Chapter 9
Insurance Models Including Expenses
9.1 Insurance Models Including Expenses
Recall the basic pricing equation:
actuarial p.v. of premiums = actuarial p.v. of benefits
e.g., for a $1000 whole life insurance policy with benefits payable at the end of the year and premiums payable at the beginning of the year, pricing equation becomes
i.e.
Generalization:
Actuarial p.v. of premiums = (actuarial p.v. of benefits) + (actuarial p.v. of expenses).
The resultant premium G = expense loaded premium
= benefit premium (P) + expense premium (e)
Typically, G is loaded for profits and contingencies and the resultant premium charged is the gross premium.
9.1.1 Types of Expenses
1. As a percentage of gross premiums, e.g. commissions, premium taxes. 2. Per policy expenses, e.g. policy fees, settlement costs, issue costs. 3. Per $1000 of insurance, e.g. underwriting costs.
Example 9.1.1 Suppose the expenses on an ordinary life policy are assumed to be as
follows:
Other expenses amount to $10 at the beginning of the first year and $2 at the beginning of each subsequent year per $10,000 of insurance.
Cost of settlement of claim is $5 per $10,000 of insurance.
Assuming death benefits are paid at the end of the year of death, determine the annual premium G payable at the beginning of each year for a $100,000 whole life policy.
Solution
actuarial p.v. of premiums
actuarial p.v. of % premium expenses
actuarial p.v. of expenses per $10,000 of insurance
actuarial p.v. of benefits
Pricing equation is then
i.e.
Example 9.1.2 For a single premium, continuous whole life insurance issued to with face amount f, you are given:
(i)
(ii) % of premium expenses are 8% of the expense loaded premium.
(iii) Per policy expenses are 75 at the beginning of the first year and 25 at the beginning of each subsequent year.
(iv) Claim expenses are 15 at the moment of death (v)
(vi) Deaths are uniformly distributed over each year of age (vii) The expense loaded premium is expressed as
Calculate h
Solution
actuarial p.v. of per premium expenses
actuarial p.v. of per policy expenses
actuarial p.v. of claim expenses
actuarial p.v. of benefits
i.e.
Equating coefficients, we get
From tables at ,
i.e.
i.e.
9.1.2 Expense reserve and benefit reserves
If net or benefit premium such that actuarial p.v. of net premium = actuarial p.v. of benefits
expense loading or expense premium and satisfies actuarial p.v. of expense loadings = actuarial p.v. of expenses
expense loaded premium
we can define a benefit reserve and an expense reserve the following way:
benefit reserve at
= (actuarial p.v. of future benefits at )-(actuarial p.v. value of future benefit premiums at )
expense reserve at
= (actuarial p.v. of future expenses at )-(actuarial p.v. value of future expense premiums at )
Since mortality increases by duration and expenses decrease by duration, with level expense and benefit premiums, expense reserves are generally negative and benefit reserves are non-negative.
,
We can similarly develop recursive formulas for benefit and expense reserves.
Assume expenses at the beginning of duration t are and upon death, settlement
costs are ,
Then
and
If we define total reserve at
then
We can also define prospective loss random variables separately for expenses and benefits in the same manner as in net level premium reserves.
E[prospective expense loss random variable at duration t]
E[prospective benefit loss random variable at duration t]
Example 9.1.3 A fully discrete three year endowment insurance of 1,000 issued to (x) has
level expense loaded premium, G equal to the net level premium plus an expense loading e.
You are given:
(i) Expenses incurred at the beginning of the year are:
First Year Renewal Years
% of G 18% 7%
Per policy 13 5
(ii) The expense reserve two years after issue equals – 16.10 (iii) G=342.86
Calculate
x x+1 x+2 x+3
actuarial p.v. of future expenses – actuarial p.v. of future expense premiums
i.e.
Example 9.1.4 The expense loaded premium G for a fully discrete, three year endowment
insurance of 1,000 issued to (x) is calculated using the equivalence principle. Expenses are paid at the beginning of each year.
You are given:
(i) (ii)
(iii)
(iv) (v) (vi)
Expenses % of premium Per policy
First Year 30% 8
Renewal 10% 4
Calculate the expense reserve at the end of the first year.
i.e.
