Present Values of Premium Streams

Why Premium Timing Matters

A level premium is a fixed amount at each scheduled payment date, but its expected present value depends on more than its amount. Each installment must be discounted to the valuation date, and a payment that is due only if the insured is alive must also be weighted by the probability that the insured survives to that date. A fixed installment due regardless of survival has no such survival adjustment.

The distinction is especially important when comparing contracts with premiums due in advance, premiums due in arrears, and fixed payment schedules. The amount $P$ may be the same in each case, but the dates and conditions of payment determine the present value.

Definition: The present value of a premium stream is the sum of the premiums payable, each multiplied by its discount factor and, when payment depends on survival, by the probability that the insured is alive at the payment date. The expected present value (EPV) is the expectation of this discounted, possibly random, sum.

Premiums Payable While Alive

Assume an insured aged $x$ is alive at time 0, the effective annual interest rate is $i$, and $v=1/(1+i)$. Suppose an annual premium $P$ is due at the beginning of each policy year, starting at time 0, and is payable only if the insured is alive on that date. For an $n$-year premium period, the potential payment dates are $0,1,\ldots,n-1$. The time-0 payment is certain because the insured is alive at issue; later payments depend on survival.

\operatorname{EPV}(\text{premiums}) = P\sum_{k=0}^{n-1} v^k\,{}_kp_x = P\ddot{a}_{x:\overline{n}|}.

Here ${}_kp_x$ is the probability that a life aged $x$ survives $k$ years, and the temporary life annuity-due factor $\ddot{a}_{x:\overline{n}|}$ is defined by the sum shown. For a whole-life stream of annual premiums due in advance, the corresponding factor is $\ddot{a}_x=\sum_{k=0}^{\infty}v^k{}_kp_x$, provided the series converges.

If installments are instead due at the end of each year and are payable only if the insured is alive then, the payment dates are $1,2,\ldots,n$. The EPV is $P\sum_{k=1}^{n}v^k{}_kp_x$, the temporary life annuity-immediate value $P a_{x:\overline{n}|}$. The payment at time $k$ is weighted by survival to time $k$, not merely by survival to the beginning of that policy year.

Theorem: For a deterministic premium $P$ due at time $k$ only if the insured is alive then, its EPV contribution is $Pv^k{}_kp_x$. If it is due regardless of survival, its EPV contribution is $Pv^k$. These statements assume the stated payment condition, a fixed effective annual interest rate, and valuation at time 0.

Why the Survival Factor Appears

The survival weighting follows directly from the definition of expectation. Let $I_k$ equal 1 if the insured is alive at time $k$, and 0 otherwise. The present value of a premium due at time $k$ only while alive is $Pv^k I_k$. Since $E[I_k]={}_kp_x$, the expected present value of that installment is $Pv^k{}_kp_x$. Adding the expected values of the installments gives the stream formula by linearity of expectation.

E\left[\sum_{k=0}^{n-1}Pv^k I_k\right] =\sum_{k=0}^{n-1}Pv^kE[I_k] =P\sum_{k=0}^{n-1}v^k{}_kp_x.

This derivation does not require the survival indicators at different times to be independent. It uses only each date’s survival probability and the fact that expectation is additive. For a payment guaranteed regardless of survival, the indicator is always 1, so its expected present value is simply $Pv^k$.

Worked Example: Annual Premiums Due in Advance

A hypothetical three-year policy requires a premium of $1,000 at the beginning of each policy year while the insured is alive. The insured is alive at issue. Assume an effective annual interest rate of 10% and a conditional probability of surviving each year of 0.90, the same for each of the three years. Find the EPV of the premiums.

Method. The payment dates are $0,1,2$. The time-0 premium is certain, and the probabilities of being alive at times 1 and 2 are $0.90$ and $0.90^2=0.81$. The discount factor is $v=1/1.10=10/11$.

The EPV factor is $1+0.90(10/11)+0.81(10/11)^2 =1+9/11+81/121 =301/121 =2.4876033$.

Therefore the EPV is $1{,}000(301/121)=\$2{,}487.6033$, or $2,487.60 to the nearest cent.

Check. The undiscounted expected number of payments is $1+0.90+0.81=2.71$, so an EPV below $2,710 is reasonable. Discounting the later payments gives $2,487.60. The method avoids treating the second and third installments as certain.

Installments Due Regardless of Survival

A schedule can instead require each installment regardless of whether the insured survives. This might represent a hypothetical fixed debt obligation payable by an estate, rather than a conventional life-contingent policy premium. The contractual condition must be read carefully: the word “premium” alone does not establish whether payments stop at death.

For $n$ guaranteed installments of $P$ at times $0,1,\ldots,n-1$, the EPV is the annuity-certain value $P\ddot{a}_{\overline{n}|}$. If the guaranteed installments are due at times $1,2,\ldots,n$, the value is $Pa_{\overline{n}|}$. Neither formula includes survival probabilities.

\ddot{a}_{\overline{n}|}=\sum_{k=0}^{n-1}v^k, \qquad a_{\overline{n}|}=\sum_{k=1}^{n}v^k.

