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Casualty Actuarial Society E-Forum, Winter 2009 289

Monitoring Renewal Rate Change on Cat-Exposed Excess

Property Business

Ira Robbin, Ph.D.

________________________________________________________________________

Abstract

This paper explains why a commonly-used metric of pricing performance, the Renewal Rate Change statistic,

might not give true indications of the real rate change on Catastrophe (Cat)-Exposed Excess Property business.

At the account level, false readings may arise when the renewal and expiring policies cover different layers or

different sets of locations. When that happens, premium changes stemming from such differences are

confounded with real changes in rate level. The paper presents a proposal to appropriately reflect coverage layer

and location schedule differences. The proposal involves use of Cat Loss Simulation models to estimate the

percentage by which the premium for an account should change in response to such differences. Once individual

policy rate changes are correctly calculated, there is a potential problem in aggregating the individual results to a

correct portfolio total. Concrete examples are presented to demonstrate that weighting with Renewal Premiums

is incorrect and will lead to an overly optimistic answer. The paper then proposes alternative weights that lead to

an unbiased result.

Keywords Rate Monitor, Renewal Rate Change, Excess Property, Catastrophe Models

1. INTRODUCTION

The Renewal Rate Change is a popular pricing metric, but it can give misleading indications when

used to measure the rate change on Catastrophe-Exposed Excess Property business. There are

potential problems both at the account level, in defining rate change for an account, and at the

portfolio level, in weighting together the individual account rate changes to get a portfolio total.

For an individual account, the nominal change in rates between the renewal and expiring policies

will not necessarily provide a valid rate change comparison when those policies cover different

excess layers or cover different sets of locations. We will estimate the relative effect such differences

should have on rate level by calculating how they impact Technical Premium

rates. This will allow

us to offset the Nominal Renewal Rate Change for the effect of these differences and thus arrive at a

better measure of the real rate change. The necessary Technical Premiums will be computed using

results from a Catastrophe (Cat) Loss Simulation model.

At the portfolio level, an optimistic bias is introduced if individual account rate changes are

aggregated into a portfolio total using Renewal Premium weights. We will show what is wrong with

Technical Premium is here used to denote an indicated risk-loaded premium that is calculated directly from a set

pricing algorithm without influence of schedule rating or other judgment modification.

A cat loss simulation model runs thousands of modeled events against input locations and insurance coverages to

arrive the estimated distribution of insurance losses from specified catastrophe perils. See Burger, Fitzgerald, White, and

Woods [2] for a description of this type of model.

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Renewal Premium weights and then propose better ones.

1.1 The Lack of Actuarial Literature on the Subject

There are many articles on rates and rate changes, but we have not found any that directly

address the concerns we have identified regarding coverage layer changes, location schedule changes,

and aggregation averaging. The one paper that is focused on rate monitors notes that “renewal rate

change reports often provide misleading indications of rate changes due to changes in the underlying

mix of business on each policy,”

but does not present a solution to that problem. Price monitors

have also been the subject of discussion at several actuarial professional meetings,

but those

discussions have not yet found their way into the actuarial literature. There also may be internal

corporate memos documenting computations similar to those we will present. Unfortunately, those

articles are not part of the public domain. Our belief is that this will be the first paper in the

literature to directly address the key questions we have identified about renewal rate change

computations for excess property business.

1.2 Industry Reported Rate Increases

In the aftermath of the Katrina, Rita, and Wilma hurricanes of 2005, many industry pricing

surveys reported significant rate increases on hurricane-exposed property business.

Since we do not

know how insurance companies computed renewal rate change figures by account or how they

aggregated them into portfolio totals, we do not know if there were any distortions in those reported

rate change figures. However, if we succeed in our objective, so that our methods, or improved

variations, are adopted as industry standard practices, it may help to dispel questions about the

computational validity or comparability of reported rate increases following the next major cat event.

1.3 Uses of the Renewal Rate Change Statistic

The Renewal Rate Change statistic is one of the most popular pricing metrics. It is used in two

major ways. First, it is used by actuaries as one of several inputs needed to compute on-level

premiums and to project rate adequacy. Second, it is used by managers as a gauge of pricing

performance.

Vaughn [8], p. 506.

See the presentations by Kundrot [4], Nyce [5], and Palisi [6].

For example, the May, 2006 Expert Commentary on the First Quarter Market Survey from The Council of Insurance

Agents and Brokers (CIAB) [3] had the quote “Overall, rates are down, except for property cat exposures (which are) up

more than 25 percent…” (Italics in original).

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1.3.1 On-Level Factors, New and Nonrenewed Business

In this paper, we will focus on measuring the Renewal Rate Change and only the Renewal Rate

Change. So our discussion will only briefly touch on two other important pieces in the measurement

of the overall rate: the contribution due to newly written business and the impact resulting from

nonrenewed business.

Since the calculation of Renewal Rate Change omits consideration of new business, it provides at

best an incomplete picture of the overall rate level.

We hypothesize that relative price adequacy

between new and renewal business varies with the insurance cycle. When the market is tight and

nonrenewed accounts go searching for a new carrier, they are in a weak bargaining position and may

pay a large premium. Then a new business account may be more adequately priced than a renewal.

On the flip side, when the market softens and companies are flush with capacity, underwriters will

agree to cut price to gain new accounts and meet volume targets. Then a new account may be

relatively underpriced.

The Renewal Rate Change statistic also neglects any consideration of nonrenewed business. In

general, the nonrenewed book is a mix of adequately priced accounts which departed for better deals

elsewhere and poorly priced accounts which were cancelled or nonrenewed by company

underwriters. The mix between the two likely varies over the insurance cycle and should change

when the insurance company revises its pricing and underwriting policies.

A detailed computation of on-level factors would require going beyond looking at rate change on

renewal accounts to also consider rate adequacy levels on new and nonrenewed accounts. However,

in practice this is seldom done. Most on-level factors are derived solely from rate change statistics on

renewal business. This is technically an incomplete treatment that might lead to inaccuracy in

computed on-level factors.

1.3.2 Management Uses of Rate Monitors

The Renewal Rate Change statistic is one of several rate change statistics that may be used by

company executives to monitor pricing. It is conceptually easy to understand, treats each

underwriter fairly, and fosters clear lines of responsibility. These are some of the reasons the

Vaughn [8] p. 506 states “there are several drawbacks associated with renewal rate change reports. For instance, the

renewal rate change report does not monitor the price level changes associated with new-business policies.”

Boor [1] describes a scenario that supports our hypothesis. He states “accounts may be ‘orphaned’ and unable to find

coverage. This produces a situation where accounts are willing to accept higher prices for the benefit of having insurance

coverage when it is hard to obtain.” p. 3.

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Renewal Rate Change statistic is generally accepted by both business line managers and

underwriters.

