Why Adjacent Whole-Life Values Reveal Interest
A whole-life insurance value reflects both the chance that a benefit will be paid and the time value of that benefit. When the mortality probabilities are known, the one-year recursion isolates the discount factor: the current-age value is the discounted combination of a death benefit in the next year and the value of continuing coverage if the insured survives. Rearranging that relation lets us infer the interest rate from two adjacent whole-life values.
The calculation is useful only when the values and mortality probabilities refer to the same insurance design and valuation basis. This tutorial uses a unit benefit payable at the end of the year of death, annual effective interest that is constant from year to year, and a discrete-year mortality model. The method is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.
The term \(q_x\) appears because a death during the first year produces a unit payment at the end of that year. The term \(p_xA_{x+1}\) accounts for surviving the year and then having the future whole-life insurance value at age \(x+1\). Both possible outcomes are valued at time \(x\), so the combined amount is multiplied by one year’s discount factor \(v\).
Solving the Recursion for Interest
Define the quantity inside the parentheses as \(C_x=q_x+p_xA_{x+1}\). If \(A_x>0\) and \(C_x>0\), the recursion gives one unique discount factor and, in turn, one unique annual effective rate. The order of the ratio matters: the current-age value is the numerator when solving for \(v\).
Since \(v=1/(1+i)\), taking the reciprocal gives \(1+i=1/v\). Thus the interest rate is obtained by taking the reciprocal of the inferred discount factor and subtracting one.
These formulas do not require a particular mortality distribution for the complete future lifetime. They use only the one-year death and survival probabilities at age \(x\), together with the adjacent whole-life values. They do require that the values represent the same unit-benefit contract with death benefits payable at the end of the year of death. Changing benefit amount or timing changes the recursion and can change the inferred rate.
Derivation and Calculation Process
The inversion follows directly from conditioning on the first year. With probability \(q_x\), death occurs during that year and the unit benefit is discounted one year. With probability \(p_x\), the life survives and the future expected present value at age \(x+1\) is also discounted one year. Therefore the expected present value is \(v q_x+v p_x A_{x+1}\), which factors as \(v(q_x+p_xA_{x+1})\). Dividing both sides by the positive quantity in parentheses isolates \(v\); converting from \(v\) to \(i\) yields the interest formula above.
Check that \(A_x\) and \(A_{x+1}\) are for the same unit whole-life benefit, payable at the end of the year of death, and that the stated probabilities are \(q_x\) and \(p_x=1-q_x\).
Calculate \(C_x=q_x+p_xA_{x+1}\). Keep the survival probability attached to the future value.
Calculate \(v=A_x/C_x\), not \(C_x/A_x\).
Calculate \(i=1/v-1\), then substitute \(v\) into \(A_x=vC_x\) to check the original recursion.
Worked Applications
Worked Example: A Zero-Interest Result
A hypothetical unit whole-life policy has \(q_x=0.10\), so \(p_x=0.90\). The values are \(A_x=0.28\) and \(A_{x+1}=0.20\). Infer the annual effective interest rate and determine whether it is compatible with nonnegative interest.
Method. Use the one-year recursion to calculate the conditional value, then recover \(v\) and convert it to \(i\). This method avoids assuming that a positive insurance value necessarily implies a positive interest rate.
First, the survival probability is \(p_x=1-q_x=1-0.10=0.90\).
Next, calculate the quantity in parentheses: \(q_x+p_xA_{x+1}=0.10+0.90(0.20)=0.10+0.18=0.28\).
Therefore, \(v=A_x/(q_x+p_xA_{x+1})=0.28/0.28=1\).
Convert the discount factor to an annual effective rate: \(i=1/v-1=1/1-1=0\), or \(0.0000\%\) when expressed as a percentage to four decimal places.
Check. Substituting into the recursion gives \(A_x=1(0.10+0.90(0.20))=0.28\), as stated. Since \(v=1\), the result is exactly zero interest and is compatible with nonnegative interest.
Common mistake avoided. Do not subtract the survival-weighted future value from \(A_x\) and treat the remainder as the discount factor. The recursion discounts the entire conditional value; here that value equals \(A_x\), giving \(v=1\), not a negative rate.
Worked Example: Inferring a Positive Rate
A hypothetical insurer is valuing a unit whole-life benefit payable at the end of the year of death. At age \(x\), \(q_x=0.08\), \(p_x=0.92\), \(A_x=0.25\), and \(A_{x+1}=0.25\). Find the annual effective interest rate.
Method. Apply the recursion directly and solve for \(v\) before converting to \(i\). This avoids interpreting the difference between the two insurance values as an interest rate.
Calculate the conditional value: \(q_x+p_xA_{x+1}=0.08+0.92(0.25)=0.08+0.23=0.31\).
The discount factor is \(v=0.25/0.31=25/31\).
Taking the reciprocal and subtracting one, \(i=31/25-1=31/25-25/25=6/25=0.24\).
