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					           Math 477 / 568
Casualty Actuarial Mathematics

                     Fall 2009
   University of Illinois at Urbana-Champaign

            Professor Rick Gorvett

                  Session 2:
            Risks and Risk Theory

                August 27, 2009
 What Did We Discuss Last Time?

• What an actuary is, in words and numbers
• What actuaries have traditionally done
• What actuaries are doing, and starting to do,
  these days
• What the P/C insurance market looks like
                 Today’s Agenda
•   Vocabulary
•   Actuarial science vs. finance
•   Elements of an insurable risk
•   Risk management functions, objectives, and
•   Contract law
•   Insurance contract provisions
•   Types of insurance coverages (summaries provided
    for your information)
•   Adverse selection
•   Risk theory and insurer solvency
    We First Must Learn to Quack
   Vocabulary is sometimes troublesome in practice.
    E.g, here are two words with multiple meanings

• “Risk”: sometimes used to mean
  – Uncertainty
  – An entity exhibiting uncertainty (e.g., policyholder)
• “Loss”: sometimes used to mean
  – An event causing damage or injury
  – A policyholder claim
  – A dollar amount associated with damage or injury
   Some Foundational Vocabulary
• Risk: uncertainty concerning loss (occurrence,
• Objective risk: variation of actual relative to
  expected loss
  – Law of Large Numbers: as the number of
    exposures increases, loss experience can be
    predicted more accurately
• Subjective risk: perception of risk or
  uncertainty – i.e., by an individual observer
Some Foundational Vocabulary (cont.)
“Risk is synonymous with uncertainty – lack of knowledge.”
       - Irving Fisher, The Theory of Interest, 1930
“But Uncertainty must be taken in a sense radically distinct from
  the familiar notion of Risk, from which it has never been
  properly separated.”
       - Frank Knight, Risk, Uncertainty, and Profit, 1921
“The term „risk,‟ as loosely used in everyday speech and in
  economic discussion, really covers two things which,
  functionally at least, in their causal relations to the phenomena
  of economic organization, are categorically different…. The
  essential fact is that „risk‟ means in some cases a quantity
  susceptible of measurement, while at other times it is
  something distinctly not of this character…. It will appear that
  a measurable uncertainty, or „risk‟ proper, as we shall use the
  term, is so far different from an unmeasurable one that it is not
  in effect an uncertainty at all. We shall accordingly restrict the
  term „uncertainty‟ to cases of the non-quantit(at)ive type.”
       - Knight, ibid
Some Foundational Vocabulary (cont.)

• Chance of loss: probability of a specified
  event occurring
  – Probabilities can be objective or subjective
  – A different concept than “risk”
• Peril: a cause of loss
  – E.g., fire, earthquake, windstorm,…
  – For example, an insurance policy may cover losses
    from “all perils” (except those specifically
    excluded) or “named perils”
Some Foundational Vocabulary (cont.)

 • Hazard: a physical or “emotional” condition
   that increases (or even creates) the chance of
   loss (frequency and/or severity)

 Physical hazard: e.g., icy roads, defective
 Moral hazard: dishonesty or character
  defects that increase loss frequency or severity
 Morale hazard: carelessness or indifference
  to loss because of existence of insurance
Some Foundational Vocabulary (cont.)

• Pure risk: involves only the possibilities of
  loss or no loss
  – No possibility of gain
  – Often one can protect against pure risks through
    insurance or another risk financing mechanism
• Speculative risk: involves the possibilities of
  loss, no loss, or gain
  – E.g., gambling
  – Generally, not considered insurable
Types of Pure Risks – Often Insurable
• Personal risks: risks that directly affect an
  – E.g., death, sickness, unemployment
• Property risks: risks involving damage to or
  theft of property
  – E.g., direct loss, indirect loss
• Liability risks: risks involving bodily injury or
  property damage to another person
  – Generally no upper limit; legal defense costs
Actuarial Science
      Bühlmann’s Classification

          Actuaries are of three kinds

• Actuaries of the first kind: life
  – Deterministic calculations
• Actuaries of the second kind: casualty
  – Probabilistic methods
• Actuaries of the third kind: financial
  – Stochastic processes
Actuaries vs. Financial Economists


• Technical, quantitative skills
• Risk considerations
• Specialized vocabulary
• Address monetary issues
Actuaries vs. Financial Economists


