The Financial Reporter - SOA...The Financial Reporter Published by the Life Insurance Company Financial Reporting Section of the Society of Actuaries 475 N. Martingale Road, Suite

The Financial ReporterThe Newsletter of the Life Insurance Company Financial Reporting Section

An International Financial Reporting Standards (IFRS)Phase II Discussion Paper Primerby Mark J. Freedman and Tara J.P. Hansen

December 2007 Issue No. 71

US GAAP and IFRS are aligning their con-cepts, principles and rules. This means U.S.insurance companies will almost certainlybe impacted by any accounting changes that takeplace whether they occur domestically or abroad.The only question is how quickly this will occur.

Recently, the SEC issued for public comment aproposal to accept from foreign private issuersfinancial statements prepared in accordance withIFRS without any reconciliation to US GAAP.The SEC has also issued a concept release toobtain information as to whether U.S. issuersshould have the option to prepare financial state-ments in accordance with IFRS. In addition, inAugust 2007, the FASB issued an invitation tocomment on the International AccountingStandards Board’s (IASB) Phase II DiscussionPaper (Discussion Paper), entitled “PreliminaryViews on Insurance Contracts,” in order to assesswhether there is a need for a project on account-ing for insurance contracts and whether or not towork with the IASB in a joint project.

The landscape of financial reporting here in the United States is changing rapidly, especiallywith the issuance of FASB Statements No. 157 and 159. Statement No. 157, Fair ValueMeasurements, applies to all existing pronounce-ments under GAAP that require (or permit) theuse of fair value. It also establishes a frameworkfor measuring fair value in GAAP, clarifies thedefinition of fair value within that framework,and expands disclosures about the use of fairvalue measurements. Statement No. 159, “TheFair Value Option for Financial Assets andFinancial Liabilities,” including an amendment

of Statement No. 115, “Accounting for CertainInvestments in Debt and Equity Securities,” per-mits entities to measure certain other items notincluded within the scope of Statement No. 157at fair value. The issuance of Statement No. 159by the FASB is a big step forward in requiringfair value reporting. Insurers looking to adoptStatement No. 159 for their insurance contractswill be faced with the challenge of determining

continued on page 3 >>

The Financial Reporter

Published by the Life Insurance CompanyFinancial Reporting Section of the Society of Actuaries

475 N. Martingale Road, Suite 600Schaumburg, Illinois 60173p: 847.706.3500f: 847.706.3599 w: www.soa.org

This newsletter is free to section members.Current-year issues are available from the communications department. Back issues of section newsletters have been placed in the SOAlibrary and on the SOA Web site (www.soa.org).Photocopies of back issues may be requested for anominal fee.

2006-2007 Section LeadershipHenry W. Siegel, ChairpersonJerry F. Enoch, Vice-ChairpersonCraig Reynolds, SecretaryMike Y. Leung, Treasurer(Annual Program Committee Rep.)Errol Cramer, BOG PartnerRod L. Bubke, Council MemberSusan T. Deakins, Council MemberJason A. Morton, Council MemberYiji S. Starr, Council MemberVincent Y.Y. Tsang, Council Member(Spring Program Committee Rep.)Kerry A. Krantz, Web Coordinator

Rick Browne, Newsletter EditorKPMG LLP303 East Wacker Drive • Chicago, IL • 60601p: 312.665.8511f: 312.275.8509e: [email protected]

Carol Marler, Associate Editore: [email protected]

Keith Terry, Associate Editore: [email protected]

Sam Phillips, Staff Editore: [email protected]

Julissa Sweeney, Graphic Designere: [email protected]

Mike Boot, Staff Partnere: [email protected]

Christy Cook, Project Support Specialiste: [email protected]

Facts and opinions contained herein are the soleresponsibility of the persons expressing themand shall not be attributed to the Society ofActuaries, its committees, the Life InsuranceCompany Financial Reporting Section or theemployers of the authors. We will promptly correct errors brought to our attention.

Copyright ©2007 Society of Actuaries.All rights reserved.Printed in the United States of America.

December 2007 Issue No. 71

1 An International Financial Reporting Standards (IFRS) Phase II Discussion PaperPrimerA summary of the provisions of the recently issued IASB Phase II Discussion Paper.

