QUESTION: How can I calculate conversion factors for a “pop-up” annuity optional form? This is something that many U.S. public pension plans have. In the most common version, the normal form of benefit is a 10-year certain and life annuity, with optional election of actuarially equivalent forms like life annuity, 100% joint & survivor annuity and 50% joint & survivor annuity. For the joint & survivor options, members may elect a “pop-up” in which if the beneficiary predeceases the member, the annual benefit reverts (or “pops-up”) to the amount payable under the normal form. The pop-up annuity may or may not include a certain period.

There are other variations in which the pop-up annuity pops up to an amount payable under a form other than the normal form. These conversions involve two factors, K1*K2. K1 converts from the normal form to the form that’s paid if the beneficiary predeceases the member (let’s call this the “member’s form”). K2 converts from the member’s form to the pop-up annuity. K2 is determined using the formulas in Case 1 and Case 2 below, but substituting “member’s form” wherever “normal form” appears.

• Example A: Pop-up annuity pops up to the amount payable under a life annuity (ax), but normal form is 10 year certain and life annuity (a:10 + 10|ax).

Conversion factor is K1*K2. K1 is the conversion from 10 year certain & life to life annuity (a:10 + 10|ax)/ax. K2 is the conversion from life annuity to pop-up annuity derived as in Case 1 below.

• Example B: Pop-up annuity with a 5 year certain period pops up to the amount payable under a 5 year certain and life annuity (a:5 + 5|ax), but normal form is life annuity (ax).

Conversion factor is K1*K2. K1 is the conversion from life annuity to 5 year certain and life annuity, ax/(a:5 + 5|ax). K2 is the conversion from 5 year certain and life annuity to pop-up annuity with a 5 year certain period derived as in Case 2 below.

ANSWER:

1. Use ProVal’s Administration Factors tool to calculate the ingredients (e.g., ax, ay and axy) for the appropriate formula below. When viewing the results, click the File button and save to an Excel workbook.

2. Use Excel to combine the ingredients and compute conversion factors per the appropriate formula below.

The formula for computing the conversion factor depends on whether the pop-up annuity includes a certain period (independent of whether the normal form includes a certain period).

 

Case 1: Pop-up annuity does not include a certain period

Setting the pop-up annuity to be actuarially equivalent to the normal form, we have:

B*ax + f*K*B*ay – B*axy + (1-f)*K*B*axy = B*NF.

Solving for conversion factor, K, we get:

K = [NF - (ax - axy)] / [f*ay + (1-f)*axy],

where

NF = annuity factor for normal form, e.g., ax,

f = J&S percentage, and

B = annual benefit amount for normal form (reduces out of the solution).

In the common case of a life annuity normal form, i.e., NF = ax, the conversion factor reduces to:

K = axy / [f*ay + (1-f)*axy].

There are three different amounts associated with this pop-up annuity:

• When both are alive, the pop-up annuity pays a reduced amount that reflects the value of the beneficiary benefit:

B*1 + f*K*B*1 – B*1 + (1-f)*K*B*1 = K*B

• When only the beneficiary is alive, the pop-up annuity pays a specified fraction of the joint benefit:

B*0 + f*K*B*1 – B*0 + (1-f)*K*B*0 = f*K*B

• When only the member is alive, the pop-up annuity pays the same amount as the normal form:

B*1 + f*K*B*0 – B*0 + (1-f)*K*B*0 = B

 

Case 2: Pop-up annuity includes a certain period

A pop-up annuity with a certain period may or may not be associated with a normal form that has a certain period. If the normal form does not have a certain period, the amount payable during the certain period under this form of pop-up annuity may be defined either as the normal form benefit or as the normal form benefit reduced for the value of the certain period.

Setting the pop-up annuity with a certain period to be actuarially equivalent to the normal form, where the benefit payable to the member after the beneficiary’s death is the normal form benefit, we have:

B*a:n + B*n|ax + f*K*B*n|ay – B*n|axy + (1-f)*K*B*n|axy = B*NF.

Solving for the conversion factor, K, we get:

K = [NF – (a:n + n|ax - n|axy)] / [f*n|ay + (1-f)*n|axy],

where

n = certain period,

a:n = certain-only annuity for n years,

n|ax = deferred-for-n-year annuity,

NF = annuity factor for normal form, e.g., a:n + n|ax,

f = J&S percentage, and

B = annual benefit amount for normal form (reduces out of the solution).

In the common case where the normal form is a certain and life annuity, i.e., NF = a:n + n|ax, the conversion factor reduces to:

K = n|axy / [f*n|ay + (1-f)*n|axy].

There are three different amounts associated with this pop-up annuity containing a certain period (across four contingencies):

• During the certain period, the pop-up annuity pays the same amount as the normal form:

B*1 + B*0 + f*K*B*0 – B*0 + (1-f)*K*B*0 = B

• After the certain period while both are alive, the pop-up annuity pays a reduced amount that reflects the value of the beneficiary’s benefit:

B*0 + B*1 + f*K*B*1 – B*1 + (1-f)*K*B*1 = K*B

• After the certain period while only the beneficiary is alive, the pop-up annuity pays a specified fraction of the joint benefit:

B*0 + B*0 + f*K*B*1 – B*0 + (1-f)*K*B*0 = f*K*B

• After the certain period while only the member is alive, the pop-up annuity pays the same amount as the normal form:

B*0 + B*1 + f*K*B*0 – B*0 + (1-f)*K*B*0 = B.