Equivalence Principle for Life Contracts

Why Equivalence Is an Expected-Value Calculation

A life insurance benefit is contingent: its amount, timing, or whether it is paid at all can depend on the insured’s future lifetime. A single premium paid at policy issue is different. It is a known amount paid at time 0. The equivalence principle sets that premium equal to the expected present value of the promised benefits, using the stated interest and mortality assumptions.

This calculation gives a net premium: it prices the benefits alone, without an allowance for expenses, commissions, profit, or other loading. It is a model-based expected-value calculation, not a guarantee that the insurer’s actual present-value outgo will equal the premium for an individual policy.

Definition: Under the equivalence principle, the net single premium $P$ for a life contract is the expected present value, at issue, of its benefit payments. If $Z$ is the random present value of benefits, then $P=E[Z]$. This form applies to a single premium paid at time 0; it does not by itself specify the value or timing of a stream of premiums.

For an annual discrete policy, a benefit paid at the end of policy year $k$ is discounted by $v^k$, where $v=1/(1+i)$ and $i$ is the effective annual interest rate. The probability of a claim in year $k$ must also reflect survival to the start of that year. If mortality rates vary by duration, define $q_{x+j}$ as the conditional probability that a life aged $x+j$ dies during the following year, given survival to that age. Then ${}_jp_x$ is the probability of surviving $j$ years from age $x$, and ${}_jp_x=\prod_{r=0}^{j-1}(1-q_{x+r})$.

Deriving the Equivalence Principle for a Single Premium

Consider a contract with a benefit $B_k$ paid at the end of policy year $k$ if death occurs during that year, for $k=1,\ldots,n$. Let $I_k$ equal 1 if death occurs during year $k$, and 0 otherwise. If there is also a survival benefit $B_n^{(s)}$ paid at time $n$ to a life that survives the term, let $I_s$ indicate survival to time $n$. The present value of the benefits is

Z=\sum_{k=1}^{n} B_k v^k I_k+B_n^{(s)}v^n I_s.

Expectation is additive: the expected value of a sum is the sum of the expected values. Also, $E[I_k]=P(I_k=1)$, since an indicator is 1 exactly when its event occurs. For death during year $k$, that probability is ${}_{k-1}p_x q_{x+k-1}$; the survival indicator has probability ${}_np_x$. Thus the expected present value, and therefore the net single premium, is

P=E[Z] =\sum_{k=1}^{n} B_k v^k\,{}_{k-1}p_x q_{x+k-1} +B_n^{(s)}v^n\,{}_np_x.

This derivation explains why a conditional mortality rate alone is not the probability of a claim in a later policy year. The life must first survive to that year. For a level-benefit term policy, set each $B_k=B$ and $B_n^{(s)}=0$. For an endowment policy with a level death benefit and an equal survival benefit at time $n$, include both the death-benefit terms and the final survival term. The two benefits at time $n$ apply to mutually exclusive outcomes.

List the possible payment events.
Separate death in each policy year from survival to the end of the term, if survival is covered.

Find each event probability.
For death in year $k$, multiply the probability of surviving the first $k-1$ years by the conditional probability of death during year $k$.

Discount each payment to issue.
A payment at the end of year $k$ has present value multiplied by $v^k$, not $v^{k-1}$.

Add expected present values.
Multiply each possible benefit by its probability and discount factor, then sum. The result is the net single premium.

Worked Calculations

Worked Example: A Two-Year Term Benefit

A hypothetical insurer issues a two-year discrete term policy to a life aged $x$. It pays $30,000 at the end of the year of death if death occurs within the two-year term. Assume an effective annual interest rate $i=5\%$, with $q_x=0.02$ and $q_{x+1}=0.05$. Find the net single premium.

Method and assumptions. The benefit is paid at the end of the year of death, and there is no payment if the insured survives two years. Mortality rates are conditional on being alive at the start of the stated year. The discount factor is

v=\frac{1}{1.05}=0.95238095\ldots

The probability of death during year 1 is $0.02$. The probability of death during year 2 is the probability of surviving year 1 multiplied by the conditional probability of death in year 2:

P(\text{death in year 2})=(1-0.02)(0.05)=0.98(0.05)=0.049.

The year-1 benefit is discounted one year, and the year-2 benefit is discounted two years. Therefore,

P=30{,}000\left(\frac{0.02}{1.05}+\frac{0.049}{1.05^2}\right) =30{,}000(0.019047619+0.044444444) =30{,}000(0.063492063) =\$1{,}904.7619\ldots

Rounded to the nearest cent, the net single premium is $1,904.76. A common mistake is to use $0.05$ as the unconditional probability of death in year 2. That incorrect calculation gives

30{,}000\left(\frac{0.02}{1.05}+\frac{0.05}{1.05^2}\right) =30{,}000(0.019047619+0.045351474) =\$1{,}931.97.

The second-year term in this expression omits survival through year 1. It is not the correct expected claim cost. The premium also passes a basic check: the two-year death probability is $0.02+0.049=0.069$, so the undiscounted expected benefit is $2,070; discounting benefits payable in the future should produce a smaller amount.

Worked Example: A Two-Year Endowment Benefit

A hypothetical two-year endowment policy pays $10,000 at the end of the year of death, or $10,000 at time 2 if the insured survives the full term. Assume $i=10\%$, $q_x=0.10$, and $q_{x+1}=0.20$. Find the net single premium.

