How to Value a Deferred Life Contract
A deferred life contract has a period during which some or all benefits are not payable. Its value depends not just on the amount of each payment but also on when it can be paid and whether the insured must survive to that time. In a contract with both deferred insurance and an annuity component, the death benefit and the survival-contingent payments must be valued separately before they are combined.
The calculations below use a discrete annual model: interest is at an effective annual rate \(i\); death benefits, when payable, are paid at the end of the year of death; and annuity payments are paid at the end of each stated year if the insured is then alive. Premiums are payable at the beginning of each stated year while the insured is alive. These timing assumptions matter: changing a payment date changes its discount factor and may change whether survival is required.
Write \(v=1/(1+i)\). For a payment \(B\) at the end of year \(k\), its value at issue is \(Bv^k\) times the probability that the payment is made. If the payment requires survival to time \(k\), use the probability of survival to that time; if it is a death benefit, use the probability of death in the covered year. When valuing from a later time, use probabilities conditional on survival to that valuation time and discount by the number of years remaining.
Separating Benefits, Premiums, and Timing
For a deferred contract, the main practical step is to make a timeline before calculating. Mark the deferral period, the dates at which benefits can begin, and every premium date. Then identify the probability attached to each cash flow. For example, a premium due at time 2 is paid only if the insured is alive at time 2, while an annuity installment due then also requires survival to time 2.
For each benefit, state its amount, payment time, and whether it requires death in a covered year or survival to the payment date.
Multiply the amount by its payment probability and the appropriate discount factor, then add the values.
Include only premiums payable after the valuation time, and weight each by the probability the insured is alive when it is due.
Use equivalence for a level net premium, or subtract the value of future premiums from the value of future benefits for a prospective reserve.
The equivalence principle sets the expected present value of premiums equal to the expected present value of benefits. It determines a net premium under the stated assumptions; expenses, profit margins, or other loadings are not included unless explicitly added.
Worked Example: Level Premium and Reserves for a Deferred Contract
A hypothetical contract has a two-year deferral. The death benefit is $50,000 at the end of year 3 or year 4 if death occurs in that year. It also pays $5,000 at the end of years 3 and 4 if the insured is alive. Premiums are payable at times 0, 1, and 2 if the insured is alive. The annual conditional death probabilities for years 1 through 4 are \(0.10, 0.20, 0.25, 0.40\), respectively, and \(i=5\%\). Find the level net premium and the prospective reserves at times 1 and 2, each immediately after any premium due at that time.
Method. First find survival and death probabilities from issue. The probabilities of surviving to times 1 through 4 are \(0.90, 0.72, 0.54, 0.324\). Thus the probabilities of death in years 3 and 4 are \(0.72(0.25)=0.18\) and \(0.54(0.40)=0.216\). Use \(v=1/1.05=0.9523809524\), with \(v^2=0.9070294785\), \(v^3=0.8638375985\), and \(v^4=0.8227024743\).
Death-benefit value at issue. The year-3 and year-4 payments have expected present values
Annuity-benefit value at issue. Survival probabilities to the payment dates are \(0.54\) and \(0.324\), so the value is
Total benefit value is \(\$16{,}659.7252+\$3{,}665.1395=\$20{,}324.8647\). The premium factor accounts for premiums at times 0, 1, and 2, conditional on survival to each premium date:
By equivalence, the unrounded premium is \(\$20{,}324.8647/2.510204082=\$8{,}096.8973\), or $8,096.90 per year to the nearest cent. As a check, multiplying the unrounded premium by the unrounded premium factor recovers the unrounded benefit value.
Reserve at time 2. The insured has survived the deferral period. This reserve is immediately after the time-2 premium, so there are no future premiums. Conditional on being alive at time 2, death in year 3 has probability \(0.25\), and death in year 4 has probability \(0.75(0.40)=0.30\). Survival to times 3 and 4, conditional on survival to time 2, is \(0.75\) and \(0.75(0.60)=0.45\). Therefore
Reserve at time 1. Conditional on survival to time 1, the covered death probabilities are \(0.80(0.25)=0.20\) in year 3 and \(0.80(0.75)(0.40)=0.24\) in year 4. Survival to the annuity dates is \(0.60\) and \(0.36\). The value of future benefits at time 1 is
Only the time-2 premium remains. Its value at time 1 is \(0.80vP=0.761904762P=\$6{,}169.06\), using the unrounded premium. Thus \({}_1V_x=\$23{,}712.3421-\$6{,}169.0646=\mathbf{\$17{,}543.28}\). As an independent check, one-year recursion gives \(v(0.80)({}_2V_x-P)=0.761904762(\$31{,}122.4490-\$8{,}096.8973)=\$17{,}543.28\), with unrounded inputs.
The method avoids a common timing error: subtracting a premium at time 2 from the time-2 reserve when that reserve is defined immediately after the premium. The premium is already paid at that instant; it is included as a future outflow when valuing at time 1, not as a future premium when valuing immediately after time 2.
Why Deferral Produces a Survival-and-Discount Factor
A deferred benefit can also be valued in two stages: first value it at the end of the deferral period, conditional on being alive then; next bring that value back to the earlier valuation time. For a deferral of \(n\) years, the earlier value equals the conditional value at time \(n\), multiplied by the probability of surviving to time \(n\) and by \(v^n\). This follows by conditioning on survival: if the insured does not survive the deferral period, the deferred life benefit has no value; if the insured does survive, its conditional value is discounted \(n\) years.
The identity applies when the later value is defined using the same benefit terms, interest assumption, and conditional survival model. It does not mean that every deferred payment has the same survival probability: each later payment still needs its own conditional probability of being paid.
