Why Mixed Reserve Problems Need a Cash-Flow Map
A policy reserve may combine a death benefit, a payment made if the insured survives to an anniversary, and premiums that change over time. The central task is to value the cash flows still possible after the valuation date, conditional on the insured being alive at that date. A reliable solution separates each benefit and premium before discounting; combining them too early makes it easy to omit a survival condition or use the wrong probability.
The examples here use annual, discrete cash flows and an effective annual interest rate. Unless stated otherwise, benefits and premiums are paid at the end of a policy year. The hypothetical contracts have no expenses, and the reserve is calculated as the present value of future benefits less the present value of future premiums. The mortality probabilities are specified for each remaining year, so no interpolation or additional mortality model is needed.
Building the Prospective Formula
Suppose a policy is valued just after any premium due at time \(t\), with \(n\) years remaining. In remaining year \(k\), let \(D_k\) be the benefit paid at the end of that year if death occurs during the year, and let \(R_k\) be the benefit paid at the end of that year if the insured survives it. Let \(G_k\) be a premium due at that year-end if the insured is alive then. These amounts may vary by year; a zero amount means that component is not payable in that year.
Write \({}_kp_{x+t}\) for the probability of surviving from the valuation date to the end of remaining year \(k\). The conditional death and survival probabilities during that year are \({}_1q_{x+t+k-1}\) and \({}_1p_{x+t+k-1}\), respectively. Because the insured must first survive to the start of year \(k\), the expected present value of its benefit is weighted by \({}_{k-1}p_{x+t}\). The premium \(G_k\), by contrast, is payable only if the insured survives all the way to the end of year \(k\).
Here \(v=1/(1+i)\), where \(i\) is the effective annual interest rate. This form allows the death benefit and the survival benefit to differ by year. If the policy has several distinct benefit components, add their amounts in the appropriate death or survival outcome before applying the formula, or value each component separately and add the results.
Record how many years remain and whether a premium due at the valuation time has already been paid.
For each year, record the death benefit, the survival benefit, and any premium payable if alive at year-end.
A death benefit requires survival to the start of its year followed by death during that year. A year-end premium requires survival through that year.
Discount each expected cash flow to the valuation time, add the benefits, and subtract the premiums.
Why the Outcome Probabilities Appear
For a particular remaining year \(k\), the insured must survive the preceding \(k-1\) years before either end-of-year outcome is possible. Conditional on reaching that year's start, death and survival during the year are mutually exclusive and exhaustive outcomes. Thus the conditional expected benefit for that year is \({}_1qD_k+{}_1pR_k\). Multiplying by the probability of reaching the year and discounting \(k\) years gives its expected present value:
For a premium due at the year-end, payment occurs only on the survival branch. The probability of being alive to pay it is \({}_kp_{x+t}\), which explains the premium term \(v^k\,{}_kp_{x+t}G_k\). This distinction is important: the death-benefit term uses the probability of reaching the start of the year and then dying, while the premium term uses the probability of surviving the entire year.
Worked Mixed-Policy Problems
Worked Example: A Premium Due Immediately
A hypothetical two-year policy is valued at time 0 immediately before a premium of $90 is due. At each year-end it pays $400 if death occurred during that year, or $40 if the insured survives to that year-end. The conditional death probabilities are 0.10 in year 1 and 0.25 in year 2. The effective annual interest rate is 25%. There are no other premiums. Find the reserve immediately before the time-0 premium.
Method. First calculate the present value of the benefits, conditional on being alive at time 0. Since the year-2 death probability is conditional on reaching year 2, the probability of surviving year 1 must also be included in the year-2 term. Then subtract the premium payable immediately at time 0.
The discount factor is \(v=1/1.25=0.80\). In year 1, the expected end-of-year benefit, conditional on being alive at time 0, is:
Its present value is \(0.80(76)=\$60.80\). The probability of surviving year 1 is \(0.90\). Conditional on reaching year 2, the expected year-2 benefit is:
The time-0 present value of that benefit is \(0.80^2(0.90)(130)=0.64(0.90)(130)=\$74.88\). Total benefit value is \(\$60.80+\$74.88=\$135.68\). The $90 premium is due immediately, so it is not discounted:
Check: The benefits exceed the premium by \(\$45.68\), as required by the prospective definition. Common mistake avoided: Do not discount the immediate premium or forget the 0.90 probability of reaching year 2. Either error changes the reserve.
Worked Example: Nonlevel Premiums at an Intermediate Duration
A hypothetical three-year policy is valued at time 1, immediately after any premium then due. There are two years remaining. In the first remaining year, the death probability is 0.20; in the second, it is 0.10, conditional on reaching that year. At the end of each remaining year, the policy pays $700 on death during that year or $60 on survival to that year-end. Premiums of $50 and $40 are due at times 2 and 3, respectively, if the insured is alive then. The effective annual interest rate is 25%. Find the time-1 reserve.
Method. Value only the cash flows after time 1. For each year, calculate its expected benefit conditional on being alive at the start of that year. Weight later-year amounts by the probability of surviving to that year. Weight each premium by the probability of being alive when it is due.
Again \(v=0.80\). For the first remaining year, the expected benefit at time 2, conditional on being alive at time 1, is:
Its time-1 present value is \(0.80(188)=\$150.40\). The probability of surviving this year is \(0.80\). In the second remaining year, the expected benefit conditional on being alive at time 2 is:
The probability of being alive at time 2 is \(0.80\), so the time-1 present value of this benefit is \(0.80^2(0.80)(124)=0.64(0.80)(124)=\$63.488\).
