[mathjax]

Basic Probability

Functions and Moments

Probability Density Function

\(f(x)=\dfrac{d}{dx}F(x)\) or \(f(x)=-\dfrac{d}{dx}S(x)\)

Hazard Rate Function

\(\mu (x)=h(x)=\dfrac{f(x)}{S(x)}=-\dfrac{d\ln S(x)}{dx}\)

Cumulative Hazard Rate Function

\(H(x)=\int_{-\infty }^{x}{h(t)dt}=-\ln S(x)\)

\(S(x)=e^{-H(x)}=e^{-\int_{-\infty }^{x}{h(t)dt}}\)

Moment of X

• • n^th Raw Moment of X: \(\mu’_{n}=E[x^n]\)
  • n^th Central Moment of X: \(\mu_n=E[{(x-\mu )}^n]\)
  • Covariance: \(Cov(X,Y)=E[(X-\mu_X)(Y-\mu_Y)]=E[XY]-E[X]E[Y]\)
  • Correlation Coefficient: \(\rho_{XY}=Cov(X,Y)/({\sigma_X}{\sigma_Y})\)
  • Coefficient of Variance: \(\sigma /\mu \)
  • Skewness: \(\gamma_1=\mu_3/\sigma^3\)
  • Kurtosis: \(\gamma_2=\mu_4/\sigma^4\)
  • Moment Generating Function: \(M_X(t)=E[e^{tX}]\)
  • Probability Generating Function: \(P_X(t)=E[t^X]\)

Differentials

• • \(\dfrac{d}{dx}{{c}^{ax}}=a\ln (c){{c}^{ax}}\)

  • \(\dfrac{d}{dx}{{\log }_{c}}x=\frac{1}{x\ln (c)}\)

Sum of Distributions

• • If \(X_1 \sim Gamma(\alpha_1,\theta)\), …, \(X_n \sim Gamma(\alpha_n,\theta)\), then \(X=\sum{X_i} \sim Gamma(\alpha=\alpha_1+\alpha_2+…+\alpha_n, \theta)\)
  • If \(X_1 \sim Exp(\theta)\), …, \(X_n \sim Exp(\theta)\), then \(X=\sum{X_i} \sim Gamma(\alpha=n, \theta)\)
  • If \(X_1 \sim Poi(\lambda_1)\), …, \(X_n \sim Poi(\lambda_n)\), then \(X=\sum{X_i} \sim Poi(\lambda=\lambda_1+\lambda_2+…+\lambda_n)\)
  • If \(X_1 \sim Bin(m_1,q)\), …, \(X_n \sim Bin(m_n,q)\), then \(X=\sum{X_i} \sim Bin(m=m_1+m_2+…+m_n,q)\)
  • If \(X_1 \sim NB(r_1,\beta)\), …, \(X_n \sim NB(r_n,\beta)\), then \(X=\sum{X_i} \sim NB(r=r_1+r_2+…+r_n,\beta)\)

Integration by Parts

\(\int{udv}=uv-\int{vdu}\)

Conditional probability and expectation

Bayes’ Theorem

\(\Pr (A|B)=\dfrac{\Pr (B|A)\Pr (A)}{\Pr (B)}\) or \(f_X(x|y)=\dfrac{f_Y(y|x)f_X(x)}{f_Y(y)}\)

Law of Total Probability

If \(B_i\) is a set of exhaustive (in other words, \(\Pr (\bigcup\nolimits_{i}{B_i})=1\)) and mutually exclusive (in other words \(\Pr (B_i\bigcap{B_j})=0\) for \(i\ne j\)) events, then for any event A,

\(\Pr (A)=\sum\limits_{i}{\Pr (A\bigcap{B_i})}=\sum\limits_{i}{\Pr (A|B_i)\Pr (B_i)}\)

Correspondingly for continuous distributions,

\(\Pr (A)=\int{\Pr (A|x)f(x)dx}\)

Conditional Mean (Double Expectation)

\(E_X[X]=E_Y[E_X[X|Y]]\)

More generally for any function \(g\),

\(E_X[g(X)]=E_Y[E_X[g(X)|y]]\)

Moment and Probability Generating Functions

Generating Function

Generating function for a usually infinite sequence \(a_0,a_1,…\) is \(f(z)\) of the form:

\(f(z)=\sum\limits_{n=0}^{\infty }{a_n{z^n}}\)

The idea of a generating function is that if you differentiate this function n times, divide by \(n!\), and evaluate at 0, you will recover \(a_n\):

\(\dfrac{{f^{(n)}}(0)}{n!}=a_n\)

Moment Generating Function (MGF)

\(M_X(t)=E[e^{tX}]\)

Property

• • • n^th Raw Moment: \(M^{(n)}(0)\)
    • If \(X\) is the sum of independent random variables, its moment generating function is the product, if the moment generating functions of those variables.

Probability Generating Function (PGF)

\(P(z)=E[Z^X]=M(\ln z)\)

Property

• • • \(p_n=\dfrac{{P^{(n)}}(0)}{n!}\)
    • \(Pr(X=0)=P(0)\)
    • n^th Factorial Moment: \({{\mu }_{(n)}}=E[X(X-1)\cdots (X-n+1)]\) and in general, \(P^{(n)}(1)={{\mu }_{(n)}}\)
    • \(P'(1)=\mu_{(1)}=E[X]\), \(P{‘}{‘}(1)=\mu_{(2)}=E[X(X-1)]\), \(P'{‘}{‘}(1)=\mu_{(3)}=E[X(X-1)(X-2)]\)

Parametric Distributions

Scaling

General Approach

1. Let \(Y=cX,c>0\)

2. \(F_Y(y)=\Pr (Y\le y)=\Pr (cX\le y)=\Pr (X\le \dfrac{y}{c})=F_X(\dfrac{y}{c})\)

Lognormal & Inverse Gaussian Distributions

To scale a lognormal: If \(X\sim LN(\mu ,\sigma )\), then \(cX\sim LN(\mu +\ln c,\sigma )\)

Normal Distribution

To scale a normal: If \(X\sim N(\mu ,\sigma )\), then \(cX\sim N(c\mu,c\sigma )\)

Transformation

Deduction

1. If \(Y=g(X)\), with \(g(x)\) a one-to-one monotonically increasing function, then

\(F_Y(y)=\Pr (Y\le y)=\Pr (X\le {g^{-1}}(y))=F_X(g^{-1}(y))\)

and differentiating,

\(F_Y(y)=F_X(g^{-1}(y))\dfrac{d{g^{-1}}(y)}{dy}\)

2. If \(g(x)\) is one-to-one monotonically decreasing, then

\(F_Y(y)=\Pr (Y\le y)=\Pr (X\ge {{g}^{-1}}(y))=S_X(g^{-1}(y))\)

and differentiating,

\(f_Y(y)=-f_X(g^{-1}(y))\dfrac{dg^{-1}(y)}{dy}\)

3. Putting both cases (monotonically increasing and monotonically decreasing) together:

\(f_Y(y)=-f_X(g^{-1}(y))\dfrac{dg^{-1}(y)}{dy}\)

Variance

Additivity

1. \(Var(aX+bY)=a^2Var(X)+2abCov(X,Y)+b^2Var(Y)\)

2. \(Var(\bar{X})=Var(\dfrac{\mathop{\sum}\nolimits_{{i}{=}{1}}\nolimits^{n}{X_i}}{n})=\dfrac{nVar(X)}{n^2}=\dfrac{Var(X)}{n}\)

Normal Approximation

1. To calculate a percentile of a random variable being approximated normally, find the value of x such that \(\Phi(x)\) is that percentile.

2. Scale x: multiply by the standard deviation.

3. Translate x: add the mean.

Continuity Correction

When a discrete distribution is approximated with the normal distribution, a continuity correction is required.

If the discrete distribution can assume values \(a\) and \(b\) but cannot assume values in between \(a\) and \(b\), the continuity correction is \((a+b)/2\) and:

1. 1. For \(Pr(X>a)\), we estimate \(Pr(X>a+(a+b)/2\).
   2. For \(Pr(X<b)\) (or \(Pr(X\le a)\)), we estimate \(Pr(X<b+(a+b)/2\) (or \(Pr(X\le a+(a+b)/2\)).
   3. For \(Pr(X\ge b)\) (or \(Pr(X>a)\)), we estimate \(Pr(X\ge b+(a+b)/2\).

Bernoulli Shortcut

If \(X\) is Bernoulli and \(Y\) can only assume the values \(a\) and \(b\), with a having probability \(q\), then:

\(Y=(a-b)X+b\)

\(Var(Y)=(a-b)^2Var(X)=(a-b)^2q(1-q)\)

Mixtures and Splices

Mixtures

\(f_X(x)=\sum\limits_{i=1}^{n}{w_i{f_{X_i}}(x)}\) and \(F_X(x)=\sum\limits_{i=1}^{n}{w_i{F_{X_i}}(x)}\)

\(E_X(x)=\sum\limits_{i=1}^{n}{w_i{E_{X_i}}(x)}\) and \(E_X(x^2)=\sum\limits_{i=1}^{n}{w_i{E_{X_i}}(x^2)}\)

Property

A mixture is not the same as a sum of random variables. On the other hand, examples of sums of random variables are:

• • • The total of a random sample of \(n\) items is a sum of random variables. For a random sample, the items are independent and identically distributed.
    • Aggregate loss on a policy with multiple coverages is a sum of random variables, one for each coverage.

