The Equivalence Principle and Fully Continuous Benefit Premiums for Whole, Term, and Endowment Life Insurance Policies: Practice Problems and Solutions

A life insurance policy is frequently purchased by means of a life annuity consisting of contract premiums that are specified in the insurance contract. The determination of contract premiums is based on some manner of premium principle.

The equivalence principle is a kind of premium principle. It requires that the expected loss of an insurance policy to the insurer be equal to zero. If L is the loss random variable for the insurer, then, under the equivalence principle,

E[L] = 0 and E[present value of benefits] = E[present value of benefit premiums].

(Note: the term benefit premiums will be used to refer to contract premiums satisfying the equivalence principle.)

We will first work with fully continuous premiums, where level annual benefit premiums (of the same amount per year) are paid on a continuous basis.

For a whole life insurance policy that pays a unit in benefits upon the death of life (x) and that has actuarial present value Ā_x, the annual fully continuous benefit premium is denoted as P^-(Ā_x) (with the line drawn directly over the "P" whenever possible) and can be found as follows:

P^-(Ā_x) = Ā_x/ā_x

Recall: Ā_x = ₀^∞∫v^t*_tp_x*μ_x(t)dt and Ā_x = μ/(μ + δ) for constant force of mortality μ and constant force of interest δ. Moreover, ā_x = ₀^∞∫v^t*_tp_x*dt and ā_x = 1/(μ + δ) for constant force of mortality μ and constant force of interest δ.

Thus, for constant force of mortality μ and constant force of interest δ, P^-(Ā_x) =

Ā_x/ā_x = [μ/(μ + δ)]/[ 1/(μ + δ)] = P^-(Ā_x) = μ.

The variance of the loss Var[L] under this kind of benefit premium can be found as follows:
Var[L] = (²Ā_x - (Ā_x)²)/(δā_x)²

For an n-year term life insurance policy that pays a unit in benefits upon the death of life (x) and that has actuarial present value Ā¹_x:n¬, the fully continuous annual benefit premium is denoted as

P^-(Ā¹_x:n¬) = Ā¹_x:n¬ / ā_x:n¬

Recall: Ā¹_x:n¬ = ₀^∞∫z_t*f_T(t)dt = ₀ⁿ∫v^t*_tp_x*μ_x(t)dt and ā_x:n¬ = ₀ⁿ∫v^t*_tp_x*dt.

For an n-year endowment insurance policy that pays a unit in benefits upon the death of life (x) and that has actuarial present value Ā_x:n¬, the fully continuous annual benefit premium is denoted as

P^-(Ā_x:n¬) = Ā_x:n¬ / ā_x:n¬.

Recall: Ā_x:n¬ = ₀ⁿ∫v^t*_tp_x*μ_x(t)dt + vⁿ*_np_x.

Source: Bowers, Gerber, et. al. Actuarial Mathematics. 1997. Second Edition. Society of Actuaries: Itasca, Illinois. pp. 167-173.

Original Problems and Solutions from The Actuary's Free Study Guide

Problem S3L44-1. The life of a triceratops has the following survival function associated with it: s(x) = e^-0.34x. The annual force of interest in Triceratopsland is 0.07. François the Triceratops is currently 3 years old has a whole life insurance policy, which will pay him 1 Triceratops Currency Unit (TCU) upon death. Under the equivalence principle, what is the annual fully continuous level benefit premium that François will pay for this policy?

Solution S3L44-1. Since triceratops lifetimes are exponentially distributed, triceratopses have a constant force of mortality of 0.34. We thus use the formula P^-(Ā_x) = μ = 0.34 TCU.

Problem S3L44-2. The life of a triceratops has the following survival function associated with it: s(x) = e^-0.34x. The annual force of interest in Triceratopsland is 0.07. François the Triceratops is currently 3 years old has a whole life insurance policy, which will pay him 1 Triceratops Currency Unit (TCU) upon death. Under the equivalence principle, what is the variance of the loss to the insurer for this policy?

Solution S3L44-2. We use the formula Var[L] = (²Ā_x - (Ā_x)²)/(δā_x)². Since triceratopses exhibit a constant force of mortality, we have Ā_x = μ/(μ + δ) = 0.34/0.41 = 34/41. Moreover, ā_x = 1/(μ + δ) = 1/0.41 = 100/41. Now we need to find ²Ā_x = μ/(μ + 2δ) for a constant force of mortality.

Thus, ²Ā_x = 0.34/0.48 = 34/48.

Hence, Var[L] = (34/48 - (34/41)²)/(0.07*100/41)² = 17/24 = about 0.70833333333.

Problem S3L44-3. The life of a giant pin-striped cockroach has the following survival function associated with it: s(x) = 1 - x/94, for 0 ≤ x ≤ 94 and 0 otherwise. Hcaorkcoc the Giant Pin-Striped Cockroach is currently 56 years old and has a whole life insurance policy which will pay 10 Golden Hexagons (GH) upon death. The annual force of interest is 0.02. Under the equivalence principle, what is the annual fully continuous level benefit premium that Hcaorkcoc will pay for this policy?

Solution S3L44-3. Here, P^-(Ā_x) = 10Ā_x/ā_x.

We use the formula Ā_x = ₀^∞∫v^t*_tp_x*μ_x(t)dt.

We want to find 10Ā₅₆. Since no giant pin-striped cockroach lives past the age of 94, our integral's upper bound will be 94-56 = 38, because Hcaorkcoc will not live for more than 38 additional years.