Rate monitors are used in different ways by executives in managing insurance companies. Some

executives set monitor targets and explicitly look at price monitor performance when evaluating

underwriters and business units. These executives believe a connection between monitor

performance and compensation will act as a powerful incentive for underwriters to push the line on

price as far as it can go. Other executives disagree. They feel that too much emphasis on a statistic

may distract underwriters from sound underwriting and may even spur them to find ways to “game”

the system. We will not argue one way or the other about how rate change statistics should be used

in managing a company. Our intent is to make actuaries aware that there may be widely divergent

philosophies in different companies. Further, our discussion of the Renewal Rate Change statistic

should not be construed as an endorsement or criticism for using that statistic, or any pricing

statistic, as a measure of underwriter performance.

1.4 Loss Ratios Instead of Rate Monitors?

Since what really matters are results, one might play devil’s advocate and question whether rate

monitors have any use at all in managing the business. Why not just look at loss ratios? Isn’t the

proof of price performance ultimately in the results? Why supplement the loss ratio perspective with

anything else? There are several general answers to this line of questioning. First, there is the matter

of timing. Rate monitors are a leading indicator that can help us predict what loss ratio to expect on

the business we are writing.

Loss ratios are a backward look at results on what has been written. So

rate monitors give management a more timely indication of possible problems ahead. Second, there

is the point that loss development introduces a lag before loss ratios results can be estimated with

reasonable accuracy. When the lag runs into years, there may be some legitimate question about who

is responsible for current results. Consider any long-tailed casualty line, where an underwriter can

switch companies every few years and, with any luck, will stay one step ahead of the loss tail. Why

should current underwriters be held to account for the shipwreck caused by a prior crew? In

contrast, a price monitor would typically show the pricing achieved by current underwriters over the

most recent prior month or quarter, with perhaps a lag of a month or two.

Wang and Faber [9] support this view stating, “What the insurance industry needs is leading indicators…” and that “…

companies need to diligently track rate level changes” (p. 59).

Price monitor values can also age over time due to late bookings, premium audits, endorsements, and retro

adjustments. In most cases, price monitor development is essentially complete after a few months.

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The final problem with focusing solely on the loss ratio is that it can be too volatile. On this

point, Cat-Exposed Excess Property is a good example. The loss ratio is usually quite low for every

insurance company and underwriter, except when a catastrophe occurs. Then everyone’s loss ratio is

in the stratosphere. With so many underwriters doing poorly that year, it is hard to legitimately single

out a specific underwriter for having had a poor showing. Of course, this is an oversimplification: a

good underwriter will carefully control exposure aggregation, will write a geographically diversified

mix of business, and utilize other underwriting practices which should translate into better results in

any particular year and over the cycle. Nonetheless, the point still stands: a good cat underwriter will

often lose money in a bad year and a poor cat underwriter will often make money in a good year.

1.5 Organization of the Paper

In Chapter 2, we will define the Renewal Rate Change statistic at the account level and show how

to make the necessary adjustments. Then, in Chapter 3 we will examine how to aggregate the

individual account rate changes into a portfolio average. All key formulas are documented in

Appendix A, while Appendix B contains a discussion of data issues.

2. RENEWAL RATE CHANGE BY ACCOUNT

In this chapter we will first explain why it is necessary to go beyond the Nominal Renewal Rate

Change on Cat-Exposed Excess Property business. Then we will walk through a hypothetical Excess

Property example and show how to make account level adjustments for coverage layer and location

schedule differences between the renewal and expiring policies. To avoid complications, we will

assume all accounts have Specified Peril catastrophe coverage.

2.1 Coverage Layer and Location Schedule Changes

For Cat-Exposed Excess Property business, it is not at all unusual for accounts to have renewal

and expiring policies that provide different layers of coverage or that have different schedules of

insured locations. For many other types of insurance business, the vast majority of accounts renew

policies with coverage layers that are the same as the expiring ones and that have minimal or

Accounts often have policies that cover Flood, Fire, Terrorism and other perils. Some polices may provide All-Risk

Coverage or they may provide Difference in Conditions (DIC) Coverage.

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predictable changes in exposure. Why is the Cat-Exposed Excess Property business different? Our

answer is that Property Cat market has inherent characteristics which push the insured and the

insurance company to initiate changes more frequently than for many other lines. In addition, much

of this business is placed through Surplus Lines brokers. It is possible that accounts in the

Nonadmitted market are subject to more frequent program changes than Admitted market accounts.

2.1.1 Changes Initiated by the Insured

The insured typically decides to revise its insurance program

in response to market price swings

or capacity fluctuations. These are endemic to the Cat-Exposed Property market. In the aftermath of

a significant cat, capacity gets tight, and the rate on Cat-Exposed Property skyrockets. The insured,

who may be strapped for cash following a cat, is in no position to pay the huge premium increase

needed to renew its expiring program. The insured may thus feel forced to retain more risk, either by

changing the layer of insurance it purchases or, less frequently, by removing some locations from its

insurance program. After a few years, prices will drop as naive capacity floods the market, and the

process will reverse itself.

2.1.2 Changes Initiated By the Insurance Company

There are several situations in which an insurance company might initiate changes. A company

faced with a downgrade by a rating agency, for instance, may need to reduce its exposed limits. To

do this, it might reduce its percentage shares of layers with large limits or write different layers with

lower limits. In another scenario, changes on individual accounts flow from a deliberate change

made in the company’s underwriting strategy. For example, the company might have been writing

large shares of high excess layers that kept it above the “working layer.” When the account is up for

renewal, the company may have adopted a “ventilation” strategy under which it aims to write

modest percentages of several disconnected layers, including some working layers.

2.1.3 Changes Initiated By the Surplus Lines Broker

A broker may sometimes be the driving force behind changes in an account’s insurance program.

It is possible that splitting a large layer in two or combining two small layers into one can result in an

overall price reduction to the insured. As well, a broker may want to introduce new carriers to help

Beyond changing coverage layers and location schedules, changes are also made with respect to per location

deductibles, per occurrence deductibles, occurrence hour range definitions (contiguous 96 hours versus 128 hours, etc.),

location or peril sublimits, Business Interruption (BI) waiting periods and limits, and so forth.

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spur competition for the account. This might necessitate cutting back on the shares of the

incumbent writers or splitting layers to make room for the new carriers. A broker also may feel the

need to make such changes in order to stave off competition from another Surplus Lines broker.

Note that some program structure revisions can be made without altering the customer’s overall

layer of insurance coverage, even though they will alter the specific layers written by particular

insurance companies.

2.2 Defining Real Rate Change

Before making adjustments to a nominal rate change figure in order to obtain the real rate

change, we should first discuss the concept of a “real” rate change. When the expiring and renewal

policies have identical coverages and location schedules, the real rate change is the same as the

nominal rate change: both equal the change in the charged rate.