;
i.e.
Example 9.1.5 For a fully continuous whole life insurance of $1 issued to (x), the
expense-augmented loss variable is given as:
where and
initial expenses
annual rate of continuous maintenance expenses
annual expense loading in the premium
You are given:
(i) (ii) (iii) (iv) (v)
Net and expense loaded premiums are calculated according to the equivalence principle.
Calculate
Substituting,
9.2 Miscellaneous Topics: Per Policy Expenses
Three ways to price for per policy expenses.
1. Policy fee method.
a. Determine a policy fee to be charged each year, satisfying
actuarial p.v. of policy fee = actuarial p.v. of per policy expenses + actuarial p.v. of percent of premium expenses applied to policy fee.
Example 9.2.1 For a whole life insurance policy on x, suppose per policy expenses are $10
in the first year and $2 in renewal years. Per premium expenses are 50% of gross premium in year one and 20% of gross premium in renewable years.
Solving,
b. Solving for a gross premium rate per 1,000 such that
actuarial p.v. of = actuarial p.v. of face amount 1,000 + actuarial p.v. of expenses excluding per policy expenses
Example 9.2.2 Suppose for the same example above, we have in addition per 1,000 of face
expenses which are $3nin year one and $1 in renewable years.
Then satisfies
Solving,
c. Gross premium charged for a face amount
Example 9.2.3 Gross premium for a 100,000 policy
Note: The policy fee method is the theoretically correct method which fully recognizes all
expenses in pricing.
2. Approximate premium rate method.
a. Assume an average policy face amount
b. Determine the gross premium for such that
c. Then gross premium rate per 1,000
d. Gross premium charged for a face amount equals
Example 9.2.4 Same example as before. Suppose assumed average policy size equals
50,000. Then satisfies
Solving,
Gross premium charged for a 100,000 policy
3. Band method or quantity discount method.
Different rates per 1,000 are charged for different bands of face amounts. a. Determine face amount bands, e.g. , ,
,
b. Solve for the gross premiums in a band for a given average face amount in the band.
c. Rate per 1,000 charged for the band
Example 9.2.5 Suppose the policy fee method calculates a premium rate per 1,000
excluding per policy expenses of $5 per 1,000 and a policy fee of $50.
Consider a banding method which charges $5.5 per 1,000 for face amounts 100,000 and $5.25 per 1,000 for face amounts >100,000.
i.e.
for second band satisfies
i.e.
i.e.
9.2.1 Relationship between methods
Since the policy fee method is the theoretically correct method for recognizing per policy expenses, every other method is an approximation to it. If the policy fee method charges per 1,000, then the correct rate per 1,000 for a face amount
Clearly, the larger is, the smaller the rate per 1,000 charged.
1. The approximation method = policy fee method for the assumed average policy size . For , the approximate premium method overcharges and for , it undercharges, relative to the policy fee method.
2. For the average face amount within a band, the band method = policy fee method. For and in the band, the band method overcharges and for and in the band, the band method undercharges, relative to the policy fee method.
3. At the minimum policy size, the policy fee method results in the highest rate per 1,000 and at the maximum policy size, the policy fee method results in the lowest rate per 1,000.
Rate per 1,000
policy fee method
XXXXXX Average size XXXXXX band method
XXXXXXXXX
XXXXXX
face amount
Min Max
Example 9.2.6 A whole life insurance has annual premiums payable at the beginning of the
year and death benefits payable at the moment of death. The following expenses are allocated to this policy at the beginning of each year.
% of premium per 1,000 face per policy
First year 30% 3.00 150
Renewal 10% 0.00 50
You are given:
(i) (ii)
A level policy fee is used to recognize per policy expenses in the expense loaded premium formula. Calculate the minimum face amount such that the policy fee doesn’t exceed 50% of the expense loaded premium.
Solution
Let face amount in 1,000’s
expense loaded premium satisfies
Solving,
Policy fee satisfies
i.e.