Worked Example: Deferred Annuity Valued at Issue and at the Deferral Date
A hypothetical life annuity pays $4,000 at the ends of years 3, 4, and 5 if the insured is alive. It has no payments during the first two years. The probabilities of survival from issue to times 1 through 5 are \(0.80, 0.60, 0.30, 0.15, 0.06\), and \(i=5\%\). Find its single premium at issue and its value at time 2 conditional on survival to that time.
Direct valuation at issue. The three payments are made with probabilities \(0.30, 0.15,\) and \(0.06\), respectively. With \(v^3=0.863837599\), \(v^4=0.822702474\), and \(v^5=0.783526166\), the value is
Conditional valuation at time 2. Given survival to time 2, survival probabilities to times 3, 4, and 5 are \(0.30/0.60=0.50\), \(0.15/0.60=0.25\), and \(0.06/0.60=0.10\). The conditional value at time 2 is
Bring this value to issue by multiplying by the probability of reaching time 2 and discounting two years: \(0.60v^2(\$3{,}157.3264)=\$1{,}718.27\). The two methods agree to the nearest cent, checking both the survival conditioning and the payment timing.
Premiums and Reserves Through the Deferral Period
A premium paid during deferral can fund benefits that begin later, even though no benefit is payable during the early years. The premium factor must reflect the policy’s actual payment pattern. For a premium at time \(j\), include the probability that the insured is alive at time \(j\); do not treat a life-contingent premium stream as certain.
Reserves also depend on the valuation instant. A reserve immediately before a premium includes that premium among future premiums; a reserve immediately after it does not. In a year with no benefit, the reserve can still change because interest is earned, the probability of surviving to future benefits changes, and a premium may become due. State the instant clearly before applying a reserve formula.
Worked Example: Checking a Deferred Contract Reserve Recursively
Consider a second hypothetical contract at zero interest. Conditional death probabilities in years 1 through 4 are \(0.10, 0.20, 0.25, 0.40\). It pays $20,000 for death in year 3 or 4 and $1,000 at the ends of years 3 and 4 if alive. Premiums are payable at times 0, 1, and 2 while alive. Calculate the level net premium and the reserves at times 1 and 2 immediately after any premium due then.
The survival probabilities to times 1 through 4 are \(0.90,0.72,0.54,0.324\). Death probabilities in years 3 and 4 are \(0.72(0.25)=0.18\) and \(0.54(0.40)=0.216\). At zero interest, the death benefits are worth \(20{,}000(0.18+0.216)=\$7{,}920\), and the annuity payments are worth \(1{,}000(0.54+0.324)=\$864\). Total benefit value is \(\$8{,}784\). The premium factor is \(1+0.90+0.72=2.62\), so \(P=\$8{,}784/2.62=\$3{,}352.6718\), or $3,352.67 to the nearest cent. Multiplication by 2.62 verifies the unrounded benefit value.
At time 2, conditional death probabilities for the covered years are \(0.25\) and \(0.75(0.40)=0.30\); survival probabilities to the annuity dates are \(0.75\) and \(0.45\). There are no premiums after time 2, so \({}_2V_x=20{,}000(0.25+0.30)+1{,}000(0.75+0.45)=\$12{,}200\).
At time 1, conditional death probabilities for years 3 and 4 are \(0.20\) and \(0.24\); survival probabilities for the two annuity payments are \(0.60\) and \(0.36\). Future benefits are worth \(20{,}000(0.20+0.24)+1{,}000(0.60+0.36)=\$9{,}760\). The time-2 premium is payable only if the insured survives year 2, so its expected value at time 1 is \(0.80P=\$2{,}682.14\). Hence \({}_1V_x=\$9{,}760-\$2{,}682.14=\mathbf{\$7{,}077.86}\). Recursion checks the result: \(0.80({}_2V_x-P)=0.80(\$12{,}200-\$3{,}352.6718)=\$7{,}077.86\).
Common Exam Traps
- Using the wrong deferral boundary. If coverage begins after two complete years, a death in year 2 is not automatically covered. Follow the exact contract wording and payment time.
- Using survival probability for a death benefit. A death claim in year \(k\) uses the probability of death in that covered year, not simply the probability of surviving to year \(k\).
- Omitting survival weighting for annuity payments. A payment at time \(k\) is contingent on being alive at time \(k\), even when earlier installments were payable.
- Misplacing premiums at the valuation date. Decide whether the reserve is before or after a premium due at that time before including it in future premiums.
- Discounting from the wrong date. When valuing at time \(t\), discount a payment at time \(k\) by \(v^{k-t}\), not \(v^k\).
These methods are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus. The central discipline is to write each cash flow with its date and contingency before combining values; this makes the premium calculation and reserve checks consistent.
Check Your Understanding
For each problem, identify the relevant payment dates and contingencies before calculating.
- At zero interest, a hypothetical contract pays $1,000 at the end of year 3 if the insured is alive. The probability of survival to time 2 is \(0.80\), and the probability of survival to time 3 is \(0.60\). Find the single premium. (Answer: $600.00)
- A deferred annuity pays $2,000 at the ends of years 2 and 3 if alive. Survival probabilities from issue to those dates are \(0.80\) and \(0.60\), and \(i=5\%\). Find its value at issue. (Answer: $2,487.85)
- At zero interest, a contract pays $10,000 only for death in year 3. Conditional death probabilities in years 1, 2, and 3 are \(0.10,0.20,0.25\). Premiums are due at times 0, 1, and 2 if alive. Find the level net premium. (Answer: $687.02 per year)
- Immediately after the deferral date, a contract pays $20,000 for death in the next year or the following year, and $1,000 at the end of each of those years if alive. At zero interest, the conditional death probabilities in those years are \(0.20\) and \(0.50\). Find the prospective reserve. (Answer: $13,200.00)