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AS3429 Ch.6 Lecture notes (W2018)

Ch. 6 Premium Calculations

6.1 Summary 6.2 Preliminaries 6.3 Assumptions

6.4 Present Value of future loss random variable

6.5 Equivalence Principle

6.6 Gross Premium Calculation

6.7 Profit

6.8 Portfolio Percentile Premium Principle

6.9 Extra Risks

Note: additions//updates will be made to posted lecture notes as we work through the material

AS3429 Ch.6 Lecture notes (W2018)

6.1 Summary • Chapter focus is on how to determine premium calculations for

life insurance and life annuities, with a greater focus on life insurance

• Two different premium principles considered o Equivalence principle (most common for traditional policies)

o Portfolio percentile principle

6.2 Preliminaries

(i) Premium Types

(a) Net Premium

o considers benefit costs only, excludes expenses (and profit)

(b) Gross Premium

o explicitly allows for expenses

AS3429 Ch.6 Lecture notes (W2018)

6.2 Preliminaries(continued)

(ii) Premium Frequency Life Insurance o Typically periodic premium payments(eg annually, monthly, bi-weekly) o Periodic premiums are usually a level $ amount but they can be

varying (e.g. step rate)

Life Annuities o Whole/temporary life annuities usually single premium purchase o Deferred life annuities may be purchased with single premium or

periodic premiums during deferred period (pension funding applications)

(iii) Premium Timing and duration • Premiums are payable in advance with first premium payable when

policy is purchased

• Periodic premiums can be paid for the life or term of an Insurance policy or shorter(e.g. Limited Pay Whole Life), and cease to be payable on death of the insured

AS3429 Ch.6 Lecture notes (W2018)

6.3 Assumptions and Table sources

• The Standard Select Survival Model(SSSM)’ is default model for

text examples and exercises. The model assumes i=5% and;

µx =A+Bc x , A=0.00022, B=2.7x10

-6 , C=1.124

µ[x]+s = 0.9 2-s

µx+s for 0 ≤ s ≤ 2

l20=100,000(radix)

• as noted previously, copy of Appendix D(SSSM) is posted - it also includes several values(e.g. select Insurance factors, pure

endowment factors, second moments for whole life insurance) - SOA LTAM tables now used are ultimate part of SSSM (copy posted)

• Standard Select Survival model(SSSM) Excel Worksheets - required for some text questions(as well as to understand some text examples) - recommend you set up own SSSM worksheet for your own use and/or use the

worksheet posted on OWL - author spreadsheet(SSSM) referenced earlier (Life Cons review notes) includes

several more factors than Table D

• ILT (copy posted)& other tables may be used for some class examples

AS3429 Ch.6 Lecture notes (W2018)

6.4 P.V. of the Future Loss Random Variable

•••• Cash flows associated with life insurance/annuity policies are generally life contingent

o This includes expenses and premiums as well as benefits •••• Can model (future outgo-future income) with the random variable

that represents the present value(P.V.) of future loss o Net Future loss- excludes expenses o Gross Future loss- includes expenses o Future loss random variable depends on time of death

Future Loss Random Variable Notation:

nL = PV benefit outgo – PV Net premium income

gL = PV benefit outgo + PV expenses – PV Gross premium income

AS3429 Ch.6 Lecture notes (W2018)

Example 1(6.4)

A whole life insurance policy is issued to [45]. The sum insured is $25,000 payable immediately on death. Interest is at i =5%. A premium of $500 is paid at the beginning of each year for the policy. The policyholder dies at the end 25.8 years. Calculate the future loss at issue.

AS3429 Ch.6 Lecture notes (W2018)

Example 2

A whole life insurance policy is issued to [45]. The sum insured is $200,000 payable immediately on death. Premiums of amount P are payable annually in advance, ‘ceasing at age 65 or on earlier of death’ (20 Pay Whole Life Policy).What is the future loss random

variable for this policy?