Mark J.Freedman and Tara J.P. Hansen

6 The Siren Call of Models—Beware of the RocksThoughts from the Section Chair.

Henry W. Siegel

8 Potential Implications of IASB’s Preliminary Views on Insurance ContractsWhat are some of the implications of the recently issued IASB Discussion Paper? This is one ofa series of articles by the American Academy of Actuaries’ Life Insurance Financial ReportingCommittee.

Leonard Reback

11 Risk Margins to the Non-Market Risks under FAS 157: Suggested ApproachA methodology for calculating risk margins related to policyholder behavior risks when meas-uring fair value under FAS 157.

Vadim Zinkovsky

17 A Critique of Fair Value as a Method for Valuing Insurance LiabilitiesA discussion of some of the challenges the insurance industry will face as we move towards fairvalue accounting for insurance products.

Darin Zimmerman

22 Embedded Value Reporting ReduxThe Academy’s Life Financial Reporting Committee announces the soon to be released practicenote on EV.

Tina Getachew

23 The Lowly Loss RatioThe mathematical underpinnings of loss ratios.

Paul Margus

32 What’s OutsideArticles appearing recently in other publications that may be of interest to financial reportingactuaries.

Carol Marler

What’s Inside

Financial Reporter | December 20072

the fair value of those insurance contract liabilitiesand are considering the principles in the DiscussionPaper.

Developing an understanding of the current direc-tion of IFRS for insurance products is imperative forU.S. accounting and actuarial practitioners. ThisIFRS Phase II Discussion Paper Primer has beendesigned to provide a summary of the key proposalsand issues that are facing U.S. insurers as this mate-rial evolves into authoritative guidance.

The Discussion Paper, issued by the IASB in May2007, contains the IASB’s preliminary views on thevarious recognition and measurement componentsconsidered in accounting for insurance contracts andidentifies issues that are still under consideration.The IASB has invited interested parties to commenton the Discussion Paper by November 16, 2007.This process will ultimately lead to an insurancestandard that will replace the current IFRS 4“Insurance Contracts.”

This article summarizes the main proposals in theDiscussion Paper, and in particular, emphasizes thoseissues where the views of the IASB are not uniform-ly accepted.

ObjectivesThe Discussion Paper reflects a principles-basedapproach with additional high-level and prescriptiveguidance. Insurance contracts are to be subject to thesame general principles as those of other financialservice entities. This approach seeks to ensure consis-tency of financial statements for insurance, assetmanagement and banking companies. The revisedplatform for insurers should lead to increased com-parability of financial statements, better identifica-tion of key value drivers, and enhanced share valuesdue to improved transparency.

Measurement ModelThe proposed measurement model for insurance lia-bilities rests on three building blocks:

• Explicit, unbiased, market-consistent, probability-weighted, and current estimates of expected futurecash flows;

• Discount rates consistent with prices observable inthe market place; and

• Explicit and unbiased estimates of the margin thatmarket participants require for bearing risk (riskmargin) and for providing any services (servicemargin).

The expected future cash flowsshould be explicit, current andconsistent with observablemarket prices, and excludeentity-specific cash flows.There is a subtle, but impor-tant, difference between mar-ket-consistent and entity-spe-cific assumptions. Claimsassumptions, for example,would presumably be the samefor both bases, but expenseassumptions, for instance, maynot be. This is true because the claims experiencerelates to the block of business and would be trans-ferred with the block of business upon sale, butany entity-specific expense savings that the currententity enjoys may not be transferred with theblock.

The IASB believes that discounting should beapplied to all liabilities in an effort to enhancecomparability of financial statements. Many U.S.and Japanese insurers believe that most non-lifeinsurance liabilities should not be discounted andthey have already expressed this view with theIASB.

Risk margins convey a level of uncertainty associatedwith future cash flows. These margins should bemarket-consistent and reassessed at each reportingdate. The IASB has given high-level guidance withrespect to risk margins, but has left the details relat-ed to their development to the insurance industry.One example of the high-level guidance is that oper-ational risk can only be provided for if it is relateddirectly to the liability itself. Acceptable approachescurrently appear to include cost of capital, per-centile, Tail Value at Risk, and multiple of standarddeviation.

