Reserves with Continuously Paid Premiums

Introduction

A prospective reserve compares two values at a policy duration: the expected present value of benefits still due, and the expected present value of premiums still payable. When premiums are paid continuously, the premium side is not a sequence of payments at anniversaries. It is a stream paid at a specified rate while the insured is alive.

The key is to value that stream from the reserve date onward, conditional on the insured being alive at that date. Premiums already paid are not future premiums, and premiums are not payable after death if the contract specifies payment only while alive. This tutorial develops the calculation for continuous premiums and applies it to hypothetical term, endowment, and whole-life policies.

Definition: For a policy in force at duration $t$, the prospective reserve ${}_tV_x$ is the conditional expected present value at time $t$ of benefits payable after time $t$, less the conditional expected present value at time $t$ of premiums payable after time $t$. The valuation is conditional on the insured being alive at time $t$.

Valuing the Remaining Premium Stream

Suppose a level premium is paid continuously at rate $\bar P$ per year while the insured is alive. Let $\tau$ be the remaining period during which premiums are payable; for whole-life premium payments, $\tau=\infty$. Write ${}_sp_{x+t}$ for the probability that a life aged $x+t$ survives a further $s$ years, and let $\delta$ be the constant force of interest. The value at time $t$ of the future premium stream is

\bar P\int_0^\tau e^{-\delta s}{}_sp_{x+t}\,ds.

The integral is the value of a continuous annuity payable while the insured survives. Its $ds$ contribution at future time $s$ is discounted by $e^{-\delta s}$, and it is received only if the insured is then alive, with probability ${}_sp_{x+t}$. If the contract continues premiums after death, or stops premiums at a date other than the one assumed in $\tau$, the premium value must be modified to match those terms.

Let $B_t$ denote the conditional expected present value at time $t$ of all future benefits. The prospective reserve is therefore

{}_tV_x = B_t-\bar P\int_0^\tau e^{-\delta s}{}_sp_{x+t}\,ds.

If the continuous premium rate is a net rate determined by the equivalence principle, it is chosen at issue so that the expected present value of premiums equals the expected present value of benefits. That issue equation determines $\bar P$; it does not mean that the reserve at every later duration is zero. Benefit patterns and survival probabilities over the remaining term can differ from those at issue.

Why the Premium Integral Is the Correct Value

The premium stream can be viewed as many small payments. In a short interval of length $ds$ at future time $s$, the premium paid, if the insured is alive, is $\bar P\,ds$. Its expected value at time $t$ is $\bar P\,ds\,{}_sp_{x+t}$, and its discounted value is $\bar P e^{-\delta s}{}_sp_{x+t}\,ds$. Adding these contributions across the remaining premium-payment period gives the integral.

\lim_{\text{maximum interval length}\to 0} \sum_s \bar P e^{-\delta s}{}_sp_{x+t}\,\Delta s = \bar P\int_0^\tau e^{-\delta s}{}_sp_{x+t}\,ds.

The same conditioning principle applies to benefits. For example, if a benefit $b$ is paid immediately on death and the force of mortality at age $x+t+s$ is $\mu_{x+t+s}$, then its value at time $t$, over a remaining term $\tau$, is

b\int_0^\tau e^{-\delta s}{}_sp_{x+t}\mu_{x+t+s}\,ds.

The factor ${}_sp_{x+t}\mu_{x+t+s}\,ds$ is the probability of surviving to the start of the small interval and dying within it, to first order in $ds$. A survival benefit at the end of a term is valued separately, using its amount times the probability of survival to maturity and the maturity discount factor.

Worked Examples

Worked Example: A Two-Year Term Policy with Changing Mortality

A hypothetical two-year term policy pays $1,000 immediately on death during the term. The force of interest is $\delta=0$. The force of mortality is $0.10$ during policy year 1 and $0.20$ during policy year 2. A level net premium is paid continuously while the insured is alive. Calculate the premium rate at issue and the reserve at the end of year 1.

Method. First equate the issue values of benefits and premiums. Then, at duration 1, condition on survival to that date and value only the one year of coverage and premiums remaining.

The probability of surviving the first year is $e^{-0.10}$. The probability of surviving the full two years is $e^{-0.10}e^{-0.20}=e^{-0.30}$. With no interest, the expected benefit at issue is

1{,}000(1-e^{-0.10})+ 1{,}000e^{-0.10}(1-e^{-0.20}) = 1{,}000(1-e^{-0.30}) = 259.1818.

The continuous premium annuity factor at issue is the expected total time the insured is alive during the two policy years:

\int_0^1 e^{-0.10s}\,ds + e^{-0.10}\int_0^1 e^{-0.20s}\,ds = \frac{1-e^{-0.10}}{0.10} + e^{-0.10}\frac{1-e^{-0.20}}{0.20} = 0.95162582+0.82009599 = 1.77172181.

Thus the net premium rate is

\bar P = \frac{259.1818}{1.77172181} \approx 146.2881.

At the end of year 1, the remaining force of mortality is $0.20$. The future benefit value is

1{,}000(1-e^{-0.20})=181.2692.

The remaining premium annuity factor is

\int_0^1e^{-0.20s}\,ds = \frac{1-e^{-0.20}}{0.20} = 0.90634623.

Therefore,

{}_1V_x = 181.2692-(146.2881)(0.90634623) \approx \$48.68.

The reserve is positive because the expected value of the remaining death benefit exceeds the value of the remaining premiums. The calculation avoids the mistake of using the issue premium annuity factor at the end of year 1.