When they are not identical, we

need to eliminate the portion of the nominal rate change which stems from differences in the layer

of coverage or differences in the schedule of locations. What remains is the “real” rate change.

To introduce some theoretical precision about what portion of a rate change should be attributed

to changes in the coverage layer and the location schedule, suppose we had a single manual showing

perfectly adequate rates for both the expiring and renewal coverage layers and location schedules. By

the phrase, “perfectly adequate rates,” we mean a set of actuarially sound indicated rates that satisfy

all rules of mathematical consistency and which within those rules faithfully reflect the risk-return

preferences of the company.

In principle, the company should be equally willing to write any

coverage layer and location schedule combination at the perfectly adequate rates listed in the

theoretical manual. In this sense, all the rates in the manual are at the same rate level.

Our proposed conceptual approach is to use rate relativities from this hypothetical manual of

perfectly adequate rates in order to ascertain the effect of coverage and location schedule differences

between the renewal and expiring policies. For instance, if the manual says there is a 10% rate

difference between the renewal and expiring layers, then we will back out that 10% rate difference

when computing the real rate change.

We should note that perfectly adequate manual pricing does not

imply the manual rates are

A rate in this context is the premium per unit of exposure.

In defining perfect adequacy, we make reasonable assumptions as needed to avoid complications due to minimum

premium rules, commission differences, and other similar factors.

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calibrated to produce the same expected loss ratio for all coverages. Depending on how we price

risk, we might reasonably have lower expected loss ratios for high excess layers than for ground-up

limited layers.

Also, it should be emphasized we are using only one manual, the current one, in evaluating

perfectly adequate premiums and in deriving the impact of coverage layer and location schedule

changes. To be specific, we will assume the current manual is perfectly adequate as of the effective

date of the renewal policy, even though it contains rates for layers and locations from the expiring

policy. Note the manual of adequate rates could change from year to year -- for instance, due to the

effect of trend. An account with no real rate change might thus end up with a renewal policy less

adequately priced than the expiring policy.

To implement this framework, we have to clear a major hurdle: the lack of a manual of perfectly

adequate rates for a Cat-Exposed Excess Property account. The proposal we will later present entails

using Cat Loss Simulation models to generate Technical Premium rates that will serve as the best

available estimates of perfectly adequate rates.

2.3 A Hypothetical Example Account

We will now look at an example in which there are both coverage layer and location schedule

differences. To begin our example, suppose Wayne’s Widgets is a major widget manufacturer and

vendor. Assume it has a factory, a warehouse, and several retail stores in two hurricane-exposed

states. Suppose our insurance company has written cat coverage on Wayne’s Widgets for two years

running. Table 2.3.1 summarizes the premiums, coverages, and exposures for the expiring and

renewal policies.

Table 2.3.1

Wayne's Widgets

Expiring Renewal

Premium $50,000 $40,000

Coverage $5m p/o $25m x $5m $2.5m p/o $10m x $15m

Company Limit $5,000,000 $2,500,000

Layer 100% Limit $25,000,000 $10,000,000

Attachment $5,000,000 $15,000,000

Exposure (TIV) $30,000,000 $25,000,000

Premium, Coverage, and Exposure Summary

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The coverage for the expiring policy is $5 million part of $25 million excess of $5m and is written

as “$5m p/o $25m x $5m,” This means the insurance company has a 20% share of the $25m x $5m

layer. The exposure is the Total Insured Value (TIV) of locations covered under the policy. This

value is gross of deductibles.

Based on this data, we calculate the Nominal Rate, where the Nominal Rate equals the 100%

Layer Premium Per $100 of TIV. The 100% Layer Premium is the premium for the full layer. To

calculate it, we take the company premium and divide by the share. The reason the rate change is

calculated with rates on a 100% basis is that this prevents changes in share from being incorrectly

counted as rate changes. The Nominal Rate Change for our sample account is shown in Table 2.3.2.

2.3.1 Coverage Layer Differences

Our initial conclusion, based on the negative Nominal Rate Change, is that pricing has slipped.

Yet this initial conclusion does not seem right. While the rate has dropped 23%, the 100% layer limit

has dropped by 60%, from $25 million to $10 million. This suggests the charged rate could also fall

by 60% without reducing rate level.

The beneficial effect of limit reduction is directly incorporated in another pricing statistic, the

Rate on Line (ROL).

For the expiring policy, the 100% premium is $250,000 and the 100% limit is

ROL is defined as the 100% Layer Premium divided by 100% of the limit of the layer to be insured.

Table 2.3.2

Wayne's Widgets

Expiring Renewal

Premium $50,000 $40,000

Coverage $5m p/o $25m x $5m $2.5m p/o $10m x $15m

Company Limit $5,000,000 $2,500,000

Layer 100% Limit $25,000,000 $10,000,000

Attachment $5,000,000 $15,000,000

Exposure (TIV) $30,000,000 $25,000,000

Company Share 20% 25%

Layer 100% Premium $250,000 $160,000

Rate per $100 TIV $0.8333 $0.6400

Nominal Rate Change -23%

Nominal Rate Change

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$25,000,000. So, the ROL is $10,000 per million. Similarly we can derive an ROL of $16,000 per

million for the renewal policy. The ROL statistic thus indicates an improvement of 60%.

Raising the attachment reduces loss in many scenarios and this also points towards an

improvement in the real rate level. Taken together, the reduction in the limit and the increase in the

attachment would lead us to conclude the real rate change is positive, not negative.

2.3.2 Location Schedule Differences

A location schedule shows the location and Total Insured Value (TIV) of covered structures.

Separate TIVs are shown for Structures, Contents, Business Interruption (BI), Extra Expense, and

possibly other coverages. The location schedule should also have age of structure, construction type,

occupancy class, protection rating, and other information about each location. The location

schedules for our fictitious account are shown in Table 2.3.2.1.

