Rounding and Precision in Life Insurance APVs

Why Precision Matters in a Discrete Insurance APV

A discrete life insurance present value is a sum of payments for different possible years of death. Each payment is weighted by both the probability of death in that policy year and the discount factor for that year. Rounding either quantity too early can change several terms in the sum, so a small local error may become noticeable in the final value, especially for a large benefit.

A reliable calculation keeps full available precision through the probability and discounting steps, sums the resulting present values, and rounds only the final answer. The main challenge is not using many digits for their own sake; it is avoiding inconsistent precision that changes intermediate quantities before the calculation is complete.

Definition: For a life aged $x$, the probability of death during policy year $k+1$, conditional on being alive at issue, is ${}_k p_x q_{x+k}$, where ${}_k p_x$ is the probability of surviving $k$ years and $q_{x+k}$ is the probability of dying in the following year. For an $n$-year discrete term insurance with benefit $B$ payable at the end of the year of death and effective annual interest rate $i$, the APV is $$ B\sum_{k=0}^{n-1} v^{k+1}{}_k p_xq_{x+k}, \qquad v=\frac{1}{1+i}. $$

This formula assumes the benefit is paid at the end of the policy year of death, the interest rate is constant and effective annually, and the stated mortality probabilities apply to the insured life. It does not describe a benefit paid immediately at death or a benefit whose amount varies by policy year without modification. For a varying benefit $B_{k+1}$, replace $B$ in the $k$th term by $B_{k+1}$.

Keep the Calculation Internally Consistent

The survival probabilities are built from the conditional one-year probabilities. In particular, ${}_0p_x=1$ and ${}_{k+1}p_x={}_kp_x(1-q_{x+k})$. The probability of death in year $k+1$ is therefore the survival probability to the start of that year multiplied by the conditional probability of death during it. A calculation that rounds the survival probability at each recursion can change later death probabilities as well as the current one.

Discount factors also depend on the year of payment. With a constant effective rate, the payment at the end of year $k+1$ is discounted by $v^{k+1}$, not by $v^k$. Keep enough digits in $v$ and in its powers; repeatedly using a rounded value of $v$ can create a systematic difference over several years.

Establish the timing and inputs.
Identify the payment time, benefit, interest rate, and mortality probabilities for each policy year.

Calculate survival and death probabilities.
Use the given probabilities at their available precision. Do not round a survival probability before using it in a later year.

Discount each possible payment.
Use the discount factor for the actual payment year and retain the available precision in the factor and its powers.

Sum, then round.
Add the present values first. Round the final monetary amount to cents, unless the question specifies a different reporting unit.

Worked Examples

Each example below compares a calculation that retains precision with a deliberately rounded calculation. The rounded calculation illustrates the size and direction of the effect for that particular set of values; premature rounding can increase or decrease an APV in other cases.

Worked Example: Rounding Death Probabilities Before Discounting

A hypothetical three-year term policy pays $100,000 at the end of the year of death. The effective annual interest rate is $5\%$. The probabilities, measured from issue, of death in years 1, 2, and 3 are $0.012345$, $0.023456$, and $0.034567$, respectively. Calculate the APV, then compare it with the result from rounding each death probability to three decimal places before valuation.

Method. Multiply each issue-based death probability by the discount factor for its payment year, then sum and multiply by the benefit. This avoids treating a probability rounded for display as though it were an exact input.

First calculate the discount factors, retaining additional digits:

v=\frac{1}{1.05}=0.9523809524,\qquad v^2=0.9070294785,\qquad v^3=0.8638375985.

The three discounted probability contributions are:

\begin{aligned} 0.012345v &= 0.0117571429,\\ 0.023456v^2 &= 0.0212752834,\\ 0.034567v^3 &= 0.0298602741. \end{aligned}

Their sum is $0.0628927004$. Multiplying by the benefit gives an APV of $100{,}000(0.0628927004)=\$6{,}289.27004$, or $6,289.27 to the nearest cent.