Worked Example: An Endowment with Continuous Premiums

A hypothetical two-year endowment pays $500 immediately on death during the term, or $1,200 at the end of year 2 if the insured survives. The force of mortality is constant at $0.20$, the force of interest is $\delta=0$, and a level net premium is paid continuously while alive. Find the net premium rate at issue and the reserve at the end of year 1.

Method. At issue, include both the death benefit and the survival benefit in the benefit value. At duration 1, include death benefits during the remaining year and the maturity benefit if the insured survives that year.

Write $q=1-e^{-0.40}=0.32967995$ for the probability of death within two years and $p=e^{-0.40}=0.67032005$ for survival to maturity. The benefit value at issue is

$$ 500q+1{,}200p = 500(0.32967995)+1{,}200(0.67032005) = 164.83998+804.38406 = 969.22403. $$

The expected time alive during the two-year period, which is the continuous premium annuity factor, is

\int_0^2 e^{-0.20s}\,ds = \frac{1-e^{-0.40}}{0.20} = 1.64839977.

The net premium rate is consequently

\bar P = \frac{969.22403}{1.64839977} \approx \$587.98\text{ per year}.

At the end of year 1, the insured has one year remaining. The probability of death during that year is $1-e^{-0.20}=0.18126925$, and the probability of survival to maturity is $e^{-0.20}=0.81873075$. Thus the future benefit value is

$$ 500(0.18126925)+1{,}200(0.81873075) = 90.63462+982.47690 = 1{,}073.11153. $$

The remaining premium annuity factor is

\int_0^1e^{-0.20s}\,ds = \frac{1-e^{-0.20}}{0.20} = 0.90634623.

Using the unrounded issue premium rate in the calculation,

{}_1V_x = 1{,}073.11153-(587.97874)(0.90634623) \approx \$540.20.

The maturity benefit is conditional on surviving the remaining year; it is not paid with certainty at the reserve date. The reserve is rounded to the nearest cent only at the final step.

Worked Example: A Whole-Life Policy with a Specified Premium Rate

A hypothetical whole-life policy pays $10,000 immediately on death. The force of mortality is constant at $\mu=0.10$, the force of interest is $\delta=0.04$, and a premium of $800 per year is paid continuously while the insured is alive. Calculate the reserve at issue and at duration 3. The premium rate is specified and is not assumed to be a net rate.

Method. Under constant force of mortality, ${}_sp_x=e^{-0.10s}$. The expected present value of the future death benefit is

10{,}000\int_0^\infty e^{-0.04s}e^{-0.10s}(0.10)\,ds = 10{,}000(0.10)\int_0^\infty e^{-0.14s}\,ds = \frac{1{,}000}{0.14} = 7{,}142.8571.

The continuous premium stream has value

800\int_0^\infty e^{-0.04s}e^{-0.10s}\,ds = 800\int_0^\infty e^{-0.14s}\,ds = \frac{800}{0.14} = 5{,}714.2857.

The reserve at issue is $7{,}142.8571-5{,}714.2857=\$1{,}428.57$. At duration 3, the same calculation applies conditional on survival: the remaining lifetime has the same constant force of mortality, and the benefit and premium amounts are unchanged. Hence ${}_3V_x=\$1{,}428.57$ as well. The equality across durations depends on the constant mortality and interest assumptions and the whole-life benefit and premium structure; it is not a general reserve pattern.

Calculation Process and Common Trap

Condition on being alive at the reserve date.
Use the future survival probabilities from age $x+t$, not the probabilities from issue.

Value future benefits.
Include each future death or survival payment according to its timing and the policy terms.

Value future premiums.
Multiply the continuous premium rate by the discounted expected future time premiums are payable.

Subtract and check the timing.
The prospective reserve is future benefit value minus future premium value; neither value includes past cash flows.

A frequent exam trap is to subtract the issue premium annuity factor when calculating a reserve later. In the two-year term example, that error would give $181.2692-(146.2881)(1.77172181)$, approximately $-\$77.91$. But at the end of year 1, only the second policy year remains, so the correct premium factor is $0.90634623$, giving the reserve of $\$48.68$. The issue factor includes the first year, which has already passed.

Continuous premiums are rates per unit of time. If a problem gives a monthly rate or a force per month, convert consistently before integrating over years. Also check whether premiums stop at death, at maturity, or at an earlier premium-cessation date. The integral must represent the contract’s actual future payment period.

Key takeaway: At duration $t$, value only future cash flows conditional on survival to $t$. For continuous premiums payable while alive, subtract the premium rate times the remaining discounted survival integral from the conditional value of future benefits.

Check Your Understanding

For each problem, assume benefits are paid as stated and continuous premiums are payable only while the insured is alive. Round money to the nearest cent.

A one-year term policy pays $2,000 immediately at death. The force of mortality is $0.10$, the force of interest is zero, and premiums are paid continuously. Find the net premium rate at issue. (Answer: $200.00 per year)
At a reserve date, one year remains on a term policy paying $1,000 immediately at death. The force of mortality is $0.20$, the force of interest is zero, and the continuous premium rate is $150 per year. Find the reserve. (Answer: $45.32)
A two-year endowment pays $400 immediately at death or $1,000 at maturity if the insured survives. The force of mortality is $0.10$, the force of interest is zero, and a level continuous net premium is payable while alive. Find the issue premium rate. (Answer: $491.67 per year)
A whole-life policy pays $5,000 immediately at death. The force of mortality is $0.08$, the force of interest is $0.02$, and the continuous premium rate is $300 per year. Find the reserve at any duration, conditional on survival. (Answer: $1,000.00)
A term policy has one-half year of coverage remaining. It pays $2,500 immediately at death; the force of mortality is $0.12$, the force of interest is $0.04$, and the premium rate is $180 per year. Find the reserve. (Answer: $57.66)

What You'll Learn