CHAPTER 8 INSURANCE PRICING Mahdzan & Boey 2015 Learning Outcomes Define insurance pricing and explain the components of insurance cost Discuss the implications of homogeneous and heterogeneous buyers in insurance pricing. Describe how investment returns, administrative cost and profit loading determine fair insurance premium. Compute fair premium based on the components of insurance cost Compute life insurance pricing for term insurance and whole-life insurance Compute life insurance pricing under single premium and level premiums. Mahdzan & Boey 2015 1.0 Insurance Pricing and Components of Insurance Cost Insurance pricing: The determination of rates (premiums) that an insurer charges for insurance. Pure premium: The minimum premium that is necessary to cover the expected losses or expected claim cost. Mahdzan & Boey 2015 1.0 Insurance Pricing and Components of Insurance Cost Fair insurance premium Expected claim cost Investment Returns Administrative costs Fair profit loading Mahdzan & Boey 2015 1.0 Insurance Pricing and Components of Insurance Cost 1.1 Expected claim cost (Pure premium) Represents the expected loss that will be faced by the insured, which then passes on to the insurer. To illustrate, we classify buyers of insurance as: (i) Homogenous buyers: Buyers of insurance who are equal in terms of their loss distribution (ii) Heterogeneous buyers: Buyers of insurance that have different loss probabilities. Mahdzan & Boey 2015 Homogeneous buyers Everyone in the risk pool has the same probabilities of loss & expected claim cost. Losses are independent of each other (uncorrelated). Large number of participants in the pool Table 1: Loss distribution of homogeneous buyers Outcome (x) Probability (p) p*x Expected claim (RM) Loss (RM1,000) 0.10 (0.10 x RM1,000) 100 No loss (RM0) 0.90 (0.90 x RM0) 0 Expected claim cost = RM100 100 By charging each buyer equally by RM100, the insurer can cover its expected losses. Mahdzan & Boey 2015 Heterogeneous buyers Everyone in the risk pool have different risk exposures depending on their own unique lifestyle and circumstances. Assume two groups of insurance buyers – low-risk buyers and high-risk buyers. Stay-at-home mums & Factory-operator mums Table 1: Loss distribution of homogeneous buyers Outcomes (x) Probability (p) p*x Expected claim Stay home mums Loss (RM1,000) 0.05 0.05 x RM1,000 = RM50 Factory-operator 50 No loss (RM0) 0.95 0.95 x RM0 = RM0 Loss (RM1,000) 0.20 0.20 x RM1,000 = mums RM200 No loss (RM0) 0.80 200 0.80 x RM0 = RM0 Mahdzan & Boey 2015 Same Rate Insurance Co Unable to differentiate the two groups Charge same premium rates to both groups Takes the average of the premiums, (RM50+200)/2 = RM125 Stay-home mums – disadvantaged Factory-operator mums – advantaged Stay-home mums would be subsidizing the premiums of the factoryoperator mums Mahdzan & Boey 2015 Selective Insurance Co Excellent team who can correctly distinguish the two groups Carefully select its policyholders to comprise only stay-home mums, and deny coverage on factory-operator mums Premiums for stay-home mums < RM125 - giving Selective a pricing advantage over SameRate Co. E.g. Selective charges RM100 to stay-home mums - still make a gain of RM50 since the expected claim cost of stay-home mums are RM50. Stay-home mums would choose buying from Selective rather than SameRate. SameRate would then be left with factory-operator mums as policyholders (factory-operators cannot switch to Selective because Selective only accepts stay-home mums as policyholders and deny coverage on factory-operators). The final outcome of this method would be that Selective would carry all low risk buyers, and SameRate would carry all high risk buyers. Mahdzan & Boey 2015 1.0 Insurance Pricing and Components of Insurance Cost The preceding situation would result in a situation called Adverse Selection. Adverse selection: A