Official source: IRS Forms, Instructions & Publications (https://www.irs.gov/pub/irs-pdf/p1457.pdf), retrieved 2026-10-03. U.S. Government work (17 U.S.C. § 105).

---

Publication 1457 (Rev. 6-2023) Catalog Number 63854M Department of the Treasury Internal Revenue Service www.irs.gov

A. Two Life Last-to-Die Factors 7

B. Income or Annuity Payable for Such Time as One Person Survives Another 7

C. First-to-Die Factors 8

D. Factors Involving One Life and a Term of Years 9

E. Factors Involving Two Lives and a Term of Years 10

F. Pooled Income Funds 11

2

The actuarial tables associated with this publication are:

3

This publication provides examples for using actuarial factors for certain income, gift, and estate tax valuations of future interests. These examples use factors from actuarial tables associated with this Publication. The actuarial tables are located at

www.irs.gov/retirement-plans/actuarial-tables

The examples illustrate methods for using the factors in the associated tables for valuations, and for finding other factors not found directly in the tables. The examples in Part A show the use of lastto-die remainder factors in Table R(2), and show the method for finding last-to-die life estate and annuity factors. Life estate factors and factors for a term of years estate may also be referred to as income factors. Part B and Part C examples illustrate the use of factors from the single life Table S and from the last-to-die Table R(2) to obtain factors for such time as one person survives another, and to obtain first-to-die factors. In Part D, commutation factors from Table H are used to find factors for one life and a term of years. Part E provides similar explanations to compute factors for two lives and term of years. Part F demonstrates the use of the remainder factors for pooled income funds. In all cases, appropriate factors may also be computed with software using the same actuarial formulas from which these factors are computed, if the final factors so computed have, respectively, at least as many significant figures as are shown in these examples.

The factors and tables associated with this publication involving life contingencies are derived from the values of lx taken from the Life Table for the Total Population appearing as Table 1, in U.S. Decennial Life Tables for 1999-2001 published by the U.S. Department of Health and Human Services, Public Health Service, National Center for Health Statistics. Values from Table 1 appear in the associated set of actuarial tables in Section 7, labeled as Table 2010CM. However, in Table 2010CM the values of lx are stated to seven digits.

4

In the IRS actuarial tables on IRS.gov associated with this publication and these examples, the factors in Sections 1 through 6 are based on interest rates ranging from 0.2% to 20.0% in intervals of 0.2%. The wide range of rates is shown pursuant to Internal Revenue Code (IRC) Section 7520, which requires the use of an interest rate of 120% of the annual mid-term applicable federal rate for the month in which the valuation date falls. This rate is referred to in this publication as the “Section 7520 rate,”, or simply the “7520 rate.” All the factors associated with this publication reflect annual compounding of interest.

Table S contains factors for the present worth of a life annuity, a life estate and a remainder interest based on a single life.

Table R(2) contains factors for the present worth of the remainder interest in $1.00 payable at the death of the last to die of two persons.

Table B contains factors for the present worth of an annuity, an income interest and a remainder interest for a term certain.

• _

Table H contains commutation tables for Dx, Nx and Mx.

Table K contains adjustment factors for annuities payable at the end of annual, semi-annual, quarterly, monthly and weekly periods.

Table J contains adjustment factors for term certain annuities payable at the beginning of annual, semi-annual, quarterly, monthly, and weekly periods.

Table 2010CM is the underlying mortality table used to calculate factors involving life contingencies.

5

*On October 22, 1988, IRC Section 7520 was enacted. It requires the use of an interest rate equal to 120% of the midterm applicable federal rate, rounded to the nearest 0.2%.

**Transition period: For valuation dates from May 1, 2019, through June 1, 2023, you may rely on the actuarial tables based either on Table 2000CM or on Table 2010CM. However, you must be consistent in using the same set of tables to value each interest (income, remainder, partial, etc.) in the same property and with respect to all transfers occurring on the same valuation date.

6

Example 1. Based on an interest rate of 4.2%, the present worth of $1.00 due at the death of the last to die of two persons age 60 and 65 is $0.35061. Find in Table R(2) Section 2 2 the older age 65 and the younger age 60. Across from those ages, under the column headed 4.2% is the remainder factor 0.35061.

A life estate factor or an annuity factor for the same ages and interest rate can be computed using examples 2 and 3.

Example 2. Based on an interest rate of 4.2%, the present worth of the right to receive the use of $1.00 until the death of the last to die of two persons age 60 and 65 is $0.64939, determined as:

Remainder Factor from Example 1 = 0.35061

Income Factor = 1.00000 - 0.35061

= 0.64939

Example 3. Based on an interest rate of 4.2%, the present worth of an annuity of $1.00 per annum payable at the end of each year until the death of the last to die of two persons age 60 and 65 is $15.4617, determined as:

Income Factor from Example 2 = 0.64939

Annuity Factor = 0.64939 ÷ 0.042

= 15.4617

Example 4. Based on an interest rate of 4.2%, the present worth of the right to receive the use of $1.00 for such time as a person age 65 survives a person age 60 is $0.07110 determined as:

Income Factor from Example 2 = 0.64939 Single Life Income Factor from Table S at 4.2%, age 60 = 0.57829

Required Income Factor = 0.64939 - 0.57829

= 0.07110

7

Example 5. Based on an interest rate of 4.2%, the present worth of an annuity of $1.00 per annum payable annually at the end of each year for such time as a person age 65 survives a person age 60 is $1.6928 determined as:

Joint and Survivor Annuity Factor from Example 3 = 15.4617 Single Life Annuity Factor, age 60, From Table S at 4.2% = 13.7689