Service margins represent what market participantsrequire for providing other services in addition tocollecting premiums and paying claims. These mar-gins should also be market-consistent. The invest-ment management function in variable (and pre-sumably other interest sensitive) contracts is anexample of a service for which an explicit servicemargin may be established under this new frame-work. This concept is generally not considered incurrent pricing and embedded value techniques usedby insurance companies.

Developing an understanding ofthe current direction of IFRS forinsurance products is imperativefor U.S. accounting and actuarialpractitioners.

Financial Reporter | December 2007 3

>> ... Financial Reporting Standards (IFRS) …

continued on page 4 >>

Mark J. Freedman, FSA,MAAA is a Principal atErnst & Young LLP inPhiladelphia, Pa. He can be reached at215.448.5012 or [email protected]

Tara J.P. Hansen, FSA,MAAA is a SeniorActuarial Advisor atErnst & Young LLP inNew York, N.Y. She can be reached at 212.773.2329 [email protected]

The primary implementation issue related to bothrisk and service margins is how to calibrate them.

Current Exit ValueThe IASB is proposing that an insurer measureinsurance liabilities at what it calls “current exitvalue.” This represents the amount another partywould reasonably expect to receive in an arm’s-lengthtransaction to accept all contractual rights and obli-gations of a liability. Under an exit value measure-ment framework, there are no requirements to breakeven at issue. Such a framework will prove challeng-ing in terms of determining an appropriate level ofrisk margin, since the calibration is a very subjectiveprocess under an exit value framework. In particular,there is not currently a secondary market to transferinsurance liabilities. While one could look to recentacquisitions and reinsurance arrangements, the spe-cific components rarely, if ever, become public infor-mation. However, these, combined with the retailmarket, could be taken into consideration in a hypo-thetical model.

A further point to note is that “current exit value”may or may not be equivalent to “fair value,” asdefined in the IASB’s discussion paper on fair valuemeasurement (to be used when fair value must beapplied under IFRS). This discussion paper on fairvalue measurement is almost a word-for-word copyof Statement No. 157, which is FASB’s version offair value that has already been adopted. At the timethis article is being written, the IASB has noted thatthere do not appear to be material differencesbetween these two concepts, but it intends to explorethis issue more thoroughly.

“Entry value” is another method of risk margin cali-bration covered in the Discussion Paper. Under the“entry value” framework, no gain or loss at issuewould arise, since risk margins would be calibratedto the premium received less acquisition expenses.The IASB has rejected this approach in favor of cur-rent exit value, even though the CFO Forum, a dis-cussion group made up of the Chief FinancialOfficers of 19 major European insurers, andGNAIE, a group made up of North American andJapanese insurance companies, lobbied for an entry-value type approach.

Own Credit StandingThe measurement of liabilities should include theeffects, if any, of the credit characteristics of the lia-bility (insurer’s credit standing). In the DiscussionPaper, the IASB wants the impact of this item to bedisclosed. This issue has been rather controversial tosome actuaries and accountants, although the IASBdoes not believe the impact will be large.

To be consistent with a fair value measurement ofother financial instruments, the IASB has said thecreditworthiness of the insurance contracts shouldbe reflected in the measurement of the liability. Theimplicit assumption is that the transfer is to a thirdparty of similar credit standing. This topic has gen-erated much discussion—many do not agree withthe rationale of reducing liabilities when a company’sfinancial condition deteriorates. However, in a well-regulated market with guaranty funds, it is notexpected that this will have a significant impact onthe level of reserves ultimately held under IFRSPhase II. There may be other offsets to this potentialreduction of liabilities that still need to be explored.

For example, if a cost of capital approach is used forderiving risk margins, an entity’s cost of capital willincrease as credit standing declines, implying a high-er risk margin and therefore higher liabilities. Also,actuarial models sometimes already reflect the effectson pricing of credit standing indirectly via lapserates, e.g., for life insurers.

Treatment of Future Renewal PremiumsThe Discussion Paper indicates that future premi-ums would be included in the liability calculationonly to the extent that either:

1. The insurer has an unconditional contractualobligation to accept premiums whose value is lessthan the value of the resulting additional benefitpayments; or

Financial Reporter | June 20074

>> ... Financial Reporting Standards (IFRS) …

2. They are required for the policyholder to contin-ue to receive guaranteed insurability at a price thatis contractually constrained.

