Why Whole-Life Reserve Recursions Matter
A whole-life policy has no fixed expiry date: premiums continue while the insured is alive, and the death benefit is payable whenever death occurs. A reserve recursion turns that long stream of possible premiums and benefits into a one-year balance equation. It is useful both for calculating a reserve at a selected duration and for checking a prospective valuation.
The recursion depends on timing. In this tutorial, premiums are annual and payable at the beginning of each policy year while the insured is alive. The level death benefit is paid at the end of the year of death. Reserves are measured immediately before the premium due at the start of the year. Keeping this convention explicit prevents a common one-premium timing error.
The reserve convention here includes the premium due immediately in the prospective value. Thus, under the equivalence principle, \(V_0=0\): at issue, the present value of future benefits equals the present value of premiums. A reserve measured immediately after the premium has been paid differs by \(P\); it is \(V_k+P\), provided the same policy and valuation assumptions apply.
The Annual Whole-Life Recursion
For a net reserve, expenses are ignored and the valuation basis uses the same benefit, premium, interest, and mortality assumptions as the policy calculation. Over the year after time \(k\), the premium \(P\) is received at time \(k\), and the resulting funds accumulate to time \(k+1\). Death during that year requires payment of \(B\) at year-end. If the insured survives, the funds must support the reserve immediately before the next premium.
The condition \({}_1p_{x+k}>0\) is needed to solve for \(V_{k+1}\) by division. If survival to the next duration has probability zero, the conditional reserve for a survivor at that duration is not determined by this equation.
Why the Recursion Holds
The equation follows by conditioning on the two mutually exclusive outcomes over the year. The funds available at time \(k\) are the reserve \(V_k\) plus the premium \(P\). Accumulating for one year gives \((V_k+P)(1+i)\). At time \(k+1\), the death branch has probability \({}_1q_{x+k}\) and requires benefit \(B\); the survival branch has probability \({}_1p_{x+k}\) and requires reserve \(V_{k+1}\). Equating the accumulated funds to the probability-weighted obligations gives the recursion.
This is an expected-value balance on the valuation basis, not a claim that the same amount of money is available in every individual outcome. The death benefit is paid in the death branch; the reserve for the survivor is needed only in the survival branch. That is why the two obligations are weighted by different probabilities.
Identify whether the reserve is measured before or after the premium due at time \(k\), and when the death benefit is paid.
Accumulate the time-\(k\) reserve plus any premium received, then equate it to the death-branch benefit and survival-branch reserve.
Multiply \(V_{k+1}\) by \({}_1p_{x+k}\), then divide by that probability when solving forward.
Substitute the calculated reserve back into the original equation; both sides must agree.
For a level net premium determined by the equivalence principle, the prospective reserve at duration \(k\), immediately before the premium then due, can also be written
Here \(A_{x+k}\) is the present value of a unit whole-life benefit paid at the end of the year of death, and \(\ddot{a}_{x+k}\) is the present value of unit annual premiums payable in advance while alive. This prospective expression and the recursion are alternative ways to obtain a reserve under the same assumptions; agreement between them is a useful check.
Worked Example: Calculate the Premium and the First Reserve
A hypothetical whole-life policy pays \(B=\$15{,}000\) at the end of the year of death. The effective annual interest rate is \(i=25\%\). The first-year death probability is \(0.10\); at every later duration it is \(0.20\). Premiums are annual, payable in advance while alive. Find the net level premium and the reserve at time 1 immediately before the premium due then.
Method: prospective values followed by the recursion. At durations 1 and later, mortality is constant, so \({}_1p=0.80\) and \(v=1/1.25=0.80\). The whole-life annuity-due factor and insurance factor at those durations are
At issue, the first-year death probability is \(0.10\), with survival probability \(0.90\). Therefore
The equivalence premium is
At time 1, the prospective reserve before the premium is
Check by recursion from \(V_0=0\): the left side is \((0+2{,}000)(1.25)=\$2{,}500\). The right side is \((0.10)(15{,}000)+(0.90)(1{,}111.11\ldots)=\$1{,}500+\$1{,}000=\$2{,}500\). The balance holds. The reserve is rounded to the nearest cent only at the final step.
Reserves Can Be Negative
A net reserve is a present-value difference, not necessarily a nonnegative account balance. A negative value means that, on the stated valuation basis and at the stated timing, the present value of future net premiums exceeds the present value of future benefits. This can occur, for example, when the premium was set using higher mortality at earlier durations but the remaining mortality is lower.
Worked Example: A Negative Reserve After Mortality Changes
A hypothetical whole-life policy pays \(B=\$11{,}500\). The effective annual interest rate is \(25\%\). The first-year death probability is \(0.20\), and at all later durations it is \(0.10\). Annual premiums are payable in advance while alive. Find the net level premium and the reserves immediately before the premiums at times 1 and 2.
