# Premium calculation. summer semester 2013/2014. Technical University of Ostrava Faculty of Economics department of Finance

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1 Technical University of Ostrava Faculty of Economics department of Finance summer semester 2013/2014

2 Content 1 Fundamentals Insurer s expenses 2 Equivalence principles Calculation principles 3 Equivalence principle s term life insurance endowment insurance whole life insurance 4

3 Fundamentals Insurer s expenses Insurers offer a policy with certain benefits under given conditions and sell it at stated price, premium. If the contract is accepted and premium is paid by the customer, the customer becomes the policyholder. The aim This lecture is focused on premium calculations.

4 Fundamentals Fundamentals Insurer s expenses Figure:

5 Fundamentals Fundamentals Insurer s expenses Step 1 The choice of statistical bases to construct the probability distribution of random (PV of the) benefits. Step 2 The choice of annual interest rate (or term structure of interest rates) - discounting is omitted if duration is short. Step 3 Calculation of other expenses not related to the benefits. Step 4 Calculation of minimal profit margin for insurer.

6 Insurer s expenses Fundamentals Insurer s expenses Initial expenses occur when the policy is issued (commissions to agents for selling and underwritting expenses). Renewal expenses occur each time when premium is payable (also cover fixed costs, for example staff salaries etc.). Termination expenses occur when policy expires (on the death, on the maturity date, etc.); are associated with the paperwork to finalize and pay a claim.

7 Equivalence principles Calculation principles Equivalence principle: P = E [S], where P is premium and E [S] is expected risk (or mean aggregated claims in portfolio). P > E [S] P E [S] = m, where m is risk (safety) loadings.

8 Calculation principles Equivalence principles Calculation principles The expected value principle (EVP) P = E [S] + αe [S] = (1 + α) E [S] for α > 0, where α is relative security loading (or pure premium loading factor) on the pure premium E [S]. The standard deviation principle (SDP) P = E [S] + αsd [S] The variance principle (VP) P = E [S] + αvar [S]

9 Calculation principles Equivalence principles Calculation principles The quantile principle (QP) Premium P is set for a risk as a certain percentile, p, of the distribution of the risk S, i.e. P satisfies Pr (S P) = p The zero utility principle (ZUP) Let W be initial wealth and let s have utility function u satisfying u (x) 0, u (x) 0 for x > 0 (it implies that u is concave) and u is not exponential function. P is stated to keep zero gain in expected utility by insuring the risk, thus u (W ) = E [u (W + P S)].

10 LIfe insurance Equivalence principle s Equivalence principle: P = PV (E [S]), where PV is present value. Remark Premium can be paid as a lump sum (single premium) or as a regular series of payments (annually, monthly,...).

11 s Equivalence principle s Single premium for term life insurance S = C (1 + i) 1 if the insured dies in the first year C (1 + i) 2 if the insured dies in the second year C (1 + i) r if the insured dies in the rth year 0 if the insured is alive in the rth year P = C(1 + i) 1 0 1q x + C(1 + i) 2 1 1q x C(1 + i) r r 1 1q x, where r 1 1 q x is the probability that the insured, age x, dies between time r 1 and r.

12 s Equivalence principle s Single premium for endowment insurance S = 0 if the insured is alive in the first year 0 if the insured is alive in the second year C (1 + i) r if the insured is alive in the rth year P = PV (E [S]) = C (1 + i) r ( 1 0 r q x ), where ( 1 0 r q x ) is probability that the insured will survive r year.

13 s Equivalence principle s Annual premium for endowment insurance Pä n = PV (E [Y ]) P = PV (E [Y ]) ä n, where ä n is present value of an anuity-certain of 1 payable annually in advance for n years, thus ä n = 1 + v + v v n = (1 + i)n 1 (1 + i) n i where v n = (1 + i) n is a discount factor. (1 + i),

14 s Equivalence principle s Single premium for whole life insurance S = Aa n (1 + i) 1 if the insured dies in the first year Aa n (1 + i) 2 if the insured dies in the second year Aa n (1 + i) r if the insured dies in the rth year 0 if the insured is alive in the rth year P = PV (E [S]) = Aa n (1 + i) 1 0 1q x + Aa n (1 + i) 2 1 1q x Aa n (1 + i) r r 1 1q x

15 s Equivalence principle s Single premium for whole life insurance a n = v + v v n = (1 + i)n 1 (1 + i) n i is present value of an annuity-certain of 1 payable annually in arrear for n years. Annual premium for whole life insurance Pä n = PV (E [Y ]) P = PV (E [Y ]) ä n

16 Setting the safety loadings Explicit safety loading approach the value of α is chosen (SDP, VP, etc.); common in practice of non-life insurance. Implicit safety loading approach no explicit safety loading parameter in the formula within life insurance; it is included in the premium calculation by setting r 1 1q x > r 1 1 q x and discount rate i < i; for instance: P = PV (E [S]) = C ( 1 + i ) r ( 1 0 r q x),

17 For further study D. C. M. Dickson, M. Hardy, and H. R. Waters, Actuarial mathematics for life contingent risks. Cambridge: Cambridge University Press, Y.-K. Tse, Nonlife actuarial models : theory, methods and evaluation. Cambridge: Cambridge University Press, 2009.

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