This is a controversial issue with respect to NorthAmerican style universal life and flexible annuityproducts, since current pricing techniques include anassumption for future expected premiums. If futureexpected premiums cannot be taken into account,there may be a significant loss at issue in today’sproducts, largely due to heaped commission struc-tures.

Unit of AccountThe liability, including the determination of riskmargins, should be determined on a portfolio basis.“Portfolio basis” refers to insurance contracts that aresubject to similarly broad risks and managed togeth-er as a single portfolio. Determining risk margins ona portfolio basis means that one must exclude thebenefits of diversification between portfolios.

Deferred Acquisition Costs (DAC)Under the framework described by the DiscussionPaper, there will be no separate DAC asset to accountfor the investment the insurer makes in the customerrelationship. Acquisition costs are to be expensedwhen incurred, as they play no direct role in deter-mining current exit value.

Discretionary Participating Features andUniversal Life ContractsIn the IASB’s current view, policyholder participa-tion rights consisting of dividends and excess interestcredits for life insurers do not create a liability untilthe insurer has an unconditional obligation to poli-cyholders. A prior claim without an obligationshould not be recognized as a liability; the amountexpected to be paid in the future should be treated aspart of equity. In assessing whether an insurer has aconstructive obligation to pay dividends to partici-pating policyholders, the IASB will rely on itsConceptual Framework and IAS 37, “Provisions,Contingent Liabilities, and Contingent Assets.” Thiscould potentially be a large issue for U.S. participat-ing and interest sensitive life and annuity products,since current pricing takes into account expectedcredits to policyholders and not just guaranteed pay-ments.

SummaryThe Discussion Paper provides for significantchanges to financial reporting for insurance contractsand related activity under IFRS.

The anticipated timeline forIFRS Phase II for insurers is asfollows:

• An Exposure Draft is expectedto be issued in late 2008.

• A final standard is expected tobe adopted in late 2009.

• Implementation is projectedto become applicable for2011.

Many European companies have been pilot testing a“current exit” value type standard for several years;therefore, most of these companies will be prepared.

However, as discussed earlier in this article, theaccounting changes in Europe are having a dominoeffect in the United States. While several of thelarge U.S.-based insurance companies have been intune with the development of the proposed guid-ance, others appear to be largely asleep at the wheel,probably because they may not have appreciatedthe significance of US GAAP aligning with IFRS.The current events should be a wake-up call to U.S.insurers, as preparation for managing under a cur-rent exit value approach can take several years. Thecompetitive landscape is changing and action isrequired.

The current events should be awake-up call to U.S. insurers, aspreparation for managing undera current exit value approach can take several years.


>>... Financial Reporting Standards (IFRS) …

5

$

As actuaries, we are uniquely concerned withboth measuring the past and projecting thefuture. We look at the past in order to getinsight into the future, but we can’t assume that thepast will always continue into the future.

One of the most common misunderstandings aboutactuaries is that we forecast the future. In fact, weproject the future, we don’t forecast it. The problemis, sometimes we forget this.

History 101Many years ago I was involved in pricing the firstGICs. Not directly, but in an oversight capacity. Welooked at various scenarios—I think it was the firststeps in a stochastic model and I can still remembersome of the conversations. We noted early that ahump pattern (an increase in interest rates followedby a decrease) was good, so long as the decrease elim-inated capital losses when the GIC ended. We alsonoted that a straight increase in interest rates wasbad. Very bad.

We discussed this: what happens if interest rates goup to 12 percent, half a percent a year from when wepriced it? “That can never happen,” was theresponse—and it seemed reasonable. After all, thiswas around 1975 and interest rates since WWII hadbeen stable, increasing slightly, for many years.Surely interest rates could never go so high—thegovernment would do something first. Well, youknow how that worked out. The government decid-ed that curbing inflation was more important thankeeping interest rates down.

