12 of the 14 male annuity-price data points.

 

2.         For the male immediate annuity at age 65, the company’s annuity price must be at least 12.5% higher (lower) than the annuity price of the company with the median annuity price.

 

3.         For the male immediate annuity at age 65, the difference between the company’s annuity price and the annuity price of the company with the second highest (lowest) price must be greater than the difference in annuity prices of the companies with the second and fourth highest (lowest) annuity prices.

After eliminating the price quotes from any outlier companies, PBGC averages the price quotes of the remaining companies for each of the 14 annuity-price data points for men.  The interest factor set PBGC selects is the set that results in calculated annuity values (the present value of future benefits) that best match the 14 averaged annuity price quotes from the companies responding to the survey.  To determine which interest factor set gives this best match, PBGC calculates an annuity value for each of the 14 annuities using its specified mortality table and thousands of interest factor sets. 

PBGC first uses a select period of 20 years and a low ultimate interest factor and tests for the best select interest factor by increasing a low initial select factor value by one basis point (one one-hundredth of one percent) for each new run until the select interest factor reaches a predetermined upper limit.  For example, the first test might begin with a select interest factor of 4.0% for the first 20 years and an ultimate interest factor of 3.0% (4.0, 20, 3.0).  Additional iterations are performed using the same ultimate interest factor and select period ((4.01, 20, 3.00), (4.02, 20, 3.00), ...).  It then increases the ultimate interest factor by one basis point and repeats the process by again adjusting the select interest factor by one basis point at a time[4].  It repeats this process until an upper bound on the ultimate factor is reached.  Finally, it repeats the entire process using a 25-year select period.  [Note: tested interest factor sets include those where the ultimate factor is greater than the select factor.]  From all these iterations using different select and ultimate interest factors and two select periods, it selects the “best” interest factor set as determined y the process discussed next.

For each interest factor set, PBGC calculates the annuity value of the 14 immediate and deferred annuities at the ages mentioned earlier.  It also calculates the error, or percentage difference between the calculated value and the average of the surveyed annuity prices.  To determine the “best fit” between the calculated values and average of the survey prices, PBGC uses the interest factor set that gives the smallest “key mean error sum”.  The “key mean error sum” is the sum of the absolute value of the mean error (the absolute value of one-fourteenth of the sum of the 14 individual errors) and the mean absolute error (one-fourteenth of the sum of the absolute values of the 14 individual errors).  The examples below use the following definitions:

 

  Key Mean Error Sum:      |ME| + MAE

                            |ME|:      absolute value of mean error, and

                           MAE:      mean absolute error.

It is not appropriate to use the interest factor set that has the smallest absolute value of the mean error because the mean error ignores the possibility of having large errors with different signs that “cancel each other out”.  If the interest factor set results in a large error for one of the data points, then PBGC will overestimate (or underestimate if the error is negative) the present value of future benefits for plans that have many participants whose characteristics put them near that data point.  Thus, it is important that the interest factor set give the “best fit” over all 14 data points.  The absolute value of the mean error shows which interest factor set gives the best result, on average, while the mean absolute error penalizes those interest factor sets where there are large errors for individual data points.  By selecting the interest factor set with the smallest sum of these two error measures, PBGC ensures it is selecting the set that will provide the most accurate estimated annuity values regardless of the age distribution of participants in the individual plans it trustees. 

A simplified example illustrating PBGC’s procedure for determining the key mean error sum follows:

Example:

For illustrative purposes, assume there are only five annuity-price data points (assume they are immediate annuities beginning at ages represented by 1, 2, 3, 4, and 5).  PBGC’s process calculates and compares the key mean error sums of literally thousands of interest factor sets.  In this example, we limit the number of iterations to three.  The particular interest factor sets used in each of the three illustrative iterations are different but unspecified here.  Using these interest factor sets and the specified mortality table, PBGC calculates annuity values for each of the five data points and compares the calculated results with the averaged annuity prices quoted by companies responding to the ACLI survey.  The steps that are used to calculate the key mean error sum are shown below each iteration.

