Why the Payment Year Determines the Discount
For an annual death-benefit contract, the benefit may be payable at the end of the policy year in which death occurs. To express that future payment as a value at issue, discount it for the number of full years between issue and payment. The key is to count those years correctly: a benefit payable at the end of policy year \(k\) is discounted for \(k\) years, not \(k-1\).
This tutorial focuses on that discounting step. It does not require a mortality probability: the calculations below value a stated benefit on the condition that it is paid at the specified year-end. Mortality probabilities are needed when combining possible payment years, but they are not needed to discount one known future payment.
The discount factor for one year is the present value of one dollar due at the end of that year. For two years, it is the present value of one dollar due at the end of the second year. Since interest accumulates once per year at the same effective rate, the factor for \(k\) years is the one-year factor multiplied by itself \(k\) times.
Deriving the Annual Discount Factor
The effective annual rate \(i\) means that one dollar invested at the start of a year grows to \(1+i\) dollars at year-end. Repeating this process for \(k\) years means that one dollar invested at issue grows to \((1+i)^k\) dollars at the end of year \(k\). Therefore, the amount that must be invested at issue to produce \(B\) dollars then is the future amount divided by the accumulation over those \(k\) years.
This also gives a useful check. If a calculated present value is multiplied by \((1+i)^k\), the result must be the stated year-\(k\) benefit. If it does not, either the discount factor or the number of periods has been entered incorrectly.
Using an Effective Annual Rate
When the problem gives \(i\) directly, first form the one-year discount factor \(v=1/(1+i)\), then raise it to the number of years until payment. Alternatively, calculate the \(k\)-year accumulation factor \((1+i)^k\) and divide the benefit by it. The two approaches are equivalent. Keep the rate in decimal form in the calculation: for example, \(4\%\) is \(0.04\), so the annual accumulation factor is \(1.04\).
Worked Example: Discounting a Benefit Paid at the End of Year 3
A hypothetical policy provides a death benefit of \( \$80{,}000 \), payable at the end of policy year 3 if death occurs during that year. The effective annual interest rate is \(4\%\). Find the value at issue of this payment. Assume the same annual effective rate applies for all three years; no mortality distribution or death probability is needed because the payment is being valued conditional on occurring at the specified time.
Method. Convert the annual rate to an accumulation factor, accumulate for exactly three years, and divide the benefit by that factor.
The one-year accumulation factor is \(1+i=1+0.04=1.04\). The three-year accumulation factor is \(1.04^3=1.124864\).
Thus the three-year discount factor is \(v^3=1/1.124864=0.88899636\) to eight decimal places. The present value is
Rounded to the nearest cent, the value at issue is \(\$71{,}119.71\). Reaccumulating gives \(\$71{,}119.7088\ldots\times1.124864=\$80{,}000\), confirming the result. The common mistake to avoid is discounting for only two years because year 3 is the payment year. The payment is at the end of year 3, so three full years have elapsed.
Using an Annual Accumulation Factor
A problem may state the annual accumulation factor rather than the interest rate. If that factor is \(a\), then \(a=1+i\), so the discount factor is \(1/a\), and the \(k\)-year discount factor is \(1/a^k\). Do not use the accumulation factor itself to discount: accumulation moves a value forward in time, while discounting moves it back.
Worked Example: Starting From the Accumulation Factor
A hypothetical insurer’s valuation basis uses an annual accumulation factor of \(1.08\). A benefit of \(\$100{,}000\) is payable at the end of policy year 3 if death occurs in that year. Find its value at issue. The annual factor is assumed constant for all three years; no probability distribution for death is required for this conditional payment value.
Method. Take the reciprocal of the given factor to obtain the one-year discount factor, then apply it three times.
The effective annual interest rate implied by the accumulation factor is \(i=1.08-1=0.08\), or \(8\%\). The one-year discount factor is \(v=1/1.08\). The three-year accumulation factor is
Therefore,
Rounded to the nearest cent, the value at issue is \(\$79{,}383.22\). As a check, \(\$79{,}383.2241\ldots\times1.259712=\$100{,}000\), subject only to displayed rounding. A frequent error is to multiply the benefit by \(1.08^3\); that would accumulate the future amount further rather than discount it to issue.
