# Premium Calculation. Lecture: Weeks Lecture: Weeks (STT 455) Premium Calculation Fall Valdez 1 / 31

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1 Premium Calculation Lecture: Weeks Lecture: Weeks (STT 455) Premium Calculation Fall Valdez 1 / 31

3 Preliminaries chapter summary Chapter summary Contract premiums net premiums gross (expense-loaded) premiums Present value of future loss random variable Premium principles the equivalence principle (or actuarial equivalence principle) portfolio percentile premiums Return of premium policies Chapter 6 of Dickson, et al. Lecture: Weeks (STT 455) Premium Calculation Fall Valdez 3 / 31

4 Net random future loss Net random future loss An insurance contract is an agreement between two parties: the insurer agrees to pay for insurance benefits; in exchange for insurance premiums to be paid by the insured. Denote by PVFB 0 the present value, at time of issue, of future benefits to be paid by the insurer. Denote by PVFP 0 the present value, at time of issue, of future premiums to be paid by the insured. The insurer s net random future loss is defined by L n 0 = PVFB 0 PVFP 0. Note: this is also called the present value of future loss random variable (in the book), and if no confusion, we may simply write this as L 0. Lecture: Weeks (STT 455) Premium Calculation Fall Valdez 4 / 31

5 Net random future loss equivalence principle The principle of equivalence The net premium, generically denoted by P, may be determined according to the principle of equivalence by setting E [ L n 0 ] = 0. The expected value of the insurer s net random future loss is zero. This is then equivalent to setting E [ PVFB 0 ] = E [ PVFP0 ]. In other words, at issue, we have APV(Future Premiums) = APV(Future Benefits). Lecture: Weeks (STT 455) Premium Calculation Fall Valdez 5 / 31

6 Net random future loss illustration An illustration Consider an n-year endowment policy which pays B dollars at the end of the year of death or at maturity, issued to a life with exact age x. Net premium of P is paid at the beginning of each year throughout the policy term. If we denote the curtate future lifetime of (x) by K = K x, then the net random future loss can be expressed as L n 0 = Bv min(k+1,n) P ä min(k+1,n). The expected value of the net random future loss is E [ L n ] [ 0 = BE v min(k+1,n)] ] P E [ä min(k+1,n) = BA x: n P ä x: n. Lecture: Weeks (STT 455) Premium Calculation Fall Valdez 6 / 31

7 Net random future loss illustration An illustration - continued By the principle of equivalence, E [ L n 0 ] = 0, we then have P = B A x: n ä x: n. Rewriting the net random future loss as ( L n 0 = B + P ) v min(k+1,n) P d d, we can find expression for the variance: Var [ ( L n ] 0 = B + P ) 2 [ 2 Ax: n d ( ) ] 2 A x: n. One can also show that this simplifies to Var [ L n 0 ] = B 2 2A x: n ( ) 2 A x: n ( ) 2. 1 Ax: n Lecture: Weeks (STT 455) Premium Calculation Fall Valdez 7 / 31

8 Net random future loss general principles Some general principles Note the following general principles when calculating premiums: For (discrete) premiums, the first premium is usually assumed to be made immediately at issue. Insurance benefit may have expiration or maturity: in which case, it is implied that there are no premiums to be paid beyond expiration or maturity. however, it is possible that premiums are to be paid for lesser period than expiration or maturity. In this case, it will be explicitly stated. Lecture: Weeks (STT 455) Premium Calculation Fall Valdez 8 / 31

9 Fully discrete whole life insurance Fully discrete annual premiums - whole life insurance Consider the case of a fully discrete whole life insurance where benefit of \$1 is paid at the end of the year of death with level annual premiums. The net annual premium is denoted by P x so that the net random future loss is L 0 = v K+1 P x ä K+1, for K = 0, 1, 2,... By the principle of equivalence, we have P x = E[ v K+1] [ ] = A x. E ä ä x K+1 The variance of the net random future loss is Var[L 0 ] = 2A x (A x ) 2 2 (dä x ) 2 = A x (A x ) 2 (1 A x ) 2. Lecture: Weeks (STT 455) Premium Calculation Fall Valdez 9 / 31

10 Fully discrete whole life insurance Other expressions You can express the net annual premiums: in terms of annuity functions P x = 1 dä x ä x = 1 ä x d in terms of insurance functions P x = A x ( 1 Ax ) /d = da x 1 A x Lecture: Weeks (STT 455) Premium Calculation Fall Valdez 10 / 31

11 Fully discrete whole life insurance with h pay Whole life insurance with h premium payments Consider the same situation where now this time there are only h premium payments. The net random future loss in this case can be expressed as {äk+1, for K = 0, 1,..., h 1 L 0 = v K+1 P ä h, for K = h, h + 1,... Applying the principle of equivalence, we have P = A x ä x: h. Lecture: Weeks (STT 455) Premium Calculation Fall Valdez 11 / 31