Table 2.3.2.1

Location Schedule Expiring Policy

Loc

Number Description

Year

Built Street Address City State ZIP TIV

1 Co HQ 2005 12 Shady Lane Pleasantville AA 12345 $3,000,000

2 Warehouse 1977 Industrial Park Center East Town AA 12222 $8,000,000

3 Factory 1995 22 Fast Lane Grime AA 12288 $7,000,000

4 Retail Store A 2001 Harbor St Marina ShoreHarbor AA 10225 $3,000,000

5 Retail Store B 1998 Pier 7 Lighthouse AA 10245 $3,000,000

6 Retail Store C 1982 Beach Lane Landing Cape Shark AA 10255 $3,000,000

7 Retail Store D 1999 Dock 15 Fishtown BB 31288 $3,000,000

7Total $30,000,000

Location Schedule Renewal Policy

Loc

Number Description

Year

Built Street Address City State ZIP TIV

1 Co HQ 2005 12 Shady Lane Pleasantville AA 12345 $3,000,000

3 Factory 1995 22 Fast Lane Grime AA 12288 $7,000,000

4 Retail Store C 1982 Beach Lane Landing Cape Shark AA 10255 $3,000,000

5 Retail Store D 1999 Dock 15 Fishtown BB 31288 $3,000,000

8 Retail Store E 1993 13 Canal Street SandyShore AA 12255 $3,000,000

9 Retail Store F 2003 81 Peninsula Drive WaveCrest BB 31224 $3,000,000

10 Retail Store G 1992 Seashore Mall Seashore BB 31288 $3,000,000

7Total $25,000,000

Location Schedules

Wayne's Widgets

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For convenience and simplicity, we have summarized the TIVs to a single number and omitted

the construction code, occupancy class, and other such information. For any real application, such

information could have a material effect on the answer and should not be omitted. In comparing

renewal and expiring schedules in our example, we note the insured removed the warehouse and two

of its retail stores from the program, and replaced them with three other stores. We also observe

that more of the TIV is concentrated in state BB for the renewal than it was for the expiring policy.

The impact of these mix changes is not knowable in advance. However, the Nominal Rate Change

will misconstrue as real rate change any rate movement that might arise from such location schedule

differences.

2.4 Technical Premium Based on Cat Loss Simulation Model Statistics

As previously outlined, our proposed solution entails quantifying the impact that coverage layer

and location schedule differences have on Technical Premium rates. In our application, a Technical

Premium denotes an indicated premium computed by machine algorithm without schedule rating or

other judgment modification. We will assume the Technical Premium includes provision for risk-

loaded loss. We will also suppose that it does not reflect any minimum premium or minimum rate

on line constraints.

Our proposal is to compute Technical Premiums using results from a cat loss simulation model.

The pure loss provision in the premium will be equal to the Average Annual Loss (AAL) from the

model and the risk load could be based on any of a variety of risk metrics generated by the model.

These risk metrics include the Variance, Standard Deviation, Probable Maximum Loss, and Tail

Conditional Expectation. We will neither define all these metrics nor discuss their particular

advantages or disadvantages here.

Instead, in order to demonstrate our proposal, we will focus on

one widely accepted risk metric, the Probable Maximum Loss (PML),

and compute our risk load as

5% of the PML. Including a loading factor of 1.50 for expenses, we arrive at the following

illustrative Technical Premium formula:

(

)

PMLAAL

•

= 05.5.1Premium Technical

(2.4.1)

See Robbin and DeCouto [8] for one discussion of various risk metrics.

In the sense that we are using it, the PML is associated with a return period. For example, the 100-year PML is the size

of cat loss that on average occurs no more frequently than 100 years.

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2.5 Cat Loss Simulation Model Runs

As previously outlined, our proposed solution entails quantifying the impact that coverage layer

and location schedule differences have on Technical Premium rates. To avoid confusing a change in

share with a change in the 100% coverage layer, we will refer to a difference in the 100% layer of

coverage as a difference in Coverage Structure. In our example, there is a Coverage Structure change

from the 25m x 5m layer to the 10m x 15m layer.

We will also decompose location schedule differences into Exposure Magnitude and Location

Mix components. In our example, there is an Exposure Magnitude movement in the TIV reduction

from $30m to $25m. There are also Location Mix changes in the types of structures covered and in

the distribution of the structures between states and within states.

We will separately quantify the effects of Coverage Structure and Location Mix differences. One

could estimate the combined impact of changes in Coverage Structure and Location Mix with one

overall factor. However, it is useful to have a breakdown of their separate effects. Especially when

these adjustments point in different directions, it may be somewhat unconvincing to provide a single

number summary without showing the offsetting contributions made by Coverage Structure and

Location Mix differences.

To implement this “separate effects” approach, we will make a series of runs with our Cat Loss

Simulation model. We will start with the Expiring Location Schedule and Expiring Coverage

Structure, then change the location schedule and finally change the 100% coverage layer. Thus, the

sequence of cat runs we are proposing is as shown in Table 2.5.1.

It is important to emphasize we are using only one version of a cat loss simulation model in

evaluating the impact of these changes. Typically we would employ the most recent version of the

model as it incorporates the latest knowledge and advances in methodology. To switch models

would be analogous to switching measuring sticks between two measurements: we would be unsure

how much of any difference was due to a real difference and how much was due to the switch in our

measuring stick.

For a limited excess layer, a uniform change in the values of all locations does not necessarily lead to the same

proportionate change in the 100% layer loss.

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2.6 Adjustment Factor Formulas

To translate the 100% Layer Technical Premiums from the three cat runs listed in Table 2.5.1

into Technical Rates, we divide by the appropriate TIVs. We will then define Location Mix and

Coverage Structure Adjustment Factors (MXAF and CSAF respectively) by taking the following

ratios of Technical Rates:

Rate based on Renewal Location Schedule and Expiring 100% Coverage Layer

Rate based on Expiring Location Schedule and Expiring 100% Coverage Layer

MXAF =

(2.6.1)

Rate based on Renewal Location Schedule and 100% Renewal Coverage Layer

Rate based on Renewal Location Schedule and 100% Expiring Coverage Layer

CSAF =

(2.6.2)

The derivation of adjustment factors for our sample account is shown in Table 2.6.1. Note under

our definitions the Location Mix Adjustment Factor will be unity if there is no change in the

location schedule and the Coverage Structure Adjustment Factor will be unity if there is no change

in the 100% layer of coverage. An adjustment factor value above unity means that some of the

Nominal Renewal Rate increase is absorbed in covering the impact of the associated location

schedule or coverage layer change. See Appendix A for more compact versions of these formulas.

Table 2.5.1

Cat Runs for Renewal Rate Change Monitoring

Number Location

Schedule

100% Coverage

Layer

Cat Model

1. Expiring Expiring Current

2. Renewal Expiring Current

3. Renewal Renewal Current

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2.7 Renewal Rate Change Formula

We are now finally ready to calculate the Renewal Rate Change (ΔR) by netting the impacts of the

Location Mix and Coverage Structure changes from the Nominal Renewal Rate Change (ΔN).

The

formula is:

(1 )

XAF CSAF

Δ= −

⋅

(2.7.1)

Here ΔN is the Nominal Rate Change, MXAF is the Location Mix Adjustment Factor, and CSAF

is the Coverage Structure Adjustment Factor.

In Table 2.7.1, we apply this formula to Wayne’s Widgets. Table 2.7.1 shows how the Nominal

Renewal Rate decrease in our example has been transformed into a sizeable (adjusted) Renewal Rate

increase. We think this more accurately represents the real rate change.

An alternative terminology is to call ΔR the Effective Renewal Rate Change and refer to ΔN as the Renewal Rate

Change. We choose to call ΔR the Renewal Rate Change and refer to ΔN as the Nominal Renewal Rate Change.