Method: value the constant-mortality tail, then step back to issue. For durations 1 and later, \({}_1p=0.90\), \(v=0.80\), and \({}_1q=0.10\). Thus
At issue, \({}_1q_x=0.20\) and \({}_1p_x=0.80\), so
Therefore
The time-1 reserve is
At time 2, the same mortality and interest assumptions still apply, so the same prospective calculation gives \(V_2=-\$1{,}000\). The recursion confirms both values. From time 0, \((0+1{,}200)(1.25)=\$1{,}500\), while \((0.20)(11{,}500)+(0.80)(-1{,}000)=\$1{,}500\). From time 1, \((-1{,}000+1{,}200)(1.25)=\$250\), while \((0.10)(11{,}500)+(0.90)(-1{,}000)=\$250\). Both equations balance exactly.
The negative reserve is not an arithmetic failure: after the first year, the remaining mortality is lower, while the level premium was set using the higher first-year mortality as well.
Forward Calculation Across Multiple Durations
When mortality changes by duration, a single constant-mortality formula may not describe the whole contract. The recursion can still be applied year by year, using the death and survival probabilities for the current duration. The next example checks two consecutive balances.
Worked Example: Step Through Two Different Mortality Rates
A hypothetical whole-life policy pays \(B=\$36{,}500\). The effective annual interest rate is \(25\%\). The death probabilities are \(0.20\) in the first year, \(0.10\) in the second year, and \(0.20\) at every later duration. Annual premiums are payable in advance while alive. Find the net level premium, then calculate \(V_1\) and \(V_2\), each measured immediately before the premium due.
Method: value the tail and work back to the issue premium. From duration 2 onward, the mortality is constant at \(0.20\). With \(v=0.80\),
At duration 1, the death probability is \(0.10\), so
At issue, the death probability is \(0.20\). Hence
The net level premium is
Now calculate the prospective reserves:
Check the first recursion: \((0+5{,}200)(1.25)=\$6{,}500\), and \((0.20)(36{,}500)+(0.80)(-1{,}000)=\$6{,}500\). Check the second: \((-1{,}000+5{,}200)(1.25)=\$5{,}250\), and \((0.10)(36{,}500)+(0.90)(16{,}000/9)=\$3{,}650+\$1{,}600=\$5{,}250\). The reserves are rounded to the nearest cent only at the final step.
Common Exam Trap: Forgetting the Survival Probability
When solving the recursion for \(V_{k+1}\), the reserve at the next duration is conditional on survival to that duration. Its contribution to the balance is therefore \({}_1p_{x+k}V_{k+1}\), not \(V_{k+1}\) alone.
For the first worked example, a mistaken calculation might subtract the expected death benefit but omit division by the survival probability:
This is not \(V_1\); it is the probability-weighted survival-branch reserve \({}_1p_xV_1\). The correct calculation divides by \(0.90\):
A separate timing error is to add \(P\) to the reserve when the premium has already been included in the stated reserve convention. Here \(V_k\) is immediately before the premium, so the funds accumulated for the year are \(V_k+P\). If a problem instead states a reserve immediately after the premium, its amount is already \(V_k+P\); adding another premium would count it twice.
Check Your Understanding
For each problem, use reserves immediately before the premium due at the stated time. Round money to the nearest cent.
- A hypothetical whole-life policy has \(V_k=\$600\), \(P=\$900\), \(B=\$20{,}000\), \(i=4\%\), and \({}_1q_{x+k}=0.03\). Calculate \(V_{k+1}\). (Answer: \(\$989.69\))
- A policy has \(V_k=-\$300\), \(P=\$1{,}500\), \(B=\$12{,}000\), \(i=5\%\), and \({}_1q_{x+k}=0.12\). Calculate \(V_{k+1}\). (Answer: \(-\$204.55\))
- For a hypothetical whole-life policy, \(i=25\%\), \(B=\$15{,}000\), and the first-year death probability is \(0.10\), followed by \(0.20\) at every later duration. Find the net level premium and \(V_1\). (Answer: \(P=\$2{,}000,\ V_1=\$1{,}111.11\))
- A hypothetical whole-life policy has \(i=25\%\), \(B=\$11{,}500\), first-year death probability \(0.20\), and death probability \(0.10\) at every later duration. Find the net level premium and the reserve immediately before the time-1 premium. (Answer: \(P=\$1{,}200,\ V_1=-\$1{,}000\))
- At duration \(k\), a hypothetical whole-life policy has \(V_{k+1}=\$500\), \(P=\$1{,}200\), \(B=\$10{,}000\), \(i=5\%\), and \({}_1q_{x+k}=0.20\). Solve backward for \(V_k\). (Answer: \(V_k=\$1{,}085.71\))