12 Fully discrete illustrative examples Illustrative example 1 Consider a special endowment policy issued to (45). You are given: Benefit of \$10,000 is paid at the end of the year of death, if death occurs before 20 years. Benefit of \$20,000 is paid at the end of 20 years if the insured is then alive. Level annual premiums P are paid at the beginning of each year for 10 years and nothing thereafter. Mortality follows the Illustrative Life Table with i = 6%. Calculate P according to the equivalence principle. Lecture: Weeks (STT 455) Premium Calculation Fall Valdez 12 / 31

13 Fully discrete illustrative examples SOA-type question Two actuaries use the same mortality table to price a fully discrete two-year endowment insurance of 1,000 on (x). You are given: Kevin calculates non-level benefit premiums of 608 for the first year, and 350 for the second year. Kira calculates level annual benefit premiums of π. d = 0.05 Calculate π. Lecture: Weeks (STT 455) Premium Calculation Fall Valdez 13 / 31

14 Fully discrete illustrative examples Illustrative example 2 An insurance company issues a 15-year deferred life annuity contract to (50). You are given: Level monthly premiums of P are paid during the deferred period. The annuity benefit of \$25,000 is to be paid at the beginning of each year the insured is alive, starting when he reaches the age of 65. Mortality follows the Illustrative Life Table with i = 6%. Mortality between integral ages follow the Uniform Distribution of Death (UDD) assumption. 1 Write down an expression for the net future loss, at issue, random variable. 2 Calculate the amount of P. 3 If an additional benefit of \$10,000 is to be paid at the moment of death during the deferred period, how much will the increase in the monthly premium be? Lecture: Weeks (STT 455) Premium Calculation Fall Valdez 14 / 31

15 Different possible combinations Different possible combinations Premium payment annually m-thly of the year continuously Benefit payment at the end of the year of death at the end of the 1 mth year of death at the moment of death at the end of the year of death at the end of the 1 mth year of death at the moment of death at the end of the year of death at the end of the 1 mth year of death at the moment of death Lecture: Weeks (STT 455) Premium Calculation Fall Valdez 15 / 31

16 Fully continuous whole life insurance Fully continuous premiums - whole life insurance Consider a fully continuous level annual premiums for a unit whole life insurance payable immediately upon death of (x). The insurer s net random future loss is expressed as By the principle of equivalence, L 0 = v T P ā T. P = Āx ā x = 1 ā x δ = δāx. 1 Āx The variance of the insurer s net random future loss can be expressed as Var[L 0 ] = [ 1 + (P/δ) ] [ 2 2 Āx ( ) ] 2 Ā x = 2Ā x ( ) 2 Ā 2 x (δā x ) 2 = Ā x ( ) 2 Ā x (1 Āx) 2. Lecture: Weeks (STT 455) Premium Calculation Fall Valdez 16 / 31

17 Fully continuous whole life insurance A simple illustration For a fully continuous whole life insurance of \$1, you are given: Mortality follows a constant force of µ = Interest is at a constant force δ = L 0 is the loss-at-issue random variable with the benefit premium calculated based on the equivalence principle. Calculate the annual benefit premium and Var[L 0 ]. Lecture: Weeks (STT 455) Premium Calculation Fall Valdez 17 / 31

18 Fully continuous SOA question Published SOA question #14 For a fully continuous whole life insurance of \$1 on (x), you are given: The forces of mortality and interest are constant. 2Ā x = 0.20 The benefit premium is L 0 is the loss-at-issue random variable based on the benefit premium. Calculate Var[L 0 ]. Lecture: Weeks (STT 455) Premium Calculation Fall Valdez 18 / 31

19 Fully continuous endowment insurance Endowment insurance Consider an n-year endowment insurance with benefit of \$1: The net random future loss is { v T P ā L = T, T n v n P ā n, T > n Net annual premium formulas: P = Ā x: n ā x: n = 1 ā x: n δ = δāx: n 1 Āx: n The variance of the net random future loss: Var [ ] L 0 = [ 1 + ( P/δ) ] [ 2 2 Āx: n ( ) ] 2 Ā x: n = 2Ā x: n ( ) 2 Ā x: n (δā x: n ) 2 = 2Ā x: n ( ) 2 Ā x: n (1 Āx: n )2 Lecture: Weeks (STT 455) Premium Calculation Fall Valdez 19 / 31

20 Additional problems Illustrative example 3 For a fully continuous n-year endowment insurance of \$1 issued to (x), you are given: Z is the present value random variable of the benefit for this insurance. E[Z] = Var[Z] = Level annual premiums are paid on this insurance, determined according to the equivalence principle. Calculate Var[L 0 ], where L 0 is the net random future loss at issue. Lecture: Weeks (STT 455) Premium Calculation Fall Valdez 20 / 31

21 Additional problems Illustrative example 4 For a fully discrete whole life insurance of 100 on (30), you are given: π denotes the annual premium and L 0 (π) denotes the net random future loss-at-issue random variable for this policy. Mortality follows the Illustrative Life Table with i = 6%. Calculate the smallest premium, π, such that the probability is less than 0.5 that the loss L 0 (π ) is positive. Lecture: Weeks (STT 455) Premium Calculation Fall Valdez 21 / 31