Frailty Model

A special type of continuous mixture is a frailty model. These models can be used to model loss sizes or survival times. Suppose the hazard rate for each individual is \({h}{(}{x}{|}\Lambda{)}{=}\Lambda{a}{(}{x}{)}\), where \(a(x)\) is some continuous function and the multiplier \(\Lambda\) varies by individual. Thus the shape of the hazard rate function curve does not vary by individual.

\({S}_{X}{(}{x}{)}{=}{E}_{\Lambda}{[}{S}{(}{x}{|}\Lambda{)]}{=}{E}_{\Lambda}{[}{e}^{{-}\Lambda{A}{(}{x}{)}}{]}{=}{M}_{\Lambda}{(}{-}{A}{(}{x}{))}\), where \(A(x)=\int\limits_{0}^{x}{a(t)dt}\)

In a frailty model, typical choices for \({h}{(}{x}{|}\Lambda{)}\) are:

• • Constant hazard rate, or exponential. This can be arranged by setting \(a(x) = 1\) (or \(a(x) = k\) for any constant \(k\)).
  • Weibull, which can be arranged by setting \({a}{(}{x}{)}{=}\mathit{\gamma}{x}^{\mathit{\gamma}{-}{1}}\) and typical choices for the distribution of A are gamma and inverse Gaussian.

Conditional Variance

\(Var_X(X)=Var_I[E_X(X|I)]+E_I[Var_X(X|I)]\)

Splices

• For a spliced distribution, the sum ofthe probabilities of being in each splice must add up to 1.
• If a spliced distribution is continous at point \(c\), then \(f(x_1)=c=f(x_2)\).

Property/Casualty Insurance Coverages

Automobile Insurance

• Liability Insurance
  Liability insurance provides coverage for the driver’s liability to others when the driver injures another person (bodily injury) or damages another person’s property (property damage).
• Uninsured, Underinsured, and Unidentified Motorist Coverage
  Under uninsured, underinsured, and unidentified motorist coverages, an Insured receives benefits from his insurance company for damage or injury from another driver who is uninsured, underinsured, or unidentified as if he sued the other driver and received those benefits from the other driver’s insurance company.
• Medical Benefits
  In a tort jurisdiction, medical benefits coverage provides benefits to the driver who is at fault. In a no-fault jurisdiction, this coverage is called personal injury protection and provided benefits to an injured driver whether at fault or not. Benefits include medical expenses and lost wages. This coverage is usually compulsory.
• Collision and Other-Than-Collision Coverage
  Collision coverage pays the policyholder the amount needed to repair the car, or the actual depreciated value of their car if less. This coverage is subject to a deductible.

The insurer has the following rights to recover its payments:

• • Subrogation. If the other party is at fault, the insurance company may attempt to recover the payment by suing the other party. Recovering the payment from the other party is called subrogation. If the entire damage is recovered, the policyholder will receive the deductible.
  • Salvage. If the insurance company pays the value of the car, it has the right to the car, which it may sell for its scrap metal value. This is called salvage.

Homeowners Insurance

Homeowners insurance includes the following:

1. First-party coverage
   1. Damage to dwelling either for all risks except for those specifically excluded or for named perils
   2. Damage to garage or other structures on premises
   3. Damage to contents
   4. Additional living expenses and loss of rental income
2. Liability coverage

Primary Dwelling Coverage

Coverage A

is usually for all risks excluding certain perils, such as flood, earthquakes, birds, terrorism and acts of war. Under the doctrine of proximate cause, a loss is covered only if both the cause (direct or indirect) and the consequence are covered.
As with auto collision insurance, the insurer is entitled to subrogation and salvage. For example, if there is a fire caused by a faulty appliance, the insurer may recover payment from the appliance manufacturer.
Dwelling coverage is often subject to:

• • a deductible (sometimes disappearing deductible) A disappearing deductible is a deductible of d that linearly decreases to 0 if the loss is greater than \(d+k\), where \(k>0\).
  • a policy limit
  • a coinsurance clause

Coinsurance Clause

For a homeowners policy with an \(c\text{%}\) coinsurance clause, if:

• • • the home is insured for an amount of \(u\)
    • an event causes \(x\) worth of damage
    • the home is worth \(b\) on the day of the event
    • deductible of \(d\)

1. 1. 1. The home is insured \(b\times c\text{%}\) on the day of the event, compared to the home was insured \(u\) before
      2. Policy only covers \(\frac{u}{b \times c\text{%}}\) of damage

The benefit is:

Other First-Party Coverage

• • Coverage for other structures on premises (Coverage B)

is normally limited to 10% of the coverage for the primary dwelling, and excludes structures used for business purposes or held for rental.

• • Coverage for contents (Coverage C)

often has inside limits for expensive items such as jewelry. The limit for coverage for contents in dwellings other than the primary dwelling is normally 10% of the limit for coverage of contents.

Example: Suppose a policy has inside limits for the following coverages: 1) Art: \(a\), 2) Jewelry: \(b\), 3) Cash: \(c\), then:

\(Y^L=Min(X_a,a)+Min(X_b,b)+Min(X_c,c)\)

• • Coverage for additional living expenses (Coverage D)

pays for rent while the dwelling is being repaired. It is normally limited to 20% of the coverage for the primary dwelling.

Liability Coverage

Liability coverage (Section II) is for damage to a third party on the premises. Like automobile liability coverage, there is a policy limit, which does not include legal defense costs. There may be a limited amount of medical coverage provided on a no-fault basis.

Tenants Package

Someone who rents dwelling rather than owning it may purchase a tenants package. Such a package is limited to providing insurance for the contents of the dwelling.

Workers Compensation

Before this coverage was introduced, it was difficult for employees to obtain compensation for injury on the job. They had to prove liability on the part of the employer, and might not be able to collect compensation due to:

• The contributory negligence doctrine, if the employee contributed in any way to the injury
• The fellow-servant doctrine, if any other worker contributed to the injury
• The assumption of risk doctrine, if the worker knew about the inherent dangers of the job

Workers compensation is now required for most employers. Regardless of fault, it provides:

• Medical care benefits with no dollar or time limits.
• Disability income benefits for lost wages. These may have a short elimination period (3-7 days) that is waived if the disability is long term. The percentage of wages paid, and minimum and maximum amounts paid, are set by state regulation.
• Death benefits: burial and cash income payments to eligible dependents.
• Rehabilitation benefits.

Workers compensation is often experience rated, meaning that employers with more claims tend to pay higher premiums. This gives employers an incentive to make their workplace safer.

In the U.S., workers compensation insurance is often provided by private insurers, but in other countries it is a government monopoly.

Business Insurance

• Fire insurance covers losses due to fire, lightning, ,windstorm, hail, explosion, and other perils.
• Ocean marine insurance covers damage to property that is transported by ship. Basic coverage applies only after the cargo is loaded, but some policies cover damage from warehouse to warehouse.
• Inland marine insurance covers damage to property that is transported over land, in trucks, railroads, motor vehicles, or ships on inland waterways.
• Liability insurance . Two characteristics of liability insurance are:
  • Cases can take a long time to report. On some insurance lines, a claims-made form is offered. These pay only for claims made after a specific date and reported during the policy period. Tail coverage pays for claims occurring during a period but reported after the period.
  • The policy limit does not include litigation costs, which have no limit. However, some forms include litigation costs in the limit.

Health Insurance

Major Medical Insurance

Major medical insurance, as its name indicates, pays for the larger health expenses. The design of this insurance was heavily affected by the Affordable Care Act (ACA). Typically, an insurance company contracts with a provider network.

Coverage Modifications

There are many coverage modifications in major medical insurance, known as cost sharing, in order to encourage the policyholder to minimize costs:

• • Allowed charges. The allowed charge is the amount the insurance company will pay for a service. Providers in the network agree to accept charges based on a schedule determined by the insurance company. They may not bill the policyholder for the excess of their charge over the allowed charge.
  • Balance billing. Billing for the excess (which out-of-network providers may do).
  • Deductibles. Insurance doesn’t pay until the policyholder meets an annual deductible. The deductible applies to the allowed charge. The company pays the allowed charge minus the deductible.
  • Coinsurance. The insurance pays a percentage of the claim. Coinsurance applies only after the deductible is met.
  • Out-of-pocket limits. Once the policyholder pays a certain amount in a year, expenses above that amount are 100% covered. The most the policyholder has to pay is called the out of pocket (OOP) limit.
  • Maximum limits. Insurance contracts used to limit the most an insurance company would pay. This limit could be an annual limit or a lifetime limit. However the ACA banned this provision for essential health benefits, except that grandfathered plans can have annual limits.
  • Internal limits. Internal limits are limits on the amount paid for certain types of services.
  • Copays. A copay is the amount the policyholder pays per service. Copays are used for two types of services. The first type is a service for which the insured has a lot of control over usage, such as office visits and emergency room care. The second type is a service for which administration of the benefit is not by the insurance company, making it hard to keep track of the deductible.