Want
i.e.
i.e.
i.e. minimum face amount = 3,000
9.3 Life Insurance Accounting
Consider an initial lives age each purchasing a whole life insurance policy.
Assume:
1. = annual gross premium charged, payable at the beginning of the year.
2. Expenses in year and are incurred at the beginning of the year. 3. death benefit paid at the end of the year of death.
Then net income at the end of the year t = (premium collected in year t) + (net investment income in year t) – (expense in year t)
– (increase in reserves)
i.e. (assets at the beginning of year t)
= (cash flow in year t) – (increase in reserves)
9.3.1 Balance Sheet
(assets)t - 1 + (cash flow)t = (assets)t
i.e. (assets)t - 1 + (NI)t + ( reserves)t = (assets)t
(Surplus)t = (assets)t - (reserves)t
= (assets)t - 1 + (NI)t + [(reserves)t -(reserves)t - 1 ] - (reserves)t
= (assets)t - 1 - (reserves)t - 1 + (NI)t
i.e. (Surplus)t = (Surplus)t - 1 + (NI)t
(Surplus)t - (Surplus)t - 1 = ( Surplus)t
= ( assets)t -( reserves)t
= (NI)t
= (cash flow)t - ( reserves)t
Then for any reserving system such that
, and (Surplus)0 = 0
(Surplus)n = ( Surplus)t
= (NI)t
= (cash flow)t
Example 9.3.1 Consider a two year endowment policy issued to x for $20,000. Assume:
1. Gross premium charged = $11,000 payable at the beginning of the year.
2. ;
;
3. Per premium expenses are 20% of gross premium in year one and 10% in year two. Per policy expenses are $10 in year one and $5 in year two.
4. Initial surplus = $10,000. 5. Earnings rate on assets = 10%
If 100 such policies are issued and the actual mortality experienced is and , develop income statements and balance sheets at the end of year one and year two. Assume benefits are payable at the end of the year and expenses are incurred at the beginning of the year.
Solution
Income Statement Year One
Revenue: Premiums = 11,000 X 100 = $1,100,000
Investment Income = 0.1[1,100,000 + 10,000 + 10 (100) – 0.2(1,100,000)]
= $88,900
Total Revenue = $1,188,900
Expenses: Per premium expenses = 0.2 (1,100,000) = $220,000
Per policy expenses = 10 X 100 = $1,000
Total expenses = $221,000
Benefits: Death Benefits = 0.1 (100) (20,000) = $200,000
Total expenses plus benefits = $421,000
Cash flow = 1,188,900 – 421,000 = $767, 900
Ending reserves
Increase in reserves = $756,000
Net Income = $11,900
Balance Sheet Year One
Beginning of year assets = $10,000
Beginning of year liabilities = 0
Beginning of year surplus = $10,000
End of year assets = 10,000 + 767,900 = $777,900
End of year liabilities = $756,000
End of year surplus = $21,900
Change in surplus = 21,900 – 10,000 = $11,900 = net income
Income Statement Year Two
Revenue: Premiums = 11,000 X 90 = $990,000
Investment Income = 0.1[777,900 + 990,000 - 5 (90) – 0.1(990,000)]
= $166,845
Total Revenue = $1,156,845
Expenses: Per premium expenses = 0.1 (990,000) = $99,000
Per policy expenses = 5 X 90 = $450
Total expenses = $99,450
Benefits: Death Benefits = 0.2 (90) (20,000) = $360,000
Endowment benefits = 0.8 (90) (20,000) = $1,440,000
Total expenses plus benefits = $1,899,450
Increase in reserves: Beginning reserves = $756,000
Ending reserves =0
Increase in reserves = -$756,000
Net Income = $13,395
Balance Sheet Year Two
Beginning of year assets = $777,900
Beginning of year liabilities = 756,000
Beginning of year surplus = $21,900
End of year assets = 777,900 – 742,605 = $35,295
End of year liabilities = $0
End of year surplus = $35,295
Change in surplus = 35,295 – 21,900 = $13,395 = net income
Note: Surplus at the end of the 2nd year = $35,295
=
= (net income)2 + (net income)1 + (initial surplus)
=13,395 + 11,900 + 10,000
=$39,295
If only benefit reserves are held, then for year one,
net income = cash flow - (benefit reserves)
= 767,900 – 90 (9,000) = -$42,100
Ending assets = $777,900
Ending liabilities = $810,000
Ending surplus = -$32,100
For year two,
net income = cash flow - (benefit reserves)
= -742,605 – (0 – 810,000) = $67,395
Ending benefits = $35,295
Ending liabilities = 0
Ending surplus = $35,295
Change in surplus = 35,295 – (- 32,100) = $67,395 = net income
Note:
1. Total surplus at the end of two years is unaffected by the reserving method used. Surplus at the end of the first year and net income in years one and two are affected.
