EXAM 2 Study Sheet FINA 341

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University of South Carolina *

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341

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Jan 9, 2024

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STUDY SHEET FOR FINA 341 EXAM 2: Lectures 10-17 Know how to calculate a Fair Premium by calculating its component parts (pure premium, loss adjustment expenses, underwriting expenses, profit loading). Be aware that some components may require discounting. Fair Premium = PV of Expected Claim Costs (pure premium) + PV of Expected Loss Adjustment Expenses + Underwriting Expenses + Fair Profit Loading Pure premium = (incurred losses + LAE) / # of exposure units LAE = Percent (given) of pure premium, PV = (LAE / (1 + interest rate)) <- discounting Underwriting costs = percent given * pure premium Fair profit = percent given * pure premium Example : loss distribution = $100,000 at 2%, $20,000 at 8%, and $0 at 90%. Find fair premium if policy provides full coverage, underwriting costs = 20% of pure premium, LAE = 10% of pure premium, fiar profit is 5%, premium is paid at beginning of year, claims paid at end of year, and interest rate = 8%. Pure premium = (100,000*.02) + (20,000*.08) + (0*.90) = $3,600 (need to discount) PV of expected claim costs = $3,600 / (1 +.08) = 3,333.33 [LAE = .10*3,600 = $360] <- need to discount = 360/1.08 = $333.33 [Underwriting costs = .20*3,600 = $720] [Fair profit = 0.05*3,600 = $180] Fair Premium = $3,333.33 + $333.33 + $720 + $180 = $4,566.67 Understand what reinsurance is, why it is useful to insurers and how to calculate losses based on the different reinsurance loss sharing arrangements we covered in class. - Easiest way to think about reinsurance is that it is insurance for insurance companies ; it is an arrangement by which the primary insurer that initially writes the insurance transfers to another insurer part or all of the potential losses associated with such insurance. The primary insurer is known as the ceding company (e.g., Allstate). The insurer that accepts the insurance from the ceding company is the reinsurer (e.g., Swiss Re). The retention limit is the amount of insurance retained by the ceding company. The amount of insurance ceded to the reinsurer is known as a cession. Retrocession is when a reinsurer insures part or all of a risk with yet another insurer (this process can go on indefinitely). - There are two basic methods for sharing losses in reinsurance agreements: Under the Pro rata method , the ceding company and reinsurer agree to share losses and premiums based on some proportion. Under the Excess (of loss) or XoL method , the reinsurer pays only when covered losses exceed a certain level. - Under a quota-share treaty (type of pro rata arrangement) the ceding insurer and the reinsurer agree to share premiums and losses based on some proportion o Example : Assume that Apex Fire Insurance and Geneva Reenter into a quota-share arrangement by which losses and premiums are shared 50-50. If a $100,000 loss occurs, Apex Fire pays $100,000 to the insured but is reimbursed by Geneva Re for $50,000 (all of this is invisible to the policyholder. The primary insurer adjusts and pays all claims irrespective of the reinsurance arrangements in place). - Under a surplus-share treaty (another type of pro rata arrangement) the reinsurer agrees to accept insurance in excess of the ceding insurer’s retention limit , up to some maximum amount.