high tendency for insurance to be purchased or selected by individuals faced with high risk (in adverse situations) as opposed to those with low risk, when charged with the same premiums. Due to information asymmetries Insurance pricing will not be optimal Mahdzan & Boey 2015 1.0 Insurance Pricing and Components of Insurance Cost Risk classification: The identification and grouping of people in accordance to their level of risk and expected claims. Done by actuaries:- use historical data and statistical methods to assess risk and estimate premiums for an insurance company. Risk classification increases the incentive to lower down risks by the policyholders. Insurance companies must invest in additional underwriting effort to classify their customers Must do so at affordable cost in order to minimize premiums charged to customers Mahdzan & Boey 2015 1.0 Insurance Pricing and Components of Insurance Cost Underwriting process: evaluating insurance applications from buyers to determine the acceptability of the risk from the perspective of the insurer and, if acceptable, whether special conditions should be applied and the level of premium that is to be charged. Class rates: standard rates that apply to insurance applicants within certain risk categories. Experience rating: applies when class rates are adjusted according to the loss experience of the insured. Non-claim discount (NCD): discount given to the insured of a motor vehicle insurance policy when no claims are made in a given policy year. Mahdzan & Boey 2015 1.0 Insurance Pricing and Components of Insurance Cost 1.2 Investment returns Reflect the ability of insurers to invest the premiums paid by its customers. The computation of fair premiums to include investment returns are based on the time value of money The resulting value: the discounted value of the expected claim cost. The higher the interest rates or the longer the duration of the policy, the lower the fair premiums, vice versa. Mahdzan & Boey 2015 Example 1 1) Claims paid at the end of one year Say that an insurer issues a policy that has an expected claim of RM1,000 (which is the future value, FV) at the end of one year. Assuming that there are no administrative costs and fair profit loading, we compute the premiums (P) the insurer should collect at the beginning of the year. At an annual rate of r pear year for n years, the premium (which is the present value, PV) that needs to be charged is as follows: Beginning Year 1 End Year 1 Premiums ? Exp. claims RM1,000 Mahdzan & Boey 2015 Example 1 Expected Claims (FV) = PV (1 + r)n = 1000 If r = 5%, Premiums (PV) = FV / (1+r)n = 1000 / (1.05) = 952.38 If r = 7%, Premiums (PV) = FV / (1+r)n = 1000 / (1.07)1 = 934.58 Mahdzan & Boey 2015 Example 2 Mahdzan & Boey 2015 Example 2 Expected claims (FV) = PV (1 + r)n = 1000 If r = 5%, PV = FV / (I + r)n = 1000 / (1 + 0.05)2 = 1000 / 1.1025 = 907.03 If r = 7%, PV = FV / (I + r)n = 1000 / (1 + 0.05)2 = 1000 / 1.1449 = 873.44 Mahdzan & Boey 2015 1.0 Insurance Pricing and Components of Insurance Cost 1.3 Administrative costs Include: Underwriting costs: costs involved to determine the acceptability of the risk, conducted by underwriters of the insurer. Agents’ renumeration: costs for the sales force that market and sell insurance policies to customers. Medical examination expenses Marketing costs Loss adjustment expenses: applies to general insurance Other expenses: policy issuance, distribution costs, etc. Mahdzan & Boey 2015 1.0 Insurance Pricing and Components of Insurance Cost 1.4 Profit loading As with other types of business, insurance co’s are business entities in pursuit of profits and long-term sustainability. To compensate its investors for their capital, a certain level of profit margin must be included, in excess of the discounted value of expected claim cost plus administrative costs. Claims could occur at any