Chapter 1 Lecture Notes (STAT4903 Actuarial Techniques for General Insurance - Fall 2024) PDF

THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT4903 Actuarial Techniques for General Insurance (Fall 2024) Chapter 1 Lecture Notes Chapter 1: Key concepts for general insurance This chapter introd...

THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT4903 Actuarial Techniques for General Insurance (Fall 2024) Chapter 1 Lecture Notes Chapter 1: Key concepts for general insurance This chapter introduces the important concepts and terminology related to general insur- ance. Some terms may have different meanings depending on the context, and hence it is important to understand the differences. 1.1 Introduction The true cost of the payments derived from an insurance policy may not be immediately known to the insurer. However, the insurer needs to report its financial results on a regular basis. Reserves must be made for events that have occurred but the corresponding claims have not yet been paid. This makes it necessary for insurers to estimate the cost of claims. 1.1.1 Internal management For the insurer, accurate estimation is essential for proper decision-making. It affects almost every area of operations: Pricing: – Inadequate estimates → Reduction in rates erroneously → Insolvency – Excessive estimates → Increase in rates erroneously → Loss of market share and revenue Underwriting: Inaccurate estimates may lead to the insurer making an incorrect decision to enter/withdraw from a certain market. Need for reinsurance, claims management procedures and policies may be affected. Capital management (allocation of surplus to lines of business) is also a decision made by the insurer. 1.1.2 Investors Inaccurate reserves may result in misstated balance sheets and income statements. In- vestors who rely on such information to make their investment decisions regarding the insurer may thus be affected. 1.1.3 Regulators If an insurer underestimates its future obligations, it may appear financially healthy when it is on the verge of becoming insolvent. Being unable to see the true picture of the insurer, regulators may become involved too late to save the company (and maybe even policyholders and investors). STAT4903 Chapter 1 1 Fall 2024  Chapter 1: Key concepts for general insurance 1.1.4 Insurance pricing Conceptually speaking, the price of any commodity in a free market can be written as Price = Cost + Profit. It is relatively easy to determine the cost for non-insurance goods and services. However, for many types of insurance, the ultimate cost is unknown when the policy is sold. This is because insurance is a promise to do something in the future contingent on the occurrence or non-occurrence of certain events. Therefore, we need to develop models and methods for ratemaking, i.e., setting insurance prices, in the general insurance (also known as property and casualty insurance in the United States and Canada) industry. 1.1.5 Rating manual A rating manual is a document that contains the necessary information to: appropriately classify each risk; calculate the premium associated with that risk. The earliest rating manuals were very basic. Over time, rating manuals have become increasingly complex. Needless to say, the information contained in a rating manual highly depends on the type of insurance policies (e.g., motor insurance vs property insurance). The premium is a price paid by insurance consumers, and is usually based on a rate per unit of risk exposed. Insurance premium can vary significantly for risks with different characteristics. 1.2 Organization and Background In this section, we take a look at the different types of general insurance coverages and lines of business around the world. 1.2.1 Overview of the basic techniques for estimating claims and expenses The actuary’s estimate of an insurer’s obligation for future payment resulting from claims due to past events is known as the unpaid claims estimate. The following will be discussed: Comparison of results from different techniques; The impact of environment changes on the methodologies; Evaluation of the methodologies. It is also necessary for the insurer to estimate the (unpaid) claim adjustment expenses. These reflect the costs of: administering claims; determining coverage for claims; settling or defending claims. STAT4903 Chapter 1 2 Fall 2024  Chapter 1: Key concepts for general insurance Note that these costs are incurred even if it is ultimately determined that the claim is invalid. The extent of expenses varies by the type of claims. For example: A claim for a house fire may be settled relatively quickly with only a few phone calls. Liability arising from the use of asbestos involves complex legal and medical issues. In extreme situations, the claim adjustment expenses may even exceed the claims payment itself. 