o Example: Assume that Apex Fire Insurance has a retention limit of $200,000 (called a “line” ) for a single policy, and that four lines, or $800,000, are ceded to Geneva Re (so total underwriting “capacity” is $1 million). Assume that a $500,000 property insurance policy is issued. Apex Fire takes the first $200,000 of insurance, or two-fifths, and Geneva Retakes the remaining $300,000, or three-fifths. - An excess-of-loss treaty (XoL) is designed for protection against a catastrophic loss. A treaty can be written to cover a single exposure (e.g., expensive building), a single occurrence (e.g., hurricane), or excess losses. o Example: Apex Fire Insurance wants protection for all windstorm losses in excess of $1 million. Assume Apex enters into an excess-of-loss arrangement with Franklin Re to cover single occurrences during a specified time period. Franklin Re agrees to pay all losses exceeding $1 million but only to a maximum of $10 million. Ex. 1: If a $5 million hurricane loss occurs, Franklin Re would pay $4 million (Apex would pay $1 million). Ex. 2: If a $12 million loss occurred, Apex would pay the first $1 million, Franklin the next $9 million…and Apex pays the final $2 million (since it exhausted its reinsurance). Understand the concept of Risk Based Capital (RBC), what it is, how it works, what is its purpose/function, etc. Included in your understanding should be the authority granted to regulators at certain RBC ratios and the benefits to the insurer of maintaining a strong capital position. - Risk-Based Capital Standards : Designed by National Association of Insurance Commissioners (NAIC) to reduce the risk of insolvency. Means that insurers must insurers must hold a certain amount of capital, depending on the riskiness of their investments and insurance operations. An insurer’s RBC depends on asset risk, underwriting risk, interest rate risk, and business risk. A comparison of the company’s total adjusted capital (TAC) to the amount of required risk-based capital (RBC Ratio) determines whether company or regulatory action is required. (RBC Ratio = Total Adjusted Capital/RBC ). o Example : Suppose an insurer has TAC = $22 billion and RBC of $10 billion, then the RBC Ratio = 22/10 = 2.2 or 220% (has 2.2x min level of capital) Know the causes of insurer insolvencies and how claims are (aren’t) paid in the event of an insolvency. - Insolvency of insurers continues to be an important regulatory concern. Insurers rarely go insolvent. Historically, deficient loss reserves and inadequate pricing are by far the leading cause of p-c insurer impairments. Investment and catastrophe losses play a much smaller role. - A.M. Best Rating Scale: As a practical matter, it’s very difficult for an insurer to operate unless it has a rating of A- or better. Many businesses will not buy insurance from an insurer with a rating of below A- (and many brokers will not place business with you). - Reasons for insolvencies include: o Inadequate rates. o Inadequate reserves for claims. o Rapid growth and inadequate surplus. o Mismanagement and fraud. o Bad investments. o Problems with affiliates.

o Overstatement of assets. o Catastrophe losses. o Failure of reinsurers to pay claims. o Litigation environment (e.g., Florida). Understand the relevance of the McCarran-Ferguson Act for the regulation of insurance. The McCarran-Ferguson Act (1945) states that continued regulation and taxation of the insurance industry by the states are in the public interest. Federal antitrust laws apply to insurance only to the extent that the insurance industry is not regulated by state law. - (Review Paul vs. Virginia (1869) before reading into this part ) 1944 SEUA Supreme Court decision effectively overturned the 1869 Paul v. Virginia decision—after 75 years. State and Federal regulation of insurance were both constitutional. This created an obvious dilemma with no obvious solution, Congress stepped into the void. - McCarran-Ferguson Act of 1945: crafted a partial exemption of the business of insurance from the Sherman, Clayton and FTC Acts to the extent it is regulated by the states. Maintained that federal antitrust laws do apply in cases of boycott, coercion, or intimidation. Widely misunderstood by industry critics (including occasionally some members of Congress) as a blanket exemption from antitrust statutes. NAIC’s 1946 All Industry Bill became the model law establishing a framework for regulation in the wake of McCarran-Ferguson. Stringent rate regulation became the norm and by 1948 all states had enacted rate regulatory laws, usually in line with the All-Industry Bill. What are the Guiding Principles related insurance rates? - State Laws Require the Guiding Principal: “Rates should not be inadequate, excessive or unfairly discriminatory.” - Rates should be adequate for paying all losses and expenses (and to earn a reasonable profit). - Rates should not be excessive, such that policyholders are paying more than the actual value of their protection. - Rates must not be unfairly discriminatory; exposures that are similar with respect to losses and expenses should not be charged significantly different rates. Be able to demonstrate mathematically the distinction between prices in insurance markets with heterogeneous risks when classification is present and when it is not, and the consequences of failing to classify. - - Equal Treatment Insurance Company is the only insurer in the market