If the death probabilities are rounded first, they become $0.012$, $0.023$, and $0.035$. The corresponding discounted contributions are $0.0114285714$, $0.0208616780$, and $0.0302343159$. Their sum is $0.0625245653$, giving an APV of $100{,}000(0.0625245653)=\$6{,}252.46$. The premature rounding reduces this answer by $36.81. The sum of the stated death probabilities is less than 1, so these probabilities are consistent with a possible survival beyond the three-year term.

Check. Each contribution is discounted for its own payment year, and the unrounded APV is the sum of the three discounted contributions multiplied by $100,000. The difference is caused by rounding the inputs, not by a change in the policy or its timing.

Worked Example: Rounding Survival During a Recursion

A hypothetical four-year term insurance pays $50,000 at the end of the year of death. The effective annual interest rate is $5\%$, and the conditional mortality probabilities are $q_x=0.10$, $q_{x+1}=0.20$, $q_{x+2}=0.15$, and $q_{x+3}=0.25$. Find the APV using unrounded survival probabilities. Then find the effect of rounding each cumulative survival probability to two decimal places before calculating the next year's death probability.

Method. Apply the survival recursion using the conditional death probability for each year. The death probabilities from issue are:

\begin{aligned} {}_0p_xq_x &= 1(0.10)=0.10,\\ {}_1p_xq_{x+1} &= 0.90(0.20)=0.18,\\ {}_2p_xq_{x+2} &= (0.90)(0.80)(0.15)=0.108,\\ {}_3p_xq_{x+3} &= (0.90)(0.80)(0.85)(0.25)=0.153. \end{aligned}

The discount factors for years 1 through 4 are $0.9523809524$, $0.9070294785$, $0.8638375985$, and $0.8227024749$. Multiplying each death probability by the discount factor for its year gives:

\begin{aligned} 0.10v &= 0.0952380952,\\ 0.18v^2 &= 0.1632653061,\\ 0.108v^3 &= 0.0932944606,\\ 0.153v^4 &= 0.1258734787. \end{aligned}

The sum is $0.4776713406$. Multiplication by $50,000 gives an APV of $23,883.5670, or $23,883.57 to the nearest cent.

Now round the cumulative survival probability before using it in the next year. The survival probabilities at the starts of years 2, 3, and 4 are $0.90$, $0.72$, and $0.612$. Rounded to two decimal places, these become $0.90$, $0.72$, and $0.61$. The first three death probabilities remain $0.10$, $0.18$, and $0.108$, but the fourth becomes $0.61(0.25)=0.1525$. Its discounted contribution is $0.1525v^4=0.1254621274$. The new total factor is $0.4772599893$, so the rounded-recursion APV is $23,863.00 to the nearest cent. It is about $20.57 lower than the calculation using unrounded survival.

Check. Only the final year's death probability changed: the rounded survival $0.61$ replaced $0.612$. The APV difference is therefore $50{,}000(0.612-0.61)(0.25)v^4=\$20.57$, confirming the comparison.

Worked Example: Rounding a Discount Factor Repeatedly

A hypothetical five-year term policy pays $100,000 at the end of the year of death. The effective annual interest rate is $5\%$, and the conditional probability of death in each year, for a life alive at that year's start, is $0.10$. Calculate the APV using the exact discount factor. Compare it with a calculation that rounds $v$ to $0.95$ and uses powers of that rounded factor.

Method. With a constant conditional mortality probability, the issue-based probability of death in year $k+1$ is $0.10(0.90)^k$. The exact APV factor is a geometric sum:

\begin{aligned} \sum_{k=0}^{4}0.10(0.90)^kv^{k+1} &=0.10v\sum_{k=0}^{4}(0.90v)^k\\ &=\frac{2}{21}\sum_{k=0}^{4}\left(\frac{6}{7}\right)^k\\ &=\frac{2}{21}\left(\frac{9031}{2401}\right) =\frac{18062}{50421} =0.358223756. \end{aligned}

Thus the APV is $100{,}000(0.358223756)=\$35{,}822.38$ to the nearest cent.