The inferred annual effective interest rate is \(24.0000\%\), to four decimal places. The discount factor \(25/31\) is positive and less than 1, as expected for a positive rate.
Check. Using \(v=25/31\), the recursion gives \(A_x=(25/31)(0.31)=0.25\). The result reproduces the supplied value.
Common mistake avoided. The rate is not \(1-v\). That expression is the effective annual discount rate \(d\), not the effective annual interest rate \(i\). Here \(1-v=6/31\), while \(i=1/v-1=6/25\).
Worked Example: Checking the Rate at Two Adjacent Ages
For a hypothetical unit whole-life policy, the mortality probabilities are \(q_x=0.10\) and \(q_{x+1}=0.20\). The whole-life values are \(A_x=0.3104\), \(A_{x+1}=0.32\), and \(A_{x+2}=0.25\). Infer the rate from each recursion and check whether the two calculations agree.
Method. Calculate the conditional value and discount factor separately at ages \(x\) and \(x+1\). This checks that the same annual rate is supported by both adjacent-age equations.
At age \(x\), \(p_x=1-0.10=0.90\). Thus \(q_x+p_xA_{x+1}=0.10+0.90(0.32)=0.10+0.288=0.388\).
The discount factor from this age is \(v_x=0.3104/0.388=0.80\), since \(0.388(0.80)=0.3104\). Therefore \(i_x=1/0.80-1=1.25-1=0.25\), or \(25.0000\%\).
At age \(x+1\), \(p_{x+1}=1-0.20=0.80\). The conditional value is \(q_{x+1}+p_{x+1}A_{x+2}=0.20+0.80(0.25)=0.20+0.20=0.40\).
The second discount factor is \(v_{x+1}=0.32/0.40=0.80\), giving \(i_{x+1}=1/0.80-1=0.25=25.0000\%\).
Check. Both adjacent-age calculations give \(v=0.80\) and \(i=25.0000\%\). Substitution confirms the recursions: \(0.80(0.388)=0.3104\) and \(0.80(0.40)=0.32\).
Common mistake avoided. At the second age, use \(q_{x+1}\), \(p_{x+1}\), and \(A_{x+2}\). Reusing age \(x\)'s mortality probabilities would not be the recursion for \(A_{x+1}\).
Validity Checks and Exam Traps
Before accepting an inferred rate, check both its algebra and its interpretation. The expression \(A_x/(q_x+p_xA_{x+1})\) is the discount factor. Its reciprocal minus one is the annual effective interest rate. Confusing those two quantities reverses the relationship and can produce an incorrect answer even when the arithmetic is otherwise accurate.
A negative inferred rate is not automatically an algebra error: it may be mathematically consistent if negative interest is allowed. If the problem assumes nonnegative interest, however, the discount factor must satisfy \(0<v\leq1\), equivalent to \(q_x+p_xA_{x+1}\geq A_x>0\). Do not declare a result impossible without checking the stated assumption.
The inputs also need sufficient precision. If the reported \(A\)-values are rounded, the inferred rate is generally an estimate based on those rounded values, and substitution may not reproduce the displayed value exactly. Retain full precision through the calculation and round only the final rate. Use the model's stated benefit timing and annual time unit throughout; the recursion does not apply unchanged to a benefit payable immediately at death or to a different benefit amount.
The key practical check is simple: calculate \(v\), verify it is positive and respects any interest restriction, then substitute it into the original recursion. That final substitution catches ratio reversals and arithmetic slips before the result is reported.
Check Your Understanding
For each hypothetical unit whole-life contract, assume the benefit is payable at the end of the year of death and interest is constant annually. Infer the annual effective rate where requested. Answers are shown for checking.
- At age \(x\), \(q_x=0.12\), \(A_{x+1}=0.25\), and \(A_x=0.30\). Calculate \(p_x\), the conditional value, and the inferred rate. (Answer: \(i=13.3333\%\).)
- At age \(y\), \(q_y=0.05\), \(A_{y+1}=0.30\), and \(A_y=0.268\). Recover the discount factor first, then the rate. (Answer: \(v=0.8000,\ i=25.0000\%\).)
- At age \(z\), \(q_z=0.20\), \(A_{z+1}=0.25\), and \(A_z=0.40\). Find the rate and state whether it is compatible with nonnegative interest. (Answer: \(i=0\%\); it is compatible with nonnegative interest.)
- At age \(r\), \(q_r=0.15\), \(A_{r+1}=0.40\), and \(A_r=0.35\). Calculate the conditional value and rate, then state whether the inferred rate is nonnegative. (Answer: \(q_r+p_rA_{r+1}=0.49,\ i=40.0000\%\); it is nonnegative.)
- At age \(s\), \(q_s=0.07\), \(A_{s+1}=0.50\), and \(A_s=0.50\). Infer the rate and verify that the discount factor is no greater than 1. (Answer: \(v=100/107,\ i=7.0000\%\); \(v\approx0.9346\leq1\).)