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o Premium for everyone = $750. Does Equal Treatment cover its costs? YES o $750 is the EV of all drivers if we combine both groups -> ($1000+ $500)/2 o BUT: the Speedy Drivers pay less than their expected cost, and the Sunday Drivers pay more than theirs - Objective of earning $50 profit per policy as follows: o Premium to Sunday Drivers = $550 (profit = $550-$500) o Premium to Speedy Drivers = $1050 (profit = $1050-$1000) - It costs $50 for each Speedy Driver identified (e.g., costs $50 to run DMV records). Assuming Equal Treatment continues to charge $750, what does Selective charge? o Sunday Drivers: $500+$50+$50 = 600 o Speedy Drivers: $1000+$50+$50 = 1,100 Understand what a loss payout pattern is and the distinction between long and short-tailed lines of insurance and the relative importance of investment income for each. - The cumulative percentage of losses paid overtime is known as the Loss Payout Pattern. E x p e c t e d C l a i m P a y m e n t s I n t e r e s t R a t e y e a r 1 y e a r 2 y e a r 3 0 . 1 0 . 0 5 $ 1 0 0 $ 0 $ 0 $ 9 0 . 9 1 $ 9 5 . 2 4 $ 5 0 $ 5 0 $ 0 $ 8 6 . 7 8 $ 9 2 . 9 7 $ 5 0 $ 2 5 $ 2 5 $ 8 4 . 9 0 $ 9 1 . 8 9 - 50/ (1.1) + 25/ (1.1) 2 + 25/ (1.1) 3 = $45.45 + $20.66 + 18.78 = $84.90 Understand the influence of investment earnings on insurance pricing. - Insurers are the 2 nd or 3 rd largest institutional investor in the world—investing trillions of dollars globally. Because premiums are paid in advance, they can be invested until needed to pay claims and expenses. - Investment earnings are the 2 nd largest source of revenue for insurers, behind premiums. - Investment income is extremely important in reducing the cost of insurance to policyowners and offsetting unfavorable underwriting experience. - Life insurance contracts are long-term; thus, safety of principal is a primary consideration. - In contrast to life insurance, property insurance contracts are short-term in nature, and claim payments can vary widely depending on catastrophic losses, inflation, medical costs, etc.. Understand the concept of parameter uncertainty. - Parameter Uncertainty (aka Uncertain Expected Losses or Ambiguity): Insurer does not know or is unsure of the true expected loss. If estimate of expected loss is too low (or too high) for one policyholder, then estimate is too low (or too high) for many policyholders (insurer’s errors in predicted losses are correlated). - Parameter uncertainty causes the distribution of avg. losses around the insurer’s estimate of expected losses to have a great variance, same effect as with correlated losses. Insurer needs to hold a large amount of capital, profit loading is high, less insurance for exposures with significant parameter uncertainty.

- Example: Property valued at $50,000, large number of policyholders, probability of loss = 0.02 or 0.04, insurer does not which probability is the true one, insurer views each probability as equally likely - Insurer’s expected claim costs = $1,500 = (1000+2000)/2 - But insurer knows that claim costs could be much higher; insurer needs a lot of capital. Understand the basic structure of insurer investment portfolios. Understand the various types of reserves held by insurance and the purpose they serve. - A loss reserve is an estimated amount for: Claims reported and adjusted, but not paid; claims reported and filed, but not adjusted; e ven claims incurred but not reported to the company. - Case reserves are loss reserves that are established for each individual claim (i.e., a “case”) o Methods for determining case reserves include:  The judgment method: a claim reserve is established for each individual claim  The average value method: an average value is assigned to each claim (e.g., Avg. auto property damage claim is $5,000).  The tabular method: loss reserves are determined for certain claims for which the amounts paid depend on data derived from mortality, morbidity, and remarriage tables (For example—Someone who is paralyzed due to a work- related injury may never work again. The size of the benefit will be based on age at the time of injury (helps determine expected life span), nature of the injury, etc.) - incurred-but-not-reported (IBNR) reserve is a reserve that must be established for claims that have already occurred but that have not yet been reported (e.g., might happen near the end of the accounting period, like Hurricane Ian, which occurred near the end of Q3 2022, so didn’t know what actual cost by end of Q3). Or when timing of loss is uncertain, as with COVID claims. - Unearned premium reserve is a liability item that represents the unearned portion of gross premiums on all outstanding policies at the time of valuation (e.g., if policyholder pays $1,200 on Jan. 1, the insurer “earns” that premium at the pace of $100 per month, not all at once. For example, as of Jan. 31, the insurer will have earned $100 and the remaining $1,100 are Unearned Premiums)