• Approach to risk
• Approach to interest rates
• Approach to profitability
• Bases of valuation
              Insurance vs. Finance
Insurance                 Finance
• Pure risk               • Speculative risk
   – Loss or no loss         – Loss or no loss or
   – No gain                   gain/profit
• Law of large numbers    • Portfolio risk
   – More exposures ==>      – Diversification potential
     more accurate
Requirements of an Insurable Risk

• Large number of exposure units
  – Preferably, very similar
  – Allows for law of large numbers
• Accidental loss
  – Outside control of insured
  – Moral hazard issues
  – Randomness ==> law of large numbers
Requirements of an Insurable Risk

• Losses are determinable and measurable
  – Definite as to cause, place, time, and amount
• Losses not catastrophic
  – Not a large proportion of exposure units
  – Pooling wouldn‟t work
  – Underwriting tools -- e.g.:
     • Geographic spread
     • Reinsurance
Requirements of an Insurable Risk

• Calculable chance of loss
  – Frequency
  – Severity
• Economically feasible premium
  – Affordable
  – Small relative to policy limit
Requirements of an Insurable Risk

• Which of the following are insurable risks?
  – War
  – Unemployment
  – Catastrophic meteor strike
  – Jobs re-training program
   Key Types of Insurable Risks
• Property
  – Direct property damage
  – Loss of use
  – Business interruption
• Liability
  – Damage / injury to others
     • From ownership (e.g., of car, home, business)
     • From personal activities
     • From professional activities
• Life
  – Protect family
  – Protect business
• Health
             Types of Insurance
• Private
  – Life and health        Fire
                           Inland marine
  – Property-casualty      Ocean marine
                           Personal auto
     • Lines of business   Commercial auto
     • Reinsurance         Commercial multi-peril
                           Commercial liability
• Government               Workers compensation ….

  – Social
     • Social security
     • Unemployment
  – Other
     • FDIC
Risk Management
        Functions of Insurance

• Risk-bearing and risk-pooling
  – Transfer risks to insurer (for premium)
  – Insurer diversifies by pooling
• Financial intermediation
  – Issue debt contracts, and invest funds
  – Compensation through yield spreads
  – Economic value from market imperfections
    Benefits of Insurance to Society

•   Indemnification
•   “Good night‟s sleep”
•   Investment funds
•   Loss prevention
•   Enhancement of credit
    Costs of Insurance to Society
          (the “dark side”)

• Cost of doing business
  – Expenses
• Fraud
• Inflated claims
      Methods Of Handling Risks
• Insurance is only one way of addressing risk
• Methods of managing risk:
  – Avoid the risk in the first place
  – Control the frequency or severity of loss
     • Loss prevention; loss reduction
  – Retain the risk
     • Active vs. passive retention
  – Transfer the risk via non-insurance mechanisms
     • Contractual transfer; hedging
  – Transfer the risk via insurance mechanisms
            Risk Management:
            Steps in the Process

• Determine the corporation‟s objectives
• Identify the risk exposures
• Quantify the exposures
• Assess the impact
• Examine financial risk management tools
• Select appropriate risk management approach
• Implement and monitor program
   Risk Management Objectives

• Pre-loss objectives
  – Economy
  – Reduction of worry / anxiety
     • I.e., get a “good night‟s sleep”
  – Meet externally imposed obligations /
  – Social responsibility
Risk Management Objectives (cont.)

• Post-loss objectives
  –   Survival of the organization
  –   Uninterrupted operations
  –   Earnings stability
  –   Continued growth
  –   Social responsibility
Risk Management Objectives (cont.)

• Issues and conflicts
  – Relationship of objectives to general corporate
    goals and objectives
  – Relationship to financial theory
     • Value maximization

  – Conflicts between the objectives themselves
  Risk Management Techniques
                Risk Control
     Reduce frequency and / or severity
             of potential losses
• Avoidance
  – Never take on the exposure in the first place
  – Abandon the exposure if already have it
• Loss control
  – Change characteristics of the exposure
  – Attempt to reduce both frequency and severity
• Separation
• Combination (pooling)
Risk Management Techniques (cont.)
                    Risk Financing
                    Post-loss funding
 • Retention
   – For non-serious, highly-predictable losses
   – What should retention level be?
   – How should losses be paid?
      •   No advance funding
      •   Earmarked accounts
      •   Captive insurers
      •   Risk retention groups
Risk Management Techniques (cont.)
               Risk financing (cont.)