Example:

Russ and his family have a major medical policy with the following cost sharing provisions:

• • The annual individual deductible is 2,500.
  • The annual family deductible is 5,000.
  • The coinsurance is 90%.

Before the family deductible is met, coinsurance applies to all healthcare costs incurred by family members whose individual deductibles have been met. After the family deductible is met, coinsurance applies to all healthcare costs incurred by the family. All out-of-pocket expenses count towards the deductible.

Over the last 12 months, Russ and his family submitted the following claims, in the order listed:

• • Husband’s annual physical: 500
  • Wife’s annual physical: 500
  • Husband’s surgery: 3,000
  • Son’s doctor visit: 300
  • Wife’s blood test: 200
  • Daughter’s X-ray: 800
  • Son’s annual physical: 2000

Calculate the insurance benefit paid to Russ and his family.

Deductible		\(X_1\)	\(X_2\)	\(X_3=3000\)		\(X_4\)	\(X_5\)	\(X_6\)	\(X_7\)
		500	500	2000	1000	300	200	800	600	1400
Insurer Payment		0	0	0	900	0	0	0	0	1260
Insured Payment		500	500	500	100	300	200	800	600	140
Family	OB	0	500	1000	3000	3100	3400	3600	4400	–
	+/-	+500	+500	+2000	+100	+300	+200	+800	+600	–
	EB	500	1000	3000	3100	3400	3600	4400	5000	–
Husband	OB	0	500	500	2500	2500	2500	2500	2500	–
	+/-	+500	–	+2000	–	–	–	–	–	–
	EB	500	500	2500	2500	2500	2500	2500	2500	–
Wife	OB	0	0	500	500	500	500	700	700	–
	+/-	–	+500	–	–	–	+200	–	–	–
	EB	0	500	500	500	500	700	700	700	–
Son	OB	0	0	0	0	0	300	300	300	–
	+/-	–	–	–	–	+300	–	–	+600	–
	EB	0	0	0	0	300	300	300	900	–
Daughter	OB	0	0	0	0	0	0	0	800	–
	+/-	–	–	–	–	–	–	+800	–	–
	EB	0	0	0	0	0	0	800	800	–

Variations on Major Medical

To some extent, these are obsolete due to ACA.

1. 1. Comprehensive major medical coverage.
      • Lower deductibles than major medical, designed to cover smaller expenses
      • Some carriers offer customizable plans so that insureds can select the benefits they want
   2. Catastrophic medical.
      • Very high deductible
      • Under ACA, OOP is limited, limiting deductibles.
   3. Short term medical
      • Period limited to 3, 6, 9, or 12 months.
      • Excludes pre-existing conditions.
      • ACA requires coverage at all times, so this type of coverage is not useful. However, ACA allows gaps of less than 3 months in coverage.
   4. High risk pool plans. Meant for people who could not qualify for health insurance, but under ACA health insurance companies cannot refuse applicants during open enrollment periods, so high risk pools are obsolete.

A 5^th variation, a consumer directed plan, is not obsolete under ACA Under a consumer-directed plan, the consumer selects a high-deductible health plan (HDHP) and deposits an amount of money (limited by tax law) in a Health Savings Account (HSA). The HSA may be used to pay qualified medical expenses. Deposits into the HSA are tax-deductible and funds grow tax free.

ACA Restrictions on Health Design

All non-grandfathered plans for individuals and small groups must include ten categories of essential health benefits (EHBs):

• • Ambulatory patient services
  • Emergency services
  • Hospitalization
  • Maternity and newborn costs
  • Mental health and substance abuse services
  • Prescription drugs
  • Rehabilitative and habilitative services and devices
  • Laboratory services
  • Preventive and wellness services and chronic disease management
  • Pediatric services including vision and dental

The actuarial value (AV) of a plan is the proportion of expenses paid by the insurer. For all health insurance plans:

• • Plans are classified by AV as bronze, silver, gold, platinum (with roughly 60%, 70%, 80%, and 90% AV respectively) and must fit into one of these categories.
  • Cost sharing is not allowed on preventative services.
  • Lifetime and annual limits are not allowed.
  • There is an OOP maximum limit; all payments other than for standalone pediatric dental are counted in this limit.

In addition to these requirements that apply to all health insurance plans, requirements for Exchange plans are:

• • There must be a meaningful difference between plans.
  • Plans must have adequate networks.
  • There are tests for discriminatory service areas.
  • There are tests for discriminatory cost sharing.
  • There are tests for outlier premium rates.

Dental Insurance

Most dental insurance plans are group rather than individual because:

• High frequency of claims and lower cost of claims minimizes value of insurance.
• Tax deduction is available for employer and employee contributions to group coverage, not for premiums paid for individual plans.

Dental coverage is in four categories:

• Preventive (X-rays, cleaning)
• Basic (fillings, extractions)
• Prosthetic (inlays, crowns)
• Orthodontia

Risks to insurer:

1. Often multiple treatments are available. This risk is handled by
   1. Lower reimbursement rates on more expensive alternatives (e.g. Type lil vs. Type II)
   2. Requiring approval for planned treatment
2. Treatment can be postponed, leading to anti-selection. Tb control this, individual policies may exclude some benefits initially.
3. Individual dental coverage is susceptible to anti-selection, since potential insureds know their condition. This risk may be hard to manage.

As mentioned before, ACA requires carriers to offer pediatric dental coverage, but this may be offered by another insurance company in a stand-alone policy. The ACA sets standards for coverage offered on exchanges.

Loss Reserving: Basic Methods

Case Reserves and IBNR reserves

Total Reserve = Total Ultimate Losses – Amount Paid to Date

Case Reserve = Total Incurred Losses – Total Paid Losses

IBNR Reserve = Total Reserve – Case Reserve

The claim adjuster estimates the future payments on a claim and sets up a reserve for the claim. Those reserves are called case reserves. However, additional reserves are needed for.

• Provision for future adjustments to known claims
• Provision for claim files that are closed but may reopen
• Provision for incurred but not reported claims (pure IBNR)
• Provision for reported but not recorded claims (RBNR)

These additional reserves are called gross IBNR reserves or bulk reserves. These reserves cannot be developed on a claim-by-claim basis but are developed on a bulk basis, by analyzing development trends for blocks of business.

An old non-actuarial method for calculating IBNR reserves is case reserves plus. The lBNR reserve is set as a percentage of the case reserve in some judgmental fashion. This method is subject to manipulation, with percentages raised or lowered as needed to smooth a company’s earnings. It is therefore rarely used.

• Written Premium
  Written premium is the amount of premium on a policy sold during a period of time.
• Earned Premium
  Earned premium for a calendar period is the written premium for a policy currently in-force regardless of when the premium was paid, times the portion of the policy’s duration that is within the calendar period (year, month, etc.) being considered.
  • Regardless of when the premium was paid:
    • If a one-year policy is sold on 9/1/CY1, and premium \(P_1\) is paid on 9/1/CY1 and \(P_2\) on 3/1/CY2, then the earned premium is
      • \((\frac{12-8}{12})(P_1+P_2)=(\frac{4}{12})(P_1+P_2)\) for CY1
      • \((\frac{12-10}{12})(P_1+P_2)=(\frac{2}{12})(P_1+P_2)\) for 11/1/CY1
• CY and AY
  CY, calendar year, refers to when a transaction occurs or is incurred.
  AY, accident year, pertains only to costs related to accidents. For an accident that occurs between January 1 and December 31 of a year, payments and reserve increases are associated with that accident year, regardless of when the payments made or the increases in reserve occur.
• Loss Ratio
  The loss ratio is the ratio of losses to earned premiums, This concept in at least three ways:
  • The permissible loss ratio is the loss ratio that is used for pricing. It is the complement of the ratio of expenses and provisions for contingencies and profit. The premium rate is computed so that expected future losses are equal to the premium rate times the permissible loss ratio.
  • The expected loss ratio is the expected losses divided by the premium. While this may be the same as the permissible loss ratio, it may be different if the premium rate was not based on the permissible loss ratio for whatever reason (marketing considerations, regulatory constraints).
  • The experience loss ratio is the actual loss ratio experienced on the block of business. Typically, it is calculated as losses for accident year \(x\) divided by earned premiums for calendar year \(x\).

For loss reserving, we will be using the expected loss ratio, The other two loss ratios will be used for ratemaking.

Three Methods for Calculating IBNR Reserves

Relationship:

\(R_{BF}=(1-\dfrac{1}{f_{ult}}){R_{LR}}+\dfrac{1}{f_{ult}}{R_{CL}}\), where

• \(R_{LR}\): Expected Loss Ratio Method Reserve
• \(R_{CL}\): Chain Ladder Method Reserve
• \(R_{BF}\): Bornhuetter-Ferguson Reserve

Expected Loss Ratio Method

Reserve = Earned Premium × Expected Loss Ratio – Amount Paid to Date

Characteristics:

• • If past cumulative payments are higher than expected, it isn’t reasonable to assume that future payments will therefore be lower than expected.
  • On a new line of business with no claims experience, it would be the only method to use.