•••• Distribution of future loss random variable used to determine

Premiums for given benefit/benefit for given premiums, using a Premium principle o Equivalence Principle (“benchmark principle”) o Portfolio Percentile Principle o Cash flow Profit Criterion Method (Ch.12)

AS3429 Ch.6 Lecture notes (W2018)

6.5 Equivalence Principle

•••• Under the equivalence principle, premiums are determined such that Expected value of the future loss is zero at contract issue

•••• Equivalence principle most common method for traditional insurances and is text default method for these policies (e.g Ch.6,7)

(i) Net Premiums

•••• Ignore expenses

•••• Benefits include either and/or both of death benefit/survivor benefits

•••• Equivalence principle means that E[L n

0] =0 , and since

L n

0 = PV benefit outgo – PV Net premium income, then under

equivalence principle EPVbenefit outgo= EPVnet premium income or PVFB = PVFP (at policy issue)

AS3429 Ch.6 Lecture notes (W2018)

6.5 Example 1 (SSSM is text default model) Consider a 20 pay whole life insurance with annual premiums issued to [50]. The death benefit(sum insured) is $500,000 payable at end of year of death.

(a) What is an expression for net future loss random variable?

(b) Calculate the net annual premium using the SSSM

AS3429 Ch.6 Lecture notes (W2018)

Example 2 (a few working pages are attached) Consider an endowment insurance with term n years and sum insured S payable at the earlier of the end of the year of death or at maturity, issued to a select life aged x. Premiums of amount P are payable annually throughout the term of the insurance (same as saying P’s are paid for the life of the policy).

(A) Derive expressions in terms of S, P and standard actuarial functions for;

(i) The net future loss, Ln 0 (ii) The mean of Ln 0, (iii) The variance of Ln 0

(B) Assume S=150,000; n=20, [x]=50 and SSSM

(iv) Calculate the annual premium (equivalence principle ) (v) Suppose now the premium is payable quarterly. What is the quarterly premium? Assume UDD within each year of age and you are provided factors for α(4) and β(4).

AS3429 Ch.6 Lecture notes (W2018)

Example 2(working page)

AS3429 Ch.6 Lecture notes (W2018)

Example 2(working page)

AS3429 Ch.6 Lecture notes (W2018)

Example 2–annuity factor calculation for B (v)

)4(

|20:]50[ a&&

= )4(

]50[a&& )4(

70]50[20 aE &&−−−−

)m(

|n:]x[ a&&

= )m(

]x[a&& )m(

n]x[]x[n aE +− &&

Under UDD : )m(Ba)m(a ]x[ )m(

]x[ −α≈ &&&&

or use: ]x[)m(

)m(

]x[ A i

i A ≈

in identity:

)m(

]x[)m(

]x[ d

)A1( a

−

=&&

Also, can show under UDD;

≈ )m(

|n:]x[ a&& )E1)(m(Ba)m( ]x[n|n:]x[ −−α &&

)4(

|20:]50[ a&& 598403.12)E1)(4(Ba)4( ]50[20|20:]50[ =−−α≈ &&

AS3429 Ch.6 Lecture notes (W2018)

Example 3 (Text 6.4)

An insurer issues a “regular premium deferred annuity contract” to a select life aged x. Premiums are payable monthly throughout the deferred period. The annuity benefit of X per year is payable monthly in advance from age x+n for the remainder of the life of (x).

(a) Write down the net future loss random variable in terms of lifetime random variables for [x]. (b) Derive an expression for the monthly net premium.

(c) Assume now that, in addition, the contract offers a death benefit of S payable immediately on death during the deferred period. Write the net future loss random variable for the contract, and derive an expression for the monthly net premium

AS3429 Ch.6 Lecture notes (W2018)

Example 3 (Text 6.4)

An insurer issues a regular premium deferred annuity contract to a select life aged x. Premiums are payable monthly throughout the deferred period. The annuity benefit of X per year is payable monthly in advance from age x+n for the remainder of the life of x.

(a) Write down the net future loss random variable in terms of lifetime random variables for [x].

Ln0 = 0 – 12P |12/1K )12(

a +&& for T[x] ≤ n or K[x] (12)< n K ≡ K[x]

(12)

Ln0 = |n12/1K )12(naXv −+&& – 12P |n

)12( a&& for T[x] >n or K[x]

(12) ≥ n (b) Derive an expression for the monthly net premium.

AS3429 Ch.6 Lecture notes (W2018)

Example 3(working page)

AS3429 Ch.6 Lecture notes (W2018)

Example 4 You are given an extract(see below) from a life table with a four year select period. A three year term insurance with death benefit of $S is purchased by [41] with a net premium of $350 payable annually. Assume i=6% and that the death benefit is payable at end of year of death.