1.2.2 General insurance coverages While the following list is not exhaustive, it provides an overview of the common types of coverages encountered in general insurance. 1. Accident benefits: A no-fault automobile coverage that provides numerous benefits following a covered accident, such as: – Medical and rehabilitation expenses; – Funeral benefits; – Death benefits; – Loss of income benefits. 2. Automobile coverages, for example: Collision: automobile physical damage coverage; protects against claims re- sulting from any damages to the insured’s vehicle caused by collision with another vehicle or object. Commercial automobile liability: provides protection from the liability arising from the business use of owned, hired or borrowed automobiles, or from the operation of an employee’s automobiles on behalf of the business. Private passenger automobile liability: provides third-party liability protection to the insured against a claim or suit for bodily injury or property damage arising out of the operation of a private passenger automobile. Private passenger automobile physical damage: provides protection against damage to or theft of a covered private passenger automobile. 3. Crime insurance: Protects individuals and organizations from the loss of money, securities or inventory resulting from crime; Examples include: – Employee dishonesty; – Embezzlement; – Forgery; – Robbery; – Safe burglary; STAT4903 Chapter 1 3 Fall 2024  Chapter 1: Key concepts for general insurance – Computer fraud; – Wire transfer fraud; – Counterfeiting. 4. General liability: An umbrella term of some coverages. Examples include the exposures of: – Premises liability; – Operations liability; – Products liability; – Completed operations liability; – Professional liability (errors and omissions). 5. Medical malpractice / medical professional liability insurance: Unique general liability exposures for insured (both individuals and organiza- tions) offering medical care and related professional service; Often separated into hospital professional and physician/surgeon professional liability insurance. 6. Primary insurance: First layer of insurance coverage. Pays compensation in the event of claims arising out of an insured event ahead of any other insurance coverages that the policyholder may have. 7. Property insurance: Provides protection against most risks to property, e.g., fire, theft, certain damage due to the weather; Specialized coverages such as fire insurance, earthquake insurance, and home- owners insurance exist. 8. Umbrella and excess insurance: Liability types of coverage available to individuals and companies; Provides protection against claims above and beyond the amounts covered by primary insurance policies; In some circumstances, it provides protection for claims not covered by the primary policies. 9. Workers compensation and employers liability insurance: Provides coverage for the benefits the employer (the insured in this case) becomes legally responsible for due to workplace injury, illness and/or disease. Also covers the cost to defend against, and possibly pay, liability claims made against the employer on account of bodily injury to an employee. STAT4903 Chapter 1 4 Fall 2024  Chapter 1: Key concepts for general insurance 1.3 Basic insurance terms This section introduces the most fundamental terms in insurance ratemaking. 1.3.1 Exposure Exposure is the basic unit of risk that underlies the insurance premium. The unit varies greatly by the line of business. Homeowners insurance: house-year (one house insured for a period of one year). Workers compensation insurance: a certain sum of the annual payroll. There are four different ways of measuring exposures: Written exposures: the total exposures arising from policies issued during a spec- ified period of time. Earned exposures: the portion of the written exposures for which coverage has already been provided as of a certain point in time. Unearned exposures: the portion of the written exposures for which coverage has not yet been provided as of a certain point in time. In-force exposures: the number of insured units that are exposed to loss at a certain point in time. An illustrative