Premiums#
When insurance company agrees to pay some life contingent benefits, the policyholder agrees to pay premiums to the insurance company to secure these benefits. The premiums will also need to reimburse the insurance company for the expenses associated with the policy.
The calculation of the premium may not explicitly allow for the insurance company’s expenses, which we refer to as a net premium (also called a risk premium or benefit premium). If the calculation does explicitly allow for expenses, the premium is called a gross premium (also called expense-loaded premium).
Premiums for life insurance are payable in advance, with the first premium payable when the policy is purchased.
Fully continuous insurance: both benefits and premiums are payable continuously
Fully discrete insurance: benefits are paid at the end of the year, premiums are paid at the beginning of the year
Semi-continuous insurance: benefits are paid at moment of death, premiums are paid at the beginning of the year
Present value of loss at issue r.v. \(_0L\)#
The loss at issue random variable is defined as the present value of the amount the insurance loses.
\(_0L\) = PV of future benefits - PV of future premiums
PV of loss at issue
Fully continuous whole life insurance:
\(_0L = v^{T_x} - P \overline{a}_{\overline{T_x|}} = v^{T_x} - P (\dfrac{1 - v^{T_x}}{\delta})\)
loss at issue is the present value of the insurance payment less the present value of the level premimum annuity of \(P\) per year
\(_0L = \overline{A}_x - P \overline{a}_x = \overline{A}_x(1 + \dfrac{P}{\delta}) - \dfrac{\pi}{\delta}\)
expected loss at issue by applying expected values to the loss at issue equation
Equivalence principle#
Under this principle, premiums are set such that the actuarial present value of the benefit premiums equals the actuarial present value of the benefits, hence expected loss at issue is zero:
\(~ E[_0L] = EPV_0 (\textsf{future benefits}) - EPV_0 (\textsf{future premiums}) = 0\)
Net premium#
For net premiums, we take into consideration outgoing benefit payments only: expenses are not a part of the calculation. The benefit may be a death benefit or a survival benefit or a combination. Under the equivalence principle, the net premium is set such that the expected value of the future loss is zero at the start of the contract, \(E[L_0] = 0\).
Whole life insurance:
\(P_{x} = \dfrac{A_{x}}{\ddot{a}_{x}}\)
fully discrete whole life insurance net premium
\(\overline{P}_{x} = \dfrac{\overline{A}_{x}}{\overline{a}_{x}}\)
fully continuous whole life insurance net premium
Term life insurance:
\(P^{1}_{x:\overline{t|}} = \dfrac{A^{1}_{x:\overline{t|}}}{\ddot{a}_{x:\overline{t|}}}\)
fully discrete term life net premium
\(\overline{P}^{1}_{x:\overline{t|}} = \dfrac{\overline{A}^{1}_{x:\overline{t|}}}{\overline{a}_{x:\overline{t|}}}\)
fully continuous term life net premium
Pure endowment:
\(P^{~~~1}_{x:\overline{t|}} = \dfrac{~_tE_x}{\ddot{a}_{x:\overline{t|}}}\)
fully discrete pure endowment net premium
\(\overline{P}^{~~~1}_{x:\overline{t|}} = \dfrac{~_tE_x}{\overline{a}_{x:\overline{t|}}}\)
fully continuous pure endowment net premium
Endowment insurance:
\(P_{x:\overline{t|}} = \dfrac{A_{x:\overline{t|}}}{\ddot{a}_{x:\overline{t|}}}\)
fully discrete endowment insurance net premium
\(\overline{P}_{x:\overline{t|}} = \dfrac{\overline{A}_{x:\overline{t|}}}{\overline{a}_{x:\overline{t|}}}\)
fully continuous endowment insurance net premium
Shortcuts for whole life and endowment insurance only:
For whole life and endowment insurance only, by plugging in the insurance or annuity twin, the following shortcuts are available for calculating net premiums from only the life insurance or annuity factor.
\(P_{x} = b ~ (\dfrac{1}{\ddot{a}_x} - d) = b ~ (\dfrac{dA_x}{1 - A_x})\ \)
fully discrete whole life shortcut
\(\overline{P}_{x} = b ~ (\dfrac{1}{\overline{a}_x} - \delta) = b ~ (\dfrac{d\overline{A}_x}{1 - \overline{A}_x})\)
fully continuous whole life shortcut
\(P_{x:\overline{t|}} = b ~ (\dfrac{1}{\ddot{a}_{x:\overline{n|}}} - d) = b ~ (\dfrac{dA_{x:\overline{n|}}}{1 - A_{x:\overline{n|}}})\ \)
fully discrete endowment insurance shortcut
\(\overline{P}_{x:\overline{t|}} = b ~ (\dfrac{1}{\overline{a}_{x:\overline{n|}}} - \delta) = b ~ (\dfrac{d\overline{A}_{x:\overline{n|}}}{1 - \overline{A}_{x:\overline{n|}}})\)
fully continuous endowment insurance shortcut
Gross premium#
When we calculate a gross premium for an insurance policy or an annuity, we take account of the expenses the insurer incurs. There are three main types of expense associated with policies – initial expenses, renewal expenses and termination or claim expenses.
