Why Equivalence Pricing and Reserve Recursions Belong Together
A level net premium is calculated at issue by equating the expected present value of premiums to the expected present value of benefits. A reserve at a later duration measures the value, on the same assumptions, of the benefits and premiums still to come. The annual recursion links those values: it balances the premium collected, interest earned, death benefits paid, and reserve for policies that continue.
This tutorial uses annual premiums payable at the beginning of each policy year while the insured is alive, and death benefits payable at the end of the year of death. There are no expenses, lapses, or other cash flows. The mortality and interest assumptions used to price the contract are also used to value its reserves. The treatment is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.
Equivalence Pricing and the One-Year Recursion
Let the death benefit be \(B\), the effective annual interest rate be \(i\), and \(v=1/(1+i)\). Write \(q_{x+k}={}_1q_{x+k}\) and \(p_{x+k}={}_1p_{x+k}=1-q_{x+k}\) for the probabilities of death and survival during the year from age \(x+k\). For a term contract with \(n\) annual premiums, premiums are due at times \(0,1,\ldots,n-1\), provided the insured is alive. A death during any covered year produces a benefit at the end of that year.
The equivalence principle sets the expected present value of premiums at issue equal to that of benefits. In the term case, the probability of surviving to time \(k\) is \({}_kp_x\). A premium due at time \(k\) therefore has expected present value \(v^k{}_kp_xP\), while a benefit paid at the end of year \(k+1\) has expected present value \(v^{k+1}{}_kp_xq_{x+k}B\). Consequently:
For whole-life insurance with the same annual timing, the corresponding compact formula is \(P=BA_x/\ddot{a}_x\). In both cases, the premium is a level annual net premium: it covers expected benefits on the pricing basis, not expenses.
To obtain the recursion, consider the prospective reserve immediately before the premium at time \(k\). The premium \(P\) is paid then. During the following year, death occurs with probability \(q_{x+k}\), in which case \(B\) is paid at year-end; otherwise, the policy continues and has reserve \({}_{k+1}V_x\) immediately before its next premium. The prospective value at time \(k\) is therefore:
Multiplying by \(1+i\) and rearranging gives the annual net premium reserve recursion:
This equation is a cash-flow balance as well as an algebraic identity. The reserve plus the premium is accumulated at interest for one year. At year-end, the expected amount needed is the death benefit for those who die plus the next reserve for those who survive. The survival branch uses \({}_{k+1}V_x\), which already includes the next premium in its prospective value; adding another premium to that branch would count it twice.
Worked Example: Whole-Life Insurance with a First-Year Mortality Change
Worked Example: Whole-Life Insurance with a First-Year Mortality Change
A hypothetical whole-life policy pays \(B=\$100\) at the end of the year of death. The annual net premium is paid while the insured is alive, beginning at issue. Assume \(i=25\%\), so \(v=0.8\). The first-year mortality probability is \(q_x=0.10\). After the first year, assume the conditional probability of death in every year is \(0.20\), so the corresponding survival probability is \(0.80\). Find the level net premium and the reserve immediately before the second premium.
Method. Calculate the tail whole-life insurance and annuity values under the constant post-first-year mortality assumption. Combine these with the first-year cash flows to price the contract, then apply the prospective reserve definition. This avoids incorrectly using the post-first-year mortality assumption in the first year.
For the tail, \(q=0.20\), \(p=0.80\), and \(pv=0.80(0.80)=0.64\). The whole-life insurance value is a geometric series: the first possible death payment has present value \(qv\), and each additional year introduces another factor \(pv\). The annuity-due value is also geometric:
The first-year benefit has expected present value \(0.10(100)(0.80)=\$8\). If the insured survives the first year, the time-0 discount and survival factor is \(0.90(0.80)=0.72\). The remaining benefits therefore have expected present value \(0.72(100)(4/9)=\$32\). Total benefit value is \(\$8+\$32=\$40\).
The premium due at issue has present value \(P\). The expected present value of later premiums is \(0.72P(25/9)=2P\). Thus, the total premium value is \(3P\), and equivalence gives \(3P=40\), so \(P=\$40/3=\$13.333\ldots\).
At time 1, before the second premium, the mortality assumption is the constant tail assumption. The benefit value is \(100(4/9)=\$44.444\ldots\), and the future premium value is \((40/3)(25/9)=\$37.037\ldots\). Hence:
Check the recursion from issue: \((0+40/3)(1.25)=\$16.666\ldots\). Its right side is \(0.10(100)+0.90(200/27)=\$16.666\ldots\). The two sides agree. Rounded to the nearest cent, the premium is \(\$13.33\) and the reserve is \(\$7.41\).
Worked Example: Two-Year Term Insurance and the Expiry Boundary
Worked Example: Two-Year Term Insurance and the Expiry Boundary
A hypothetical two-year term policy pays \(B=\$1{,}000\) at the end of the year of death. Premiums are due at times 0 and 1 while the insured is alive. Assume \(i=25\%\), \(q_x=0.20\), and \(q_{x+1}=0.25\). Find the level net premium and the reserve immediately before the premium at time 1.