diagram is as follows. Written exposures Total exposures for policies issued during this period Period Earned/Unearned exposures Earned exposure Unearned exposure for this policy for this policy Policy issue Specified time Policy expiry In-force exposures Number of insured units exposed to loss at this time Policy 1 Policy 2 Policy 3 Policy 4 Specified time STAT4903 Chapter 1 5 Fall 2024  Chapter 1: Key concepts for general insurance 1.3.2 Premium Premium is the amount paid by the insured for insurance coverage. We have similar classifications as exposures: Written premium: the total premium associated with policies that were issued during a specified period. Earned premium: the portion of the written premium for which coverage has already been provided as of a certain point in time. Unearned premium: the portion of the written premium for which coverage has not yet been provided as of a certain point in time. In-force premium: the full-term premium for policies that are in effect at a given point in time. 1.3.3 Claim Claim is a demand made by the insured to the insurer for indemnification under a policy. Some important terms are as follows: Policy: an agreement through which the insured pays premium to the insurer in exchange for a promise to indemnify the insured for the financial consequences of a covered event. Claimant: the insured or a third party making the demand for payment. Accident/Occurrence date: the date of occurrence of the covered event. – The event may be sudden or the result of continuous or repeated exposure until the event is apparent. – In the latter case, the date at which the event becomes apparent is generally regarded as the accident date. Report date: the date on which the claimant reports the claim to the insurer. Unreported claims/Incurred but not reported (IBNR) claims: claims that are not currently known by the insurer. Open claim: a claim that has not been settled. Closed claim: a claim that has been settled. Re-opened claim: a previously closed claim on which further activity occurs. 1.3.4 Loss Loss is the amount of compensation paid or payable to the claimant under the terms of the insurance policy (whereas a claim is a demand for compensation). Some terms regarding losses are as follows: Paid claims/losses: amounts that have been paid to claimants. STAT4903 Chapter 1 6 Fall 2024  Chapter 1: Key concepts for general insurance Case estimate/outstanding/reserve: estimate of the amount of money required to ultimately settle a claim, net of paid claims. Case estimates are adjusted over time, and payment is expected to be made in the future. Reported claims/losses, also known as case incurred claims or incurred on reported claims: sum of paid claims and case estimate: Reported claims = Paid claims + Case estimate. Note that this pertains to reported cases only — IBNR claims are not included. Ultimate claims/losses: total dollar value of all claims for a defined group of policies, when they are all settled and closed without any chance of reopened claims. The ultimate claims consist of the following components: – Cumulative claim payments (paid claims), e.g., amount paid between January 1 and December 31 of 2021 for claims with accident dates in the same period of time. – Case estimates for reported claims, e.g., estimate of the amount still yet to be paid for claims (that are reported) with accident dates between January 1 and December 31 of 2021. – Development on known claims, also known as incurred but not enough reported (IBNER) estimate, which is the change in reported claims that may occur in the future. We can write IBNER estimate = Total amount estimated to ultimately settle reported claims − Total reported claims at evaluation time. – Incurred but not yet reported (IBNYR) claims, often referred to as pure IBNR. This is the narrow definition of IBNR. Hence, we can write Estimated ultimate claims = Paid claims + Case estimate +IBNER + IBNYR. | {z } Reported claims (1.1) Note that IBNER arises because there may be some future development of existing claims, which have not yet been reported to the insurer. Hence, a broad definition of IBNR claims includes provision for future development on known claims, estimate for reopened claims, provision for claims incurred but not yet reported, and provision for claims in transit. For the purposes of our discussion, we write (Broad) IBNR = IBNER + IBNYR. Hence, equation (1.1) can also be rewritten as Estimated ultimate claims = Reported claims + (Broad) IBNR. This decomposition