• Non-insurance transfers
  – Contracts
  – Leases
  – Hold-harmless agreements
• Insurance
  –   Voluntary vs. mandatory
  –   Deductibles
  –   Excess insurance      Issues:
                            - Price
  –   Selection of insurer  - Service
                              - Solidity
   Contract Law
Insurance Contracts
   Requirements Of A Contract

• Offer

• Acceptance

• Consideration

• Legally competent parties

• Legal purpose
   Insurance Contracts – Special
• Aleatory contract
  – Contractual exchange may be unequal in value
  – Contingent on the occurrence / non-occurrence
    of a specific event
• Unilateral contract
  – A legally enforceable promise is made by just
    one party
• Conditional contract
  – There are qualifications on the insurer‟s promise
   Insurance Contracts – Special
       Characteristics (cont.)
• Personal contract
  – Contract is between the insurer and insured
  – Property/casualty policies generally cannot be
    “assigned” to others (although life policies can)
• Contract of adhesion
  – Contract is generally written by just one of the
    parties (the insurer)
  – Thus, any ambiguities in the contract are
    generally resolved against the insurer
     Other Insurance Concepts
• Indemnity
  – Restore insured to (financial) condition before
    the loss
  – Purpose of “indemnity” principle:
     • Do not allow the insured to profit
     • Reduce moral hazard
  – Actual Cash Value (ACV)
  – Exceptions:
     • Valued policies
     • Replacement cost coverage
     • Life insurance
 Other Insurance Concepts (cont.)

• Insurable interest
  – Insured has an “insurable interest” if suffers in
    event of a loss
  – Life insurance: required at contract inception
  – P/C insurance: required at time of loss
• Subrogation
  – The insured‟s right to recover against another
    (responsible) party is transferred to the insurer
Elements of an Insurance Contract

• (Application)
  – Applicant requests and offers to buy coverage
  – Insurer accepts or rejects the request
• (Binder)
  – Temporary agreement providing coverage
• Declarations
  – Identifies the insured and critical information
  – Describes coverage in broad terms
Elements of an Insurance Contract
• Definitions
• Insuring agreement(s)
  – Summarizes major promises agreed to
  – Named-perils: only specifically named perils
    are covered by the policy
  – All-risks: policy covers any losses unless from a
    peril which is specifically excluded
Elements of an Insurance Contract
• Exclusions and exceptions
  – Reasons for exclusions:
     • Perils are uninsurable (refer back to requirements of
       an insurable risk)
     • Avoid duplication of coverage
     • Coverage is not typically needed
• Conditions
  – Identifies the duties of the insured
Elements of an Insurance Contract

• Miscellaneous
  – Relationships among insurer / insured / third
  – E.g., cancellation, subrogation
• Endorsements (P/C) and riders (life)
  – Change the provisions of the original policy
• Insured retains “first” part of loss
• Purposes of deductibles:
   – Eliminate small claims
   – Reduce premiums
   – Reduce moral / morale hazard
• Types of deductibles:
   –   Straight
   –   Aggregate
   –   Franchise
   –   Disappearing
• Versus self-insured retentions (SIRs)
• In health insurance: insured and insurer split
  responsibility for losses on a percentage
  basis (e.g., 20 / 80)
• In property insurance: gives insured an
  incentive to insure property for at least a
  certain amount
  – “Insurance-to-value”
  – If insufficient coverage is purchased, insured
    must share in the loss
• Criminal law
     • Wrongs against society
• Breach of contract
• Torts – wrongs against others
  – Intentional
     • Assault, battery, trespass, slander
  – Absolute / strict
     • Liability imposed regardless of negligence / fault
     • Manufacturing explosives, owning dangerous animals
  – Negligence
     • Failure to exercise a legal standard of care
        Elements of Negligence

• A legal duty to act (or not act) in a way that
  protects others from harm
• Breach of that duty
• Damage or injury to the one owed the duty
• Causal relationship (“proximate cause”)
  between the breach and the injury
           Types of Damages
• Special damages
  – “Economic” damages
  – Clearly determined and measured
     • Medical expenses
     • Property damage
     • Loss of earnings
• General damages
  – Cannot be specifically measured
     • Pain and suffering
     • Loss of companionship
• Punitive damages
  – Intended to punish or deter
    Defenses Against Negligence