The Chain-Ladder or Loss Development Triangle Method

Loss Development: For a claim with a long time to settlement, the cumulative amount paid usually grows from year to year. This growth is called loss development.

Age-to-Age Development Factor: The ratio of cumulative amount paid through year \(x\) to cumulative amount paid through year \(x-1\) is called an age-to-age development factor, where the age of a claim is the amount of time since the loss occurred; the age is 0 in the year it occurred, 1 in the next year, and so on.

Characteristics:

• • The chain-ladder method ignores the loss ratio, and projects future payments purely off past payments.
  • The chain ladder method is not stable, large payments or large case reserves in a year usually generate even higher reserves, since paid-to-date or incurred-to-date is usually multiplied by factors greater than 1. It is the opposite of the loss ratio method, which lowers future projected payments in response to higher earlier payments.

Methodology

1. Cumulative Payment Loss Triangle

	DY0	DY1	DY2	DY3	DY4
AY1	\(C_{1,0}\)	\(C_{1,1}\)	\(C_{1,2}\)	\(C_{1,3}\)	\(C_{1,4}\)
AY2	\(C_{2,0}\)	\(C_{2,1}\)	\(C_{2,2}\)	\(C_{2,3}\)
AY3	\(C_{3,0}\)	\(C_{3,1}\)	\(C_{3,2}\)
AY4	\(C_{4,0}\)	\(C_{4,1}\)
AY5	\(C_{5,0}\)

2. Loss Development Factors

	DY 0 to 1	DY 1 to 2	DY 2 to 3	DY 3 to 4
AY1	\(S_{1,0}=\dfrac{C_{1,1}}{C_{1,0}}\)	\(S_{1,1}=\dfrac{C_{1,2}}{C_{1,1}}\)	\(S_{1,2}=\dfrac{C_{1,3}}{C_{1,2}}\)	\(S_{1,3}=\dfrac{C_{1,4}}{C_{1,3}}\)
AY2	\(S_{2,0}=\dfrac{C_{2,1}}{C_{2,0}}\)	\(S_{2,1}=\dfrac{C_{2,2}}{C_{2,1}}\)	\(S_{2,2}=\dfrac{C_{2,3}}{C_{2,2}}\)
AY3	\(S_{3,0}=\dfrac{C_{3,1}}{C_{3,0}}\)	\(S_{3,1}=\dfrac{C_{3,2}}{C_{3,1}}\)
AY4	\(S_{4,0}=\dfrac{C_{4,1}}{C_{4,0}}\)

3. Linked Ratios

Arithmetic Average Method

	DY 0 to 1	DY 1 to 2	DY 2 to 3	DY 3 to 4
Linked Ratios	\(R_{0,1}=\dfrac{\sum\nolimits_{i=1}^{4}{{S_{i,0}}}}{4}\)	\(R_{1,2}=\dfrac{\sum\nolimits_{i=1}^{3}{{S_{i,1}}}}{3}\)	\(R_{2,3}=\dfrac{\sum\nolimits_{i=1}^{2}{{S_{i,2}}}}{2}\)	\(S_{i,3}\)

Volume-Weighted Average Method

	DY 0 to 1	DY 1 to 2	DY 2 to 3	DY 3 to 4
Linked Ratios	\(R_{0,1}=\dfrac{\sum\nolimits_{i=1}^{4}{{C_{i,1}}}}{\sum\nolimits_{i=1}^{4}{{C_{i,0}}}}\)	\(R_{1,2}=\dfrac{\sum\nolimits_{i=1}^{3}{{C_{i,2}}}}{\sum\nolimits_{i=1}^{3}{{C_{i,1}}}}\)	\(R_{2,3}=\dfrac{\sum\nolimits_{i=1}^{2}{{C_{i,3}}}}{\sum\nolimits_{i=1}^{2}{{C_{i,2}}}}\)	\(R_{3,4}=\dfrac{C_{i,4}}{C_{i,3}}\)

4. Use linked ratios to evaluate loss reserves for future DY

	DY0	DY1	DY2	DY3	DY4
AY1	\(C_{1,0}\)	\(C_{1,1}\)	\(C_{1,2}\)	\(C_{1,3}\)	\(C_{1,4}\)
AY2	\(C_{2,0}\)	\(C_{2,1}\)	\(C_{2,2}\)	\(C_{2,3}\)	\(C_{2,3}R_{3,4}\)
AY3	\(C_{3,0}\)	\(C_{3,1}\)	\(C_{3,2}\)	\(C_{3,2}R_{2,3}\)	\(C_{3,2}R_{2,3}R_{3,4}\)
AY4	\(C_{4,0}\)	\(C_{4,1}\)	\(C_{4,1}R_{1,2}\)	\(C_{4,1}R_{1,2}R_{2,3}\)	\(C_{4,1}R_{1,2}R_{2,3}R_{3,4}\)
AY5	\(C_{5,0}\)	\(C_{5,0}R_{0,1}\)	\(C_{5,0}R_{0,1}R_{1,2}\)	\(C_{5,0}R_{0,1}R_{1,2}R_{2,3}\)	\(C_{5,0}R_{0,1}R_{1,2}R_{2,3}R_{3,4}\)

• To use more recent data, we may average only recent years. For example, a 2-year arithmetic average method would use average the 2 most recent years to compute the link ratios.
• This method may also be used with an incurred losses triangle. Incurred losses include case reserves. We would expect link ratios to be lower, possibly even below 1, since case reserves increase the earlier numbers. After performing the calculation, the difference between the developed ultimate numbers and the incurred-to-date numbers is the IBNR reserve.

The Bornhuetter-Ferguson Method

The Bornhuetter-Ferguson method is a compromise between the loss ratio method and the chain ladder method. The idea of the method is:

1. 1. Suppose based on the link ratios the development factor to ultimate levels is \(f_{ult}\).
   2. Then \(\dfrac{1}{f_{ult}}\) of the ultimate loss has been paid (or incurred) so far and \(1-\dfrac{1}{f_{ult}}\) of the ultimate loss remains to be paid (or incurred).

Reserve = Earned Premium × Expected Loss Ratio × \(\boldsymbol{(1-\dfrac{1}{f_{ult}})}\), where

• • \(f_{ult}\): The development factor to ultimate levels, \({f_{ult}}=\prod_{j}{{f_j}}\), where \(f_j\) and \(f_{j-1}\) are link ratios from year \(j-1\) to year \(j\)
  • \(\dfrac{1}{f_{ult}}\): The ultimate loss has been paid (or incurred) so far
  • \(1-\dfrac{1}{f_{ult}}\): The ultimate loss remains to be paid (or incurred)

Loss Reserving: Other Methods

Projecting Frequency and Severity Separately

The projection of counts of closed claims is more stable and reliable. Dividing payments by counts yields a cumulative severity triangle which will probably increase with development year since the largest cases take the longest to close.

Methodology

Cumulative Payment Loss Triangle (Table 1)

	DY0	DY1	DY2	DY3	DY4
AY1	\(C_{1,0}\)	\(C_{1,1}\)	\(C_{1,2}\)	\(C_{1,3}\)	\(C_{1,4}\)
AY2	\(C_{2,0}\)	\(C_{2,1}\)	\(C_{2,2}\)	\(C_{2,3}\)
AY3	\(C_{3,0}\)	\(C_{3,1}\)	\(C_{3,2}\)
AY4	\(C_{4,0}\)	\(C_{4,1}\)
AY5	\(C_{5,0}\)

Cumulative Closed Claims Triangle (Table 2)

	DY0	DY1	DY2	DY3	DY4
AY1	\(N_{1,0}\)	\(N_{1,1}\)	\(N_{1,2}\)	\(N_{1,3}\)	\(N_{1,4}\)
AY2	\(N_{2,0}\)	\(N_{2,1}\)	\(N_{2,2}\)	\(N_{2,3}\)
AY3	\(N_{3,0}\)	\(N_{3,1}\)	\(N_{3,2}\)
AY4	\(N_{4,0}\)	\(N_{4,1}\)
AY5	\(N_{5,0}\)

1. Link Ratios for Claim Counts

	DY 0 to 1	DY 1 to 2	DY 2 to 3	DY 3 to 4
Linked Ratios	\(R_{0,1}=\dfrac{\sum\nolimits_{i=1}^{4}{{N_{i,1}}}}{\sum\nolimits_{i=1}^{5}{{N_{i,0}}}}\)	\(R_{1,2}=\dfrac{\sum\nolimits_{i=1}^{3}{{N_{i,2}}}}{\sum\nolimits_{i=1}^{4}{{N_{i,1}}}}\)	\(R_{2,3}=\dfrac{\sum\nolimits_{i=1}^{2}{{N_{i,3}}}}{\sum\nolimits_{i=1}^{3}{{N_{i,2}}}}\)	\(R_{3,4}=\dfrac{\sum\nolimits_{i=1}^{1}{{N_{i,4}}}}{\sum\nolimits_{i=1}^{2}{{N_{i,3}}}}\)