places an emphasis on whether the amount is associated with a claim that has been reported to the insurer or not. STAT4903 Chapter 1 7 Fall 2024  Chapter 1: Key concepts for general insurance Ultimate claims Broad IBNR (IBNER + IBNYR) Case estimate Paid claims Reported claims Unpaid claims 1.3.5 Loss adjustment expense Loss adjustment expenses (LAE) are those incurred by the insurer in the claim settlement process. It is typically broken down into two types: Allocated loss adjustment expenses (ALAE): claim-related expenses that are di- rectly attributable to a specific claim, such as legal and expert witness expenses. Unallocated loss adjustment expenses (ULAE): claim-related expenses that cannot be directly assigned to a specific claim, such as the salaries of claims department personnel. The two types of expenses add up to the total loss adjustment expense, i.e., LAE = ALAE + ULAE. 1.3.6 Underwriting expense Underwriting expenses are those incurred in the acquisition and servicing of policies. Typical categories of underwriting expenses are: Commissions and brokerage: amounts paid to insurance agents or brokers as compensation for generating business; usually paid as a percentage of premium written. Factors that may affect the commission percentage include: – Whether it is a new or renewal business; – Quality of the business written; – Volume of business written. Other acquisition costs: expenses other than commissions and brokerage expenses paid to acquire business, such as advertisements. General expenses: remaining expenses associated with the insurance operations, such as costs related to the general upkeep of the home office. Taxes, licenses and fees: all taxes and miscellaneous fees paid by the insurer (excluding federal income taxes), such as premium taxes and licensing fees. STAT4903 Chapter 1 8 Fall 2024  Chapter 1: Key concepts for general insurance 1.3.7 Profit The insurer expects to make a profit on average for assuming the risk that the premiums received may not be sufficient to pay claims and expenses. There are two main sources of profit: Underwriting profit or operating income: sum of the profits (income minus outgo) generated from the individual policies. Investment income: money earned by investing funds held by the insurance com- pany. 1.4 Life of a claim Many steps are involved from filing a claim to its eventual settlement. The different types of claim transactions over the life of a claim can include the following: 1. Establishment of the initial case outstanding estimate; 2. Notification to the reinsurer if the claim is expected to exceed the insurer’s retention; 3. Partial claim payment; 4. Expense payment to independent adjuster (the adjuster is someone hired by the insurer to deal with specific types of claims that the insurer does not have expertise in); 5. Change in case outstanding estimate; 6. Assumed final claim payment; 7. Elimination of the case estimate and closure of the claim. Some claims that are closed may have further activities: 8. Reopening of the claim and establishment of a new case outstanding estimate; 9. Partial payment for defense litigation; 10. Final claim payment; 11. Final payment for defense litigation; 12. Closure of claim. The following example provides a sample claim with an illustration of the above events. Example. A sample claim with all transactions listed. STAT4903 Chapter 1 9 Fall 2024  Chapter 1: Key concepts for general insurance Claim fact summary Policy period: 2017/12/01 — 2018/11/30 Date of accident: 2018/11/15 Date of claim report: 2019/02/20 Claim transactions Reported value of Cumulative claim to paid to Case Date Transaction date date estimate 2019/02/20 Case estimate of $15,000 established for $15,000 $0 $15,000 claim only 2019/04/01 Claim payment of $1,500 — case estimate $15,000 $1,500 $13,500 reduced to $13,500 (change of −$1,500) 2019/05/01 Expense payment to independent adjuster $15,500 $2,000 $13,500 of $500 — no change in case estimate 2019/09/01 Case estimate for claim increased to $32,000 $2,000 $30,000 $30,000 (change of $16,500) 2020/03/01 Claim thought to be settled with additional $26,000 $26,000 $0 payment of $24,000 and claim closed — case estimate reduced to $0 2021/01/25 Claim reopened; case estimate of $10,000 $46,000 $26,000 $20,000 for claim and $10,000 for defense costs 2021/04/15 Partial payment of $5,000 for defense $46,000 $31,000 $15,000 litigation — case estimate for defense costs reduced to $5,000; no change in case estimate for claim 2021/09/01 Final claim