Solving for the Discount Rate from Related Values

Why Related Present Values Reveal the Discount Rate

An insurance value and a life annuity value can be linked because their payments occur at complementary times. If both present values are supplied on the same mortality and interest basis, their relationship may determine the discount rate without requiring the underlying survival probabilities. The key is to identify the contract type and payment timing before rearranging an identity.

This tutorial uses discrete annual contracts: insurance benefits are paid at the end of the year of death, and life annuity-due payments are made at the beginning of each year while the insured is alive. All values in an identity must be calculated using the same interest and mortality assumptions. These relationships are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

The Whole-Life Identity and Rate Conversion

For whole-life insurance with a unit benefit payable at the end of the year of death, and a whole-life annuity-due with unit payments at times \(0,1,2,\ldots\) while alive, the present values satisfy:

This identity is useful for rate recovery because the unknown rate appears in the discount rate \(d\), which can be isolated directly:

Once \(d\) is found, convert it to the effective annual interest rate. Since \(d=i/(1+i)\), solving for \(i\) gives:

The annuity value in the denominator must be positive. For a nonnegative effective annual interest rate, the recovered discount rate must satisfy \(0\le d<1\), and the corresponding interest rate is also nonnegative. A result outside this range is a signal to check the input values, the contract definitions, and whether all values share the same basis.

Why the Whole-Life Identity Holds

The identity follows from a pathwise relationship between the insurance payment and the annuity-due payments. Let \(K=\lfloor T_x\rfloor\) be the curtate future lifetime. Death during year \(K+1\) produces a unit insurance benefit at time \(K+1\), with present value \(v^{K+1}\). The annuity-due payments occur at times \(0,\ldots,K\), so their present value is \(\sum_{j=0}^{K}v^j\). By the finite geometric-series formula:

Thus, for each such lifetime outcome, the insurance present value plus \(d\) times the annuity present value is \(v^{K+1}+(1-v^{K+1})=1\). Taking expectations gives \(A_x+d\ddot{a}_x=1\). The timing matters: the death benefit is paid at the end of the year of death, while the annuity payments are due at the beginning of each year the person is alive.

Finite-Term Contracts: Include the Pure Endowment

The finite-term relationship requires attention to what happens if the insured survives the term. A unit term insurance pays only on death during the \(n\)-year term. A pure endowment pays a unit amount at time \(n\) if the insured survives to that time. Together, these two benefits make an endowment insurance. The finite-term identities are:

where \(A_{x:\overline{n}|}=A^1_{x:\overline{n}|}+{}_nE_x\) is the present value of unit endowment insurance. Therefore, if the term insurance value, pure endowment value, and temporary annuity-due value are supplied on the same basis:

Alternatively, if the endowment insurance value is supplied directly, use \(d=(1-A_{x:\overline{n}|})/\ddot{a}_{x:\overline{n}|}\). If only term insurance and temporary annuity values are given, with no pure endowment value or equivalent information, the displayed identity does not determine \(d\) by itself: the survival benefit at time \(n\) has not been accounted for.

Common Traps and Consistency Checks

A rate-recovery calculation can be algebraically simple and still use the wrong identity. Before solving, confirm whether the insurance is whole life, term, or endowment; whether benefits are paid at the end of the year of death; and whether the annuity is due. Also verify that values are for compatible benefit and payment amounts. If the insurance benefit is not unit-sized, first ensure that the value used in the identity is expressed per unit benefit or is scaled consistently.

• Omitting the pure endowment. Term insurance alone does not cover survival to the term’s end. For term insurance, include \({}_nE_x\) in the identity.
• Confusing \(d\) with \(i\). The identity first gives \(d\). Convert using \(i=d/(1-d)\), not \(i=d\).
• Mixing payment timings. The identities here use end-of-year death benefits and annuity-due payments. A different timing convention requires a different relationship.
• Combining values from different bases. Values generated using different mortality assumptions, terms, benefit amounts, or payment frequencies need not satisfy the same identity.

For example, suppose term insurance has value \(0.24\), its pure endowment value is \(0.46\), and the temporary annuity-due value is \(6.0\). The incorrect calculation omits the survival benefit:

This incorrectly implies \(i=0.1267/(1-0.1267)\approx 0.1450\), or about 14.50%. The correction includes the pure endowment:

Thus the correct effective annual rate is 5.26%. The large difference arises from omitting a contract benefit, not from a rounding issue.