2. Ultimate Claim Counts

	DY0	DY1	DY2	DY3	DY4
AY1	\(N_{1,0}\)	\(N_{1,1}\)	\(N_{1,2}\)	\(N_{1,3}\)	\(N_{1,4}\)
AY2	\(N_{2,0}\)	\(N_{2,1}\)	\(N_{2,2}\)	\(N_{2,3}\)	\(N_{2,3}R_{3,4}\)
AY3	\(N_{3,0}\)	\(N_{3,1}\)	\(N_{3,2}\)	\(N_{3,2}R_{2,3}\)	\(N_{3,2}R_{2,3}R_{3,4}\)
AY4	\(N_{4,0}\)	\(N_{4,1}\)	\(N_{4,1}R_{1,2}\)	\(N_{4,1}R_{1,2}R_{2,3}\)	\(N_{4,1}R_{1,2}R_{2,3}R_{3,4}\)
AY5	\(N_{5,0}\)	\(N_{5,0}R_{0,1}\)	\(N_{5,0}R_{0,1}R_{1,2}\)	\(N_{5,0}R_{0,1}R_{1,2}R_{2,3}\)	\(N_{5,0}R_{0,1}R_{1,2}R_{2,3}R_{3,4}\)

3. Divide the table 2 into Table 1 to obtain the Cumulative Severity Triangle

	DY0	DY1	DY2	DY3	DY4
AY1	\(S_{1,0}=\dfrac{C_{1,0}}{N_{1,0}}\)	\(S_{1,1}=\dfrac{C_{1,1}}{N_{1,1}}\)	\(S_{1,2}=\dfrac{C_{1,2}}{N_{1,2}}\)	\(S_{1,3}=\dfrac{C_{1,3}}{N_{1,3}}\)	\(S_{1,4}=\dfrac{C_{1,4}}{N_{1,4}}\)
AY2	\(S_{2,0}=\dfrac{C_{2,0}}{N_{2,0}}\)	\(S_{2,1}=\dfrac{C_{2,1}}{N_{2,1}}\)	\(S_{2,2}=\dfrac{C_{2,2}}{N_{2,2}}\)	\(S_{2,3}=\dfrac{C_{2,3}}{N_{2,3}}\)
AY3	\(S_{3,0}=\dfrac{C_{3,0}}{N_{3,0}}\)	\(S_{3,1}=\dfrac{C_{3,1}}{N_{3,1}}\)	\(S_{3,2}=\dfrac{C_{3,2}}{N_{3,2}}\)
AY4	\(S_{4,0}=\dfrac{C_{4,0}}{N_{4,0}}\)	\(S_{4,1}=\dfrac{C_{4,1}}{N_{4,1}}\)
AY5	\(S_{5,0}=\dfrac{C_{5,0}}{N_{5,0}}\)

4. Link Ratios for Claim Sizes

	DY 0 to 1	DY 1 to 2	DY 2 to 3	DY 3 to 4
Linked Ratios	\(K_{0,1}=\dfrac{\sum\nolimits_{i=1}^{4}{{C_{i,1}}}}{\sum\nolimits_{i=1}^{5}{{C_{i,0}}}}\)	\(K_{1,2}=\dfrac{\sum\nolimits_{i=1}^{3}{{C_{i,2}}}}{\sum\nolimits_{i=1}^{4}{{C_{i,1}}}}\)	\(K_{2,3}=\dfrac{\sum\nolimits_{i=1}^{2}{{C_{i,3}}}}{\sum\nolimits_{i=1}^{3}{{C_{i,2}}}}\)	\(K_{3,4}=\dfrac{\sum\nolimits_{i=1}^{1}{{C_{i,4}}}}{\sum\nolimits_{i=1}^{2}{{C_{i,3}}}}\)

5. Ultimate Severities

	DY0	DY1	DY2	DY3	DY4
AY1	\(S_{1,0}\)	\(S_{1,1}\)	\(S_{1,2}\)	\(S_{1,3}\)	\(S_{1,4}\)
AY2	\(S_{2,0}\)	\(S_{2,1}\)	\(S_{2,2}\)	\(S_{2,3}\)	\(S_{2,3}K_{3,4}\)
AY3	\(S_{3,0}\)	\(S_{3,1}\)	\(S_{3,2}\)	\(S_{3,2}K_{2,3}\)	\(S_{3,2}K_{2,3}K_{3,4}\)
AY4	\(S_{4,0}\)	\(S_{4,1}\)	\(S_{4,1}K_{1,2}\)	\(S_{4,1}K_{1,2}K_{2,3}\)	\(S_{4,1}K_{1,2}K_{2,3}K_{3,4}\)
AY5	\(S_{5,0}\)	\(S_{5,0}K_{0,1}\)	\(S_{5,0}K_{0,1}K_{1,2}\)	\(S_{5,0}K_{0,1}K_{1,2}K_{2,3}\)	\(S_{5,0}K_{0,1}K_{1,2}K_{2,3}K_{3,4}\)

6. Total reserve = Ultimate Frequency × Ultimate Severity – Amount Paid to Date

In this case, Amount Paid to Date = \(C_{2,3}+C_{3,2}+C_{4,1}+C_{5,0}\)

Closure Method

The closure method is a method that projects frequency and severity separately. However, it is based on incremental counts and payments rather than cumulative counts and payments.

Methodology

Cumulative Payment Loss Triangle

	DY0	DY1	DY2	DY3	DY4
AY1	\(C_{1,0}\)	\(C_{1,1}\)	\(C_{1,2}\)	\(C_{1,3}\)	\(C_{1,4}\)
AY2	\(C_{2,0}\)	\(C_{2,1}\)	\(C_{2,2}\)	\(C_{2,3}\)
AY3	\(C_{3,0}\)	\(C_{3,1}\)	\(C_{3,2}\)
AY4	\(C_{4,0}\)	\(C_{4,1}\)
AY5	\(C_{5,0}\)

Cumulative Closed Claims Triangle

	DY0	DY1	DY2	DY3	DY4
AY1	\(N_{1,0}\)	\(N_{1,1}\)	\(N_{1,2}\)	\(N_{1,3}\)	\(N_{1,4}\)
AY2	\(N_{2,0}\)	\(N_{2,1}\)	\(N_{2,2}\)	\(N_{2,3}\)
AY3	\(N_{3,0}\)	\(N_{3,1}\)	\(N_{3,2}\)
AY4	\(N_{4,0}\)	\(N_{4,1}\)
AY5	\(N_{5,0}\)

1. Incremental Payments Triangle

	DY0	DY1	DY2	DY3	DY4
AY1	\(IC_{1,0}=C_{1,0}\)	\(IC_{1,1}=C_{1,1}-C_{1,0}\)	\(IC_{1,2}=C_{1,2}-C_{1,1}\)	\(IC_{1,3}=C_{1,3}-C_{1,2}\)	\(IC_{1,4}=C_{1,4}-C_{1,3}\)
AY2	\(IC_{2,0}=C_{2,0}\)	\(IC_{2,1}=C_{2,1}-C_{2,0}\)	\(IC_{2,2}=C_{2,2}-C_{2,1}\)	\(IC_{2,3}=C_{2,3}-C_{2,2}\)
AY3	\(IC_{3,0}=C_{3,0}\)	\(IC_{3,1}=C_{3,1}-C_{3,0}\)	\(IC_{3,2}=C_{3,2}-C_{3,1}\)
AY4	\(IC_{4,0}=C_{4,0}\)	\(IC_{4,1}=C_{4,1}-C_{4,0}\)
AY5	\(IC_{5,0}=C_{5,0}\)

2. Incremental Closed Claims Triangle

	DY0	DY1	DY2	DY3	DY4
AY1	\(IN_{1,0}=N_{1,0}\)	\(IN_{1,1}=N_{1,1}-N_{1,0}\)	\(IN_{1,2}=N_{1,2}-N_{1,1}\)	\(IN_{1,3}=N_{1,3}-N_{1,2}\)	\(IN_{1,4}=N_{1,4}-N_{1,3}\)
AY2	\(IN_{2,0}=N_{2,0}\)	\(IN_{2,1}=N_{2,1}-N_{2,0}\)	\(IN_{2,2}=N_{2,2}-N_{2,1}\)	\(IN_{2,3}=N_{2,3}-N_{2,2}\)
AY3	\(IN_{3,0}=N_{3,0}\)	\(IN_{3,1}=N_{3,1}-N_{3,0}\)	\(IN_{3,2}=N_{3,2}-N_{3,1}\)
AY4	\(IN_{4,0}=N_{4,0}\)	\(IN_{4,1}=N_{4,1}-N_{4,0}\)
AY5	\(IN_{5,0}=N_{5,0}\)