payment for an additional $48,000 $43,000 $5,000 $12,000 — case estimate for claim reduced to $0 (change of −$10,000) 2022/03/01 Final defense cost payment for an $49,000 $49,000 $0 additional $6,000 — case estimate for defense costs reduced to $0 (change of −$5,000) and claim closed From the above example, we can observe the characteristics of insured claims: Claim activity extends over a period of time. The value of a claim may change over its life and is not finally established until closure. A claim can have many different types of associated payments (payment to the injured party, to the independent adjuster, and for defense litigation). A claim has many associated dates: – Policy effective date: the date the insurer issues the insurance policy (2017/12/01). STAT4903 Chapter 1 10 Fall 2024  Chapter 1: Key concepts for general insurance – Accident date or date of loss: the date the covered injury occurs (2018/11/15). – Report date: the date the insurer receives notice of the claim (2019/02/20). – Transaction date: the date on which either a case estimate transaction takes place or a payment is made (all the dates in the above table). – Closing date: the date on which a claim is closed either initially or finally (2020/03/01 and 2022/03/01). – Reopening date: the date the insurer reopens the claim (2021/01/25). Some further comments on the example: It is important to differentiate between cumulative paid claims and incremental paid claims. Incremental paid claims refer to the sum of all claim payments made during a specified time interval, while the cumulative paid claims refer to the sum of all claim payments made to date. The case estimate is the estimated amount of future payment for a claim at a specified time. Similar to paid claims, we also need to differentiate between cumulative reported claims (case incurred) and incremental reported claims. We can see that Cumulative reported claims = Cumulative paid claims + Case estimate, all based on the same point in time. Also, Incremental reported claims = End-of-period cumulative reported claims − Beginning-of-period cumulative reported claims = Sum of incremental paid claims over the period + Change in case estimate over the period, where Change in case estimate = Ending case estimate − Beginning case estimate. The following presents another illustration on the effect of claim transactions on reported claims. Example. The following table provides ten illustrative scenarios on how claim trans- actions during a year affect reported claims. STAT4903 Chapter 1 11 Fall 2024  Chapter 1: Key concepts for general insurance At 2017/12/31 During 2018 At 2018/12/31 # Cum. Case Reprt. Paid Chg. in Reprt. Cum. Case Reprt. paid est. claims claims case claims paid est. claims claims est. claims 1 0 0 0 100 0 100 100 0 100 2 200 0 200 50 0 50 250 0 250 Making payments where there had been no previous case estimate increases reported claim. 3 0 0 0 0 1,000 1,000 0 1,000 1,000 Establishing a case estimate increases reported claim by the amount of the case estimate. 4 0 1,000 1,000 100 −100 0 100 900 1,000 Payment with offsetting case estimate reduction has no effect on reported claim. 5 500 5,000 5,500 200 −1,000 −800 700 4,000 4,700 If case estimate is reduced by a larger amount than the claim payment, the impact is a reduction to reported claim. 6 5,000 10,000 15,000 12,000 −10,000 2,000 17,000 0 17,000 If payment on closing exceeds case estimate, reported claim transaction is positive. 7 5,000 10,000 15,000 6,000 −10,000 −4,000 11,000 0 11,000 If payment on closing is less than case estimate, reported claim transaction is negative. 8 5,000 15,000 20,000 4,500 0 4,500 9,500 15,000 24,500 Claim payment with no change in case estimate increases the reported claim. 9 3,000 10,000 13,000 0 −4,000 −4,000 3,000 6,000 9,000 No payment and decrease in case estimate decreases the reported claim. 