2. Assets accumulated are unaffected by the reserving method used.
3. The use of an expense as well as a benefit reserve reduces fluctuation in net income.
Example 9.3.2
Premiums and expenses are paid at the beginning of the year. Claims and investment income are paid at the end of the year.
Calculate the interest rate earned on assets for year t.
Solution
(Surplus)t-1 (Reserve)t
t-1 t
(Expenses)t (Investment income)t (Premiums)t (Net income)t
(Net income)t =(Premiums)t+(Investment income)t - (Expenses)t- (Claims)t- ( Reserve)t
i.e. 552,000 = 850,000 +512,000 – 350,000 – 335,000 - ( Reserve)t
Solving, ( Reserve)t = (Reserve)t - (Reserve)t - 1 = 125,000
i.e., (Reserve)t - 1 = (Reserve)t - 125,000 = 4,680,000
(Assets)t - 1 = (Surplus)t - 1 + (Reserve)t - 1 = 1,220,000 + 4,680,000 = 5,900,000
(Investment income)t = i [ (Assets)t - 1 + (Premiums)t - (Expenses)t ]
i.e. 512,000 = i [5,900,000 + 850,000 - 350,000]
Solving, i = (512,000/6,400,000) = 0.08
9.3.2 Surplus and Net Income when Actual Experience Equals
Reserving Assumptions
Suppose = annual gross premium
= expense loaded premium =
1. premiums payable at the beginning of year 2. expenses incurred at beginning of year 3. benefits paid at the end of year
4. initial surplus = 0; initial lives
5. actual experience = reserve assumptions
We know
……….(1)
………..(2)
(1)+(2) ………(3)
where
t=1
Premiums
Expenses
Investment income
Claims
Reserves
(Net Income)1
(Net Income)1
= 0 from (3) since
i.e. (Net Income)1 = 0
(Assets)1 = (Reserves)1 + (Surplus)1
t = 2
Premiums
Expenses
Investment income
Claims
Reserves
(Net income)2
(Net income)2
= 0 from (3)
General:
(Net income)t
= 0
(Assets)t = (Reserves)t
Example 9.3.3 Consider a two year term insurance policy for $18,000. Assume
1. expenses = 40% of gross premium in year one. = 20% of gross premium in year two.
3. Expenses and premiums paid at beginning of year; claims paid at end of year.
If gross premium GP = expense loaded premium, then GP satisfies
Net premium P satisfies
e = expense loading
;
i.e.
i.e.
;
i.e.
i.e.
i.e. ;
Assume initial number of lives
Then
t=1
Premiums = 80(6,000) = 480,000
Expenses = 80(0.4)(6,000) = 192,000
Investment income = 0.5[480,000-192,000] = 144,000
Claims = 18,000(0.25)(80) = 360,000
Cash flow = 72,000
Reserves = 1,200(80)(0.75) = 72,000
Net income = 0
(Assets)1 = (Reserves)1 = 72,000
Premiums = 60(6,000) = 360,000
Expenses = 60(0.2)(6,000) = 72,000
Investment income = 0.5[360,000-72,000 + 72,000] = 180,000
Claims = 18,000(0.5)(60) = 540,000
Cash flow = -72,000
Reserves = 0 – 72,000
Net income = 0
9.3.3 Analysis of Surplus with Premium Loading
Suppose gross premium where
benefit premium, expense premium, loading.
