Level Premium Equivalence Using Annuity Values

Introduction

A level annual premium is a fixed amount paid at stated intervals, but its present value is not simply the premium multiplied by the number of years in the contract. The number and timing of payments depend on how long the insured survives. To find a premium under the equivalence principle, value the benefit as life insurance and value the premiums as a life annuity.

This tutorial focuses on a discrete whole-life contract with a benefit payable at the end of the year of death and premiums payable annually in advance while the insured is alive. The premium derived is a net premium: it balances expected present values of benefits and premiums, without an allowance for expenses, profit, or other loads.

Definition: Let $K_x$ be the curtate future lifetime of a life aged $x$, so death occurs between durations $K_x$ and $K_x+1$. For a unit whole-life benefit payable at the end of the year of death, $A_x=E[v^{K_x+1}]$. For annual premiums of 1 payable at times $0,1,2,\ldots$ while the insured is alive at each payment time, the expected present value is $\ddot{a}_x=\sum_{j=0}^{\infty}v^j\,{}_jp_x$. Here $v=1/(1+i)$, and ${}_jp_x$ is the probability of surviving $j$ years from age $x$.

At time 0 the first annual premium is certain. A premium at time 1 is paid only if the insured survives one year; a premium at time 2 is paid only if the insured survives two years, and so on. The life-annuity-due factor $\ddot{a}_x$ captures those survival-contingent payments and their discounting.

Deriving the Level Premium

Suppose the policy pays a benefit $B$ at the end of the year of death and charges a level annual premium $P$ in advance while the insured is alive. The expected present value of the benefit is $B A_x$. The expected present value of all premiums is $P\ddot{a}_x$. The equivalence principle sets these two expected present values equal.

B A_x=P\ddot{a}_x \qquad\Longrightarrow\qquad P=\frac{B A_x}{\ddot{a}_x}.

The formula follows by valuing the random cash flows directly. For an individual who dies in year $K_x+1$, the benefit has present value $Bv^{K_x+1}$. A premium is paid at time $j$ exactly when the insured is alive then, so the present value of the premium stream is $P\sum_{j=0}^{K_x}v^j$. Taking expectations gives $B E[v^{K_x+1}]=B A_x$ for the benefit and $P\sum_{j=0}^{\infty}v^j{}_jp_x=P\ddot{a}_x$ for the premiums. Equating them gives the stated premium formula.

This is an expected-value balance, not a claim that the present value of premiums equals the present value of benefits for every policyholder. The realized number of premiums depends on the actual lifetime.

Theorem: For a discrete whole-life insurance with benefit $B$ payable at the end of the year of death and level premiums $P$ payable annually in advance while the insured is alive, the net level annual premium under the equivalence principle is $P=B A_x/\ddot{a}_x$. The formula uses the annuity-due factor because the first premium is payable immediately.

A Useful Consistency Check

For a whole-life contract with certain eventual death and the timing just defined, the insurance and annuity-due factors satisfy a useful identity. It can check supplied factors before they are used in a premium calculation.

A_x+d\ddot{a}_x=1, \qquad d=1-v=\frac{i}{1+i}.

To see why, consider a life that dies in year $K_x+1$. The present value of a unit benefit plus $d$ times the present value of a unit premium stream is

v^{K_x+1}+d\sum_{j=0}^{K_x}v^j = v^{K_x+1}+(1-v)\frac{1-v^{K_x+1}}{1-v} =1.

Taking expectations gives $A_x+d\ddot{a}_x=1$. The identity assumes the whole-life benefit is payable at the end of the year of death and the annuity-due payments occur at times $0$ through $K_x$. It does not apply unchanged if the contract has a different benefit timing or a different premium payment pattern.

Worked Example: Annual Premium from Supplied Whole-Life Factors

A hypothetical whole-life policy pays $100,000 at the end of the year of death. Premiums are paid annually in advance while the insured is alive. At an effective annual interest rate of 4%, the mortality basis gives $A_x=0.30$ and $\ddot{a}_x=18.2$. Find the net level annual premium.

Method and assumptions. Use the equivalence principle with a discrete end-of-year death benefit and an annuity-due premium stream. The supplied factors summarize the mortality assumption at the stated interest rate.

Check the factors. At $i=0.04$, $v=1/1.04=25/26$, so $d=1-v=1/26$. Then

A_x+d\ddot{a}_x =0.30+\frac{1}{26}(18.2) =0.30+0.70 =1.00.