time from the inception of policy, investors of the insurance company must provide capital that is sufficient to cover claims and avoid insolvency. A higher profit loading is required for claims that are less predictable. Mahdzan & Boey 2015 1.0 Insurance Pricing and Components of Insurance Cost Example of Fair Premium Computation Suppose a group of policyholders has the following loss distribution as shown below. Assume a time period of two (2) years after which claims will be made, and interest rate of 6%. Administrative expenses are 15% of expected claim costs, and profit loading is 5% of expected claim cost. Compute fair premiums for the following loss distribution: Outcome (x) Probability (p) Loss (RM50,000) 0.10 No loss (RM0) 0.90 Loss distribution Mahdzan & Boey 2015 Mahdzan & Boey 2015 2.0 Life Insurance pricing Mortality risk: The probability of dying for individuals based on their age and gender, hence, the expected claim cost. To compute life insurance pricing, we will differentiate between: (i) Term insurance (ii) Whole life insurance. Mahdzan & Boey 2015 Mahdzan & Boey 2015 2.0 Life Insurance pricing One-year term For a one year term insurance, an insurer needs to consider the probability of a person dying at his current age. E.g.: Mr. Isaac, 35 year old, purchases a one year term life insurance policy with a coverage of RM200,000. Should death occur during the tenure of the policy, Mr. Isaac’s family will receive RM200,000. The probability of dying is 0.001592. What this means is that the probability of a 35 year old male surviving for another year is 0.998408 (1-0.001592). The loss distribution is summarized in the following table. For simplicity, we ignore administrative expense and profit loading. Mahdzan & Boey 2015 Loss distribution for Mr. Isaac, age 35 Outcome (x) Step 1: Probability (p) Loss (RM200,000) 0.001592 No loss (RM0) 0.998408 Compute expected claim cost Exp. claim cost = (0.001592 x RM200,000) + (0.998408 x RM0) = RM318.40 Step 2: Compute discounted value of expected claim cost Discounted value of exp. claim cost (PV) = FV / (1+r)n = RM318.40 / (1.05) = RM303.24 Hence, the premium of a one year term policy is RM303.24. Mahdzan & Boey 2015 If Mr. Isaac were to purchase another one-year policy the following year at age 36, the loss distribution would be: Outcome (x) Probability (p) Loss (RM200,000) 0.001660 No loss (RM0) 0.998340 Loss distribution for Mr. Isaac, age 36 Step 1: Compute expected claim cost Exp. claim cost = (0.001660 x RM200,000) + (0.998340 x RM0) = RM332.00 Step 2: Compute discounted value of expected claim cost Discounted value of exp. claim cost (PV) = FV / (1+r)n = RM332.00 / (1.05) = RM316.19 Hence, the present value of premiums of a one-year term policy at Mr. Isaac’s age of 36 is RM316.19. Mahdzan & Boey 2015 Two year term To show how premiums differ according to the duration of a policy, we now assume that Mr. Isaac, age 35, purchases a two-year term insurance policy. The investment return is assumed to be 5%, and other loadings (administrative and profit) are ignored. There are two types of payment methods that can be charged by the insurer, single premium and level premium. Computation of the two payment methods will be demonstrated below. (i) Single premium- Policyholders pay a one-time premium at the inception of the policy. Since we are ignoring administrative and profit loading, the inflow (premium) must equal the outflow (which is the pure premium or discounted expected claim cost), as shown in Column v in Table 4. Mahdzan & Boey 2015 i) Single premium- Policyholders pay a one-time premium at the inception of the policy. Since we are ignoring administrative and profit loading, the inflow (premium) must equal the outflow (which is the pure premium or discounted expected claim cost), as shown in Column v in Table 4. Mahdzan & Boey 2015 Table 8.7: Summary of computation