3. Dividing incremental claim counts into incremental payments to get the incremental severity

	DY0	DY1	DY2	DY3	DY4
AY1	\(IS_{1,0}=\dfrac{IC_{1,0}}{IN_{1,0}}\)	\(IS_{1,1}=\dfrac{IC_{1,1}}{IN_{1,1}}\)	\(IS_{1,2}=\dfrac{IC_{1,2}}{IN_{1,2}}\)	\(IS_{1,3}=\dfrac{IC_{1,3}}{IN_{1,3}}\)	\(IS_{1,4}=\dfrac{IC_{1,4}}{IN_{1,4}}\)
AY2	\(IS_{2,0}=\dfrac{IC_{2,0}}{IN_{2,0}}\)	\(IS_{2,1}=\dfrac{IC_{2,1}}{IN_{2,1}}\)	\(IS_{2,2}=\dfrac{IC_{2,2}}{IN_{2,2}}\)	\(IS_{2,3}=\dfrac{IC_{2,3}}{IN_{2,3}}\)
AY3	\(IS_{3,0}=\dfrac{IC_{3,0}}{IN_{3,0}}\)	\(IS_{3,1}=\dfrac{IC_{3,1}}{IN_{3,1}}\)	\(IS_{3,2}=\dfrac{IC_{3,2}}{IN_{3,2}}\)
AY4	\(IS_{4,0}=\dfrac{IC_{4,0}}{IN_{4,0}}\)	\(IS_{4,1}=\dfrac{IC_{4,1}}{IN_{4,1}}\)
AY5	\(IS_{5,0}=\dfrac{IC_{5,0}}{IN_{5,0}}\)

4. Assuming annual effective trend is x%, all average severity numbers are trended to AY5 and then calculate their averages for each development year

	DY0	DY1	DY2	DY3	DY4
AY1	\(IS_{1,0}{(1+x)}^4\)	\(IS_{1,1}{(1+x)}^4\)	\(IS_{1,2}{(1+x)}^4\)	\(IS_{1,3}{(1+x)}^4\)	\(IS_{1,4}{(1+x)}^4\)
AY2	\(IS_{2,0}{(1+x)}^3\)	\(IS_{2,1}{(1+x)}^3\)	\(IS_{2,2}{(1+x)}^3\)	\(IS_{2,3}{(1+x)}^3\)
AY3	\(IS_{3,0}{(1+x)}^2\)	\(IS_{3,1}{(1+x)}^2\)	\(IS_{3,2}{(1+x)}^2\)
AY4	\(IS_{4,0}{(1+x)}^1\)	\(IS_{4,1}{(1+x)}^1\)
AY5	\(IS_{5,0}\)
AVG	\(AVG_0\)	\(AVG_1\)	\(AVG_2\)	\(AVG_3\)	\(AVG_4\)

5. Project severity by de-trending the averages (present value) by dividing them by (1+x%)

	DY0	DY1	DY2	DY3	DY4
AY1	\(IS_{1,0}{(1+x)}^4\)	\(IS_{1,1}{(1+x)}^4\)	\(IS_{1,2}{(1+x)}^4\)	\(IS_{1,3}{(1+x)}^4\)	\(IS_{1,4}{(1+x)}^4\)
AY2	\(IS_{2,0}{(1+x)}^3\)	\(IS_{2,1}{(1+x)}^3\)	\(IS_{2,2}{(1+x)}^3\)	\(IS_{2,3}{(1+x)}^3\)	\(\dfrac{AVG_4}{{(1+x)}^4}\)
AY3	\(IS_{3,0}{(1+x)}^2\)	\(IS_{3,1}{(1+x)}^2\)	\(IS_{3,2}{(1+x)}^2\)	\(\dfrac{AVG_3}{{(1+x)}^3}\)	\(\dfrac{AVG_4}{{(1+x)}^3}\)
AY4	\(IS_{4,0}{(1+x)}^1\)	\(IS_{4,1}{(1+x)}^1\)	\(\dfrac{AVG_2}{{(1+x)}^2}\)	\(\dfrac{AVG_3}{{(1+x)}^2}\)	\(\dfrac{AVG_4}{{(1+x)}^2}\)
AY5	\(IS_{5,0}\)	\(\dfrac{AVG_1}{{(1+x)}^1}\)	\(\dfrac{AVG_2}{{(1+x)}^1}\)	\(\dfrac{AVG_3}{{(1+x)}^1}\)	\(\dfrac{AVG_4}{{(1+x)}^1}\)

6. Calculate the link ratios for claim counts using volume-weighted averages, and then calculate the ultimate claim counts based on the link ratios for all accident years

	DY 0 to 1	DY 1 to 2	DY 2 to 3	DY 3 to 4
Linked Ratios	\(R_{0,1}=\dfrac{\sum\nolimits_{i=1}^{4}{{N_{i,1}}}}{\sum\nolimits_{i=1}^{4}{{N_{i,0}}}}\)	\(R_{1,2}=\dfrac{\sum\nolimits_{i=1}^{3}{{N_{i,2}}}}{\sum\nolimits_{i=1}^{3}{{N_{i,1}}}}\)	\(R_{2,3}=\dfrac{\sum\nolimits_{i=1}^{2}{{N_{i,3}}}}{\sum\nolimits_{i=1}^{2}{{N_{i,2}}}}\)	\(R_{3,4}=\dfrac{\sum\nolimits_{i=1}^{1}{{N_{i,4}}}}{\sum\nolimits_{i=1}^{1}{{N_{i,3}}}}\)

	DY4
AY1	\(UN_1=N_{1,4}\)
AY2	\(UN_2=N_{2,3}R_{3,4}\)
AY3	\(UN_3=N_{3,2}R_{2,3}R_{3,4}\)
AY4	\(UN_4=N_{4,1}R_{1,2}R_{2,3}R_{3,4}\)
AY5	\(UN_5=N_{5,0}R_{0,1}R_{1,2}R_{2,3}R_{3,4}\)

7. Generate Claim Closure Percentage table. For claim counts, we divide incremental claims closed in development year \(t\) by claims still open: ultimate claims – cumulative claims through year \(t-1\) to obtain claim closure ratios. These ratios are averaged.

	DY0	DY1	DY2	DY3	DY4
AY1	\(\dfrac{IN_{1,0}}{UN_1}\)	\(\dfrac{IN_{1,1}}{UN_1-IN_{1,0}}\)	\(\dfrac{IN_{1,2}}{UN_1-\sum\nolimits_{i=0}^{1}{IN_{1,i}}}\)	\(\dfrac{IN_{1,3}}{UN_1-\sum\nolimits_{i=0}^{2}{IN_{1,i}}}\)	100%
AY2	\(\dfrac{IN_{2,0}}{UN_2}\)	\(\dfrac{IN_{2,1}}{UN_2-IN_{2,0}}\)	\(\dfrac{IN_{2,2}}{UN_2-\sum\nolimits_{i=0}^{1}{IN_{2,i}}}\)	\(\dfrac{IN_{2,3}}{UN_2-\sum\nolimits_{i=0}^{2}{IN_{2,i}}}\)
AY3	\(\dfrac{IN_{3,0}}{UN_3}\)	\(\dfrac{IN_{3,1}}{UN_3-IN_{3,0}}\)	\(\dfrac{IN_{3,2}}{UN_3-\sum\nolimits_{i=0}^{1}{IN_{3,i}}}\)
AY4	\(\dfrac{IN_{4,0}}{UN_4}\)	\(\dfrac{IN_{4,1}}{UN_4-IN_{4,0}}\)
AY5	\(\dfrac{IN_{5,0}}{UN_5}\)
AVG	\(AVG_0\)	\(AVG_1\)	\(AVG_2\)	\(AVG_3\)	\(AVG_4\)

8. Use the above averaged claim closure rates to develop claim counts for all years.

	DY1	DY2	DY3	DY4
AY1	\(IN_{1,1}=N_{1,1}-N_{1,0}\)	\(IN_{1,2}=N_{1,2}-N_{1,1}\)	\(IN_{1,3}=N_{1,3}-N_{1,2}\)	\(IN_{1,4}=N_{1,4}-N_{1,3}\)
AY2	\(IN_{2,1}=N_{2,1}-N_{2,0}\)	\(IN_{2,2}=N_{2,2}-N_{2,1}\)	\(IN_{2,3}=N_{2,3}-N_{2,2}\)	\(UN_2-\sum\nolimits_{i=0}^{3}{IN_{2,i}}\)
AY3	\(IN_{3,1}=N_{3,1}-N_{3,0}\)	\(IN_{3,2}=N_{3,2}-N_{3,1}\)	\(IN_{3,3}=(UN_3-\sum\nolimits_{i=0}^{1}{IN_{3,i}}){AVG_3}\)	\(UN_3-\sum\nolimits_{i=0}^{3}{IN_{3,i}}\)
AY4	\(IN_{4,1}=N_{4,1}-N_{4,0}\)	\(IN_{4,2}=(UN_4-\sum\nolimits_{i=0}^{1}{IN_{4,i}}){AVG_2}\)	\(IN_{4,3}=(UN_4-\sum\nolimits_{i=0}^{2}{IN_{4,i}}){AVG_3}\)	\(UN_4-\sum\nolimits_{i=0}^{3}{IN_{4,i}}\)
AY5	\(IN_{5,1}=(UN_5-IN_{5,0}){AVG_1}\)	\(IN_{5,2}=(UN_5-\sum\nolimits_{i=0}^{1}{IN_{5,i}}){AVG_2}\)	\(IN_{5,3}=(UN_5-\sum\nolimits_{i=0}^{2}{IN_{5,i}}){AVG_3}\)	\(UN_5-\sum\nolimits_{i=0}^{3}{IN_{5,i}}\)

9. Multiply incremental frequency by incremental severity to obtain loss payments in the year, then add up projected loss payments to obtain the total reserve.