Suppose initial surplus = 0; death benefit;
expenses in year,
initial number of ives
Then satisfies and
benefit reserve satisfies
……….(1)
satisfies and
expense reserve satisfies
(1) + (2) gives
where
i.e. ……….(3)
The net income in year one for an initial lives
from (3)
assets at the end of year one
from (3)
In general,
Surplus in year t per initial life
x x+1 ………. x+t-1 x+t
If initial surplus , then
In general,
Surplus in year t per initial life
= accumulated value of premium loading + accumulated value of initial surplus
If only benefit reserves are held, with initial surplus ,
i.e., surplus in year t per initial life
=
= accumulated value of [premium loading + excess of expense premium over actual expenses]+ accumulated value of initial surplus
9.3.4 Analysis of Surplus with Premium Loading and per
Premium Expenses
Suppose gross premium where
benefit premium, expense premium, loading.
initial surplus
expense in year t excluding per premium expenses
% of premium expense factor in year t
i.e. % of premium expense in year t
death benefit; initial number of lives
Then satisfies
and benefit reserves satisfies ;
………..(1)
and expense reserve satisfies ;
……….(2)
(1)+(2):
where
i.e. ………(3)
The net income in year one
Similarly,
Surplus in year t per initial life
=accumulated value of initial surplus + accumulated value of premium loading
If only benefit reserves are held, one can similarly show
Surplus in year t per initial life
= accumulated value of [premium loading + excess of expense premium over actual expenses] + accumulated value of initial surplus
Example 9.3.4 For a fully discrete whole life insurance, you are given:
(i) Gross annual premium = 10.0 (ii) Net annual premium = 9.0
(iii) Expenses incurred at the beginning of each year are 0.5 in the first year and increase at a compound rate of 10% each year.
(iv) for all x (v)
Calculate the expected surplus at the end of year three for each initial insured.
Solution
If only benefit reserves are held,
surplus per initial insured at end of year three
i.e.
Note:
This is the model solution given for this question. If we assume both benefit and expense reserves are held, then expense premium e satisfies
;
loading (i.e. negative loading)
accumulated value of premium loading
9.4 Modified Reserve Systems
Consider a general benefit reserving system for a whole life insurance plan having benfit
premiums such that
Then and by the formula connecting successive terminal reserves,
etc.
If level gross premium charged and
expenses incurred in the beginning of year
then expense loading in the gross premium for year .
one can show that for an initial lives,
surplus at the end of year
= accumulated value to end of year of excess of expense loadings over expenses
In particular, surplus at the end of year 1
Clearly the smaller is, the larger the value of . The smallest value of occurs under the net level premium (NLP) reserve system.
i.e. and satisfies
Modified reserve systems are modifications to the net level premium reserve system. They are set up to increase the expense loading . This is to recognize the higher first year expenses and help decrease the first year surplus strain.
The general structure of a modified reserve system has three levels of benefit premiums. For an h-pay policy with a j-year modification period,
first year benefit premium renewal premium for years
net level premium for years and they satisfy
………(1)
Since the modified reserve increases the first year expense loading,
i.e.
Since ,
i.e. ……….(2)
i.e.
Also, from (1)
i.e. ……….(3)
For an h-pay whole life insurance policy,
modified tth year reserve
tth year NLP reserve
For ,
For ,
i.e. over the modification period j,
modified reserves < net level premium reserves
Since (surplus)t = (assets)t - (reserves)t
and (assets)t (cash flow)k is unaffected by the reserving method,
Alternative interpretation
Suppose extra first year expense
annual expense
i.e. total first year experience.
If expense loaded premium,
then
and ………(1)
……….(2)
Equating (1) and (2), we get
Extra first year expense
annual expense
Now consider an expense reserve system over the modification period such that the
expense premium satisfies
i.e.
For , actuarial p.v. of future expenses - actuarial p.v. of future expense premiums
Then for
=(NLP reserve) + (“expense reserve”)
=(NLP reserve) – (unamortized portion of extra first year expense)
Note: extra first year expense = first year expense allowance
9.4.1 Full Preliminary Term (FPT)
Allows for the largest expense loading in the first year such that resulting first year reserve
i.e.
i.e.
For the FPT method,
1.