Required Annuity Factor = 15.4617 – 13.7689

= 1.6928

Example 6. Based on an interest rate of 4.2%, the present worth of $1.00 due at the death of the first to die of two persons age 60 and 65 is $0.55816, determined as:

Single Life Remainder Factor, age 60, from Table S at 4.2% = 0.42171 Single Life Remainder Factor, age 65, from Table S at 4.2% = 0.48706

Joint and Survivor Remainder Factor from Example 1 = 0.35061

Required Remainder Factor = 0.42171 + 0.48706 - 0.35061

= 0.55816

Example 7. Based on an interest rate of 4.2%, the present worth of the right to receive the use of $1.00 until the death of the first to die of two persons age 60 and 65 is $0.44184, determined as:

First-to-Die Remainder Factor from Example 6 = 0.55816

Income Factor = 1.00000 - 0.55816

= 0.44184

Example 8. Based on an interest rate of 4.2%, the present worth of an annuity of $1.00 per annum payable annually at the end of each year until the death of the first to die of two persons age 60 and 65 is $10.5200, determined as:

Income Factor from Example 7 = 0.44184

Required Annuity Factor = 0.44184 ÷ .042

= 10.5200

8

Example 9. Based on an interest rate of 2.8%, the present worth of a temporary annuity of $1.00 per annum payable annually at the end of each year for 10 years or until the prior death of a person age 60 is $8.1791, determined as:

Initial Age = 60 plus Term of Years = 10

Terminal Age = 70

N-Factor, Table H(2.8), age 60 = 271,994.3 minus N-Factor, Table H(2.8), age 70 = 133,677.8

Difference = 138,316.5

D-Factor, Table H(2.8), age 60 = 16,911.03

Required Annuity Factor = 138,316.5 / 16,911.03

= 8.1791

Example 10. Based on an interest rate of 2.8%, the present worth of $1.00 due at the death of a person age 60 provided the death occurs within the first 10 years is $0.10392, determined as:

Initial Age = 60 plus Term of Years = 10

Terminal Age = 70

M-Factor, Table H(2.8), age 60 = 9,295.187 minus M-Factor, Table H(2.8), age 70 = 7,537.826

Difference = 1,757.361

D-Factor, Table H(2.8), age 60 = 16,911.03

Desired Remainder Factor = 1,757.361 / 16,911.03

= 0.10392

Example 11. Based on an interest rate of 2.8%, the present worth of the right to receive the income from $1.00 for 10 years or until the prior death of a person age 60 is $0.22901, determined as:

Temporary Annuity Factor, Example 9 = 8.1791

Required Income Factor = 8.1791 x .028

= 0.22901

9

Example 12. Based on an interest rate of 2.8%, the present worth of the right of a person age 21, if living, to receive $1.00 upon attaining age 30 is $0.77336, determined as:

D-Factor, Table H(2.8), age 30 = 42,794.49 divided by D-Factor, Table H(2.8), age 21 = 55,336.02

Required Factor = 0.77336

Example 13. The probability that a person age 21 will be alive at age 30 is 0.991558, determined as:

lx value, Table 2010CM, age 30 ^{= 97,989.90}

lx value, Table 2010CM, age 21 ^{= 98,824.20}

Required Probability = 97,989.90 / 98,824.20

= 0.991558

Example 14. Based on an interest rate of 2.8%, the present worth of a temporary annuity of $1.00 per annum payable in equal monthly installments at the end of each month for 10 years or until the prior death of a person age 60 is $8.2838, determined as:

Annuity Factor from Example 9 = 8.1791 Monthly Adjustment Factor, Table K at 2.8% = 1.0128

Required Annuity Factor = 8.1791 x 1.0128

= 8.2838

Note : Factors from Table J are used in the same manner as factors from Table K, for application to situations where the annuity is a term certain annuity payable at the beginning of each period.

Example 15. Based on an interest rate of 2.8%, the present worth of the right to receive $1.00 at the end of 10 years provided at least one of two persons age 60 and 65 alive is $0.74233, determined as:

lx value, Table 2010CM, age 60 ^{= 88,665.95}

lx value, age 70 ^{= 77,957.53}

77,957.53 1 - = 0.120773 88,665.95

10

lx value, Table 2010CM, age 65 ^{= 84,221.59}

lx value, age 75 ^{= 69,174.83}

69,174.83 1 - = 0.178657 84,221.59

Remainder Factor, Table B(2.8), 10-Year Term = 0.758698

Required Remainder Factor = (1 - 0.120773 x 0.178657) x 0.758698

Required Factor = 0.74233

Table R(2) may be used to obtain joint and survivor remainder factors for pooled income funds. The interest rate to be used to determine the remainder factor is the yearly rate of return for the fund as defined in IRS Notice 89-60. If the yearly rate of return falls between two interest rates for which the factors are given in Table R(2), a linear interpolation may be made.

Example 16. Based on a yearly rate of return of 3.636%, the present worth of the remainder interest in a pooled income fund of $1.00 payable at the death of the last to die of two persons age 60 and 65 is $0.40038 determined as:

Remainder Factor, Table R(2), at 3.6% = 0.40374 minus Remainder Factor, Table R(2), at 3.8% = 0.38507

Difference = 0.01867

3.636 % - 3.600 % X

3.800 % - 3.600 % 0.01867

X = 0.00336

Remainder Factor at 3.6% = 0.40374

minus X = 0.00336 Interpolated Remainder Factor at 3.636% = 0.40038

As an alternative to using an interpolation method, it is also acceptable to compute the remainder factor directly from the underlying actuarial formulas using an interest rate of 3.636%, provided that the resulting remainder factor is expressed to at least 5 decimal places, and that this method is applied consistently in valuing all interests in the same property.

11