Table 2.6.1

Expiring Exposure

irin

Layer

Renewal Exposure

irin

Layer

Renewal Exposure

Renewal Layer

Coverage $5m p/o $25m x $5m $5m p/o $25m x $5m $2.5m p/o $10m x $15m

Exposure (TIV) $30,000,000 $25,000,000 $25,000,000

Company Share 20% 20% 25%

100% AAL $50,000 $40,000 $15,000

100% PML $15,000,000 $11,000,000 $5,000,000

Technical Premium $300,000 $225,000 $97,500

Technical Rate $10.00 $9.00 $3.90

Adjustment Factor 0.900 0.433

Technical Premiums, Technical Rates and Adjustment Factors

Wayne's Widgets

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2.8 Premium Reconciliation

It is useful to reconcile the Expiring and Renewal Premiums. The reconciliation proceeds

mathematically as follows:

=⋅⋅ ⋅ ⋅ ⋅+Δ(1 )

REN REN

REN EXP

EXP EXP

sTIV

P P MXAF CSAF R

sTIV

(2.8.1)

In words, to go from the Expiring Premium to the Renewal Premium, we need to adjust for

changes in Share, Exposure Magnitude, Exposure (location) Mix, Coverage Structure (100% layer),

and then reflect the true Renewal Rate Change.

For our example account we get:

.25 25

40, 000 50, 000 .90 .433 (1.97)

.20 30

=⋅⋅⋅⋅⋅

(2.8.2)

From this perspective, we see the overall premium change has been split between a real rate level

change component and other components that are not counted as real rate change contributors.

These non-rate change factors include volume scaling factors such as change in Share and change in

Exposure Magnitude. They also include rate movement factors due to Location Mix and Coverage

Structure changes.

Table 2.7.1

Wayne's Widgets

Expiring Renewal

Premium $50,000 $40,000

Coverage $5m p/o $25m x $5m $2.5m p/o $10m x $15m

Exposure (TIV) $30,000,000 $25,000,000

Company Share 20% 25%

Layer 100% Premium $250,000 $160,000

Rate per $1000 TIV $8.33 $6.40

Nominal Rate Change -23%

Location Mix Adjustment Factor (MXAF) 0.9000

Coverage Structure Adjustment Factor (CSAF) 0.4333

Renewal Rate Change 97%

Renewal Rate Change

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2.9 Notional Expiring Premium

We can also infer what the Expiring Premium would have been if it were based on the same

location schedule and layer of coverage as the Renewal policy. We will call this the Notional

Expiring Premium and denote it as P

NXP.

Following the definition, it is derived by starting with the

Expiring Premium and adjusting for Share, Exposure Magnitude, Location Mix, and Coverage

Structure changes. It is not hard to see this is equivalent to the Renewal Premium net of the Renewal

Rate Change.

(1 )

REN REN REN

NXP EXP

EXP EXP

TIV s P

P P MXAF CSAF

TIV s R

=⋅ ⋅⋅ ⋅ =

(2.9.1)

For our Wayne’s Widgets example, we find that P

NXP

=40,000/ (1.97) = $20,312. This is far less

than the actual Expiring Premium of $50,000.

2.10 Account Renewal Rate Change Summary

To summarize our derivation of Renewal Rate Change for an account, we first put the rates on a

100% basis to eliminate any distortion due to change in share. Then we adjusted the Nominal Rate

Change for Location Mix and Coverage Structure changes. The adjustments were derived by using a

Cat Loss simulation model to compute, in sequence, the impact such changes would have on

Technical Premium rates. We assumed the relationship between Technical Premiums rates should

apply to the charged premium rates. So, for example, if changing the 100% coverage layer moved

the Technical Premium Rate by 10%, then we assumed 10 points of any Nominal Rate Increase

would be attributable to the difference in Coverage Structure. The final Renewal Rate Change is the

Nominal Rate Change net of these Location Mix and Coverage Structure adjustment factors. We

have also seen how the Renewal Rate Change thus defined can be reconciled with the absolute

change in premium.

2.11 Alternative Renewal Rate Change Estimates

Once we go beyond the simple calculation of Nominal Renewal Rate Change and attempt to

reflect the impact of differences between the Renewal and Expiring policies, we introduce questions

about how to evaluate the rate impact of such differences. So, while all actuaries should arrive at the

Another name for what we call the Notional Expiring Premium is the Renewal Premium at Expiring Rates.

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same Nominal Rate Change for an account, they might arrive at alternative estimates of the Renewal

Rate Change. Such divergent estimates could arise from the use of alternative Cat Simulation Models

and Technical Premium formulas.

On working excess layers, we have found that most models and formulas produce percentage

adjustments often within a few points of one another, and seldom more than five to ten points

apart. In contrast, on very high excess layers, we may see significantly different results when

different models and formulas are used. As Cat Simulation Models improve over time, we would

expect the range of differences to become narrower and we would also expect major differences to

be found on smaller sets of accounts.

A key point is that all actuarially sound models and formulas

should produce consistent results, even for high excess layers. For example, if the 100% layer limit is

reduced and all else remains unchanged, the Coverage Structure Adjustment Factor will always be

less than unity since the reduced coverage and reduced risk imply a reduced rate.

While it is disheartening not to have an indisputable exact answer, the use of estimates is

common in many aspects of current actuarial practice. Part of the art inherent in actuarial science is

in selecting appropriate parameters, formulas, and models in order to derive a reasonable range of

estimates and then in selecting a final pick within that range. Such is the case in the procedure we are

proposing for estimating the real rate change.

3. PORTFOLIO AVERAGE RATE CHANGE

Once we have the Renewal Rate Change for each policy, the question then is how to aggregate

results to get the portfolio average Renewal Rate Change. It is a common practice to simply take a

weighted average of the rate changes, where Renewal Premiums are used as weights. However, as we

will demonstrate, this practice leads to an overly optimistic estimate of rate change. The bias goes

one way: rate level improvements are not as substantial as indicated and rate level decreases are

worse than indicated. Weights based on Expiring Premiums are better, but they give an out-of date

and potentially distorted picture. We will present an alternative in which the weights are based on

the Notional Expiring Premiums defined in Chapter 2. We will walk through a series of hypothetical

This problem is not unique to Cat-Exposed Property insurance. Using increased limits factors, one could derive

coverage structure adjustments for casualty accounts and different sets of factors would lead to different rate change

estimates.

Other implementation and data issues are discussed in Appendix B.

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scenarios comparing the weighted average rate changes that result from these weights.

3.1 Renewal Premium Weighting Bias

Suppose we have a portfolio of four risks. Assume for the base case that the Exposure

Magnitude stays constant, as does the Share, Location Mix, and Coverage Structure. Only the rates

change as shown in Table 3.1.1.