10 2,000 10,000 12,000 1,000 5,000 6,000 3,000 15,000 18,000 Payment and increase in case estimate result in increase in reported claim. Note that the reported claims in an interval of time can be positive or negative. However, the cumulative reported claims at a point in time is rarely negative. The claim process can be summarized by the following diagram: Ultimate claim IBNYR IBNER + Reported Claim open Claim closed Claim reopened Accident Report Claim Claim Further Claim date date recorded settled activity ultimately closed 1.5 Fundamental insurance equation Based on the concepts mentioned above, we can formulate the fundamental insurance equation: Premium = Claims (or losses) + LAE + UW Expenses + UW Profit. | {z } | {z } | {z } Price Cost Profit STAT4903 Chapter 1 12 Fall 2024  Chapter 1: Key concepts for general insurance The goal of ratemaking is to balance both sides of the equation. The CAS’s ratemaking principle states that “a rate provides for all costs associated with the transfer of risk”. There are two key points to consider in the ratemaking process: 1. Ratemaking is prospective. The ratemaking process involves estimation of the various components of the equation. It uses relevant historical experience, but not to set a premium to recoup past losses. The CAS’s principle states that “a rate is an estimate of the expected value of future costs”. Historical experience is used to fit such an estimate. Some factors to consider in adjusting historical experience are: Rate changes; Operational changes; Inflationary pressures; Changes in the mix of business written; Law changes. 2. Overall and individual balance. The insurance equation should be balanced at both an overall as well as an individual or segment level. The CAS’s principle also states that “a rate provides for the costs associated with an individual risk transfer”. Failure to recognize differences in individual risk produces inequitable rates. For example, in workers compensation insurance, steel worker on high-rise buildings is generally classified as high-risk while a clerical office employee is generally classified as low-risk. The rates for insuring these two workers should be different. 1.6 Basic insurance ratios In this section, we will introduce some important insurance ratios that are commonly used in general insurance. 1.6.1 Frequency Frequency is a measure of the rate at which claims occur: Number of claims Frequency =. Number of exposures Usually, the formula uses reported claims in the numerator and earned exposures in the denominator. For example, if there are 100,000 claims with 2 million earned exposures, the frequency is 5%. Uses: – Identify general industry trends; – Measure the effectiveness of specific underwriting actions. STAT4903 Chapter 1 13 Fall 2024  Chapter 1: Key concepts for general insurance 1.6.2 Severity Severity is a measure of the average cost of claims: Claim amount Severity =. Number of claims It can be calculated based on paid or reported claims: Amount of paid claims (closed) Paid severity = ; Number of closed claims Amount of reported claims Reported severity =. Number of reported claims It may include or exclude ALAE. Uses: – Provide information on the trend of claim sizes; – Identify changes in claims handling practices. 1.6.3 Pure premium Also known as loss (claim) cost or burning cost, the pure premium measures the average claim amount per unit of exposure: Claim amount Pure premium = = Frequency × Severity. Number of exposures It is “pure” in the sense that it measures the cost that is purely attributable to the loss. It is usually based on reported or ultimate claims and earned exposures. It may include or exclude ALAE and/or ULAE. Use: Identify industry trends attributable to both frequency and severity. 1.6.4 Average premium Unlike the pure premium which measures the “cost” or loss, the average premium analyzes the premium side of the insurance equation: Total premium Average premium =. Number of exposures Both premium and exposures should be of the same type: written, earned, or in-force. Use: Identify shifts in the mix of business written, e.g., toward higher or lower risk characteristics reflected in rates. STAT4903 Chapter 1 14 Fall 2024  Chapter 1: Key concepts for general insurance 1.6.5 Claim ratio Also commonly known as the loss ratio, it measures the proportion of each premium dollar used to pay claims: Claim amount Pure premium Claim ratio = =. Premium Average premium To understand the last equality, recall that the pure premium actually measures the loss or claim amount (per exposure unit) while the average premium measures the premium received (per exposure unit) by the insurer. For example, if the total claim dollars are $300 million and the total premium is $400 million, then the claim ratio is 75%. Usually, the formula uses total reported claims in the numerator and total earned premium in the denominator. It may include or exclude LAE. Use: Measure the adequacy of the overall and segment rates. 