      Iteration 1

Age	Average Survey Price	Calculated Annuity Value	Percentage Difference (Error)
1	1000	1140	14.0%
2	1150	1245	8.3%
3	1329	1315	-1.1%
4	1511	1402	-7.2%
5	1700	1449	-14.8%

 

Total Error             E                -0.8%       Total Abs Error      TAE     45.4%

Mean Error            ME            -0.2%         Mean Abs Error     MAE    9.1%

Abs Value of ME   |ME|           0.2%

 

Key Mean Error Sum               |ME| + MAE    =  0.2% + 9.1%  = 9.3%

 

      Iteration 2

Age	Average Survey Price	Calculated Annuity Value	Percentage Difference (Error)
1	1000	1030	3.0%
2	1150	1163	1.1%
3	1329	1331	0.2%
4	1511	1471	-2.6%
5	1700	1636	-3.8%

 

Total Error             E                -2.1%         Total Abs Error        TAE     10.7%

Mean Error            ME            -0.4%          Mean Abs Error       MAE    2.1%

Abs Value of ME   |ME|           0.4%

 

Key Mean Error Sum               |ME| + MAE    =  0.4% + 2.1%  =  2.5%

 

      Iteration 3

Age	Average Survey Price	Calculated Annuity Value	Percentage Difference (Error)
1	1000	1090	9.0%
2	1150	1205	4.8%
3	1329	1341	0.9%
4	1511	1479	-2.1%
5	1700	1597	-6.1%

 

Total Error             E                6.5%           Total Abs Error        TAE     22.9%

Mean Error            ME            1.3%            Mean Abs Error       MAE    4.6%

Abs Value of ME   |ME|           1.3%

 

Key Mean Error Sum               |ME| + MAE    =  1.3% + 4.6%  =  5.9%

 

In the first iteration, the absolute mean error is small at only 0.2%, but the individual average survey price vs. calculated annuity value errors are large.  The second iteration shows a slightly larger absolute mean error, at 0.4%, but none of the individual errors are large, so the key mean error sum is small.  The third iteration, with the largest absolute mean error of 1.3% also has small individual errors.  Because it has the smallest key mean error sum, the interest factor set used in the second iteration is selected as the best fit.  The interest factor set used in the third iteration is preferred to that used in the first iteration, even though it has a much larger absolute value of the mean error, because it has a smaller key mean error sum.

Selecting the Interest Factor Set—Specifics

Even though the ACLI survey results are received quarterly, PBGC only uses them to determine its interest factor set once each year.  Between these “recalibrations”, the interest factor set is adjusted each month by matching the changes in the Lehman “Long Corporate A or Higher” bond index value on the last day of the month.  A description of the index and how it is used appears below.

PBGC uses two quarterly surveys, the March 31st survey and the June 30th survey when recalibrating its financial statement interest factor set.  This interest factor set is used for valuing PBGC’s financial statement obligations as of September 30th, the end of its fiscal year.  All subsequent monthly factors for the year then follow the moves in the Lehman index. 

There are several steps in determining the September interest factor set.  These steps are:

1. determine the best interest factor set based on the June 30th ACLI survey;
2. determine the best interest factor set based on the March 31st ACLI survey, with the constraints that the select period for the March-based set match the select period for the June-based set and the ultimate interest factor for the March-based set be within 25 basis points of the ultimate interest factor determined for the June-based interest factor set;
3. bring the March 31st select and ultimate interest factors forward to June 30th by adding or subtracting from both interest factors the same basis point change experienced by the Lehman index between March 31st and June 30th;
4. average the two select interest factors and, separately, the two ultimate interest factors determined for June 30th and the March 31st rolled forward to June 30th;
5. finally, roll the averaged select and ultimate factors forward to September 30th by adding or subtracting the same basis point change experienced by the Lehman index between June 30th and September 30th. 

It is unlikely that the March interest factor set determined by this procedure would be the best fit of the March survey data were the constraints not imposed.  This is especially true since PBGC started testing for the best interest factor set fit using single-basis-point increments. 

