Diagnosing Inconsistent Whole-Life Values

Why Check Adjacent-Age Values?

A whole-life insurance value at age $x$ depends on what can happen during the next year and, if the insured survives, on the value at age $x+1$. That link provides a direct consistency check: values supplied for adjacent ages must agree with the interest rate, mortality probability, and benefit timing used to calculate them. A mismatch does not, by itself, identify which input is wrong, but its size and pattern can point to a likely error.

Throughout this tutorial, $A_x$ denotes the actuarial present value of a unit whole-life insurance benefit payable at the end of the year of death, with annual effective interest rate $i$. Write $v=1/(1+i)$, $q_x$ for the probability that a life aged $x$ dies within one year, and $p_x=1-q_x$ for the probability of surviving that year. The standard one-year recursion applies to this benefit timing and a consistent mortality basis.

Definition: The one-year whole-life recursion is $$ A_x=v\left(q_x+p_xA_{x+1}\right) =v\left(q_x+(1-q_x)A_{x+1}\right). $$ It assumes a unit benefit paid at the end of the year of death, annual effective interest $i$, and one-year death and survival probabilities $q_x$ and $p_x$ for a life aged $x$. It does not automatically apply unchanged to benefits paid at the moment of death, at the beginning of the year, or in a different amount.

What the Recursion Checks

The recursion divides the contract’s possible outcomes during the first year into two mutually exclusive cases. If death occurs, the unit benefit is paid at the end of that year. If the life survives, the contract continues and its value at that time is $A_{x+1}$. Both outcomes are discounted back one year.

A_x = q_x(v\cdot 1)+p_x(vA_{x+1}) = v\left(q_x+p_xA_{x+1}\right).

This gives a practical test. Calculate the right-hand side using the stated $i$, $q_x$, and $A_{x+1}$, then compare it with the supplied $A_x$. A consistency residual can be defined as the supplied value minus the value predicted by the recursion:

R_x=A_x-v\left(q_x+(1-q_x)A_{x+1}\right).

With exact inputs under the stated model, $R_x=0$. If a table reports rounded values, a small residual can arise from rounding; compare at the precision of the displayed data before concluding that the values are inconsistent. A material residual means that at least one supplied value, assumption, or timing convention does not match the others.

Key takeaway: For the stated end-of-year benefit convention, both the death benefit and the surviving policy value are discounted by $v$. The death probability in the first-year term is $q_x$, the probability for the age-$x$ year.

Recovering an Implied Input

When adjacent-age values are supplied, the recursion can also be rearranged to test the mortality or interest rate they imply. These rearrangements are useful diagnostic tools, but their answers must still be plausible and consistent with the model.

q_x=\frac{(1+i)A_x-A_{x+1}}{1-A_{x+1}} \qquad\text{if }A_{x+1}\ne 1,

i=\frac{q_x+(1-q_x)A_{x+1}}{A_x}-1 \qquad\text{if }A_x\ne 0.

For the mortality rearrangement, multiply the recursion by $1+i$, substitute $p_x=1-q_x$, and collect the terms involving $q_x$:

(1+i)A_x=q_x+(1-q_x)A_{x+1} =A_{x+1}+q_x(1-A_{x+1}).

Subtracting $A_{x+1}$ and dividing by $1-A_{x+1}$ gives the stated expression for $q_x$. The division requires $A_{x+1}\ne 1$. For the interest rearrangement, multiply the original recursion by $1+i$, divide by $A_x$, and subtract 1; this requires $A_x\ne 0$.

An implied $q_x$ must be a probability, so it must lie between 0 and 1. An implied interest rate should also agree with the stated interest assumption. A result outside a valid range is not a rate to “fix” by rounding: it signals that the supplied values cannot all satisfy the stated model.

Worked Consistency Checks

Worked Example: Checking a Rounded Adjacent-Age Value

A hypothetical insurer reports $A_{x+1}=0.4000$ and $A_x=0.3924$ for a unit whole-life benefit payable at the end of the year of death. The annual effective interest rate is $i=5\%$, and $q_x=0.02$. Determine whether the displayed values are consistent to four decimal places.