for a single premium two-year term policy i ii Time Claim cost End of Year 1 End of Year 2 RM200,000 RM200,000 iii Probability of dying (subject to survival at start of policy) 0.001592 (160/96736) = 0.001657 iv v Exp.claim cost PV of Exp.claim cost 318.40 331.40 303.24 300.60 603.84 Total Step 1: Compute expected claim cost Year 1: (0.001592 x RM200,000) + (0.998408 x RM0) = RM318.40 Year 2: (0.001657 x RM200,000) + (0.998343 x RM0) = RM331.40 Step 2: Compute discounted value of expected claim cost (PV of exp. claim cost) Discounted value of exp. claim cost (PV) = FV / (1+r)n Year 1: RM318.40 / (1.05) = RM303.24 Year 2: RM331.40 / (1.05)2 = RM300.60 Step 3: Total up the discounted exp.claim cost for Year 1 and year 2 Total discounted exp.claim cost = RM603.84 Hence, the single premium to be charged for a two-year term policy is RM603.84. Mahdzan & Boey 2015 ii) Level premium: Policyholders pay periodic, level premiums (P) over a fixed number of years. Mahdzan & Boey 2015 Table 8.7: Computation of expected premium for a level premium two-year term policy Premium 1 + PV of Premium 2 = PV of Exp.claim1 + PV of Exp. Claim2 Mahdzan & Boey 2015 Step 1: Compute expected premium for Year 1 and Year 2. Year 1: P Year 2: 0.998408P Step 2: Compute discounted value (PV) of expected premium and total for Year 1 and 2. Discounted value of exp. premium (PV) = FV / (1+r)n Year 1: P Year 2: 0.998408P/1.05 = 0.95086P Total (Year 1 & 2): P + 0.95086P = 1.95086P Step 3: Compute the discounted expected claim cost. Computation is the same as in the single premium method, hence it is RM603.84. Step 4: Equate the discounted expected claim cost and discounted expected premium 1.95086P = RM603.84 P = RM603.84 / 1.95086 P = RM309.53 Hence, the level premium that is charged at the beginning of Year 1 and 2 is RM309.53 per annum. Mahdzan & Boey 2015 Table 8.9: Comparison of premiums paid for different term policies Type of policy Purchase one-year term policies, renewed yearly Purchase a two-year policy, single premium Purchase a two-year policy, level premium Premiums (Year 1) Premiums (Year 2) Premiums (Total Yr 1&2) RM303.24 RM316.19 RM619.43 RM603.84 RM309.53 RM603.84 RM309.53 RM619.06 Observations Premium increases from Year 1 to Year 2, indicating higher mortality risks as one increase in age. The disadvantage of purchasing renewable term policies is that the insurer may deny coverage if the individual is no longer insurable (e.g. due to health conditions). By paying a single premium, premiums appear to be the cheapest, although a higher lumpsum amount is required at the beginning of the two years. Premiums are the same for both years. In a way, Year 1 premium subsidizes Year 2 premium. The total is almost the same as the one-year renewable term policy premiums. Mahdzan & Boey 2015 2.3 Pricing of Whole-life insurance Single Premium E.g.: Mr. Isaac, age 35, who wants to purchase a whole life policy with a face amount of RM200,000. The computation of expected claim cost is summarized in the following table. I ii iii iv v vi vii Age Death probability No.of living No.of deaths Prob.of death Exp. Claim Cost PV of Exp.Claim Cost 35 0.001592 96,736 154 0.001592 318.40 303.24 36 0.001660 96,582 160 0.001657 331.40 300.65 37 0.001741 96,422 168 0.001735 347.00 299.81 38 0.001837 96,254 177 0.001828 365.60 300.75 39 0.001953 96,077 188 0.001940 388.00 303.96 . . . . . . . . . . . . . . . . . . . . . 