1.6.6 Loss adjustment expense (LAE) ratio The LAE ratio compares the claim-related expenses as a proportion of the claim amount: Loss adjustment expenses LAE ratio =. Claim amount LAE includes both ALAE and ULAE. Uses: – Indicate whether claim settlement costs are stable; – Compare with ratios of other insurers as a benchmark for its claims settlement procedures. Note that there is another ratio, Claim and LAE ratio, that combines the claim amount and LAE, and is calculated as a proportion of premium: Claim amount + LAE Claim & LAE ratio = Premium Claim amount Claim amount LAE = + × Premium Premium Claim amount = Claim ratio × (1 + LAE ratio). The claim ratio and LAE ratio cannot be added up directly, since they are calculated based on different denominators. STAT4903 Chapter 1 15 Fall 2024  Chapter 1: Key concepts for general insurance 1.6.7 Underwriting expense ratio The UW expense ratio measures the proportion of premium paid as UW expenses: UW expenses UW expense ratio =. Premium To better match the nature of UW expenses to the corresponding premium measure, often UW expenses are divided into two categories: – Expenses incurred at policy inception (commissions; other acquisition ex- penses; taxes, licenses and fees): these are measured as a ratio to written premium. – Expenses incurred throughout the policy period (general expenses): these are measured as a ratio to earned premium. The UW expense ratio is then calculated by summing the above two components: Policy inception expenses General expenses UW expense ratio = +. Written premium Earned premium Uses: – Compare actual to expected changes based on inflation. – Compare with ratios of other insurers. 1.6.8 Operating expense ratio The OE ratio measures the proportion of premium paid as loss adjustment and underwriting expenses: LAE OE ratio = UW expense ratio +. Earned premium Use: Monitor operational expenditures and overall profitability. 1.6.9 Combined ratio The combined ratio is a combination of the claim and expense ratios: LAE UW expenses Combined ratio = Claim ratio + +. Earned premium Written premium For companies that use a finer division on the types of UW expenses, the combined ratio can also be calculated as Combined ratio LAE Policy inception expenses General expenses = Claim ratio + + + Earned premium Written premium Earned premium | {z } UW expense ratio = Claim ratio + OE ratio = Claim ratio × (1 + LAE ratio) + UW expense ratio. STAT4903 Chapter 1 16 Fall 2024  Chapter 1: Key concepts for general insurance In the calculation of the combined ratio, the claim ratio should not include LAE. Use: Primary measure of profitability. Example. You are given the following information for a certain policy year: Item Amount (million) Written premium 125 Earned premium 100 Claims incurred 77 LAE incurred 8 UW expenses incurred 25 Increase in prepaid expenses 5 Calculate the combined ratio. Solution. From the given information, we have: Claims incurred LAE UW expenses Combined ratio = + + Earned premium Earned premium Written premium | {z } Claim ratio 77 8 25 = + + = 105%. 100 100 125 Example. You are given the following information for a certain policy year: Item Amount (million) Written premium 10 Earned premium 9 Investment income 1 UW expenses incurred 2.5 Claims and LAE incurred 6 Calculate the combined ratio. Solution. From the given information, we have: Claims and LAE incurred UW expenses 6 2.5 Combined ratio = + = + = 91.7%. Earned premium Written premium 9 10 | {z } Claim & LAE ratio STAT4903 Chapter 1 17 Fall 2024  Chapter 1: Key concepts for general insurance 1.6.10 Retention ratio The retention ratio is a measure of the rate at which existing insureds renew their policies upon expiration: Number of policies renewed Retention ratio =. Number of potential renewal policies For example, if there are 100,000 potential renewal policies and 85,000 choose to renew, then the retention ratio is 85%. The denominator may exclude certain policies, such as those cancelled because of death or not renewed by the underwriter. Uses: – Measure the competitiveness of rates; – Input for projecting future premium volume. 1.6.11 Close ratio Also known as the hit ratio, quote-to-close ratio or the conversion rate, the close ratio measures the rate at which prospective insureds accept a new business quote: Number of accepted quotes Close ratio =. Number of quotes For example, if the insurer generates 60,000 new policies in 300,000 quotes made, then the close ratio is 20%. The denominator may count multiple quotes received by a prospective insured separately or as one. Use: Determine the competitiveness of rates for new business. STAT4903 Chapter 1 18 Fall 2024