In the Base Case we have rate changes by account that end up generating the same overall

premium for the expiring and renewal portfolios. Despite the fact that we are getting the same exact

money to cover the same exact exposures overall, the weighting of rate changes on Renewal

Premiums incorrectly indicates there is a sizeable overall rate increase. This happens because the

Renewal Premium weighting algorithm inherently gives undue emphasis to those policies that had

rate increases while deemphasizing those that had decreases. In other words, it overcounts rate

increases and undercounts rate decreases. As a consequence, the result from weighting with Renewal

Premium is biased. It paints an overly optimistic picture of any actual portfolio level increase and

masks the true extent of any portfolio level decrease.

Table 3.1.1

Expiring Renewal

Ris

Prem

(000) Share

TIV

mill

Prem

(000) Share

TIV

mill

Nominal

Rate

Chan

Rate

Chan

NXP

Prem

(000)

A $200 25% 100 $280 25% 100 40% 40% $200

B $200 25% 100 $120 25% 100 -40% -40% $200

C $50 25% 200 $70 25% 200 40% 40% $50

D $50 25% 50 $30 25% 50 -40% -40% $50

Total $500 25% $500 25% $500

Weights

Renewal Prem 16% 16%

Expiring Prem 0% 0%

NXP Prem 0% 0%

Portfolio Weighted Average Rate Change

Base Case

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3.2 Expiring Premium Weighting Inaccuracy

In the Base Case, there was no difference between the actual Expiring Premium and the Notional

Expiring Premium. Now we will modify our example by increasing our share on the accounts that

had rate increases. These share increases boost Renewal Premiums, but they do not impact the

individual account rate increases. This “Shares Up on Accounts with Rate Increases” scenario is

shown in Table 3.2.1.

Since more business is from accounts that had increases, we should expect to see a rate increase

for the overall portfolio. However, because changing shares has no impact on the Expiring

Premium, a weighting on Expiring Premium is the same as in the Base Case and yields no portfolio

rate increase.

Table 3.2.1

Expiring Renewal

Ris

Prem

(000) Share

TIV

mill

Prem

(000) Share

TIV

mill

Nominal

Rate

Chan

Rate

Chan

NXP

Prem

(000)

A $200 25% 100 $560 50% 100 40% 40% $400

B $200 25% 100 $120 25% 100 -40% -40% $200

C $50 25% 200 $140 50% 200 40% 40% $100

D $50 25% 50 $30 25% 50 -40% -40% $50

Total $500 25% $850 43% $750

Weights

Renewal Prem 26% 26%

Expiring Prem 0% 0%

NXP Prem 13% 13%

Portfolio Weighted Average Rate Change

Shares Up on Accounts with Rate Increases

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Next, in Table 3.2.2, we look at an example in which the TIVs increase only for the accounts that

had rate increases.

In Table 3.2.2, we assume the TIV changes only impact the Exposure Magnitudes. Therefore

these exposure differences do not alter the rate change for any account, yet they do change the

relative weight of an account within the portfolio. With more weight now given to accounts that had

rate increases, we should now expect to see an overall rate increase. This overall rate increase does

appear when Notional Expiring Premium weights are used. In contrast, the weighted average as

computed with Expiring Premium weights incorrectly shows no overall rate increase. This is not

surprising. By definition, the Expiring Premium weights cannot respond to this TIV-driven change

in portfolio mix.

Finally, in Table 3.2.3 we look at a scenario where accounts have Renewal Rate Changes that

offset one another in such a way that the total Notional Expiring Premium equals the total Renewal

Premium. This means there has been no overall rate change.

Table 3.2.2

Expiring Renewal

Ris

Prem

(000) Share

TIV

mill

Prem

(000) Share

TIV

mill

Nominal

Rate

Chan

Rate

Chan

NXP

Prem

(000)

A $200 25% 100 $504 25% 180 40% 40% $360

B $200 25% 100 $120 25% 100 -40% -40% $200

C $50 25% 200 $126 25% 360 40% 40% $90

D $50 25% 50 $30 25% 50 -40% -40% $50

Total $500 25% $780 25% $700

Weights

Renewal Prem 25% 25%

Expiring Prem 0% 0%

NXP Prem 11% 11%

Portfolio Weighted Average Rate Change

TIVs Up on Accounts with Rate Increases

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When we weight on Notional Expiring Premiums, we correctly end up with that 0% overall

portfolio rate change. Weighting with either Expiring or Renewal Premium incorrectly overstates the

rate increase. Further, this scenario demonstrates that a weighted average taken with Notional

Expiring Premiums does not have to fall between the corresponding weighted averages computed

with Renewal Premiums and Expiring Premiums respectively.

3.3 Notional Expiring Premium Weighting

We have seen through concrete examples that Notional Expiring Premium weights are

demonstrably superior to either Renewal Premium or actual Expiring Premium weights. They do not

overcount rate increases and undercount rate decreases like Renewal Premiums do. They do not

ignore changes in the weight of an account within the portfolio as actual Expiring Premiums do.

There is an intuitive appeal to derive weights by taking the Renewal Premiums and backing out the

Renewal Rate Changes. The resulting weights will be appropriately sensitive to relative importance

of an account within the current portfolio. As noted in Equation (2.9.1), the resulting premiums are

algebraically equivalent to the Notional Expiring Premiums. To summarize, our proposal is to

compute the overall portfolio rate change (ΔR(TOT)) using Equation (3.3.1).

Table 3.2.3

Expiring Renewal

Ris

Prem

(000) Share

TIV

mill

Prem

(000) Share

TIV

mill

Nominal

Rate

Chan

Rate

Chan

NXP

Prem

(000)

A $200 25% 100 $280 25% 100 40% 75% $160

B $200 25% 100 $120 25% 100 -40% -50% $240

C $50 25% 200 $70 25% 200 40% 17% $60

D $50 25% 50 $30 25% 50 -40% -25% $40

Total $500 25% $500 25% $500

Weights

Renewal Prem 16% 31%

Expiring Prem 0% 9%

NXP Prem -5% 0%

Portfolio Weighted Average Rate Change

Offsetting Coverage Strucuture Changes

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()

() ()

1()

()

1()

REN

NXP

REN

NXP

PiRi

RTOT

⋅Δ

+Δ

Δ= =

+Δ

∑

(3.3.1)

3.4 Weighted Harmonic Average Interpretation

We have felt it intuitively appealing to present our procedure as the computation of an arithmetic

weighted average of rate changes and to focus our attention on arriving at the proper weights. This

allowed us to directly demonstrate the error in using Renewal Premium weights when computing the

arithmetic weighted average rate change and to further argue that Notional Expiring Weights should

be used instead. Another approach is to retain the Renewal Premium Weights, but to use a different

type of average, called the harmonic average. This interpretation is mathematically presented in the

derivation shown in Equation (3.4.1).