Method. First calculate the issue equivalence premium using the unconditional probabilities of claims and premiums. Then value the remaining one-year contract prospectively and verify both annual recursions. The unconditional probability of dying in year 2 is \(0.80(0.25)\), not \(0.25\).
Here \(v=0.80\), and the probability of surviving year 1 is \(p_x=0.80\). The benefit expected present value is:
The expected present value of premiums per dollar of premium is \(1+0.80(0.80)=1.64\). Therefore, \(P=288/1.64=7200/41=\$175.609756\ldots\).
At time 1, before the final premium, the only remaining benefit is a payment at the end of the next year if death occurs during it. Its value is \(1000(0.25)(0.80)=\$200\). The premium due at time 1 is \(P\), so:
For the first year, the recursion gives \((0+P)(1.25)=\$219.512195\ldots\). The other side is \(0.20(1000)+0.80(1000/41)=\$219.512195\ldots\). In the final year, the reserve after the covered term is \({}_2V_x=0\), so \(({}_1V_x+P)(1.25)=200(1.25)=\$250\), matching \(0.25(1000)+0.75(0)=\$250\). Rounded to the nearest cent, \(P=\$175.61\) and \({}_1V_x=\$24.39\).
Worked Example: A Negative Whole-Life Net Premium Reserve
Worked Example: A Negative Whole-Life Net Premium Reserve
A hypothetical whole-life policy pays \(B=\$1{,}150\) at the end of the year of death. Assume \(i=25\%\), with \(q_x=0.20\) in the first year and \(q=0.10\) in every later year. Find the level net premium and the reserve at time 1, immediately before the next premium.
Method. Price the initial year separately from the constant-mortality tail. Then calculate the time-1 prospective reserve on the tail assumptions. This makes clear why the reserve can be negative after the initially high-mortality year.
For the tail, \(p=0.90\), \(v=0.80\), and \(pv=0.72\). Thus \(A_{\text{tail}}=0.10(0.80)/(1-0.72)=2/7\), and \(\ddot{a}_{\text{tail}}=1/(1-0.72)=25/7\). At issue, the first-year benefit value is \(1150(0.20)(0.80)=\$184\). The later benefit value is \(0.80(0.80)(1150)(2/7)=\$210.285714\ldots\). Total benefit value is \(\$394.285714\ldots\).
The premium value per dollar of premium is \(1+0.80(0.80)(25/7)=23/7\). Therefore \(P=(394.285714\ldots)/(23/7)=\$120\). At time 1, the benefit value on the tail basis is \(1150(2/7)=\$328.571429\ldots\), while the value of future premiums, including the one due immediately, is \(120(25/7)=\$428.571429\ldots\). Hence \({}_1V_x=-\$100\).
The recursion checks: \((0+120)(1.25)=\$150\), and \(0.20(1150)+0.80(-100)=\$150\). The negative reserve is not an arithmetic error: conditional on surviving the higher-mortality first year, the future level premiums have greater prospective value than the remaining benefits on this basis.
Common Timing Errors and Practical Checks
A reserve recursion is only as reliable as its timing convention. Here \({}_kV_x\) is immediately before a premium due at time \(k\). If a problem instead defines reserves immediately after premium payment, the values and recursion must be adjusted consistently; the before-premium formula above cannot simply be reused with a different reserve definition.
A frequent mistake is to omit the premium at time \(k\) from the reserve definition, even though it is due immediately. For the two-year term example, this would incorrectly report the time-1 reserve as \(\$200\), the value of the remaining benefit alone. The correct before-premium reserve is \(\$200-\$175.61=\$24.39\), because the premium due at time 1 is part of future premiums.
- Use conditional mortality for the year being valued, and multiply by earlier survival probabilities when calculating issue values.
- Discount an end-of-year death benefit for a full year from the beginning of that year.
- Include each premium only when it is due and the insured is alive.
- For a term contract, set the terminal reserve from the contract’s post-expiry obligations; it is zero here because no benefits or premiums remain.
- Check a computed reserve by substituting it into the recursion. A negative net premium reserve can be valid when the remaining expected premiums exceed the remaining expected benefits.
Check Your Understanding
For each problem, use annual premiums payable at the beginning of each covered year while alive, end-of-year death benefits, no expenses, and the stated interest and mortality assumptions.
- A one-year term policy pays \(\$2{,}000\) at the end of the year of death. If \(i=5\%\) and \(q_x=0.10\), find its net premium. (Answer: \(\$190.48\))
- A two-year term policy pays \(\$500\), with \(i=10\%\), \(q_x=0.10\), and \(q_{x+1}=0.20\). Find the level net premium. (Answer: \(\$65.91\))
- For the policy in Question 2, find the reserve at time 1 immediately before the second premium. (Answer: \(\$25.00\))
- A whole-life policy pays \(\$1{,}000\), with \(i=5\%\) and a constant annual death probability \(q=0.10\) at every duration. Find the level net premium. (Answer: \(\$95.24\))
- A three-year term policy pays \(\$1{,}000\). Assume \(i=25\%\), with annual death probabilities \(0.10\), \(0.20\), and \(0.30\) in successive years. Find the level net premium and the reserve at time 2 immediately before the third premium. (Answer: premium \(\$140.22\); reserve \(\$99.78\))