2. modified period = entire premium paying period.
For a whole life insurance plan, premiums payable for h years,
= annual premium for a whole life insurance policy on (x+1) with premiums payable for (h-1) years.
In general, for an h-pay, n year benefit term insurance,
Then for ,
=NLP reserve at duration (t-1) for an (h-1) pay, (n-1) year benefit term insurance on (x+1)
For ,
Example 9.4.1 You are given:
(i) (ii)
(iii)
(iv)
Calculate
Solution
i.e.
i.e.
9.4.2 Commissioner’s Reserve Valuation Method (CRVM)
Modification of the FPT method and separates policies into two classes:
1. “low premium” policies, and 2. “high premium” policies.
For “low premium” policies, FPT method allowed, i.e., maximum first year expense loading
allowed and .
For “high premium” policies, first year expense loading allowed is less than under the FPT
and .
Definition
(renewal premium under FPT for this policy) (renewal premium under FPT for a 20-pay life).
i.e.
For a “high premium” policy, allowable first year expense loading satisfies
Since ,
we have where
h= premium paying period = modification period.
Then
Note: One can show that for “high premium” policies, first year expense loading
first year expense loading under FPT
i.e. and
Example 9.4.2 A fully discrete whole life insurance is issued to (x)
You are given:
(ii) (iii) (iv)
Calculate the first year net premium under the Commissioner’s valuation standard.
Solution
Assuming premiums are payable for life,
i.e.
Solving
i.e.
9.4.3 Policies with Non-level Premium and Non-level Benefits
Suppose we have an h-pay, n year benefit policy with gross premiums ,
The CRVM standard is defined as follows:
Define ELRA = equivalence level renewal amount such that
actuarial p.v. at (x+1) of future level ELRA
= actuarial p.v. at (x+1) of future policy benefits.
i.e.
i.e.
Note
If the policy has an endowment benefit, it is excluded in calculating the ELRA.
Define renewal net premiums
, such that
actuarial p.v. at (x+1) of future renewal net premiums
= actuarial p.v. at (x+1) of future policy benefits (including any endowment benefit)
i.e.
i.e.
FPT is allowed if:
Then
and ,
1. actuarial p.v. of modified net premiums
= actuarial p.v. of policy benefits
2.
i.e.
like in level premium, level benefit case.
If ,
let excess first year expense allowance (i.e. )
Then since , we equivalently have in the non level benefit, non level premium case,
and
Note: In calculating and , any endowment benefit in the policy must be included in calculating actuarial p.v. of benefits.
Example 9.4.3 For a fully discrete three year endowment insurance with non-level
benefits and non-level premiums issued to (x), you are given for policy year j:
1 1 0.5 1/10
2 2 1.0 1/9
3 1 0.5 1/8
You are also given:
(i)
(ii) The maturity benefit is 1
Calculate
Solution
ELRA satisfies
p.v. of level ELRA future benefits at (x+1)
= p.v. of future policy benefits (excluding endowment) at (x+1)
i.e.
i.e.
satisfies = net premium and
p.v. at (x+1) of future net premiums=p.v. at (x+1) of future benefits (including endowment)
i.e. i.e.
i.e.
9.4.4 Canadian Standard
Modified reserve method applies over the entire premium paying period with modified
premiums , satisfying
……….(1)
………..(2)
where
and 150% of net level premium
actuarial acquisition expenses
actuarial p.v. at (x) of renewal less actuarial p.v. of renewal administrative expenses plus policyholder expenses.
where administrative expense incurred at beginning of year t
policyholder dividend paid at end of year t
Note
From (2),
= actuarial p.v. of benefits
Example 9.4.4 A fully discrete 30 year term insurance of 10,000 is issued to (x).
You are given:
(i) Net premium = 250 (ii) Gross premium = 320 (iii) Acquisition expenses = 350
(iv) Renewal administrative expenses in years 2-30 equal 7.5% of gross premium (v) Policyholder dividends = 0
(vi)
Calculate the initial reserve at time zero under the New Canadian Modified Reserve Method for this policy.
Solution
Initial reserve at time zero
i.e.