o Largest of all reserve categories. Its purpose is to pay for losses that occur during the policy period. It is also needed so that refunds can be paid to policyholders that cancel their coverage. It also serves as the basis for determining the amount that must be paid to a reinsurer for carrying reinsured polices. o The annual pro rata method is one method of calculating the reserve. Understand the difference between rate and premium. - A rate is the price per unit of insurance. - The pure premium is the portion of the rate needed to pay losses and loss adjustment expenses (i.e., costs arising directly from the claim). o The gross premium paid by the insured consists of the gross rate multiplied by the number of exposure units. Know how to adjust rates based on the (i) Pure Premium Method, and (ii) Loss Ratio Method. - Under the pure premium method , the pure premium can be determined by dividing the dollar amount of incurred losses and loss-adjustment expenses by the number of exposure units. o Example: Assume USC Insurance insures 500,000 autos in a given underwriting class (e.g., male teenage drivers). Over the past year, USC incurred losses and loss adjustment expenses of $33 million in this class of business. Pure Premium = (Incurred Losses and LAE)/(# Exposure Units)—need to add a loading for expenses (usually expressed as a % of the “gross rate”). o $33,000,000/500,000 = $66 (i.e., Pure Premium = $66 per insured auto) o If expenses = 40% of the gross rate, then to “gross up” the rate such that 40% of the gross rate is the expense loading and 60% the pure premium:  Gross Rate = (Pure Premium)/(1-Expense Ratio) = $66/(1-.40) = $110 - Under the loss ratio method , the actual loss ratio is compared with the expected loss ratio, and the rate is adjusted accordingly. o Example: Actual loss ratio is compared with the expected loss ratio and adjusted accordingly. Assume USC Insurance has incurred losses and loss adjustment expenses of $800,000 and earned premium of $1 million. Actual Loss Ratio = $800,000/$1,000,000 = 0.80. Based on the past 10 years of experience, expected LR = 0.70. How much do rates need to be increased to reflect the new, higher LR? Calculate % difference between actual and expected:  (0.80 – 0.70)/0.70 = 14.3% -> Implies rates need to rise by 14.3% Know the difference between the combined ratio and the investment income ratio. - The investment income ratio compares net investment income to earned premiums. o Inv. Income Ratio = Net investment income / Earned premiums - The combined ratio is the sum of the loss ratio and the expected ratio.

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 Understand the major techniques used by insurers to adjust price to more accurately reflect the risk associated with underwriting specific policyholders Know why rates might differ from rates calculated by the Fair Premium Model - In the “real world,” insurance pricing is more complex, fair premium model does not explain everything. Premiums and coverage appear to follow cycles: o “Hard” Markets: (prices high, coverage restricted) o “Soft” Markets: (prices low, coverage available) - Cycles and price increases following capital shocks are difficult to explain with fair premium model. - Fair premium implies that prices are forward looking, prices depend on expected future costs, prices depend on uncertainty about future costs, what happened in past is irrelevant.  Know and understand the requirements for a risk to be insurable Know and understand why it is important for an insurer to have a strong capital position. - Capital is the life’s blood of insurance companies. - The more capital a company has, the less likely it is to become insolvent, all else equal. - Benefits of Increasing Capital o Consumer Preference for Strength  Consumers prefer to insure with insurers that are financially strong.  Effect is particularly true for commercial (corporate) insurance customers.  Risk managers and brokers investigate strength before purchase.  Usually rely on rating from ratings agencies like A.M. Best.  All else equal, a strong insurer can charge more for its products. o Reduce Likelihood of Regulatory Action  If regulators have to take action to stave off a capital shortfall or insolvency, that could be the end of the insurer. Understand the concepts of moral hazard and adverse selection and how they differ / Understand the challenges moral hazard and adverse selection play in insurance markets. - Adverse selection is the tendency of people with a higher-than-average chance of loss to seek insurance at standard rates. If not controlled by underwriting, this will result in higher-than- expected loss levels. o Note that being the lowest cost insurer is NOT an underwriting principle—if you are always the lowest cost provider, likely means you’ll be adversely selected against (i.e., your product is underpriced) o If not controlled by underwriting (screening of applicants), adverse selection results in higher-than-expected loss levels. - Moral Hazard: Refers to the effect of insurance on the insured’s incentives to reduce losses.