These are annuity-certain factors: they describe the timing and discounting of payments, not whether a life survives. By contrast, life annuity factors include survival probabilities. The timing convention matters in both cases: an installment at time 0 is not discounted, whereas an installment at time 1 is multiplied by $v$.

Worked Example: End-of-Year Premiums While Alive

A hypothetical policy charges $600 at the end of each of the next three years, provided the insured is alive on the payment date. The insured is alive at issue. Assume an effective annual interest rate of 5% and a conditional probability of surviving each year of 0.80. Find the EPV.

Method. The payment dates are $1,2,3$, and the probabilities of being alive on those dates are $0.80$, $0.80^2=0.64$, and $0.80^3=0.512$. Here $v=1/1.05=20/21$.

The three discounted survival-weighted factors are $0.80(20/21)=16/21$, $0.64(20/21)^2=256/441$, and $0.512(20/21)^3=4096/9261$. Their sum is $7056/9261+5376/9261+4096/9261=16528/9261=1.7846885$.

The EPV is $600(1.7846885)=\$1{,}070.8131$, or $1,070.81 to the nearest cent.

Check. The total undiscounted expected amount is $600(0.80+0.64+0.512)=\$1{,}171.20$. Discounting each payment reduces that value to $1,070.81. The method avoids using survival to the beginning of each year for a payment due at year-end.

Worked Example: Comparing a Life-Contingent Stream With Guaranteed Installments

A hypothetical arrangement specifies three installments of $500 at times $0,1,2$. Compare the EPV if installments after time 0 are payable only while the insured is alive with the EPV if all three installments are due regardless of survival. The insured is alive at issue. Assume an effective annual interest rate of 4% and a conditional annual survival probability of 0.80.

Life-contingent installments. The probabilities of being alive at times 0, 1, and 2 are $1$, $0.80$, and $0.80^2=0.64$. Since $v=1/1.04=25/26$, the factor is $1+0.80(25/26)+0.64(25/26)^2 =1+10/13+100/169 =399/169 =2.3609467$. The EPV is $500(399/169)=\$1{,}180.4734$, or $1,180.47.

Guaranteed installments. There are no survival conditions, so the factor is $1+25/26+(25/26)^2 =1+25/26+625/676 =1951/676 =2.8860947$. The EPV is $500(1951/676)=\$1{,}443.0473$, or $1,443.05.

The guaranteed schedule has an EPV higher by $\$1{,}443.0473-\$1{,}180.4734=\$262.5739$, or $262.57. This difference comes from the survival condition, not from the installment amount or timing. The comparison avoids assuming that a fixed payment obligation automatically ends at death.

Common Mistakes and a Practical Check

A frequent error is to use an annuity-certain factor for premiums that stop at death. For example, in the first worked example, an incorrect calculation would use $1+v+v^2=1+10/11+100/121=331/121$ and report $1{,}000(331/121)=\$2{,}735.54$. That calculation treats every installment as certain. The correct factor is $1+0.90v+0.81v^2=301/121$, giving $2,487.60.

The reverse error is to apply survival probabilities to payments that are guaranteed. Doing so understates their EPV because it removes installments that remain contractually due. Before calculating, write down each payment date and state whether payment depends on being alive at that date. Then apply discounting and survival weighting separately.

Beginning versus end of year: A beginning-of-year payment at time $k$ receives the discount factor $v^k$. An end-of-year payment at time $k$ also uses $v^k$, but its survival condition is survival to that end-of-year date.
Time 0: If the insured is stipulated alive at issue, a premium due immediately is certain and has discount factor 1.
Guaranteed schedule: Use an annuity-certain factor only when the stated payment obligation continues regardless of survival.
Model consistency: Use the survival probability to the actual payment date. If only conditional annual survival probabilities are supplied, multiply them to obtain the probability of surviving multiple years.

These distinctions matter whenever a premium stream is evaluated as part of a life contract or compared with another cash-flow schedule. A reliable calculation makes the timing and payment condition explicit before substituting any annuity factor.

Check Your Understanding

For each question, identify the payment dates and survival condition before calculating. Round money to the nearest cent.

A life is alive at issue and pays $900 at times 0 and 1 only if alive at each payment date. The effective annual interest rate is 5%, and the probability of surviving one year is 0.90. Find the EPV. (Answer: $1,671.43)
A hypothetical policy requires $400 at the end of each of the next three years while the insured is alive. The effective annual interest rate is 4%, and the conditional annual survival probability is 0.90. Find the EPV. (Answer: $904.94)
Four installments of $250 are guaranteed at times 1, 2, 3, and 4, regardless of survival. The effective annual interest rate is 5%. Find the EPV. (Answer: $886.49)
A life is alive at issue and pays $700 at times 0, 1, and 2 only if alive at each payment date. The effective annual interest rate is 6%; the probabilities of being alive at times 1 and 2 are 0.85 and 0.70. Find the EPV. (Answer: $1,697.42)
A schedule has a guaranteed $1,000 installment at time 0, followed by $1,000 installments at times 1 and 2 only if the insured is alive. The effective annual interest rate is 10%, and the conditional annual survival probability is 0.80. Find the EPV. (Answer: $2,256.20)

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