Table 6.2. Annuity values and premiums.
m=4 m=12
Method a¨(4)
[45]:20 P a¨(12)
[45]:20 P Exact 12.69859 3 022.11 12.64512 3 034.89 UDD 12.69839 3 022.16 12.64491 3 034.94
W3 12.69859 3 022.11 12.64512 3 034.89
The values of a¨(m)
[45]:20 for m = 4 and 12 can either be calculated exactly or froma¨[45]:20 using one of the approximations in Section5.11. Notice that the approximation labelled W3∗ in that section is not available sincep[x]−1
is meaningless and so we cannot estimateµ[45] from the life table tabulated at integer ages. Table6.2shows values obtained using the UDD assumption and Woolhouse’s formula with three terms. The ordering of these premiums form=1, 4, 12 reflects the ordering of EPVs of 1/mthly annuities which we observed in Chapter5. In this example, Woolhouse’s formula provides a very good approximation, whilst the UDD assumption gives a reasonably accurate premium.
6.6 Gross premium calculation 151 annuity payment is made. These costs arise in a variety of ways. The processing of renewal and annuity payments involves staff time and investment expenses.
Renewal expenses also cover the ongoing fixed costs of the insurer such as staff salaries and rent for the insurer’s premises, as well as specific costs such as annual statements to policyholders about their policies.
Initial and renewal expenses may be proportional to premiums, proportional to benefits or may be ‘per policy’, meaning that the amount is fixed for all policies, and is not related to the size of the contract. Often, per policy renewal costs are assumed to be increasing at a compound rate over the term of the policy, to approximate the effect of inflation.
Termination expenses occur when a policy expires, typically on the death of a policyholder (or annuitant) or on the maturity date of a term insurance or endowment insurance. Generally these expenses are small, and are largely associated with the paperwork required to finalize and pay a claim. In calculating gross premiums, specific allowance is often not made for termination expenses.
Where allowance is made, it is usually proportional to the benefit amount.
In practice, allocating the different expenses involved in running an insurance company is a complicated task, and in the examples in this chapter we simply assume that all expenses are known.
The equivalence principle applied to the gross premiums and benefits states that the EPV of the gross future loss random variable should be equal to zero.
That means that
E[Lg0] =0, that is
EPV of benefit outgo+EPV of expenses
−EPV of gross premium income=0.
In other words, under the equivalence premium principle,
EPV of benefits+EPV of expenses=EPV of gross premium income.
(6.3) We conclude this section with three examples in each of which we apply the equivalence principle to calculate gross premiums.
Example 6.6An insurer issues a 25-year annual premium endowment insur- ance with sum insured $100 000 to a select life aged 30. The insurer incurs initial expenses of $2000 plus 50% of the first premium, and renewal expenses of 2.5% of each subsequent premium. The death benefit is payable immediately on death.
(a) Write down the gross future loss random variable.
(b) Calculate the gross premium using the Standard Select Survival Model with 5% per year interest.
Solution 6.6 (a) LetS =100 000,x=30,n=25 and letPdenote the annual gross premium. Then
Lg0=Svmin(T[x],n)+2000+0.475P+0.025Pa¨min(K
[x]+1,n)
−Pa¨min(K
[x]+1,n)
=Svmin(T[x],n)+2000+0.475P−0.975Pa¨min(K
[x]+1,n). Note that the premium related expenses, of 50% of the first premium plus 2.5% of the second and subsequent premiums are more conveniently written as 2.5% of allpremiums, plus an additional 47.5% of the first premium.
By expressing the premium expenses this way, we can simplify the gross future loss random variable, and the subsequent premium calculation.
(b) We may look separately at the three parts of the gross premium equation of value. The EPV of premium income is
Pa¨[30]:25 =14.73113P.
Note thata¨[30]:25 can be calculated from Tables3.7and6.1.
The EPV of all expenses is
2000+0.475P+0.025Pa¨[30]:25 =2000+0.475P+0.025×14.73113P
=2000+0.843278P.