• Assumption of risk
• Contributory negligence
• Comparative negligence
• Last clear chance
• Sovereign immunity
  Quick Summaries of Some Coverages

• Personal                     • Commercial
  –   Auto (liab & phys dam)     –   General liability
  –   HO (liab & property)       –   Business income
  –   Inland marine              –   Commercial multi-peril
  –   Umbrella                   –   Businessowners
                                 –   Workers compensation
• Government                     –   Commercial auto
  – Federal flood insurance      –   Professional liability
  – FAIR plans                   –   Boiler & machinery
  – Fed. crop insurance          –   Marine (ocean & inland)
                                 –   Title
                                 –   Umbrella
              Personal Auto
• Liability coverage
  – BI: bodily injury
  – PD: property damage
  – Split versus combined single limits (CSL)
• Medical payments coverage
• Uninsured / underinsured motorists coverage
• Physical damage (coverage for damage to
  “your” auto)
  – Collision
  – Comprehensive (other than collision)
• Various policy forms – e.g.,
   – HO-3: special form (very common)
   – HO-4: contents (for renters)
   – HO-6: condominium owners
• Section I: property coverages
   –   Dwelling
   –   Other structures
   –   Personal property                HO-3 policy form
   –   Loss of use
• Section II: liability coverages
   – Personal liability
   – Medical payments to others
       Personal Inland Marine
• Protection for property subject to movement
• Can tailor coverage to specific nature of
• Can get higher limits than in HO policy
• Generally, coverage is worldwide
• Personal Articles Floater (PAF)
  – E.g., can cover jewelry, furs, silverware, golf
    equipment, fine arts, stamp / coin collections
• Scheduled Personal Property Endorsement
             Personal Umbrella
• Protection against catastrophic lawsuit or
• Typically, $ 1-10 million liability limits
• Two coverages:
  – Excess liability insurance: applies on top of (in
    addition to) underlying coverage (e.g., auto and HO)
  – Other coverage: covers certain losses not covered
    by underlying policies (after deductible or self-
    insured retention)
     • E.g., personal injury: libel, slander,…
Federal Property Insurance Programs
 • Certain types of perils are difficult for
   private insurance companies to insure – at
   least at affordable premiums
 • Examples:
   – Flood insurance in flood zones
   – Property insurance in riot-prone areas
   – Crop insurance
 • For each of these, the government got into
   the insurance business
      Federal Flood Insurance
• Problems in flood-prone areas
  – Potentially catastrophic loss
  – Adverse selection
• 1968 National Flood Insurance Act
  – Provide insurance in flood zones
  – At subsidized rates
• 1983 Write-Your-Own Program
  – Private insurers write the business
  – Backed up by federal government
                  FAIR Plans
• FAIR: Fair Access to Insurance Requirements
• Property insurance: difficulties due to urban
• 1968 Urban Property and Reinsurance Act
• Make property insurance available to urban
  property owners unable to obtain coverage in
  normal markets
• 31 states
• Pool / syndicate
  – Operated by private insurers
  – Private insurers share in program loss experience
       Federal Crop Insurance

• Losses to crops – e.g., hail, wind, drought,
  excessive rain, plant disease
• Catastrophic crop insurance:
  – Guarantees 50% of average yield
  – Indemnified at 60% of expected market price
• Multiple-peril crop insurance
  – 65% or 75% yield guarantee
  – Can elect up to 100% of expected market price
            General Liability

• Exposures
  – Premises and operations
     • Premises: ownership and maintenance
     • Operations: on or off premises
  – Products and completed operations liability
  – Contractual liability
  – Directors and officers liability
Commercial General Liability (CGL)
           Loss Dates