o Examples: Drive less carefully, don’t lock car door when insured. Less concerned about buying a home in a flood-prone area. Consume more health care when insured. Less diligent about workplace safety. o Main Point: Insurers understand insurance gives rise to moral hazard. Insurance market responds to this in several ways. But central point is that as a result of Moral Hazard, insurance contracts will generally offer less than full coverage (deductibles, co-pays) causes less than full insurance coverage. Insured will therefore have to bear some risk. Know and understand why deductibles, co-payments, policy limits, etc., exist and the impacts they have on demand for insurance. - Deductibles: Share of loss for which the policyholder is responsible. Insurer pays nothing if claim is less than equal to the deductible. o Example : policy with a $500 deductible, then policyholder pays first $500 of losses  E.g.: I’m in an auto accident and the cost to repair my car is $3,000. I pay the first $500, insurer pays $2,500. o Types of deductibles: per occurrence (i.e., deductible paid on each claim) or aggregate (i.e., total of claims must reach a specific deductible amount before coverage kicks in) - Coinsurance: insured pays a proportion (the coinsurance rate) of any loss. o Example: Insured pays 20% of all medical costs o Coinsurance likely causes fewer services to be consumed (fewer claims) because the marginal cost of consumption >0 - Policy limit: the maximum amount that the insurer will pay. In property insurance, policy limits prevent people from paying for coverage in excess of losses they could sustain (i.e., don’t want to be “over insured”). o Limits also place an upward bound on what an insurer will need to pay. Therefore, insurers can estimate maximum possible losses. Know the factors that influence the supply and demand for insurance. - If an insurance contract’s premium equals the value of expected claim costs, risk-averse person (business) will likely demand full insurance coverage. o But because premiums (almost) always have a positive loading (loading > $0), then demand will be less than full coverage. o Bottom Line: Any factor (e.g., administrative costs, profit loadings/capital costs) that increases the price of coverage above the expected value of claims (pure premium) will limit the amount of private insurance coverage. This is one reason people are chronically underinsured. - In addition to reducing the demand for coverage, higher loadings affect supply. o Insurance for some types of exposures (with a high loading) is likely to be extremely limited (or nonexistent) o The administrative costs of underwriting could be so large (e.g., small employers with a history of injuring many workers) or the required risk loading so high that insurers may try to limit exposure to certain markets.

EXAMPLE QUESTION RELATED THE CALCULATION OF A “FAIR PREMIUM” What is the fair premium for a 1-year insurance policy for a policyholder with the following probability distribution? Loss Amount Probability $0 0.60 $60000 0.20 $75,000 0.15 $100,000 0.05 Note: Assume all claims and loss adjustment expenses are paid at the end of the year and the interest rate is 9%. Loss Adjustment Expenses are 10% of pure premium, Underwriting Expenses are 20% of pure premium and that the fair profit loading is 5% of pure premium. Round to the nearest dollar. Pure premium = (0*.6) + (60,000*.2) + (75,000*.15) + (100,000*.05) = $28,250 PV of expected claims = 28.250/1.09 = 25,917.4 LAE = (.10*28,250)/1.09 = 2,592 Underwriting = .20*28,250 = 5,650 Fair profit = .05*28,250 = 1,412.5 Fair premium = 25,917 + 2,592 + 5,650 + 1,412.5 = $35,572 SOLUTION Fair Premium = PV of Expected Claims + PV LAE + UW Cost + Profit Loading Pure Premium = ($0)(0.60)+$60000(.20)+$75000(.15)+$100,000(.05) = $0 + $12,000 + $11,250 + $5,000 = $28,250 LAE = $28,250(.10) = $2,825 PV of Pure Premium + PV LAE = $28,250/1.09 = $25,917 + PV LAE = $2,825/1.09 = $2,592 = $25,917 + 2,592 = $28,509 or equivalently (28250 +2825)/1.09 = $28,509 UW Costs = 0.2*$28,250 = $5650 Fair Profit Loading = .05*$28,250 = $1413. Fair Premium = $28,509 + $5650 + $1413 = $35,572

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