95 0.266454 5,387 1,435 0.014838 2,967.60 151.31 96 0.286625 3,952 1,133 0.011710 2,342.00 113.72 97 0.305869 2,819 862 0.008913 1,782.60 82.44 98 0.323783 1,957 634 0.006550 1,310.00 57.70 99 0.339972 1,323 450 0.004650 930.00 39.01 198,839 30,447.52 Computation summary of a single premium whole-life policy Mahdzan & Boey 2015 Step 1: Compute the probability of death of a 35 year old occurring in the respective subsequent years Age 36: 160/96,736 = 0.001657 Age 37: 168/96,736 = 0.001735 . Step 2: Age 98: 634/96,736 = 0.006550 Age 99: 450/96,736 = 0.004650 Compute expected claim cost for each year (age 35-100) Age 35: 0.001592 x 200,000 = 318.40 Age 36: 0.001657 x 200,000 = 331.40 . . Age 98: 0.006550 x 200,000 = 1310.00 Age 99: 0.004650 x 200,000 = 930.00 Mahdzan & Boey 2015 Step 3: Compute discounted value (PV) of expected premium for each year and total up the amounts. Discounted value of exp. claim cost (PV) Age 35: 318.40/1.05 = 303.24 Age 36: 331.40/1.052 = 300.65 = FV / (1+r)n . . Age 98: 1310/1.0564 = 57.70 Age 99: 930.00/1.0565 = 39.01 Total of discounted values of exp. claim cost = RM30,447.52 Hence, the single premium for a whole-life policy (with coverage of RM200,000) that must be paid by a 35 year old individual at the inception of the policy is RM30,447.52. Mahdzan & Boey 2015 Continuous Level premium: The insured pays a level premium from the inception of the policy to its maturity (age:100). i ii iii iv v vii Age 35 Prob.of death 0.001592 No.of living 96,736 Prob of 35 surviving till specified age 1.000000 Exp.premium payment P PV of exp.premium 36 0.001660 96,582 0.998408 0.998408P 0.950865P 37 0.001741 96,422 0.996754 0.996754P 0.904085P 38 0.001837 96,254 0.995017 0.995017P 0.859533P 39 0.001953 96,077 0.993188 0.993188P 0.817098P . . . . . . . . . . . . 95 0.266454 5,387 0.055688 0.055688P 0.002981P 96 0.286625 3,952 0.040853 0.040853P 0.002083P 97 0.305869 2,819 0.029141 0.029141P 0.001415P 98 0.323783 1,957 0.020230 0.020230P 0.000936P 99 0.339972 1,323 0.013676 0.013676P 0.000602P TOTAL 17.19506P P Table 8: Computation summary of a continuous level premiums whole-life policy Mahdzan & Boey 2015 Step 1: Compute expected premium for all years (from age 35-99), which is the number of living at each respective year divided by the number of living at age 35. Age 35: 96736/96736 = 1 Age 36: 96582/96736 = 0.998408 . . Age 98: 1957/96736 = 0.020230 Age 99: 1323/96736 = 0.013676 Step 2: Denote the expected premium payment as probability of surviving (column iv) multiplied by the unknown level premiums, P. Age 35: P Age 36:0.998408P . . Age 98: 0.020230P Age 99: 0.013676P Mahdzan & Boey 2015 Step 3: Compute discounted value (PV) of expected premium (age 35-99) and total up. Discounted value of exp. premium (PV) = FV / (1+r)n Age 35: P Age 36: 0.998408P/1.05 = 0.95086P . . Age 98: 0.02030/1.0564 = 0.000936P Age 99: 0.013676/1.0565 = 0.000602P Total is 17.9506P. Step 3: Compute the discounted expected claim cost. Computation is the same as in the single premium method, hence it is RM30,447.51. Step 4: Equate the discounted expected claim cost and discounted expected premium 17.9506P = RM30,447.51 P = RM30,447.51/17.9506 P = RM1,696.18 Hence, the continuous level premium that is must be paid at the beginning of each year is RM1,696.18 per annum. Mahdzan & Boey 2015 Limited payment premiums: Only take into consideration the expected premium payment and its discounted value E.g.: Say Mr. Isaac wants to pay premiums for 20 years only instead of until his age of 99. We only take into consideration the expected premium payment and its discounted value till Mr. Isaac’s age of 54. The total discounted value of the expected premium is 12.836409P Mahdzan & Boey 2015 We then equate the discounted premium to the discounted expected claim cost. 12.836409P = RM30,447.51 P = RM2371.96 Hence, the limited payment premiums that is to be paid us RM2,371.96 per annum for 20 years. Table 9: Comparison of premiums (single, continuous level and limited payment premium) Type of premium paid Amount of Duration of payment Total premiums paid premium Single premium RM30,447.51 Once RM30,447.51 Continuous level premium RM1,696.18 Age 35-99 = 65 years RM1696.18 x 65 = RM110,251.70 Limited payment premium RM2,371.96 Age 35-54 = 20 years RM2371.96 x 20 RM47,439.30 Mahdzan & Boey 2015