The EPV of the death benefit can be found using numerical integration or using Woolhouse’s formula, and we obtain
100 000A¯[30]:25 =100 000×0.298732=29 873.2 . Thus, the equivalence principle gives
P= 29 873.2+2 000
14.73113−0.843278 =$2 295.04 .
2 Example 6.7 Calculate the monthly gross premium for a 10-year term insurance with sum insured $50 000 payable immediately on death, issued to a select life
6.6 Gross premium calculation 153 aged 55, using the following basis:
Survival model: Standard Select Survival Model Assume UDD for fractional ages
Interest: 5% per year
Initial Expenses: $500 +10% of each monthly premium in the first year Renewal Expenses: 1% of each monthly premium in the second and
subsequent policy years
Solution 6.7LetPdenote the monthly premium. Then the EPV of premium income is 12Pa¨(12)
[55]:10. To find the EPV of premium related expenses, we can apply the same idea as in the previous example, noting that initial expenses apply to each premium in the first year. Thus, we can write the EPV of all expenses as
500+0.09×12Pa¨(12)
[55]:1 +0.01×12Pa¨(12)
[55]:10
where the expenses for the first year have been split as 9% plus 1%, so that we have 9% in the first year and 1% every year. The EPV of the insurance benefit is 50 000A¯ 1
[55]:10 and so the equivalence principle gives 12P
0.99¨a(12)
[55]:10 −0.09a¨(12)
[55]:1
=500+50 000A¯[1
55]:10. We find thata¨(12)
[55]:10 = 7.8341,a¨(12)
[55]:1 = 0.9773 and A¯ 1
[55]:10 = 0.024954, givingP=$18.99 per month.
Calculating all the EPVs exactly gives the same answer for the premium to
four significant figures. 2
Example 6.8Calculate the gross single premium for a deferred annuity of
$80 000 per year payable monthly in advance, issued to a select life now aged 50 with the first annuity payment on the life’s 65th birthday. Allow for initial expenses of $1 000, and renewal expenses on each anniversary of the issue date, provided that the policyholder is alive. Assume that the renewal expense will be $20 on the first anniversary of the issue date, and that expenses will increase with inflation from that date at the compound rate of 1% per year. Assume the Standard Select Survival Model with interest at 5% per year.
Solution 6.8The single premium is equal to the EPV of the deferred annuity plus the EPV of expenses. The renewal expense on thetth policy anniversary
is 20 1.01t−1
fort=1, 2, 3,. . .so that the EPV of renewal expenses is 20
∞ t=1
1.01t−1vttp[50]= 20 1.01
∞ t=1
1.01t vt tp[50]
= 20 1.01
∞ t=1
vtj tp[50]
= 20
1.01(¨a[50]j−1)
where the subscriptjindicates that the calculation is at rate of interestjwhere 1.01v=1/(1+j), that is j = 0.0396. The EPV of the deferred annuity is 80 00015|¨a([1250]), so the single premium is
1 000+ 20
1.01(¨a[50]j−1)+80 00015|¨a([1250]).
Asa¨[50]j =19.4550 and15|¨a[(5012)] =6.04129, the single premium is $484 669.
2 We end this section with a comment on the premiums calculated in Examples 6.6and6.7. In Example6.6, the annual premium is $2295.04 and the expenses at time 0 are $2 000 plus 50% of the first premium, a total of $3146.75, which exceeds the first premium. Similarly, in Example6.7the total premium in the first year is $227.88 and the total expenses in the first year are $500 plus 10%
of premiums in the first year. In each case, the premium income in the first year is insufficient to cover expenses in the first year. This situation is common in practice, especially when initial commission to agents is high, and is referred to as new business strain. A consequence of new business strain is that an insurer needs to have funds available in order to sell policies. From time to time insurers get into financial difficulties through pursuing an aggressive growth strategy without sufficient capital to support the new business strain. Essentially, the insurer borrows from shareholder (or participating policyholder) funds in order to write new business. These early expenses are gradually paid off by the expense loadings in future premiums. The part of the premiums that funds the initial expenses is called thedeferred acquisition cost.