()

1()

()

1()

(1 ( ))

()

1()

() ()

1() 1()

REN

REN REN

RTOT

Pi Pi

Ri Ri

⋅Δ

+Δ

Δ=

+Δ

⋅+Δ

+Δ

=−=−

+Δ +Δ

∑

∑∑

(3.4.1)

The latter ratio of sums in Equation (3.4.1) is the Weighted Harmonic Average of the Renewal

Rate Change Factors with weights based on Renewal Premiums. Some actuaries may be more

comfortable with this interpretation.

We may also use a mathematical argument based on harmonic averages to buttresses our intuitive

reasoning about why the Renewal Premium Weighted Average of Rate Changes is biased upward. It

can be easily shown that the Weighted Harmonic Average is always less than or equal to the

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Weighted Arithmetic Average. The inequality in Equation (3.4.2) holds.

()

() 1

()

1()

()(1 ()) () ()

() ()

REN

REN REN

RTOT

Pi Ri PiRi

Pi Pi

Δ= −

+Δ

⋅+Δ ⋅Δ

≤−=

∑

∑∑

(3.4.2)

It follows from Equation (3.4.2) that weighting Renewal Rate Changes with Renewal Premium

weights produces an overly optimistic result.

4. CONCLUSION

We have presented solutions to the two major actuarial problems that can cause the Renewal

Rate Change metric to be misleading on Cat-Exposed Excess Property business. Our first problem

was to figure out how to adjust the nominal rate changes so as to properly account for differences in

the layer structure and location mix of the renewal and expiring policies. Our solution was to derive

Technical Premiums with the latest cat loss simulation model, first changing the locations and then

changing the coverage layer. From these we computed Technical Premium rates and then derived

adjustment factors by taking appropriate ratios between these rates.

Our second major problem was to find which weights to use when computing the overall

portfolio rate change as a weighted average of the individual account rate changes. We found there is

an optimistic bias in the results when Renewal Premium weights are used. We also demonstrated

that weighting on Expiring Premiums is flawed, as it is insensitive to changes in Share and to

portfolio level TIV mix changes. Our solution was to weight on premiums adjusted to be at the

expiring rate level, but based on the renewal location schedule, share, and layer structure. We called

these the Notional Expiring Premiums. We have seen these are equivalent to Renewal Premiums

Net of Rate Change. We have also shown our approach is equivalent to taking the Weighted

Harmonic Average of Rate Changes while using the Renewal Premiums as weights.

We hope some practical benefit will come from our efforts to resolve actuarial questions about

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how to calculate Renewal Rate change statistics on Cat-Exposed Excess Property business.

Accounts in this business have been subject to large rate swings, the magnitude of which may have

been obscured by changes in location schedules and coverage layers and by actuarially incorrect ways

of aggregating individual account rate changes. It is our intent to foster development of a uniform

and actuarially valid approach to computing the Renewal Rate change on this business so as to

increase the accuracy and public credibility of reported rate change statistics. While the use of

different Cat Simulation models may lead to different Renewal Rate Change estimates, a reasonable

range of estimated effects due to changes in Coverage Structure and Location Mix is better than

ignoring the effects of such changes altogether. We would encourage others to write on these and

other actuarial issues inherent in rate monitoring statistics so that a more extensive literature on price

monitors develops over time.

Acknowledgments

The author wishes to acknowledge Michael Coca, whose calm, incisive comments helped guide development of the

methods presented in this paper. The author also acknowledges the contributions of Neil Bodoff and Yves Provencher.

Much of the inspiration for the paper and many of the concepts developed in the paper emerged from discussions with

Neil and Yves. Finally, the author wishes to thank David Clark and Kris Bagchi, among others, for reviewing preliminary

drafts and providing comments that helped to substantially improve the paper.

Disclaimers

The opinions expressed are solely those of the author and are not presented as a statement of the position of

Endurance American Insurance Company or any parent or affiliated company. The reader is warned that the author

assumes no liability for any damages that may result directly or indirectly from use of the methods presented in this

paper.

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Appendix A – Notation and Formulas

In Table A.1, we define our basic notation:

Table A.1

Basic Notation

P Premium at Company Share

P100 Premium at 100%

K Company Limit

K100 100% Layer Limit

T Total Insured Value

S Company Share

N Nominal Rate

For the i

policy, Equations (A.1), )A.2), (A.3), and (A.4) hold:

s(i) = K(i)/K100(i)

(A.1)

P(i) = s(i) ⋅P100(i)

(A.2)

N(i)= P100(i)/T(i)

(A.3)

P(i) = N(i)⋅s(i)⋅T(i)

(A.4)

Use subscripts EXP and REN to denote whether a variable is for the expiration or renewal policy

respectively. Define the Nominal Renewal Rate Change for the i

policy as:

()

REN EXP i

EXP

NiN

−

Δ=

(A.5)

We will suppress the policy “(i)” notation to simplify the formulas and derivations that follow

unless it is needed for clarity.

To make mix and coverage change adjustments, we define the following 100% Technical

Premium Rates:

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Table A.2

Technical Premium Rate 100% Layer Exposures

REN,REN

Renewal Renewal

REN,EXP

Renewal Expiring

EXP,EXP

Expiring Expiring

The Location Mix and Coverage Structure adjustment factors are defined as follows:

NEXP

XP EXP

MXAF

(A.6)

NREN

REN EXP

CSAF

(A.7)

The Renewal Rate Change, ΔR, is given as:

(

)

−

•

=Δ

CSAFMXAF

(A.8)

In words, to get the Renewal Rate Change we start with the Nominal Rate Change and net out

the Mix Change and Coverage Structure Change adjustments. Next we define the Notional Expiring

Premium by taking the Expiring Premium and adjusting for changes in Exposure Magnitude, Share,

Location Mix, and Coverage Structure.

REN REN

NXP EXP

EXP EXP

TIV s

PP MXAFCSAF

TIV s

=⋅ ⋅⋅ ⋅

(A.9)

It follows from these definitions that:

(1 )

REN

NXP

(A.10)

Weighting with the Notional Expiring Premium, we define the overall portfolio Renewal Rate

Change as:

() ()

()

NXP

PiRi

RTOT

⋅Δ

Δ=

∑

(A.11)

The overall portfolio Renewal Rate Change can be equivalently express as the Weighted

Harmonic Average of the Renewal Rate Change Factors using Renewal Premium weights:

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()

() 1

()

1()

REN

RTOT

Δ= −

+Δ

∑

(A.12)

To see this consider that

()

() ()

1()

()

1()

()

((1 ( )) 1)

()

1()

() ()

1() 1()

REN

NXP

REN

NXP

REN

REN REN

PiRi

Pi Pi

Ri Ri

⋅Δ

+Δ

⋅+Δ −

+Δ

==−

+Δ +Δ

∑

∑∑

(A.13)

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APPENDIX B - DATA AND MODELING ISSUES

The reliability of the Renewal Rate Change statistic depends heavily on having accurate data. The

database fields for the calculations we have outlined include premium, coverage layer (attachment,

company limit, and layer limit) parameters, and location schedule information for “matched”

renewal and expiring policies. Most experienced actuaries will have their own ways of checking the

data for reasonability and flagging questionable data items. Our purpose here is to point out a few

data issues beyond the usual ones. We also want to mention a few of issues that might arise is using

Cat Loss simulation models in the rate monitoring process.