Expenses:
\(e_i = \) initial_per_policy + initial_per_premium \(\times\) gross_premium
initial expenses at the beginning of year 1 when a policy is issued, which may be both proportional to premiums or may be ‘per policy’, meaning that the amount is fixed for all policies, and is not related to the size of the contract.
\(e_r = \) renewal_per_policy + renewal_per_premium \(\times\) gross_premium
renewal expenses in the beginning of each year 2+, may be both per policy or percent of premium.
\(E =\) settlement expense
is paid along with death benefit (\(b\)); hence total claim cost upon death is:
claim cost \(= b + E = \) death benefit + settlement expense.
Return of premiums paid without interest upon death:
\(EPV_0(\mathrm{return~of~premiums~paid}) = \sum_{k=0}^{t-1} ~ P (k+1) ~ v^{k+1} ~ _{k|}q_x = P \cdot (IA)^{1}_{x:\overline{t|}}\)
an additional benefit in some insurance policies, whose EPV can be calculated using an increasing insurance factor
Equivalence principle:
If gross premiums are set under equivalence principle, then expected gross future loss at issue equals zero:
\(~ E[_0L^g] = EPV_0 (\textsf{future benefits}) + EPV_0 (\textsf{future expenses})- EPV_0 (\textsf{future premiums}) = 0\)
Portfolio Percentile Premium#
The portfolio percentile premium principle is an alternative to the equivalence premium principle. Using the mean and variance of the future loss random variable, the portfolio percentile premium principle can be used to determine a premium. We assume a large portfolio of \(N\) identical and independent policies. The present value of the total future loss \(\overline{L}\) of the portfolio can be approximated by a normal distribution over the sum of the individual losses \(L_i\)
Note \(E[\overline{L}]\) and \(Var[\overline{L}]\) are functions of the unspecified premium \(P\). A probability percentile \(q\) (say, 95% confidence) and a threshold \(L^*\) (say, 0) are chosen, then \(P\) is solved for implicitly from the following equation, such that the probability of \(\overline{L}\) not exceeding \(L^*\) is \(q\)
Methods#
The Premiums class implements methods for computing net and gross premiums under the equivalence principle
import numpy as np
from actuarialmath import Premiums
import describe
describe.methods(Premiums)
class Premiums - Compute et and gross premiums under equivalence principle
Methods:
--------
net_premium(x, s, t, u, n, b, endowment, discrete, return_premium, annuity, initial_cost):
Net level premium for special n-pay, u-deferred t-year term insurance
insurance_equivalence(premium, b, discrete):
Compute whole life or endowment insurance factor, given net premium
annuity_equivalence(premium, b, discrete):
Compute whole life or temporary annuity factor, given net premium
premium_equivalence(A, a, b, discrete):
Compute premium from whole life or endowment insurance and annuity factors
gross_premium(a, A, IA, discrete, benefit, E, endowment, settlement_policy, initial_policy, initial_premium, renewal_policy, renewal_premium):
Gross premium by equivalence principle
Examples#
When net premiums are set by the equivalence principle, then the three class methods insurance_equivalence, annuity_equivalence or premium_equivalence can be called to compute the insurance, annuity or net premium respectively given any one of the other value, for whole life and endowment insurances. For other general life insurance or annuity benefits, net_premium computes the net premium under EPP given any term (t), deferral period (u), endowment benefit amount (endowment), other initial expected costs (initial_cost), or refund of premium without interest at death (return_premium).
life = Premiums().set_interest(delta=0.06)\
.set_survival(mu=lambda x,s: 0.04)
P = life.net_premium(x=0, discrete=False)
A = life.whole_life_insurance(x=0, discrete=False)
a = life.whole_life_annuity(x=0, discrete=False)
print('Insurance:', A, life.insurance_equivalence(premium=P, discrete=False))
print('Annuity:', a, life.annuity_equivalence(premium=P, discrete=False))
print('Net Premium:', P,
life.premium_equivalence(A=A, discrete=False),
life.premium_equivalence(a=a, discrete=False))
Insurance: 0.4 0.4
Annuity: 9.999999999999996 9.999999999999996
Net Premium: 0.040000000000000015 0.040000000000000015 0.040000000000000015
The gross_premium class method computes the level premium under the equivalence principles given expense and claim amounts (initial and renewal expenses per policy or per $1 premium, settlement expense, death benefit, and endowment) and corresponding actuarial present value factors (A for insurance benefits, a for premium annuity, IA for refund of premium without interest death benefit, and E for endowment benefit).