The $50 premium at time 2 is payable if alive then. Its present value is \(0.80(0.80)(50)=\$32.00\). The $40 premium at time 3 is payable only if the insured survives both remaining years. Its present value is \(0.80^2(0.80)(0.90)(40)=0.64(0.80)(0.90)(40)=\$18.432\).
The benefit present value is \(\$150.40+\$63.488=\$213.888\). The premium present value is \(\$32.00+\$18.432=\$50.432\). Therefore:
rounded to the nearest cent. Check: The second-year benefit and premium both require survival through the first remaining year, but their year-end conditions differ: the benefit may be paid on death in year 2, while the time-3 premium requires survival through year 2. Common mistake avoided: Using the time-0 horizon or discounting from time 0 would value the wrong reserve date.
Worked Example: Death Cover, Survival Payments, and Two Premiums
A hypothetical three-year contract is valued at time 0 just after any premium due then. At each year-end it pays $500 if death occurred during that year, or $100 if the insured survives to that year-end. The conditional death probabilities in years 1, 2, and 3 are 0.20, 0.10, and 0.25, respectively. Premiums of $60 at time 1 and $80 at time 2 are payable if the insured is alive at those times; no premium is due at time 3. The effective annual interest rate is 25%. Find the reserve.
Method. Separate the three benefit years, then value the two premiums using their own survival conditions. Here \(v=0.80\). In year 1, the expected benefit is:
The probability of surviving year 1 is \(0.80\). Conditional on reaching year 2, its expected benefit is:
Thus the time-0 present value for year 2 is \(0.80^2(0.80)(140)=0.64(0.80)(140)=\$71.68\). The probability of surviving the first two years is \(0.80(0.90)=0.72\). Conditional on reaching year 3, the expected benefit is:
Its time-0 present value is \(0.80^3(0.72)(200)=0.512(0.72)(200)=\$73.728\). The total benefit present value is \(\$144.00+\$71.68+\$73.728=\$289.408\).
The time-1 premium is payable with probability \(0.80\), so its present value is \(0.80(0.80)(60)=\$38.40\). The time-2 premium is payable with probability \(0.72\), so its present value is \(0.80^2(0.72)(80)=0.64(0.72)(80)=\$36.864\). Total premium value is \(\$75.264\). Therefore:
rounded to the nearest cent. Check: The time-2 premium uses \(0.72\), the probability of survival through both years, not merely the probability of survival through year 1. Common mistake avoided: Treating a premium due at an anniversary as payable after death during the preceding year would incorrectly include it for policies that did not survive to that anniversary.
Timing Checks and Common Traps
A reserve is tied to a precise instant. At a premium date, “immediately before” and “immediately after” are different valuations: the before-premium reserve includes the premium due at that instant, while the after-premium reserve does not. If it is due for certain given that the insured is alive at the valuation date, that immediate premium is subtracted without a discount factor. By contrast, a premium due at a future anniversary is discounted and weighted by the probability the insured is alive when it falls due.
The same timing discipline applies to benefits. Under the end-of-year convention used here, death during year \(k\) produces a payment at the end of that year. A survival benefit is paid at that same year-end only on survival. Do not treat a survival payment as certain merely because the insured is alive at the valuation date. Similarly, later cash flows must be weighted by survival to the time they can occur, not simply by their probability conditional on reaching the relevant year.
- Wrong: Weight a year-2 cash flow only by the year-2 mortality probability. Correction: Also include survival to the start of year 2.
- Wrong: Weight a year-end premium by the probability of reaching the start of that year. Correction: The premium is payable only if the insured survives to its due date.
- Wrong: Include a premium already paid in an after-premium reserve. Correction: Use only contractual cash flows strictly after the valuation instant.
- Wrong: Use one premium amount for every remaining year when premiums are nonlevel. Correction: list and value each scheduled premium separately.
Check Your Understanding
For each problem, use the stated annual timing and calculate the reserve at the specified instant. Round monetary answers to the nearest cent.
- A two-year policy is valued at time 0 just after any premium due then. The interest rate is 25%, with death probabilities 0.20 in year 1 and 0.10 in year 2. It pays $600 on death during a year or $100 on survival to that year-end. No future premiums are due. Find the reserve. (Answer: $236.80)
- A one-year policy is valued immediately before a $50 premium due at time 0. At the end of the year it pays $500 on death or $20 on survival. The death probability is 0.10, and the interest rate is 25%. Find the reserve. (Answer: $4.40)
- A policy is valued at time 1 just after any premium due then, with two years remaining. The interest rate is 25%; death probabilities in the remaining years are 0.25 and 0.20. It pays $800 on death or $50 on survival at each year-end. Premiums of $30 and $40 are due at times 2 and 3 if alive. Find the reserve. (Answer: $252.64)
- A three-year policy is valued at time 0 just after any premium due then. The interest rate is 25%, and the conditional death probabilities are 0.10, 0.20, and 0.25. It pays $1,000 on death during each year and no survival benefit. Premiums of $100 at time 1 and $80 at time 2 are payable if alive; no premium is due at time 3. Find the reserve. (Answer: $178.50)
- A two-year policy is valued at time 0 just after any premium due then. The interest rate is 25%, and the death probabilities are 0.20 in year 1 and 0.25 in year 2. At each year-end it pays $300 on death or $60 on survival. Premiums of $40 at time 1 and $20 at time 2 are payable if alive. Find the reserve. (Answer: $114.56)