Method and assumptions. Death benefits are paid at the end of the year of death. The survival benefit is paid at time 2. At the end of year 2, death during that year and survival to time 2 are separate outcomes.

The probability of death in year 1 is $0.10$. The probability of death in year 2 is $0.90(0.20)=0.18$. The probability of surviving both years is $0.90(0.80)=0.72$. These probabilities sum to $0.10+0.18+0.72=1$, which checks that all outcomes are included.

Discount the year-1 benefit by $1.10$, and both possible year-2 benefits by $1.10^2=1.21$:

\begin{aligned} P &=10{,}000\left(\frac{0.10}{1.10} +\frac{0.18}{1.21} +\frac{0.72}{1.21}\right)\\ &=10{,}000(0.090909091+0.148760331+0.595041322)\\ &=10{,}000(0.834710744) =\$8{,}347.1074\ldots \end{aligned}

Rounded to the nearest cent, the net single premium is $8,347.11. The year-2 probabilities combine to $0.18+0.72=0.90$, the probability of being alive at the start of year 2. That is appropriate because the policy pays $10,000 at time 2 whether death occurs in year 2 or the insured survives. The payments are not counted twice for any one outcome.

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

A hypothetical three-year discrete term policy pays $20,000 at the end of the year of death and nothing on survival to the end of year 3. Assume $i=4\%$, $q_x=0.10$, $q_{x+1}=0.20$, and $q_{x+2}=0.25$. Find the net single premium.

Method and assumptions. The benefit is constant, but the conditional mortality rate varies by policy year. First calculate each unconditional probability of death during the term:

\begin{aligned} P(\text{death in year 1})&=0.10,\\ P(\text{death in year 2})&=(0.90)(0.20)=0.18,\\ P(\text{death in year 3})&=(0.90)(0.80)(0.25)=0.18. \end{aligned}

The probability of surviving three years is $(0.90)(0.80)(0.75)=0.54$. The death probabilities sum to $0.10+0.18+0.18=0.46$; adding the survival probability gives $0.46+0.54=1$. There is no survival benefit, so only the three death outcomes contribute to the premium. With $v=1/1.04$,

\begin{aligned} P &=20{,}000\left(\frac{0.10}{1.04} +\frac{0.18}{1.04^2} +\frac{0.18}{1.04^3}\right)\\ &=20{,}000(0.096153846+0.166420118+0.160019345)\\ &=20{,}000(0.422593309) =\$8{,}451.8662\ldots \end{aligned}

Rounded to the nearest cent, the net single premium is $8,451.87. A common error is to use the year-3 conditional mortality rate, $0.25$, as the probability of a year-3 claim. The correct probability is $0.90(0.80)(0.25)=0.18$, because the insured must survive the first two years to be exposed to the third-year mortality rate.

Checks and Common Traps

The equivalence principle is simple to state, but a correct calculation depends on matching three things: the event that triggers payment, its probability, and its payment time. Before evaluating a premium, check each term against the contract wording.

Conditional versus unconditional mortality. $q_{x+k-1}$ applies only among lives who reach the start of year $k$. Multiply by ${}_{k-1}p_x$ to get the probability of death in that year from issue.
Payment timing. A benefit payable at the end of year $k$ is discounted by $v^k$. A benefit paid at time $n$ on survival is discounted by $v^n$.
Term versus endowment coverage. A term policy has no survival benefit unless specified. An endowment policy can pay on survival as well as on death during the term; include the survival outcome only when the contract promises it.
Net versus loaded premium. Equating the premium to expected benefit present value gives a net premium under the stated assumptions. It does not automatically include expenses or other additions.

A useful probability check is that the probabilities of all mutually exclusive outcomes over a finite term should sum to 1. For a term policy, the death probabilities over the term plus the probability of survival to the end should sum to 1, even though survival contributes no benefit. This check can reveal a missing survival factor or an omitted outcome before it distorts the premium.

The equivalence principle is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus because it connects a contract’s contingent benefits to a premium through expected present values. In every calculation, keep the contract’s payment conditions explicit; a correct mortality rate attached to the wrong payment event or time still gives the wrong premium.

Check Your Understanding

For each hypothetical contract, calculate the requested amount using the stated assumptions. Round monetary answers to the nearest cent.

A one-year discrete term policy pays $25,000 at the end of the year of death. Assume $i=5\%$ and $q_x=0.04$. Find its net single premium. (Answer: $952.38)
A two-year term policy pays $12,000 at the end of the year of death. Assume $i=4\%$, $q_x=0.03$, and $q_{x+1}=0.06$. Find the probability of death in year 2 and the net single premium. (Answer: year-2 death probability $0.0582$; premium $991.86)
A three-year endowment pays $8,000 at the end of the year of death or at time 3 on survival. Assume $i=5\%$, $q_x=0.10$, $q_{x+1}=0.20$, and $q_{x+2}=0.25$. Find the net single premium. (Answer: $7,043.73)
A two-year term policy pays $15,000 at the end of the year of death. Assume $i=5\%$, $q_x=0.08$, and $q_{x+1}=0.10$. Find the correct net single premium and the overstatement produced by using $0.10$, rather than the unconditional year-2 death probability, in the second-year term. (Answer: correct premium $2,394.56; incorrect premium $2,503.40; overstatement $108.84)

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