We find _tp₅₆ = s(x + t)/s(x) = s(56 + t)/s(56) = (1 - (56+t)/94)/(1 - 56/94) = (38 - t)/38

We find μ₅₆(t) = -s'(x)/s(x) = (1/94)/(1 - x/94) = (-1/94)/[(94 - x)/94] = 1/(94 - x) =

1/(94 - (56+t)) = 1/(38 - t). Conveniently enough, _tp₅₆* μ₅₆(t) = ((38 - t)/38)(1/(38 - t)) = 1/38. We find v^t = e^-0.02t.

Thus, 10Ā₅₆ = 10*₀³⁸∫e^-0.02t*(1/38)dt

10Ā₅₆ = 10*(-50/38)e^-0.02t│₀³⁸ = 10(50/38)(1 - e^-0.76) = 10Ā₅₆ = about 7.004389118 GH.

Now we find ā₅₆ = ā_56:38¬ = ₀³⁸∫v^t*_tp_x*dt, since Hcaorkcoc will not live for more than 38 additional years.

Thus, ā₅₆ = ₀³⁸∫e^-0.02t*(38 - t)/38*dt = ₀³⁸∫e^-0.02tdt - ₀³⁸∫(1/38)te^-0.02tdt = 26.61667865 - 11.63862424

Thus, ā₅₆ = 14.97805441 GH and so P^-(Ā₅₆) = 10Ā₅₆/ā₅₆ = 7.004389118/14.97805441 = about 0.4676434553 GH.

Problem S3L44-4. The life of a triceratops has the following survival function associated with it: s(x) = e^-0.34x. The annual force of interest in Triceratopsland is 0.07. Jerry the Triceratops is currently 3 years old has a 6-year term life insurance policy, which will pay him 1 Triceratops Currency Unit (TCU) upon death. Under the equivalence principle, what is the annual fully continuous level benefit premium that Jerry will pay for this policy?

Solution S3L44-4.

We use the formula P^-(Ā¹_x:n¬) = Ā¹_x:n¬ / ā_x:n¬.

We find Ā¹_x:n¬ = ₀ⁿ∫v^t*_tp_x*μ_x(t)dt.

Since triceratopses exhibit a constant force of mortality, we have μ_x(t) = 0.34. Moreover, since triceratops lifetimes are exponentially distributed, we have _tp_x = e^-0.34t. Moreover, v^t = e^-0.07t.

Thus, Ā¹_3:6¬ = ₀⁶∫0.34e^-0.07t*e^-0.34tdt = ₀⁶∫0.34e^-0.41tdt = (-34/41)e^-0.41t│₀⁶ = (34/41)(1 - e^-0.41*6) = about 0.7584197968 TCU.

We find ā_3:6¬ = ₀ⁿ∫v^t*_tp_x*dt = ₀⁶∫e^-0.07t*e^-0.34tdt = ₀⁶∫e^-0.41tdt = (-100/41)e^-0.41t│₀⁶

= (100/41)(1 - e^-0.41*6)

But P^-(Ā¹_3:6¬) = Ā¹_3:6¬ / ā_3:6¬ = [(34/41)(1 - e^-0.41*6)]/[(100/41)(1 - e^-0.41*6)] = 34/100 = 0.34 = μ.

Note: We have just seen an example showing that for a constant force of mortality, it does not matter whether the life insurance policy in question is a whole or a term life insurance policy. The benefit premium will be μ in either case.

Problem S3L44-5. The life of a triceratops has the following survival function associated with it: s(x) = e^-0.34x. The annual force of interest in Triceratopsland is 0.09. Hasdrubal the Triceratops is currently 7 years old has a 2-year endowment insurance policy, which will pay him 1 Triceratops Currency Unit (TCU) upon death. Under the equivalence principle, what is the annual fully continuous level benefit premium that Hasdrubal will pay for this policy?

Solution S3L44-5. We use the formula P^-(Ā_x:n¬) = Ā_x:n¬ / ā_x:n¬.

We first find Ā_x:n¬ = ₀ⁿ∫v^t*_tp_x*μ_x(t)dt + vⁿ*_np_x.

We first need to find

Ā¹_x:n¬ =₀ⁿ∫v^t*_tp_x*μ_x(t)dt and

A¹_x:n¬ = vⁿ*_np_x

We know that v = e^-0.09, x = 7, and n = 2.

We find _tp_x = s(x+t)/s(x) = s(7+t)/s(7) = e^-0.34t. So _np_x = ₂p₇ = e^-0.34*2.

We find μ_x(t) = -s'(x)/s(x) = 0.34e^-0.34t/e^-0.34t = 0.34.

Thus, Ā¹_7:2¬ =₀²∫e^-0.09t*e^-0.34t*0.34dt = ₀²∫0.34e^-0.43tdt = (-34/43)e^-0.43t│₀² = (34/43)(1 - e^-0.43*2) =

Ā¹_7:2¬ = 0.4561044

Also, A¹_7:2¬ = e^-0.09*2e^-0.34*2 = e^-0.86 = A¹_7:2¬ = 0.4231620823.

Thus, Ā_7:2¬ = A¹_7:2¬ + Ā¹_7:2¬ = 0.4231620823 + 0.4561044 = Ā_7:2¬ = about 0.8792664823.

Now we find ā_x:n¬ = ₀ⁿ∫v^t*_tp_x*dt.

ā_7:2¬ = ₀²∫e^-0.09t*e^-0.34tdt = (-100/43)e^-0.43t│₀² = (100/43)(1 - e^-0.43*2) = 1.341483529 GH.

Thus, P^-(Ā_7:2¬) = Ā_7:2¬ / ā_7:2¬ = 0.8792664823/1.341483529 = P^-(Ā_7:2¬) = 0.6554433677 GH.

Note: Even though the force of mortality in the above problem was constant, we have seen an example where the fully continuous benefit for an endowment insurance policy is not equal to μ.

See other sections of The Actuary's Free Study Guide for Exam 3L.