The supplied values are consistent with the whole-life identity.

Apply equivalence. The expected present value of the benefit is

$$ 100{,}000(0.30)=30{,}000. $$

The expected present value of premiums is $P(18.2)$. Therefore,

P=\frac{30{,}000}{18.2} =1{,}648.3516\ldots

Rounding only at the final step to the nearest cent, the net annual premium is $1,648.35. The check is $18.2(1{,}648.3516\ldots)=30{,}000$, matching the benefit present value.

Common mistake avoided. Dividing the benefit by the annuity factor alone would give $100{,}000/18.2=$5{,}494.51$, which treats the benefit as if its present value were $100,000. The benefit must first be multiplied by $A_x$, its insurance present-value factor.

Calculating the Factors from a Mortality Model

When mortality probabilities are given rather than $A_x$ and $\ddot{a}_x$, construct each factor from the cash-flow timing. If the probability of death in year $k+1$ is ${}_kp_xq_{x+k}$, then the unit benefit is discounted $k+1$ years. A premium due at time $j$ is discounted $j$ years and paid only if the insured survives $j$ years.

A_x=\sum_{k=0}^{\infty}v^{k+1}{}_kp_xq_{x+k}, \qquad \ddot{a}_x=\sum_{j=0}^{\infty}v^j{}_jp_x.

These expressions make the timing distinction explicit: the benefit is paid at the end of the year of death, whereas a premium due at the start of that year is still paid if the insured was alive at its start.

Worked Example: Two-Year Mortality Model

A hypothetical policy pays $10,000 at the end of the year of death. Premiums are payable annually in advance while the insured is alive. The effective annual interest rate is 5%. The insured has probability $0.50$ of dying in the first year; anyone surviving that year dies in the second year. Find the net level annual premium.

Method and assumptions. The death-year probabilities are $P(K_x=0)=0.50$ and $P(K_x=1)=0.50$. Thus ${}_1p_x=0.50$, and survival to the start of year 2 determines whether the second premium is paid. The discount factor is $v=1/1.05=20/21$.

Calculate the insurance factor. The benefit is paid at time 1 for a first-year death and at time 2 for a second-year death:

A_x=0.50v+0.50v^2 =\frac{1}{2}\frac{20}{21} +\frac{1}{2}\frac{400}{441} =\frac{210+200}{441} =\frac{410}{441}.

Calculate the premium factor. A premium is certain at time 0. A second premium at time 1 is paid only if the insured survives the first year:

\ddot{a}_x=1+0.50v =1+\frac{10}{21} =\frac{31}{21}.

Apply equivalence.

P=10{,}000\frac{410/441}{31/21} =10{,}000\frac{410}{651} =6{,}298.0030\ldots

Rounding only at the final step to the nearest cent, $P=$6,298.00$. The factor check gives $A_x+d\ddot{a}_x=410/441+(1/21)(31/21)=441/441=1$. The calculation avoids the timing error of discounting the second-year death benefit for only one year.

Premium Timing Changes the Annuity Factor

The formula $P=B A_x/\ddot{a}_x$ depends on premiums being payable in advance. If a contract instead charges at the end of each year, and only when the insured is alive at that payment time, the expected present value of one unit of premium is $a_x=\sum_{j=1}^{\infty}v^j{}_jp_x$. The benefit factor is unchanged if its amount and payment timing are unchanged, but the premium factor is not.

Premium timing	Expected present value of one unit of premium	Level premium under equivalence
Annual in advance while alive	$\ddot{a}_x$	$B A_x/\ddot{a}_x$
Annual at year-end, only if alive then	$a_x$	$B A_x/a_x$

The year-end convention in the second row means no premium is collected at the end of a year in which death has already occurred. A contract that collects a premium at the time of a death benefit would need its own explicit payment-order convention.

Worked Example: Comparing Premiums Paid in Advance and at Year-End

A hypothetical whole-life policy pays $30,000 at the end of the year of death. At an effective annual interest rate of 5%, death occurs in year 1 with probability $0.25$, in year 2 with probability $0.25$, and in year 3 with probability $0.50$. Compare the net annual premium if premiums are paid (a) in advance while alive and (b) at each year-end only if alive then.

Method and assumptions. Use the same end-of-year benefit factor for both designs, but calculate a separate premium annuity factor for each payment pattern. Here $v=20/21$, survival to times 1 and 2 has probabilities $0.75$ and $0.50$, and survival to time 3 has probability 0.