Note the constraints: that the select period for the March result match that for the June result, and that the March ultimate factor is within 25 basis points of the June ultimate factor.  These constraints model the assumption that the two annuity price quotes, only three months apart, will reflect similar views of the extremely long -term interest environment.

 

The following example demonstrates this 6 step process
1.  Assume the best interest factor set for the June 30th ACLI annuity price data points is:

Select interest factor:  6.08%     Select period:  25 years     Ultimate interest factor: 5.91%

2.   The March 31st interest factor set will be the best fit of the March 31st ACLI survey data such that the select period is 25 years and the ultimate interest factor lies between 5.66% and 6.16%.  Assume the best March 31st interest factor set is:

Select interest factor:  6.56%     Select period:  25 years     Ultimate interest factor: 6.11%

3.   Assume further that Lehman Long Corporate A or Higher index value was 6.35% on March 31st and 5.97% on June 30th.  The Lehman index fell by 38 basis points over the period (5.97%-6.35% = -0.38%).  Therefore, PBGC subtracts 38 basis points from each of the March 31st select and ultimate interest factors to roll these factors forward to June 30th.  Their June 30th values are 6.18% and 5.73%, respectively.

4.   The two select factors and two ultimate factors are now averaged.  The average select factor is 6.13% ((6.08%+6.18%)/2) and the average ultimate interest factor is 5.82% ((5.91%+5.73%)/2).

5.   These values must now be rolled forward to September 30th.  Assume the Lehman Long Corporate A or Higher index continues to fall and is 5.68% on September 30th.  This is 29 basis points lower than its June 30th value, so 29 basis points are subtracted from both the select interest factor and the ultimate interest factor.  The interest factor set to value PBGC’s obligations as of September 30th is:

Select interest factor:  5.84%     Select period:  25 years     Ultimate interest factor: 5.53%

 

Monthly Adjustments to Interest Factors

 

In subsequent months, the select and ultimate factors move in tandem with the Lehman’s index.  An example showing the select and ultimate interest factors over a few months illustrates this movement.  The select period will remain frozen at 25 years until the next recalibration of the interest factor set.

 

Month	Lehman’s Index on Last Day of Month	Diff. From Previous Month’s Index	Select Interest Factor	Ultimate Interest Factor
Sep	5.68%	n/a	5.84%	5.53%
Oct	5.51%	- 0.17%	5.67%	5.36%
Nov	5.64%	+ 0.13%	5.80%	5.49%
Dec	5.71%	+ 0.07%	5.87%	5.56%
Jan	5.57%	- 0.14%	5.73%	5.42%

 

The Lehman Long Corporate A or Higher Bond Index

The Lehman Long Corporate A or Higher Index used to generate monthly interest factor changes for and after the September 30 recalibration consists of US corporate bonds that have an investment rating of A or better[5].  This index was chosen for two reasons.  First, the duration of the bonds in this index closely matches the duration of PBGC’s current benefit payment obligations.  Second, the changes in the end-of-quarter spot value for this index tracked the quarter-to-quarter changes in the select and ultimate interest factors based on the ACLI quarterly annuity price survey as well or better than other tested indices. 

 

Conclusions

The PBGC sets the interest factors it uses to value its benefit obligations for financial statement purposes using recent surveys of annuity prices.  The agency’s obligations are the future benefits it must pay to participants in the plans it has trusteed.  The best way to value these liabilities is to make their value comparable to the price private sector annuity providers would charge for annuitizing these benefits.  PBGC’s procedure for setting its interest factor set ensures that the valuation of its obligations will closely approximate the private sector’s valuation of these same obligations.  It must be stressed that the interest factor set selected is highly sensitive to the mortality table PBGC uses.  Using a different mortality table would result in the selection of a different interest factor set.  However, both interest factor sets, when used in conjunction with their respective mortality tables, would result in valuations of PBGC’s liabilities that closely approximate the valuation the private sector would determine for these obligations.

The purpose of this note is to shed light on the procedures PBGC uses when selecting its financial statement interest factor set.  Comments and questions are welcome and can be directed to: Marc Ness in the Policy Research and Analysis Dept. at (202) 326-4080 ext 3227.