1

Risk and Insurance Foundation Concepts

• Definition of Risk

– Expected Value

– Variability around the Expected Value

Definitions

Risk - Unpredictable Outcome

Pure Risk Loss No Loss

Definitions

Speculative RiskLossNo LossGain

2

Definitions (Cont.)

Risk Averse - Prefer to Avoid RiskWilling to Pay More than Expected Loss to Avoid Risk

Risk Seeker - Prefer RiskWould Pay More than Expected Return to Engage in Risky Situation

Definitions (Cont.)

Exposures

- Person or Property Facing Pure Risk

Definitions (Cont.)

Personal Loss Exposures

- Affect Life, Health or Income of an Individual

Definitions (Cont.)

Property Loss Exposure - Damage to Items

Direct Loss - Cost of Repair (Replacement)Consequential Loss -Additional Costs

3

Definitions (Cont.)Liability Loss

Exposure

- Responsible for Damages to Others

Peril

• A peril is defined as a cause of loss.

– If your house burns because of fire, the peril or cause of loss is “fire”.

– If your car is damaged in a collision with another vehicle, the peril is “collision”.

Example of PerilsNaturalPerils

Human Perils SystematicPerils

FireWindHailEarthquakeFloodCollisionIceCollapse

EmbezzlementDiscriminationKidnapPollutionTheftTerrorismRiotVandalism

Changes inConsumerTastesDepressionInflationObsolesceTechnologicalAdvancesWar

Exposure Analysis: Restaurant Example

Loss in building value

Loss of income

Cost to defend and settle claim

3. Financial Consequence

FireCustomer slips and injures herself

2. Peril

BuildingFreedom from legal liability

1. Valued Thing

4

Definitions (Cont.)

Hazards - Conditions

Affecting Perils

Definitions (Cont.)

Physical Hazards - Property Conditions

Intangible Hazards - Attitudes and Culture

Definitions (Cont.)

Intangible Hazards -

Moral Hazard - Fraud

Societal Hazards - Legal and Cultural

Probability Distribution

• Identifies all possible outcomes for a random variable and the probability of the outcome

• What is a random variable?– A variable whose outcome is uncertain.

• How does this relate to the definition of risk?

5

Key Point

• Sum of the probabilities must equal 1• Example of flipping a coin

½ or 0.5 or 50%-$1

½ or 0.5 or 50%$1

ProbabilityPossible Outcomes for X

Two Types of Distributions

• Discrete

• Continuous

Presenting Probability Distributions

• Two ways of presenting probability distributions:

– Numerical listing of outcomes and probabilities

– Graphically

Example of a Discrete Probability Distribution

– Random variable = damage from auto accidents

Possible Outcomes for Damages Probability$0 0.50$200 0.30$1,000 0.10$5,000 0.06$10,000 0.04

6

Example of a Discrete Probability Distribution

Example of a Continuous Distribution

Continuous Distributions

• Important characteristic of continuous distributions– Area under the entire curve equals one

– Area under the curve between two points gives the probability of outcomes falling within that given range

Probabilities with Continuous Distributions

• Find the probability that the loss > $5,000• Find the probability that the loss < $2,000• Find the probability that $2,000 < loss < $5,000

Possible Losses

Probability

$5,000$2,000

7

Summary Measures of Loss Distributions

• Instead of comparing loss distributions, we often work with summary measures of distributions:

– Frequency

– Severity

– Expected loss

– Standard deviation or variance

– Maximum probable loss (Value at Risk)

• How does RM affect each of these measures?

Frequency• Frequency measures the number of losses in a

given period over the number of exposures to loss.

• If Sharon Steel Corp. had 10,000 employees in each of the last five years…

• And, there were 1,500 injuries over the 5-year period…

• Then, the loss frequency would be 0.03 = 1500/50,000 employee-years.

Severity

• Severity measures the magnitude of loss per occurrence.

• If the 1,500 injuries cost a total of $3,000,000…

• Then, the expected severity of loss would be $2,000.

Expected Loss• The Expected Loss is simply the product of

the frequency and the severity

• Expected loss = Frequency x Severity

• In our example, the Expected Loss equals $60

0.03 x $2,000 = $60Also known as the “pure premium”

8

Expected Value• Formula for a discrete distribution:

– Expected Value = μ= x1 p1 + x2 p2 + … + xM pM .– Example:

Possible Outcomes for Damages Probability$0 0.50$200 0.30$1,000 0.10$5,000 0.06$10,000 0.04

Expected Value = $0 + $60 +$100 + $300 + $400 = $860

Expected Value

Standard Deviation and Variance– Standard deviation indicates the expected magnitude of

the error from using the expected value as a predictor of the outcome

– Variance = (standard deviation) 2

– Standard deviation (variance) is higher when

• when the outcomes have a greater deviation from the expected value

• probabilities of the extreme outcomes increase

Variance and Standard Deviation

• Variance = Σpi(xi - μ)2

• Standard Deviation = Square Root of the Variance

i=1

N

9

Standard Deviation and Variance

– Comparing standard deviation for three discrete distributions

Distribution 1 Distribution 2 Distribution 3

Outcome Prob Outcome Prob Outcome Prob$250 0.33 $0 0.33 $0 0.4$500 0.34 $500 0.34 $500 0.2$750 0.33 $1000 0.33 $1000 0.4

Standard Deviation Calculation for Distribution 1

1. Calculate difference between each outcome and expected value

$250-500=-$250$500-500= $0$750-500 =$250

2. Square the results$62,500$0$62,500

Standard Deviation Calculation (cont.)