Insurance factor.

A_x=0.25v+0.25v^2+0.50v^3 =\frac{5}{21}+\frac{100}{441}+\frac{4000}{9261} =\frac{8305}{9261}.

(a) Premiums in advance. Payments occur at time 0, at time 1 if the insured survives year 1, and at time 2 if the insured survives year 2:

\ddot{a}_x=1+0.75v+0.50v^2 =1+\frac{15}{21}+\frac{200}{441} =\frac{956}{441}.

The annual premium is

P_{\text{due}} =30{,}000\frac{8305/9261}{956/441} =\frac{249{,}150{,}000}{20{,}076} =12{,}410.3407\ldots

(b) Premiums at year-end. A payment at time 1 is made only after survival through year 1, and a payment at time 2 only after survival through year 2. There is no payment at time 3 because no one survives to that time:

a_x=0.75v+0.50v^2 =\frac{15}{21}+\frac{200}{441} =\frac{515}{441}.

The annual premium is

P_{\text{immediate}} =30{,}000\frac{8305/9261}{515/441} =\frac{249{,}150{,}000}{10{,}815} =23{,}037.4480\ldots

Rounding only at the final step to the nearest cent, the premiums are $12,410.34 in advance and $23,037.45 at year-end. The year-end version has a smaller expected present value per dollar of premium because it has no certain payment at time 0 and fewer expected payments. Its higher level premium balances the same expected benefit present value. Using $\ddot{a}_x$ for both designs would miss this timing difference.

Common Exam Traps

Using the wrong numerator. The expected present value of a benefit $B$ is $B A_x$, not $B$. Apply the insurance factor before dividing by the premium annuity factor.
Using the wrong annuity factor. Premiums in advance require $\ddot{a}_x$; premiums at year-end, payable only if alive then, require $a_x$.
Misplacing the death benefit. A benefit payable at the end of the year of death is discounted $K_x+1$ years, not $K_x$ years.
Charging premiums after death. Premiums stop when the insured is no longer alive under the payment conditions. Their expected number cannot be treated as a fixed term without regard to survival.
Ignoring the basis of the factors. The interest rate and mortality basis used for $A_x$ and the annuity factor must match. When possible, check $A_x+d\ddot{a}_x=1$ for a whole-life benefit with the stated timing.

Key takeaway: Under the equivalence principle, a level annual net premium is the expected present value of benefits divided by the expected present value of one unit of premium. Identify both payment timings first: the benefit determines the insurance factor, and the premium schedule determines the life-annuity factor.

Check Your Understanding

For each hypothetical contract, identify the applicable factors and calculate the requested amount. Round monetary answers to the nearest cent.

At an effective annual interest rate of 4%, a whole-life policy pays $80,000 at the end of the year of death. Its factors are $A_x=0.22$ and $\ddot{a}_x=20.28$. Find the annual net premium payable in advance. (Answer: $867.85)
At an effective annual interest rate of 10%, death occurs in year 1 with probability $0.40$ and in year 2 with probability $0.60$. A $25,000 benefit is paid at the end of the year of death, and level premiums are paid in advance while alive. Find the annual net premium. (Answer: $13,903.74)
At an effective annual interest rate of 4%, $A_x=0.30$ and $\ddot{a}_x=18.2$. A whole-life policy has premiums payable at the end of each year only if the insured is alive then. If its benefit payable at the end of the year of death is $75,000, find the annual net premium. (Answer: $1,308.14)
At an effective annual interest rate of 5%, death occurs in year 1 with probability $0.25$, and otherwise in year 2. A $14,000 benefit is paid at the end of the year of death. Find the annual net premium payable in advance while alive. (Answer: $7,500.00)
At an effective annual interest rate of 5%, death occurs in year 1 with probability $0.50$ and in year 2 with probability $0.50$. Premiums are paid at the end of each year only if the insured is alive then. A benefit of $12,000 is paid at the end of the year of death. Find the annual net premium. (Answer: $7,557.60)

Premium timing	Expected present value of one unit of premium	Level premium under equivalence
Annual in advance while alive	\(\ddot{a}_x\)	\(B A_x/\ddot{a}_x\)
Annual at year-end, only if alive then	\(a_x\)	\(B A_x/a_x\)

What You'll Learn