Method. Predict $A_x$ from the one-year recursion, then compare the result with the reported value. This avoids comparing the supplied values without accounting for the one-year discount.

First calculate the discount factor and survival probability:

v=\frac{1}{1.05}=0.95238095\ldots, \qquad p_x=1-0.02=0.98.

The undiscounted value inside the recursion is:

q_x+p_xA_{x+1} =0.02+(0.98)(0.4000) =0.02+0.3920 =0.4120.

Discounting one year gives:

A_x^{\text{predicted}} =\frac{0.4120}{1.05} =0.39238095\ldots \approx 0.3924.

The residual, using the unrounded prediction, is $R_x=0.3924-0.39238095\ldots=0.00001905\ldots$. The predicted value rounds to $0.3924$ to four decimal places, so the reported adjacent-age values are consistent at that precision.

Check. Substitution confirms the rounded result: $0.95238095\ldots(0.4120)=0.39238095\ldots$, which rounds to the supplied $A_x$. The common mistake avoided here is calling a tiny residual an inconsistency without checking whether ordinary display rounding explains it.

Worked Example: Testing the Mortality and Interest Implied by Values

A hypothetical set of values gives $A_x=0.4000$ and $A_{x+1}=0.4500$. The stated annual effective interest rate is $4\%$, and the stated one-year death probability is $q_x=0.03$. Check the values, find the mortality probability implied by the two reported values at $4\%$, and find the interest rate implied if $q_x=0.03$ is retained.

Method. First calculate the value predicted by the stated assumptions. Then use each rearranged recursion as a separate diagnostic. This distinguishes a failure of the stated inputs from a possible alternative input that would make the values fit.

At the stated interest rate, $v=1/1.04$ and $p_x=1-0.03=0.97$. Thus:

q_x+p_xA_{x+1} =0.03+(0.97)(0.4500) =0.03+0.4365 =0.4665, $$ $$ A_x^{\text{predicted}} =\frac{0.4665}{1.04} =0.44855769\ldots \approx 0.4486.

The reported value $0.4000$ is not the predicted value. Its residual is $0.4000-0.44855769\ldots=-0.04855769\ldots$, a substantial discrepancy at the displayed precision.

Now infer $q_x$ from the two reported values and the stated interest rate:

q_x =\frac{(1.04)(0.4000)-0.4500}{1-0.4500} =\frac{0.4160-0.4500}{0.5500} =\frac{-0.0340}{0.5500} =-0.061818\ldots.

This is not a valid probability. Therefore the two reported values cannot be reconciled at $4\%$ by choosing another valid $q_x$.

If instead $q_x=0.03$ is retained, the interest rate implied by the reported values is:

i=\frac{0.03+(0.97)(0.4500)}{0.4000}-1 =\frac{0.4665}{0.4000}-1 =1.16625-1 =0.16625.

That is an implied effective annual rate of $16.625\%$, not $4\%$. The reported values, stated mortality, and stated interest are therefore mutually inconsistent under the specified benefit timing.

Check. Using $i=0.16625$ with $q_x=0.03$ gives $v=1/1.16625$, so $v(0.4665)=0.4000$. This confirms the algebraic interest diagnostic; it does not establish that $16.625\%$ was actually the source of the discrepancy. The common mistake avoided is treating an impossible implied mortality probability as an acceptable estimate.

Worked Example: Detecting an Omitted Discount on the Death Benefit

For a hypothetical contract, $i=10\%$, $q_x=0.10$, and $A_{x+1}=0.50$. All benefits are specified as payable at the end of the year of death. A calculation reports $A_x=0.55$. Diagnose the likely calculation error and compute the corrected value.

Method. Compare the reported value with the correct recursion and with the expression that results if the death benefit is not discounted. This identifies a numerical signature of the mistake while respecting the stated payment timing.

The survival probability is $p_x=1-0.10=0.90$, and the discount factor is $v=1/1.10$. The correct value is:

A_x =\frac{0.10+(0.90)(0.50)}{1.10} =\frac{0.10+0.45}{1.10} =\frac{0.55}{1.10} =0.50.