For the rounded calculation, the five issue-based death probabilities are $0.10$, $0.09$, $0.081$, $0.0729$, and $0.06561$. Using $0.95$ and its successive powers gives the discounted contributions:

\begin{aligned} 0.10(0.95)&=0.095,\\ 0.09(0.95)^2&=0.081225,\\ 0.081(0.95)^3&=0.069447375,\\ 0.0729(0.95)^4&=0.059377506,\\ 0.06561(0.95)^5&=0.050767767. \end{aligned}

These sum to $0.355817648$, giving a rounded-factor APV of $35,581.76. This is $240.62 lower than the result using $v=1/1.05$ without premature rounding.

Check. The rounded calculation uses the correct five death probabilities and the correct payment years. Its difference comes from replacing $1/1.05=0.9523809524$ with $0.95$ throughout the five-year discounting calculation.

Rounding Practice and Common Traps

In a hand calculation, a practical rule is to retain at least six significant digits in survival probabilities, death probabilities, and discount factors when available. More digits may be needed when the benefit is large, the term is long, or the requested answer is sensitive to small differences. A calculator or spreadsheet should retain its internal precision; display formatting should not be confused with changing the stored value.

State the reporting precision separately from the working precision. For example, it is reasonable to display an intermediate probability as $0.1235$ for readability while using its full value in subsequent calculations. At the end, report monetary APVs in dollars to the cent unless a different unit is requested. If a question explicitly instructs you to round intermediate inputs, follow that instruction and make the effect of the specified rounding clear.

Rounding each year's probability independently. Rounded death probabilities can shift the total probability mass in the covered years. Keep the original values through the APV sum.
Rounding survival before the next recursion. A small change in survival can alter the death probabilities for every remaining year.
Using a rounded discount factor repeatedly. This creates a difference in each discounted term, not just in the final term.
Mixing payment years. A benefit payable at the end of year $k+1$ uses $v^{k+1}$. Using $v^k$ is a timing error, not a precision choice.
Rounding the final answer too early. Keep the unrounded sum until the last step, then round once to the requested monetary precision.

These checks support calculations aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus. The key is to distinguish a displayed approximation from a value actually substituted into later steps: maintain one consistent set of working values, preserve the timing, and round only when reporting the result.

Check Your Understanding

For each problem, keep full precision through the calculation and round only the requested final monetary amount.

A hypothetical two-year term policy pays $20,000 at the end of the year of death. The effective annual interest rate is $4\%$, and the issue-based probabilities of death in years 1 and 2 are $0.04$ and $0.06$. Find the APV to the nearest cent. (Answer: $1,878.70)
A hypothetical three-year term policy pays $40,000 at the end of the year of death. The effective annual interest rate is $5\%$, with conditional death probabilities $0.08$, $0.12$, and $0.10$ in years 1 through 3. Find the APV to the nearest cent. (Answer: $9,850.51)
A two-year policy pays $100,000 at the end of the year of death, and the effective annual interest rate is $0\%$. The issue-based death probabilities are $0.1264$ and $0.2031$. Find the APV using the given probabilities, then find it if each probability is rounded to two decimal places first. (Answer: $32,950.00 using the given probabilities; $33,000.00 after premature rounding, a $50.00 increase)
A hypothetical four-year term policy pays $100,000 at the end of the year of death. The effective annual interest rate is $0\%$, and the conditional death probability is $0.05$ in each year. Find the APV using unrounded survival, then using survival probabilities rounded to three decimal places before each later year's death probability is calculated. (Answer: $18,549.38 using unrounded survival; $18,550.00 after rounding survival)

What You'll Learn