22 Life insurance contract expenses types Types of Life Insurance contract expenses Investment-related expenses (e.g. analysis, cost of buying, selling, servicing) Insurance-related expenses: acquisition (agents commission, underwriting, preparing new records) maintenance (premium collection, policyholder correspondence) general (research, actuarial, accounting, taxes) settlement (claim investigation, legal defense, disbursement) Lecture: Weeks (STT 455) Premium Calculation Fall Valdez 22 / 31

23 Life insurance contract expenses first year vs. renewal First year vs. renewal expenses Most life insurance contracts incur large losses in the first year because of large first year expenses: agents commission preparing new policies, contracts records administration These large losses are hopefully recovered in later years. How then do these first year expenses spread over the policy life? Anything not first year expense is called renewal expense (used for maintaining and continuing the policy). Lecture: Weeks (STT 455) Premium Calculation Fall Valdez 23 / 31

24 Life insurance contract expenses gross premiums Gross premium calculations Treat expenses as if they are a part of benefits. The gross random future loss at issue is defined by L g 0 = PVFB 0 + PVFE 0 PVFP 0, where PVFE 0 is the present value random variable associated with future expenses incurred by the insurer. The gross premium, generically denoted by G, may be determined according to the principle of equivalence by setting E [ L g 0] = 0. This is equivalent to setting E [ PVFB 0 ] + E [ PVFE0 ] = E [ PVFP0 ]. In other words, at issue, we have APV(FP 0 ) = APV(FB 0 ) + APV(FE 0 ). Lecture: Weeks (STT 455) Premium Calculation Fall Valdez 24 / 31

25 Life insurance contract expenses illustration Illustration of gross premium calculation A 1,000 fully discrete whole life policy issued to (45) with level annual premiums is priced with the following expense assumptions: % of Premium Per 1,000 Per Policy First year 40% Renewal years 10% In addition, assume that mortality follows the Illustrative Life Table with interest rate i = 6%. Calculate the expense-loaded annual premium. Lecture: Weeks (STT 455) Premium Calculation Fall Valdez 25 / 31

26 Life insurance contract expenses illustration Published SOA question #239 - modified For a 20-year endowment insurance of 25,000 on (x) with benefit payable at the moment of death, you are given: Expenses are incurred at the beginning of the year: Ā x: 20 = ä x: 20 = i = 0.05 Percent of Per 1,000 of Per Premium Insurance Policy First year 25% Renewal 5% Level annual premiums are determined using the equivalence principle. Calculate the level annual expense-loaded premium. Lecture: Weeks (STT 455) Premium Calculation Fall Valdez 26 / 31

27 Portfolio percentile premiums Portfolio percentile premium principle Suppose insurer issues a portfolio of N identical and independent policies where the PV of loss-at-issue for the i-th policy is L 0,1. The total portfolio (aggregate) future loss is then defined by Its expected value is therefore L agg = L 0,1 + L 0,2 + + L 0,N = N i=1 L 0,i N E[L agg ] = E [ ] L 0,i i=1 and, by independence, the variance is Var[L agg ] = N Var [ ] L 0,i. i=1 Lecture: Weeks (STT 455) Premium Calculation Fall Valdez 27 / 31

28 Portfolio percentile premiums - continued Portfolio percentile premium principle The portfolio percentile premium principle sets the premium P so that there is a probability, say α with 0 < α < 1, of a positive gain from the portfolio. In other words, we set P so that Pr[L agg < 0] = α. Note that loss could include expenses. Consider Example 6.11 (1st edition) or Example 6.12 (2nd edition) Lecture: Weeks (STT 455) Premium Calculation Fall Valdez 28 / 31

29 Portfolio percentile premiums illustrative example Illustrative example 5 An insurer sells 100 fully discrete whole life insurance policies of \$1, each of the same age 45. You are given: All policies have independent future lifetimes. Mortality follows the Standard Select Survival Model with i = 5%. Using the Normal approximation: 1 Calculate the annual contract premium according to the portfolio percentile premium principle with α = Suppose the annual contract premium is set at per policy. Determine the smallest number of policies to be sold so that the insurer has at least a 95% probability of a gain from this portfolio of policies. Lecture: Weeks (STT 455) Premium Calculation Fall Valdez 29 / 31

30 Return of premium Illustration of return of premium To illustrate the concept of return of premium policies, consider Question #22 from Fall 2012 SOA MLC Exam: You are given the following information about a special fully discrete 2-payment, 2-year term insurance on (80): Mortality follows the Illustrative Life Table. i = The death benefit is 1000 plus a return of all premiums paid without interest. Level premiums are calculated using the equivalence principle. Calculate the benefit premium for this special insurance. For practice: try calculating the benefit premium if the return of all premiums paid comes with an interest of say Lecture: Weeks (STT 455) Premium Calculation Fall Valdez 30 / 31

31 Other terminologies Other terminologies and notations used Expression net random future loss L 0 net premium equivalence principle generic premium (P ) Other terms/symbols used loss-at-issue 0 L benefit premium actuarial equivalence principle π Lecture: Weeks (STT 455) Premium Calculation Fall Valdez 31 / 31

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