Calculate

(i) S(sum insured) assuming the equivalence principle (ii) standard deviation of L0 (iii) Pr [L0 >0]

[[[[x] ] ] ] l[[[[x]]]] l [[[[x]+]+]+]+1 l[[[[x]+2]+2]+2]+2 l[[[[x]+3]+3]+3]+3 lx++++4444 x++++4444 [40] 100,000 99,899 99,724 99,520 99,288 44 [41] 99,802 99,689 99,502 99,283 99,033 45 [42] 99,597 99,471 99,268 99,030 98,752 46

AS3429 Ch.6 Lecture notes (W2018)

Example 4(working page)

AS3429 Ch.6 Lecture notes (W2018)

Example 4(working page

(ii) standard deviation of L0 or √Var[L0] calculations : (using S=216,326.38 from (i))

k L0 Pr(K41=k) 0

49.731,203a350Sv 1

=− &&

q[41] = 0.0011322 =1— l[41]+1/ l[41]

52.849,191a350Sv 2

2 =− &&

1│q[41]= 0.0018737

11.640,180a350Sv 3

3 =− &&

2│q[41] = 0.0021943

≥3

69.991a3500 3

−=− && 3p[41] = 0.9947997

Var[L0]=E[L0 2 ]–[E[L0]]

2 =E[L0

2 ], when equivalence principle used (to determine P)

can show Var[L0] =188,537,738 � √Var[L0] = 13,731

Table above provides Lo values for given values of k. General expression for Lo is;

Ln0 = Sv K+1 – 350 |1Ka ++++&& for K[41]< 3 K ≡ K[41]

Ln0 = 0 – 350 |3a&& for K[41] ≥ 3

AS3429 Ch.6 Lecture notes (W2018)

Example 5 A select life aged 45 buys a policy with a single premium(P). The policy provides an annuity of $30,000 per year payable annually in advance from age 65. In the event of death before age 65, the premium is returned at the end of year of death. Assume SSSM applies (used Appendix D for calculations)

(a) Give an expression for Ln0

(b) Calculate P assuming the equivalence principle

(c) Now assume annuity is guaranteed to be paid for at least 5 years if insured survives to 65. How much does P increase by?

AS3429 Ch.6 Lecture notes (W2018)

Example 5 continued

A select life aged 45 buys a policy with a single premium(P). The policy provides an annuity of $30,000 per year payable annually in advance from age 65. In the event of death before age 65, the premium is returned at the end of year of death. Assume SSSM applies ( Appendix D used)

(a) Ln0 = Pv K+1 – P for K[45] < 20 (K=K[45])

Ln0 = |201K 20 av000,30

−−−−++++

&& – P for K[45] ≥ 20

(b) Calculate P assuming the equivalence principle

(c ) Now assume annuity is guaranteed to be paid for at least 5 years if insured survives to 65. How much does P increase by?

AS3429 Ch.6 Lecture notes (W2018)

Example 5(working page)

AS3429 Ch.6 Lecture notes (W2018)

Example 6

A 20-pay $250,000 whole life insurance policy is purchased on (45) where the premiums are payable monthly and the death benefit is payable at moment of death. Assume that i=6% and ILT mortality applies. Also assume UDD within each year of age. Calculate the monthly premium(P).

Answer: P = $384.34

AS3429 Ch.6 Lecture notes (W2018)

Example 6 (working page)

AS3429 Ch.6 Lecture notes (W2018)

Notation used for net annual premiums for fully discrete Insurances (IAN)

x x

A P

&& ====

|n:x

x xn

A P

&& ====

|n:x

|n:x a

P 1

&& ====

, |m:x

|n:x

|n:x m

P 1

&& ====

|n:x

a&&

|n:x1

|n:x

A P ====

|n:x a&&

|n:x

A P ====

|m:x

|n:x

|n:x m

P &&

====

Default assumption: Periodic Premiums are payable for the life of the policy

Terminology “Fully discrete” : both premiums at benefits payable at discrete time points

“semi-continous” : DB paid at moment of death, but Premiums paid at discrete points “Fully continuous” : premiums paid continuosly, DB paid at moment of death