3. Multiply by results of step 2 by the respective probabilities

(0.33)($62,500) = $20,833(0.34)($0) = $0(0.33)($62500) = $20,833

4. Sum the results 20,833 + 0 + 20,833= $41,666This is the Variance

5. Take the Square Root = $204.12

Standard Deviation Calculation for Distribution 2

1. Calculate difference between each outcome and expected value

$0-500=-$500$500-500= $0$100-500 =$500

2. Square the results$250,000$0$250,000

10

Standard Deviation Calculation (cont.)

3. Multiply by results of step 2 by the respective probabilities

(0.33)($250,000) = $82,500(0.34)($0) = $0(0.33)($250,000) = $82,500

4. Sum the results $82,500 + 0 + $82,500 = $165,000This is the Variance

5. Take the Square Root = $406.20

Standard Deviation and Variance

Sample Mean and Standard Deviation

– Sample mean and standard deviation can and usually will differ from population expected value and standard deviation

– Coin flipping example

$1 if headsX =

-$1 if tails

• Expected average gain from game = $0• Actual average gain from playing the game 5 times =

Maximum Probable Loss– Maximum Probable Loss at the 95% level is the number,

MPL, that satisfies the equation:

• Probability (Loss < MPL) < 0.95

– Losses will be less than MPL 95 percent of the time

11

Correlation

– Correlation identifies the relationship between two probability distributions

– Uncorrelated (Independent)

– Positively Correlated

– Negatively Correlated

Correlation Examples

• What is the likely correlation?– Example 1:

Random variable 1: Auto accidents of an CSU student in 2007Random variable 2: Auto accidents of student in Australia in 2007

Correlation Examples

• What is the likely correlation?– Example 2:

Property damage due to hurricanes in Miami in Aug & Sept, 2008Property damage due to hurricanes in Ft. Lauderdale in Aug & Sept, 2008

Correlation Examples (cont.)

– Example 3:Property damage due to hurricanes in Miami in Sept. 2008Property damage due to hurricanes in Miami in Sept. 2005

12

Correlation Examples (cont.)

– Example 4:Number of people who die from AIDS in New York in 2008Number of people who die from AIDS in London in 2008

Calculating the Frequency and Severity of Loss

– Example:

• 10,000 employees in each of the past five years• 1,500 injuries over the five-year period• $3 million in total injury costs

• Frequency of injury per year = 1,500 / 50,000 = 0.03

• Average severity of injury = $3 m/ 1,500 = $2,000

• Annual expected loss per employee = 0.03 x $2,000 = $60

Question 1: Buckeye Brewery-Property Losses

• L = Expected property losses

• L has the following property loss distribution$3,000,000 with probability of 0.004$1,500,000 with probability of 0.010$800,000 with probability of 0.026$0 with probability of 0.96

• What is the expected loss?

Question 2: Buckeye Brewery-Liability Losses

• L = Expected liability losses

• L has the following liability loss distribution$5,000,000 with probability of 0.004$1,500,000 with probability of 0.025$500,000 with probability of 0.03$0 with probability of 0.941

• What is the expected loss?

13

Question 3 – Buckeye Brewery

• Are Buckeye Brewery’s property losses:– Independent– Positively correlated– Negatively correlated

• With its liability losses?

Law of Large Number

• As a sample of observations is increased in size, the relative variation about the mean declines.

35100298022601440820

3499297922591439719

3498297822581338718

3497287722571237617

3496277622561236616

349527 7521551135615

3494267421541034514

3393267320531033513

3392267220521032512

3391267119511031511

32902670 18501030510

328926691749102959

328825681748102848

31872567174792737

31862566164682626

30852565154582525

30842464144482414

30832363144382313

29822362144282202

29812361144182101

Number of successes

ExptNumber of successes

ExptNumber of successes

ExptNumber of successes

ExptNumber of successes

Expt

0.3437182

Tossing a Coin -- An experiment of 100 trials

n

np

14

Methods of Handling Pure Risk

• Avoidance

Methods of Handling Pure Risk

• Loss Control– Loss Prevention

Reduce Loss Frequency– Loss Reduction

Lower Loss Severity

Methods of Handling Pure Risk

• Retention

Methods of Handling Pure Risk

• Transfer– Corporations

15

Corporate StructureSole Proprietorships

Corporations

Partnerships

Unlimited Liability

Personal tax on profits

Limited Liability

Corporate tax on profits +

Personal tax on dividends

Methods of Handling Pure Risk

• Transfer– Contractual Agreements

Definition of Insurance(Page 52 of text)

Insurance is a social device in which a group of individuals (insureds)

transfer risk to another party (insurer) in order to combine loss experience,

which permits statistical prediction of losses and provides for payment of

losses from funds contributed(premiums) by all members who

transferred risk.

16

Insurance Versus Gambling

Gambling Creates RiskInsurance Transfers Existing RiskGambling is Speculative RiskInsurance Deals with Pure Risk

Ideal Requisites for Insurability

1 Large Number of Similar Exposure Units2 Fortuitous Losses3 Catastrophe Unlikely4 Definite Losses5 Determinable Probability Distribution6 Economic Feasibility

Which, if any, requisites are violated by the following?

• Mental Health Insurance• Flood Insurance• Satellite Launch Insurance• Tornado Insurance• Inflation Insurance• Divorce Insurance• Mutual Fund Insurance

Types of Insurance

• Personal or Commercial• Life-Health or Property-Liability• Private or Government• Voluntary or Involuntary

17

Types of Insurance - Examples

Social Security• Personal• Life-Health • Government• Involuntary

Medical Malpractice• Commercial• Property-Liability• Private• Voluntary