SOA Question 6.2
For a fully discrete 10-year term life insurance policy on (x), you are given:
Death benefits are 100,000 plus the return of all gross premiums paid without interest
Expenses are 50% of the first year’s gross premium, 5% of renewal gross premiums and 200 per policy expenses each year
Expenses are payable at the beginning of the year
\(A^1_{x:\overline{10|}} = 0.17094\)
\((IA)^1_{x:\overline{10|}} = 0.96728\)
\(\ddot{a}_{x:\overline{10|}} = 6.8865\)
Calculate the gross premium using the equivalence principle.
print("SOA Question 6.2: (E) 3604")
life = Premiums()
A, IA, a = 0.17094, 0.96728, 6.8865
print(life.gross_premium(a=a,
A=A,
IA=IA,
benefit=100000,
initial_premium=0.5,
renewal_premium=.05,
renewal_policy=200,
initial_policy=200))
SOA Question 6.2: (E) 3604
3604.229940320728
SOA Question 6.16
For a fully discrete 20-year endowment insurance of 100,000 on (30), you are given:
d = 0.05
Expenses, payable at the beginning of each year, are:
First Year |
First Year |
Renewal Years |
Renewal Years |
|
|---|---|---|---|---|
Percent of Premium |
Per Policy |
Percent of Premium |
Per Policy |
|
Taxes |
4% |
0 |
4% |
0 |
Sales Commission |
35% |
0 |
2% |
0 |
Policy Maintenance |
0% |
250 |
0% |
50 |
The net premium is 2143
Calculate the gross premium using the equivalence principle.
print("SOA Question 6.16: (A) 2408.6")
life = Premiums().set_interest(d=0.05)
A = life.insurance_equivalence(premium=2143, b=100000)
a = life.annuity_equivalence(premium=2143, b=100000)
p = life.gross_premium(A=A,
a=a,
benefit=100000,
settlement_policy=0,
initial_policy=250,
initial_premium=.04+.35,
renewal_policy=50,
renewal_premium=.04+.02)
print(A, a, p)
SOA Question 6.16: (A) 2408.6
0.3000139997200056 13.999720005599887 2408.575206281868
SOA Question 6.20
For a special fully discrete 3-year term insurance on (75), you are given:
The death benefit during the first two years is the sum of the net premiums paid without interest
The death benefit in the third year is 10,000
\(x\) |
\(p_x\) |
|---|---|
75 |
0.90 |
76 |
0.88 |
77 |
0.85 |
\(i = 0.04\)
Calculate the annual net premium.
print("SOA Question 6.20: (B) 459")
l = lambda x,s: dict(zip([75, 76, 77, 78],
np.cumprod([1, .9, .88, .85]))).get(x+s, 0)
life = Premiums().set_interest(i=0.04).set_survival(l=l)
a = life.temporary_annuity(75, t=3)
IA = life.increasing_insurance(75, t=2)
A = life.deferred_insurance(75, u=2, t=1)
print(life.solve(lambda P: P*IA + A*10000 - P*a, target=0, grid=100))
SOA Question 6.20: (B) 459
458.83181728297285
SOA Question 6.29
(35) purchases a fully discrete whole life insurance policy of 100,000. You are given:
The annual gross premium, calculated using the equivalence principle, is 1770
The expenses in policy year 1 are 50% of premium and 200 per policy
The expenses in policy years 2 and later are 10% of premium and 50 per policy
All expenses are incurred at the beginning of the policy year
\(i = 0.035\)
Calculate \(\ddot{a}_{35}\).
print("SOA Question 6.29 (B) 20.5")
life = Premiums().set_interest(i=0.035)
def fun(a):
return life.gross_premium(A=life.insurance_twin(a=a),
a=a,
benefit=100000,
initial_policy=200,
initial_premium=.5,
renewal_policy=50,
renewal_premium=.1)
print(life.solve(fun, target=1770, grid=[20, 22]))
SOA Question 6.29 (B) 20.5
20.480268314431726
SOA Question 6.33
An insurance company sells 15-year pure endowments of 10,000 to 500 lives, each age x, with independent future lifetimes. The single premium for each pure endowment is determined by the equivalence principle.
You are given:
\(i\) = 0.03
\(\mu_x(t) = 0.02 t, \quad t \ge 0\)
\(_0L\) is the aggregate loss at issue random variable for these pure endowments.
Using the normal approximation without continuity correction, calculate \(Pr(_0L) > 50,000)\).
print("SOA Question 6.33: (B) 0.13")
life = Premiums().set_survival(mu=lambda x,t: 0.02*t).set_interest(i=0.03)
var = life.E_x(x=0, t=15, moment=life.VARIANCE, endowment=10000)
p = 1 - life.portfolio_cdf(mean=0, variance=var, value=50000, N=500)
print(p)
SOA Question 6.33: (B) 0.13
0.12828940905648634
SOA Question 5.6:
For a group of 100 lives age x with independent future lifetimes, you are given:
Each life is to be paid 1 at the beginning of each year, if alive
\(A_x = 0.45\)
\(_2A_x = 0.22\)
\(i = 0.05\)
\(Y\) is the present value random variable of the aggregate payments.
Using the normal approximation to Y, calculate the initial size of the fund needed in order to be 95% certain of being able to make the payments for these life annuities.
print("SOA Question 5.6: (D) 1200")
life = Premiums().set_interest(i=0.05)
var = life.annuity_variance(A2=0.22, A1=0.45)
mean = life.annuity_twin(A=0.45)
fund = life.portfolio_percentile(mean, var, prob=.95, N=100)
print(fund)
SOA Question 5.6: (D) 1200
1200.6946732201702