The reported $0.55$ equals the expression $q_x+p_xA_{x+1}=0.10+(0.90)(0.50)$, before discounting. That is the value that would result from omitting the one-year discount factor from the entire first-year expression. Since the contract specifies an end-of-year death benefit, the corrected value is $0.50$, not $0.55$.

Check. Substituting $A_x=0.50$ into the recursion gives $1.10(0.50)=0.55=0.10+0.90(0.50)$. This verifies the corrected value. The common mistake is to treat the death probability as though it were already a present value; the death-benefit term is $vq_x$, not $q_x$.

Separating Mortality Errors from Timing Errors

A failed check is a reason to inspect assumptions, not a proof of a particular error. Use the supplied information systematically:

Confirm the contract convention.
Check that the value is for a unit whole-life benefit payable at the end of the year of death. If payment is at another time, the standard recursion above is not automatically the correct model.

Use the correct age’s mortality.
The first-year death probability is $q_x$, for a life aged $x$ at the start of the year. A value for age $x+1$ is used only in the survival branch, weighted by $p_x$.

Check discounting and arithmetic.
Apply $v=1/(1+i)$ to the death benefit and the surviving value. Compare the full-precision prediction with the displayed $A_x$.

Use implied inputs as diagnostics.
Rearrange the recursion only when its denominator is nonzero. Reject implied probabilities outside the interval from 0 to 1, and distinguish a mathematically implied rate from evidence about the actual source of an error.

Benefit timing deserves particular care. If death benefits are paid at the moment of death rather than at the end of the year, the discounting within the year depends on when death occurs. The one-year probability $q_x$ alone does not specify that timing. A different payment convention therefore requires the corresponding model assumptions; it cannot be handled by simply deleting $v$ from the end-of-year recursion.

Common Exam Traps

Using $q_{x+1}$ in the first-year death term. The first-year outcomes start from age $x$, so the death probability is $q_x$. The survival branch uses $p_x=1-q_x$ and then the age-$x+1$ value.
Discounting only the survival branch. The end-of-year death benefit is also paid one year after the valuation date, so its present value contribution is $vq_x$.
Confusing a plausible value with a consistent value. Even if $A_x$ lies between 0 and 1, it may not satisfy the recursion with the stated inputs.
Over-interpreting a mismatch. A nonzero residual shows that the supplied data and model do not all agree. It does not uniquely identify whether interest, mortality, benefit timing, or a reported value is wrong.
Ignoring precision. Compute with the stated inputs before rounding. A difference that disappears at the reported number of decimal places may be explained by display rounding; a material difference should not be dismissed that way.

Key takeaway: Check adjacent-age whole-life values by predicting $A_x$ from $v\{q_x+(1-q_x)A_{x+1}\}$. Then use the residual and, when useful, implied mortality or interest to diagnose a mismatch without confusing a diagnostic clue with proof of its cause.

Check Your Understanding

For each question, use the end-of-year death-benefit convention and keep intermediate calculations unrounded.

A hypothetical whole-life value has $i=4\%$, $q_x=0.05$, and $A_{x+1}=0.6000$. Calculate the recursion-predicted $A_x$, to four decimal places. (Answer: $0.5962$)
Values $A_x=0.4800$ and $A_{x+1}=0.5000$ are reported with $i=5\%$. Calculate the implied $q_x$ and state whether it is a valid probability. (Answer: $q_x=0.0080$, valid)
For $i=8\%$, $q_x=0.04$, and $A_{x+1}=0.4000$, calculate the predicted $A_x$ to four decimal places. (Answer: $0.3926$)
Suppose $q_x=0.06$, $A_{x+1}=0.5000$, and a supplied value is $A_x=0.5000$. Calculate the interest rate implied by the values and mortality. (Answer: $i=0.0600$, or $6.00\%$)
At $i=5\%$, reported values are $A_x=0.4400$ and $A_{x+1}=0.4600$. Calculate the implied $q_x$ to four decimal places, then determine whether those values are consistent with a separately stated $q_x=0.03$. (Answer: implied $q_x=0.0037$; not consistent with $q_x=0.03$, which predicts $A_x=0.4535$)

What You'll Learn