Insurance Pricing

1
Insurance Pricing

• Lecture 19

2
Ratemaking Objectives

• Regulatory
• Adequate high enough to pay losses leave room
  for profits contingencies (avoid insolvency)
• Not excessive
• Not discriminatory
• Business
• Simple
• Stable
• Responsive
• Encourages loss control (frequency severity)

3
Definitions

• Rate price per unit of insurance
• Exposure unit unit of measure used in insurance
  pricing
• Ex fire exposure, unit is 100 of coverage life
  unit is 1000 of coverage
• Pure premium amount needed to pay losses loss
  adjustment expenses
• Loading amount added for other expenses,
  profits and contingencies

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Definitions

• Gross rate consists of pure premium and loading
• Gross premium gross rate multiplied by number
  of exposure units
• Ex gross rate of 10 cents per 100 of property
  insurance ? gross premium for 500,000 building
  would be 500

5
Ratemaking Property Liability

• Judgment rating
• Each exposure is individually evaluated
• Rate determined largely by underwriters judgment
• Use
• Diverse loss exposures prevents class rating
• Credible loss statistics unavailable

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Ratemaking Property Liability

• Class rating
• Exposures with similar characteristics are placed
  in same underwriting class charged same rate
• Simple to apply Quick premium quotations ideal
  for personal lines
• Two methods
• Pure premium method
• Loss ratio method

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Class Rating pure premium

• Divide dollar amount of incurred losses and
  loss-adjustment expenses by number of exposure
  units
• Incurred equals all losses actually paid and
  reserves for future payment of losses that have
  occurred (incurred losses are all that have
  occurred, regardless of whether they have been
  paid)
• Add a loading for expenses and profit

8
Class Rating pure premium

• Ex Suppose 500,000 automobiles incur losses and
  loss-adj. expenses of 30,000,000
• Pure premium 30,000,000 / 500,000 60
• Expense ratio - of gross rate available for
  expenses and profit
• Assume 40 expense ratio
• Gross rate pure premium/ (1 expense ratio)
• 60 / (1 - .4) 100

9
Class Rating loss ratio method

• Actual loss ratio is compared to expected loss
  ratio and rate adjusted accordingly
• Actual loss ratio ratio of incurred losses
  loss adj. exp. to earned premiums
• Expected loss ratio percent of premiums
  expected to pay for loss expenses
• Ex incurred loss exp. 800,000 earned premiums
  1,000,000 ? actual .80
• If expected .70, then rate change (.8 - .7)/
  .7 14.3

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Merit Rating

• Class rates are adjusted up or down based on
  individual loss experience
• Schedule rating basis rate is modified
  according to certain physical characteristics
• Construction
• Occupancy
• Protection
• Exposure
• Maintenance

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Merit Rating

• Experience rating
• Class rate is adjusted up or down based on past
  loss experience
• Incentive to engage in loss control
• Retrospective rating
• Loss experience during period determines actual
  premium paid for that period min and max stated

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Life Insurance Ratemaking

• Net single premium present value of the future
  death benefit
• Only mortality and investment income considered
  loading later
• Assumes
• Premiums paid at beginning of policy
• Death claims paid at end of policy year
• Death rate uniform throughout year

13
Life Insurance Ratemaking

• Mortality data developed independently by
  insurers
• Examples herein use Commissioners 1980 Standard
  Ordinary Mortality Table
• See next slide

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CSO 1980 Mortality Tables, Male Lives
15
CSO 1980 Mortality Tables, Male Lives
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Term Insurance

• Yearly renewable assume 1000 issued to male at
  age 45
• Cost of insurance is probability of death x
  present value of amount of insurance
• Assume investment rate is 5
• From CSO table 9,210,289 out of 10,000,000 still
  alive at age 45 41,907 will die
• Probability of death in year 45 41,907/9,210,289
  .00455

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Term Insurance

• .00445 x 1,000 4.45 amount of money needed on
  hand at end of year to pay expected claim
• Beginning of year present value
• 4.45 / 1.05 4.33 per 1,000 of insurance
• Cost of insurance is probability of death x
  present value of amount of insurance

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Term Insurance

• Suppose renewed at age
• Probability of death 45,108 / 9,168,382
  .00492
• .00492 x 1,000 4.92 needed on hand at end of
  year to pay expected claim
• Beginning of year present value
• 4.92 / 1.05 4.69 per 1,000 of insurance

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Consider 5-yr Term
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Ordinary Life

• Same as for five year policy, except that
  calculation is carried to end of mortality table
  age 99
• In our example, the amount would be 270.84

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Net Annual Level Premium

• Most not purchased with single premium
• Net annual level premium (NALP) is determined by
  dividing the net single premium by the present
  value of a life annuity due (PVLAD) of 1

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Annuity Due Factor

• Annuity Due factor is Annuity Factor x (1R)
• Example for 5 years
• PYMT 1
• N 5
• R .05
• PV compute this ? 4.329477
• Convert to due 4.329477 x 1.05 4.545951

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Life Annuity Due Factor

• PV Payment x Annuity Factor
• If payments were certain, then
• Payment PV / Annuity Factor
• Payment 22.74 / 4.54595 5.00
• However, have to compute probability that a
  person age 45 will still be alive at 46 to make
  second payment, at age 47 to make 3rd, etc.

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Life Annuity Due Factor

• Age 45 1.000
• 46 9,168,382 / 9,210,289 x 1 x .9524 .948
• 47 9,123,274 / 9,210,289 x 1 x .9070 .898
• 48 9,074,738 / 9,210,289 x 1 x .8638 .851
• 49 9,022,649 / 9,210,289 x 1 x .8227 .806
• 4.503
• Payment 22.74 / 4.503 5.05

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Life Insurance, cont.

• Ordinary life same as previous except that the
  premiums and probabilities are calculated through
  age 99
• Gross Premium pure premium loading
• Loading
• Profit (or dividend), operating expenses,
  contingencies
• Expenses
• Production expenses
• Distribution expenses (selling expenses)
• Maintenance expenses (renewal commissions)

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Reserves Property Liability

• Unearned premium reserve liability that
  represents the unearned portion of gross premiums
  on all outstanding policies at time of valuation
• Purpose to pay for losses that occur during
  policy period (matching principle)
• Provide for premium refunds in event of
  cancellation
• Reinsurance pricing

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Reserves Property Liability

• Calculation annual pro rata method
• Assumes policies are issued on July 1
• Ex 2 year policy will have ¾ outstanding after
  end of 1st year
• Equity in unearned premium reserve
• Most expenses are incurred when policy written
• Creates overstatement of unearned premium reserve
  ? equity in unearned premium reserve

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Example

• Assume loss ratio of 60 (losses/earned prem)
• Assume expense ratio of 40 (exp/gross prem)
• Writes 10,000,000 of insurance