Ultimate Frequency (From Step 5)

	DY1	DY2	DY3	DY4
AY1
AY2				\(IS_{2,4}=\dfrac{AVG_4}{{(1+x)}^4}\)
AY3			\(IS_{3,3}=\dfrac{AVG_3}{{(1+x)}^3}\)	\(IS_{3,4}=\dfrac{AVG_4}{{(1+x)}^3}\)
AY4		\(IS_{4,2}=\dfrac{AVG_2}{{(1+x)}^2}\)	\(IS_{4,3}=\dfrac{AVG_3}{{(1+x)}^2}\)	\(IS_{4,4}=\dfrac{AVG_4}{{(1+x)}^2}\)
AY5	\(IS_{5,1}=\dfrac{AVG_1}{{(1+x)}^1}\)	\(IS_{5,2}=\dfrac{AVG_2}{{(1+x)}^1}\)	\(IS_{5,3}=\dfrac{AVG_3}{{(1+x)}^1}\)	\(IS_{5,4}=\dfrac{AVG_4}{{(1+x)}^1}\)

Ultimate Severity (From Step 8)

	DY1	DY2	DY3	DY4
AY1
AY2				\(UN_2-\sum\nolimits_{i=0}^{3}{IN_{2,i}}\)
AY3			\(IN_{3,3}=(UN_3-\sum\nolimits_{i=0}^{1}{IN_{3,i}}){AVG_3}\)	\(UN_3-\sum\nolimits_{i=0}^{3}{IN_{3,i}}\)
AY4		\(IN_{4,2}=(UN_4-\sum\nolimits_{i=0}^{1}{IN_{4,i}}){AVG_2}\)	\(IN_{4,3}=(UN_4-\sum\nolimits_{i=0}^{2}{IN_{4,i}}){AVG_3}\)	\(UN_4-\sum\nolimits_{i=0}^{3}{IN_{4,i}}\)
AY5	\(IN_{5,1}=(UN_5-IN_{5,0}){AVG_1}\)	\(IN_{5,2}=(UN_5-\sum\nolimits_{i=0}^{1}{IN_{5,i}}){AVG_2}\)	\(IN_{5,3}=(UN_5-\sum\nolimits_{i=0}^{2}{IN_{5,i}}){AVG_3}\)	\(UN_5-\sum\nolimits_{i=0}^{3}{IN_{5,i}}\)

Total Reserve = Closed Claims in AY_z x Incremental Severity in AY_z, for all DY

Total Reserve = \(\sum\nolimits_{i=2}^{5}{V_i}\), where:

\(V_2=IS_{2,4}IN_{2,4}\)

\(V_3=IS_{3,3}IN_{3,3}+IS_{3,4}IN_{3,4}\)

\(V_4=IS_{4,2}IN_{4,2}+IS_{4,3}IN_{4,3}+IS_{4,4}IN_{4,4}\)

\(V_5=IS_{5,1}IN_{5,1}+IS_{5,2}IN_{5,2}+IS_{5,3}IN_{5,3}+IS_{5,4}IN_{5,4}\)

Discounted Loss Reserves

We need an assumption as to how payments are distributed within each year. We will assume that they are made in the middle of the year. However:

• In development year 1, perhaps an assumption that payments are made earlier than the midpoint of the year is more reasonable, as small cases will settle rapidly.
• If there are projected payments made in an unspecified year after the final development year in the projection, in other words in the interval \((k,\infty )\) for some k, then some assumption is needed about when they are made.

Methodology

0. Paid Loss Triangle

	DY0	DY1	DY2	DY3	DY4
AY1	\(C_{1,0}\)	\(C_{1,1}\)	\(C_{1,2}\)	\(C_{1,3}\)	\(C_{1,4}\)
AY2	\(C_{2,0}\)	\(C_{2,1}\)	\(C_{2,2}\)	\(C_{2,3}\)	\(C_{2,4}\)
AY3	\(C_{3,0}\)	\(C_{3,1}\)	\(C_{3,2}\)	\(C_{3,3}\)	\(C_{3,4}\)
AY4	\(C_{4,0}\)	\(C_{4,1}\)	\(C_{4,2}\)	\(C_{4,3}\)	\(C_{4,4}\)
AY5	\(C_{5,0}\)	\(C_{5,1}\)	\(C_{5,2}\)	\(C_{5,3}\)	\(C_{5,4}\)

1. Incremental Payments Triangle

	DY0	DY1	DY2	DY3	DY4
AY1
AY2					\(IC_{2,4}=C_{2,4}-C_{2,3}\)
AY3				\(IC_{3,3}=C_{3,3}-C_{3,2}\)	\(IC_{3,4}=C_{3,4}-C_{3,3}\)
AY4			\(IC_{4,2}=C_{4,2}-C_{4,2}\)	\(IC_{4,3}=C_{4,3}-C_{4,2}\)	\(IC_{4,4}=C_{4,4}-C_{4,3}\)
AY5		\(IC_{5,1}=C_{5,1}-C_{5,0}\)	\(IC_{5,2}=C_{5,2}-C_{5,1}\)	\(IC_{5,3}=C_{5,3}-C_{5,2}\)	\(IC_{5,4}=C_{5,4}-C_{5,3}\)

2. Total Reserve (at 12/21/CY5) = Sum of the discounted incremental payments

Remark: Incremental payments are discounted based on the length of time spent to settle (development year)

Incurred Loss Triangle

1. 1. Use linked ratios to calculate ultimate payment.
      Ultimate Payment = (Paid-to-Date + Case Reserve) (Linked Ratios for all left-over development years)
   2. Develop (incremental) paid loss triangle by calculating incremental payments for all development years and total future payments.
   3. Discount the incremental payments (mid of year) of all development years.
   4. Discounted Loss Reserve for Current Accident Year = (Ultimate Payment – Paid-to-Date Payment) (PV of incremental payments / (Total Future Payments Based on the Paid Loss Triangle – Paid-to-Date Payment)

Ratemaking: Preliminary Calculations

Basic Concepts of Ratemaking

• Incurred Losses in Calendar Year (CY).
  Incurred losses in the calendar year equal paid losses plus the change in reserve for calendar year z

Incurred Losses for CY_z = Payments in CY_z + Reserve on 12/31 /CY_z – Reserve on 12/31 /CY_z-1

Loss Development and Trend

Accident claims can take years to close. We receive data with incurred claims and paid-to-date claims, but we must project these numbers to their ultimate level. Optionally we can discount future payments. Loss development is the projection of claims to their ultimate level. Rates must be based on the total amount that eventually gets paid.

Usually age-to-ultimate development factors are greater than 1, but they may be less than 1 if:

• Projection is based on incurred claims and case reserves were too conservative.
• Some losses are recovered through subrogation and salvage.

The Number of Years of Trend

If we use accident year date, we assume:

• • loss occurred in the middle of the calendar year of the accident => average accident ocrurrs on \(7/1/CY\)
  • policies are sold uniformly throughout any period => average policy will be sold \(\frac{n}{2}\) months after the effective date
  • policy accident occurrences to occur uniformly throughout any period => average accident will occur \(\frac{n}{2}\) months after the effective date

Projected Average Accident Date = Effective date + n/2 years of policy effective period + n/2 years of policy term

Then the number of years of trend is:

• • The Number of Years of Trend = Projected Average Accident Date – \(7/1/AY\)
  • The Number of Years of Trend = Projected Average Accident Date – \(1/1/CY\)
  • The Number of Years of Trend = Projected Average Accident Date – \(1/1/PY\)

Expenses

• Loss Adjustment Expenses (LAE): Some expenses are considered to be associated with the losses and are combined with the losses when ratemaking.
• Allocated Loss Adjustment Expenses (ALAE): Some loss adjustment expenses, such as legal costs to defend a policyholder, are tied with specific losses and are called allocated loss adjustment expenses.
  
• Un-Allocated Loss Adjustment Expenses (ULAE): Other loss adjustment expenses, such as the salary of the manager of the home office claim department, are not associated with specific losses.

Let \(L\) be the loss cost, developed and trended and including \(LAE\). Let \(V\) be the proportion of premium needed for expenses, contingencies, and profit. Then

• Let \(R=1-V\) is the permissible loss ratio, then Gross Premium Rate = \(\dfrac{L}{R}\)
• Let \(F\) be the fixed expense per policy, then Gross Premium Rate = \(\dfrac{L+F}{R}\)
• If \(F\) is an amount fixed by state regulation and the regulation does not allow \(F\) to be grossed up by the loss ratio, then Gross Premium Rate = \(\dfrac{L}{R}+F\)

Credibility

The weight assigned to data is called the credibility of the data, and is denoted by \(Z\). If \(Z=1\), we say that the data is fully credible. Otherwise it is partially credible.