 • Occurrence policy
   – Covers losses occurring during the policy period
   – Regardless of when claim is “made” (reported)
 • Claims-made policy
   – Covers claims reported during policy period
   – Provided that the loss occurred after the
     “retroactive date”
   – “Tail” coverage
      Business Income Insurance
• Also called “business interruption” insurance
• Covers loss of “business income” and “extra
  expenses” that result from physical damage
  to covered property
• “Business income”: pre-tax profit or loss that
  would have been earned, plus continuing
  normal operating expenses
• “Extra expenses”: would not have been
  incurred without the loss – e.g., cost of
  temporary relocation, rent of substitute
   Commercial Multi-Peril (CMP)
• Also called “commercial package policy” (CPP)
• Package policy: combination of two or more
  coverages into a single policy
• Some advantages of a package policy:
  – Convenience
  – Reduce potential gaps in protection
  – Don‟t need to deal with multiple insurers – a e.g.,
    determining which insurer provides coverage in a
    given situation
  – Potentially lower premiums (vs. purchasing
    coverages separately)
         Businessowners Policy
• Appropriate for small- to medium-sized
  businesses: stores, office buildings,
  apartment buildings
• Package policy
  – Building
  – Business personal property
  – Business income and extra expense
  – Business liability
Workers Compensation & Employers
       Liability (WC & EL)
 • Part One: Workers compensation insurance
   – Legal obligation for employers to provide
     benefits to injured (on-the-job) employees
   – Work-related injury or occupational disease
   – No “policy limit”
   – Statutory benefits per WC law in each state
 • Part Two: Employers liability insurance
   – For work-related injury/disease not compensated
     under state WC law
               WC & EL (cont.)

• Some of the objectives of WC
  – Sure, prompt, reasonable income -- regardless of fault
  – Single remedy  reduce delays and costs
  – Encourage employer interest in safety / rehabilitation
• Other issues:
  – Who is an “employee”?
  – What constitutes “employment”?
           Commercial Auto

• Liability coverage
  – BI and PD
  – Accident caused by ownership / maintenance /
    use of covered auto
• Physical damage coverage

• Numerical classifications and symbols
         Professional Liability

• Medical malpractice
  – Hospital and physicians & surgeons policies
  – Policy limits on per-incident and aggregate bases
  – Does not cover “general liability exposures”
• Errors and omissions
  – Architects and engineers
  – Insurance agents and brokers
           Boiler & Machinery

• Direct damage from covered cause of loss –
  e.g., explosions
• Can also cover indirect loss (business
  income and extra expense)
• Emphasis on loss prevention and safety
  – Significant service / engineering activities
               Ocean Marine

• Protection for goods transported over water
• Significant history – e.g., Lloyd‟s of London
• Classes of coverage:
  – Hull – physical damage to ship
  – Cargo – covers shipper for damage or loss of
  – Protection and indemnity (P&I) – liability
  – Freight – covers ship owner for loss of earnings
    if goods damaged / lost
              Inland Marine

• Protection for goods transported on land
• Some classes of coverage:
  – Domestic goods in transit
  – Property held by bailees
  – Mobile equipment and property
  – Means of transportation and communication
              Title Insurance

• Protection against unknown defects in title
• Characteristics:
  – Protects against title defects that occurred prior
    to effective date of the policy
  – Insurer “assumes” that no losses will occur
  – Policy runs indefinitely
  – Single premium
  – Insured is indemnified in dollars (up to limit)
        Commercial Umbrella

• Required underlying coverages
  – E.g., CGL, business auto, EL
• Types of losses covered
  – BI and PD
  – Personal and advertising injury
• Excess liability insurance
  – Over underlying coverage, or
  – Over self-insured retention (SIR)
A Ratemaking Issue: Adverse Selection

  • Examples
    –   Flood plain
    –   Health and life insurance
    –   High-risk drivers
    –   Non-insurance: used cars (the “lemons
  • Solution: underwriting; pricing
  Risk Theory
Insurer Solvency
                  Risk Theory
• Typically, risk theory involves the
  mathematical specification of the evolution
  of an insurer‟s financial condition

             E.g., U(t) = u + c t – S(t)

• Annual aggregate losses are generally
  estimated by specifying two stochastic
  – Frequency (number of claims)
  – Severity (size of claim, given there is a claim)
   A.M. Best’s Insolvency Study

• June, 1991
• Property / casualty insurers
• 1969-1990
• 372 total insolvencies (302 for which a
  primary cause of insolvency could be
  Best’s Insolvency Study (cont.)
• Primary causes of insolvencies
  – Deficient loss reserves          28%
  – Rapid growth                     21
  – Alleged fraud                    10
  – Overstated assets                10
  – Significant change in business   9
  – Reinsurance failure              7
  – Catastrophe losses               6
  – Miscellaneous                    9
                  Next Time

• Economics and the insurance market
• References:
  – Boor, 1998, “A Macroeconomic View of the
    Insurance Marketplace,” CAS Study Note
  – Boor, 2004, “The Impact of the Insurance Economic
    Cycle on Insurance Pricing,” CAS Study Note