B.1 Matching

The matching is often more difficult than it might first appear. Sometime an account will have a

check mark in a renewal status field, yet its expiring policy may be missing. The account name and

number may have changed, and there may be no easy automated way to pair up wayward accounts.

Coverages must also be matched, as a renewal and expiring account may sometimes have different

coverages. For example, Business Interruption (BI) may be covered on the expiring policy but not

the renewal. In that case, the premium allocated for BI on the expiring policy should be removed.

B.2 Annualized Premium

Another issue is that the expiring and renewal policy may have different durations. One way to

handle this is to compute rates with annualized premiums and exposures. Though it is an

unnecessary duplication for many accounts, it is worth having a separate annualized premium field.

This field can also be used to adjust for midterm endorsements. While annualized premiums are

recommended for use in computing rates and rate changes, the actual and not the annualized

premiums should be used when weighting rate changes to obtain the portfolio average rate change.

B.3 Aging of Statistics

The actuary may also find it instructive to look at how the monitor statistics “age” over time. In a

rush to get the latest pricing statistics, management may insist on getting monitor statistics produced

as quickly as possible after a period closes. Yet this may miss many pricing changes, new business

bookings, and nonrenewals that come in later. There may also be a spate of endorsements,

extensions, cancellations, or back-outs of endorsements. Producing refreshed versions of monitor

statistics may reveal a characteristic pattern where, for example, a 10% increase on first look declines

to 2% over the next two months as booking is “trued up” in the system. Studying a series of

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refreshed monitors may also turn up specific offices, lines of business, or underwriters with stellar

initial statistics that evaporate over time.

B.4 Quality of Location Schedule Data

The data on location schedules is usually checked by cat modelers during the cat modeling

process. However, as with any data from whatever source, the actuary would be well advised to

check it. The cat modeler will typically do statistical data checks to tell, for example, that 90% of the

locations are geo-coded to the zip code level or better. With large location schedules, having another

set of human eyes review the data can often turn up potential anomalies the machine may miss. For

example, duplicate entry errors may be masked by street addresses with names or numbers that

differ slightly. The actuary should also check the TIV data. Locations with small TIVs should not be

ignored. The TIVs for some of those locations may have been incorrectly entered in units of

thousands. A comparison should also be done of the new schedule against the prior one. If both

schedules have identical location addresses and values, the actuary should request an update. If some

locations in the new schedule are missing street addresses, the actuary may find them in the prior

schedule. Data fields on Construction, Occupancy, and other characteristics are also shown on the

location schedule. Having accurate entries will usually have a material impact on the quality of

answers coming out of any cat loss simulation model. It is thus worthwhile to ask the underwriter to

contact the broker for more definitive information if for example half the construction types are

listed as “unknown.” Overall, the key to achieving better data quality is to work cooperatively with

underwriters and cat modelers.

B.5 Implementation Issues

Before attempting to implement the procedures suggested in this paper, the actuary would be

well advised to run sensitivity tests and sample computations for a large set of renewal accounts.

When evaluated on actual accounts, a Technical Premium formula that is superior to another in

theory may turn out to be inferior in practice. A robust formula that always yields sensible results

that move in the right direction may be preferable to one that for unknown reasons produces bizarre

results on a small percentage of accounts even while performing in stellar fashion on the others.

Part of the problem is that current cat loss simulation models are not quite up to the task we have

set for them. They were designed for portfolio analysis and they do a reasonably credible job costing

reinsurance treaties. However, anomalous results can sometimes crop up for individual insureds,

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especially for those with a small number of locations. With currently available models, portfolio

impact statistics

on individual accounts also tend to misbehave more than comparable stand-alone

statistics. Advances in cat loss simulation modeling should mitigate problems of this sort over time.

The portfolio impact version of a statistic is the difference between its value on the portfolio after adding the

additional insured and its value on the initial portfolio.

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REFERENCES

[1] Boor, Joseph, “The Impact of the Insurance Economic Cycle on Insurance Pricing,” CAS Study Note, 1998.

[2] Burger, George, Beth E. Fitzgerald, Jonathon White, and Patrick B. Woods, “Incorporating a Hurricane Model

into Property Ratemaking,” Casualty Actuarial Society Forum, 1996, Winter, 129-190.

[3] Expert Commentary, “First Quarter Market Survey,” The Council of Insurance Agents and Brokers, May 2006.

[4] Kundrot, Jason, “Umbrella and Excess Liability – Understanding and Quantifying Price Movement,” CARe

Seminar Presentation, June 2006.

[5] Nyce, Chris, “Price Governance I: Price Monitoring for Standard and Middle Market Commercial Lines,” CAS

Ratemaking Seminar Presentation, March, 2007.

[6] Palisi, Gerard, “Umbrella Price Monitoring,” CARe Seminar Presentation, June 2006 .

[7] Robbin, Ira, and Jesse DeCouto, “Coherent Capital for ROE Pricing Calculations,” Casualty Actuarial Society

Forum, 2005, Spring, 255-296

[8] Vaughn, Trent R., “Commercial Lines Price Monitoring,” Casualty Actuarial Society Forum, 2004, Fall, 497-519

[9] Wang, Shaun, and Robert Faber, “Enterprise Risk Management for Property-Casualty Insurance Companies,” CAS

and SOA Jointly Sponsored Research Project, August 2006.

Abbreviations and Notations

Cat, Catastrophe

ROL, Rate on Line

PML, Probable Maximum Loss

TIV, Total Insured Value

AAL, Average Annual Loss

NXP, Notional Expiring Premium

Biography of the Author

Ira Robbin is currently Senior Vice-President and Chief Pricing Actuary for the US and London at Endurance

Worldwide Insurance. Ira received a Bachelor’s Degree in Math from Michigan State University and a PhD in Math

from Rutgers University. Then he took an actuarial research position with the Insurance Company of North America

(INA) and has been working at insurance and reinsurance companies ever since. He has headed large risk property and

casualty pricing units, developed pricing algorithms, and produced a number of price monitors. He has authored several

Proceedings, Forum, and Study Note papers on a range of subjects, taught exam preparation classes, and made

numerous presentations at actuarial meetings.