Counting Periods From the Payment Date
Policy-year wording identifies the number of annual periods when issue is at time 0 and the benefit is payable at a year-end. A payment at the end of year 1 uses \(v\); one at the end of year 2 uses \(v^2\); one at the end of year 4 uses \(v^4\). The year number is not reduced just because the benefit is payable at the end of that year. A simple timeline helps prevent this off-by-one error:
The value being sought is measured at issue.
The end of year 1 is one year after issue; the end of year \(k\) is \(k\) years after issue.
For an end-of-year-\(k\) payment, use \(v^k\).
Worked Example: Avoiding an Off-by-One Discount
A hypothetical policy pays \(\$100{,}000\) at the end of policy year 4 if death occurs in that year. The effective annual interest rate is \(5\%\), constant for the four years. Find the value at issue. The payment is treated as certain conditional on death occurring in year 4; no mortality probability is supplied or required for this calculation.
Method. Identify the payment time first. The end of year 4 is four years after issue, so discount the benefit by \(v^4\).
The one-year discount factor is \(v=1/1.05=0.95238095\ldots\). Successive powers are \(v^2=0.90702948\ldots\), \(v^3=0.86383760\ldots\), and \(v^4=0.82270247\ldots\). Equivalently, the four-year accumulation factor is \(1.05^4=1.21550625\).
Rounded to the nearest cent, the value at issue is \(\$82{,}270.25\). Reaccumulating the unrounded value for four years returns \(\$100{,}000\). The common wrong calculation is \(100{,}000v^3=\$86{,}383.76\), which discounts for three years. That value would correspond to a payment at the end of year 3, not year 4.
Common Checks and Exam Traps
Discount factors are positive and, when \(i>0\), less than 1. For a fixed positive benefit and positive interest rate, a later payment has a smaller value at issue than an earlier payment. These observations provide quick reasonableness checks, but they do not replace counting the periods.
| Given information | Correct conversion | Common error |
|---|---|---|
| Effective annual rate \(i\) | \(v=1/(1+i)\) | Using \(i\) itself as the discount factor |
| Annual accumulation factor \(a\) | \(v=1/a\) | Multiplying by \(a\) when moving a future amount back to issue |
| Payment at end of year \(k\) | Use \(v^k\) | Using \(v^{k-1}\) because payment occurs during year \(k\) |
The phrase “during policy year \(k\)” describes when death occurs; the end-of-year payment timing determines when the benefit is paid. Under the convention in this tutorial, the payment is at the end of year \(k\), so the full \(k\)-year interval is discounted. If a contract specifies a different payment time, such as payment immediately upon death, this year-end formula does not automatically apply: the payment time must be represented correctly before selecting a discount factor.
The formulas here assume an effective annual rate that is constant over the stated years and annual payment timing. If annual rates vary, the accumulation over \(k\) years is the product of the year-specific accumulation factors, and the discount factor is the reciprocal of that product. Do not replace such varying rates with one rate unless the problem supplies or justifies an equivalent rate.
Check Your Understanding
For each question, treat any stated benefit as payable at the specified year-end, conditional on the stated event. No mortality probability is needed unless explicitly requested. Round money to the nearest cent.
- A hypothetical policy pays \(\$30{,}000\) at the end of year 2. The effective annual interest rate is \(6\%\), constant for both years. Find the value at issue. (Answer: \(\$26{,}699.89\))
- The annual accumulation factor is \(1.03\). A benefit of \(\$42{,}000\) is payable at the end of year 4. Find its value at issue. (Answer: \(\$37{,}316.46\))
- A benefit of \(\$10{,}000\) is payable at the end of year 2. The annual accumulation factor is \(1.10\). Find the effective annual interest rate and the value at issue. (Answer: \(i=10\%\); \(\$8{,}264.46\))
- A hypothetical contract pays \(\$115{,}762.50\) at the end of year 3. The effective annual interest rate is \(5\%\), constant for all three years. Find the value at issue and verify it by reaccumulating. (Answer: \(\$100{,}000.00\); reaccumulating for 3 years gives \(\$115{,}762.50\))
- A benefit of \(\$100{,}000\) is due at the end of year 4, but an analyst discounts it using \(v^3\) at \(5\%\). State the error and give the correct value at issue. (Answer: The analyst used one fewer year; the correct value is \(\$82{,}270.25\), using \(v^4\))