Credibility will be used in the following ways:

• To estimate loss cost, we use a weighted average of our data and intercompany data.
• To estimate class differentials, we use a weighted average of our recent experience and our previous class differentials.

Premium at Current Rates

In order to use recent data, we compare AY_z data to earned premium in CY_z. However, premiums may have increased since CY_z. Even if z is the current year, premiums may have changed during years z – 1 and z, and earned premiums in CY_z for one-year policies are based on written premiums in years z – 1 and z. Thus it is necessary to adjust the (pre-rate-change) earned premiums to bring them up to current rates. To do this exactly, we go through every combination of rate level, class, and territory and multiply the earned exposure for each  combination by the current premium rate for that combination, then add up the products. Earned exposure is derived from exposure the same way earned premium is derived from written premium.

For example, for one-year policies that were sold during CY_z, if there was a rate increase of \(c\text{%}\) on \(8/1/CY_z\) the earned premium at current rates during CYz is the sum of the earned premiums in the right column.

Date Sold	Written Premium	Current Rate Factor	Part of Year Factor	Earned Premium at Current Rates
\(2/15/CY_z\)	\(P_1\)	\((1+c\text{%})\)	\(1-\frac{1.5}{24}\)	\(P_1(1+c\text{%})(1-\frac{1.5}{24})\)
\(6/1/CY_z\)	\(P_2\)	\((1+c\text{%})\)	\(1-\frac{5}{12}\)	\(P_2(1+c\text{%})(1-\frac{5}{12})\)
\(9/1/CY_z\)	\(P_3\)	\(1\)	\(1-\frac{8}{12}\)	\(P_3(1-\frac{8}{12})\)
\(10/1/CY_z\)	\(P_4\)	\(1\)	\(1-\frac{9}{12}\)	\(P_4(1-\frac{9}{12})\)

Parallelogram Method

An approximate method for adjusting earned premium is the parallelogram method. It assumes that earned premium at each rate level flows in uniformly throughout the year, as it would if the volume of business did not grow or shrink.

1. 1. Calculate the average premium rate in each calendar year by calculating the areas in which each rate level is effective.
   2. Divide Ultimate rate by the average rate level in each year to obtain the multiplier for that calendar year’s earned premiums. That multiplier is called the on-level factor.

Ratemaking: Rate Changes and Individual Risk Rating Plans

Rate Changes

Rate changes are calculated in three steps:

1. Calculate the overall average rate change
2. Update class differentials
3. Balance back

For each step, there are two methods: loss cost and loss ratio.

Calculating the Overall Average Rate Change

Loss Cost Method (Pure Premium Method)

The loss cost method develops the new average gross rate directly.

\(\text{Average Loss Cost} = \dfrac{\text{Expected Losses, Trended and Developed}}{\text{Number of Earned Exposures}}\)

Adding in expenses to obtain the premium:

\(\text{Average Gross Rate}=\dfrac{\text{Average Loss Cost} +\text{Fixed Expense Per Exposure}}{\text{Permissible Loss Ratio}}\)

The rate change is:

\(\text{Indicated Avg. Rate}=\dfrac{\dfrac{\text{Expected Losses, Trended and Developed}}{\text{Exposures}}+\dfrac{\text{Fixed Expense}}{\text{Exposures}}}{1-\text{Variable Expense Ratio}-\text{Profit Provision}}=\dfrac{\bar{L}+\bar{E}}{1-V-Q_T}\)

Loss Ratio Method

\(\text{Effective Loss Ratio}=\dfrac{\text{Expected Losses, Trended and Developed}}{\text{Earned Premiums at Current Rates}}\)

\(\text{Fixed Expense Ratio}=\dfrac{\text{Fixed Expenses per Exposure}}{\text{Earned Premiums at Current Rates / Exposures}}\)

The loss ratio method compares the loss ratio experienced with current rates to the permissible loss ratio.

\(\text{Indicated Avg. Rate Change}=\dfrac{\text{Effective Loss Rate}+\text{Fixed Expense Rate}}{1-\text{Variable Expense Ratio}-\text{Profit Provision}}-1=\dfrac{L+F}{1-V-Q_T}-1\)

\(\text{Indicated Average Rate}=\text{Current Average Rate}(1+\text{Indicated Rate Change})\)

Remark:

• If weighted percentages of \(c_i\) and \(c_j\) are given to \(AY_i\) and \(AY_j\) correspondingly, then

\(\text{Effective Loss Ratio}=c_i \times \text{Ultimate Loss for AY}_i + c_j \times \text{Ultimate Loss for AY}_j\)

• Exposures can be obtained by:

\(\text{Exposures} = \dfrac{\text{Earned Premium at Current Rates}}{\text{Existing Differential}}\)

Updating Class Differentials

Rates vary by territory and class. One class and territory is selected as the base territory. Usually it is the one with the most exposures.

• The ratio of the premium for a class to the premium for the base class is called the class differential or relativity.
• We will assume that differentials are multiplicative.
• The rate for a class and territory is the rate for the base class and base territory times the class differential times the territory differential.
• If you are not told which territory or class is base, assume that the territory or class having a differential of 1 is the base territory or class.
• if average rates in a territory and average class differentials are given, the base rates in the territory be backed out and compare them to obtain the existing differentials.

\(\text{Base Rate}_i=\dfrac{\text{Earned Premium at Current Rates}_i}{\text{Exposures}_i}\)

\(\text{Existing Differential}_i = \dfrac{\text{Base Rate}_i}{\text{Base Rate}_{Base}}\) or \(\text{Base Rate}_i={\text{Base Rate}_{Base}} \times \text{Existing Differential}_i\)

1. Indicated Differential for Territory i

Loss Cost Method (Pure Premium Method)

\(\text{Indicated Differential}_i = \dfrac{\text{Adjusted Loss Cost}_i}{\text{Adjusted Loss Cost}_{Base}}\)

Where:

• • • \(\text{Adjusted Loss Cost}_{i}=\dfrac{\text{Expected Losses, Trended and Developed}_{i}}{\text{Adjusted Exposures}_{i}}\)
    • \(\text{Adjusted Exposures}_{i}=\text{Average Class Differential}_{i}\times\text{Exposures}_{i}\)
    • \(\text{Average Class Differential}_{i} = \dfrac{\text{Average Loss Cost}_{i}}{\text{Base Rate}_{i}}\)

The loss cost method will not lead to the same answer as the loss ratio method if the cross-variables in the classes are not homogeneous.

For both methods, since we are only interested in the ratio of loss ratios and not their magnitudes, the two loss ratios or loss costs do not have to be trended or developed.

Loss Ratio Method

\(\text{Indicated Differential}_i = \text{Existing Differential}_i \times \dfrac{\text{Effective Loss Ratio}_i}{\text{Effective Loss Ratio}_{Base}}\)

2. Indicated Territory Differential

Partial Credibility

If there are not that many claims in a cell, the data is not fully credible. The calculated differential is weighted with the credibility factor and the existing differential gets the balance of the weight. If using square root rule, \(Z=\sqrt{N_i/N_F}\), and the indicated territory differential is:

\(\text{Indicated Territory Differential}_i = Z \times \text{Indicated Differential}_i + (1 – Z) \times \text{Existing Differential}_i\)

Balancing Back

After changing the differentials, the resulting loss cost will not balance back to the expected loss cost because the average of the differentials is not 1. We thus must multiply the rates by a factor. The numerator of the factor is the weighted average of existing differentials and the denominator is the weighted average of the proposed differentials. The weights are the earned exposures.

\(\text{Balance-Back Factor} = \dfrac{\sum{w_i \times \text{Existing Differential}_i}}{\sum{w_i \times \text{Indicated Differential}_i}}\)

The new rate for territory i is:

Individual Risk Rating Plans

Prospective rating plans change future premiums only and do not retroactively adjust premiums. 4 types of prospective rating plans are:

• Schedule rating, where the underwriter adjusts the premium for individual characteristics based on a schedule. The schedule (c%) specifies discounts or additional charges based on features of the risk.
  • Schedule Rating Debits:  \(P’=P \times (1+c\text{%})\)
  • Schedule Rating Credits: \(P’=P \times (1-c\text{%})\)
• Experience rating adjusts future premiums (P) based on past experience. If Z is the credibility factor for the insured’s experience, set the modification factor \(M\) equal to

\(M=Z\dfrac{Loss_{Actual}}{Loss_{Expected}}+(1-Z)\)

\(P’=P \times M\)

possibly with maximum and minimum bounds for M .

• Composite rating adjusts the premium for exposures. Initially a deposit of the premium for anticipated exposures is made. At the end of the period, the premium is adjusted based on actual exposures. The exposure measure is a composite measure: it may consider size of property, annual revenue, number of employees, or anything else relevant to the risk; hence the name “composite rating”. This rating system can be combined with schedule and experience rating.

\(P’=P \times M \times (1+c\text{%})\) or \(P’=P \times M \times (1-c\text{%})\)

• Large deductible policies place most of the risk on the insured. Sometimes the insurer takes care of claim adjustment regardless of loss size.