There have been lots of other times when the crystalball was cloudy:

• Buy Junk Bonds (Executive Life)• Invest in Mortgages and Real Estate

(Confederation Life)• Assume normal lapses on lapse supported business

(Long-term Care and Canadian Term-to-100)

But the classic example is Long-Term CapitalManagement. Every actuary should read “WhenGenius Failed,” the story of this failure. For thoseunfamiliar with this, the Long-Term CapitalManagement hedge fund had Nobel Prize winners

(among them Scholes, as in Black-Scholes) andmany other extremely bright and successful people.Their fatal flaw was they believed their models.

They thought they had diversified their risks. Theyhad invested in securities in many countries; if onewent down, the others would provide stability. Butthen came the Asian Flu and the Russian crisis andsuddenly EVERY country’s securities went under.Then their competitors got wind and started movingagainst them. So the company would have collapsedif the banks hadn’t bailed them out. There was amore complete article on this in The Actuary back inAugust.

The FutureSo you might be wondering why this all matters tofinancial reporting actuaries. It matters because thereare proposals in the market that don’t seem to havelearned these lessons.

For instance, the International AccountingStandards Board is suggesting that it’s OK to accruefuture expected profits on the day a policy is issued.The American Insurance Industry generally opposesgains at issues, the Academy Task Force on IFRSgenerally opposes it in all but the rarest circum-stances, but some accountants and regulators on theinternational front seem less concerned.

“Surely there are times you know you’re going tomake a profit” some have said, even though thefuture remains as cloudy as ever. “Our models showthat we’ll have gains” others say, stubbornly believingtheir models will turn out to be real.

We’ve seen gains at issue before. Enron used it.Many sub-prime lenders used it. What makes usthink it makes sense for insurance policies? If I wereallowed to issue only one required piece of guidancefor life accounting, “no gain at issue” would be it.Another issue involves adjusting for portfolios withdiversified risk. “We can reduce our required capitalbecause we have diversified our risks. If we losemoney on mortality on our life insurance, our annu-ities will bale us out!” they claim, ignoring that thepeople with annuities have different demographicsthan those with life insurance and that the mortalitychanges may occur at ages that affect the two busi-

The Siren Call of Models—Beware of theRocksby Henry W. Siegel

Henry W. Siegel, FSA,MAAA, is vice president,Office of the ChiefActuary with New YorkLife Insurance Companyin New York, N.Y. He may be reached [email protected].


nesses differently. “When things go wrong, all corre-lations approach 1” should be a lesson learned fromLong-Term Capital Management but some regula-tors seem willing to go along with a diversity credit.

The Society announced on the day I am writing thisthat they’ve completed “a research project thatexplored the covariance and correlation among vari-ous insurance and non–insurance risks in the contextof risk based capital.” I was happy when I saw thisannouncement since I think it’s an area that shrieksfor further research, but further examination showeda problem. There’s a bibliography and a theoreticalmathematical model, but there is no data to actuallydetermine if any covariance or correlation exists!This is not a criticism of the SOA, but how can oneargue for a diversity credit if there’s no data to deter-mine if it exists?

As actuaries, we need to avoid falling in love with ourmodels. It’s easy to become entranced with thembecause as actuarial students, our job was often tofeed those models and we don’t always step back andlook at the whole picture. How many times have youlamented the inability of beginning students tonotice when the model has produced somethingobviously wrong?

So as actuaries we need to be conscious that thefuture is uncertain. No surprise there, I hope. Also,as actuaries we are uniquely qualified to explain thatto those who do believe a model is real. I urge allSection Members to adopt this as a personal goal.

Given all the above, it may surprise you that I amconfident about one aspect of the future. This is myfinal column as Section Chair and I am confidentthat the Section will continue to thrive. As I havesaid before, we are in a period of exhilarating change,when all the financial reporting paradigms we’veknown will be changing. Whatever replaces them,financial reporting actuaries will be even more essen-tial in the future.

Finally, I want to thank the intrepid editor of theFinancial Reporter, Rick Browne, for putting upwith my “Just in time” style of writing. If you haven’twritten for the Financial Reporter, pick up a key-board and send Rick an article.

And always remember:

Insurance accounting is too important to be leftto the accountants!


$

>> ... Chairperson’s Corner

T he IASB Discussion Paper on InsuranceContracts presents the preliminary view thatinsurance liabilities should be valued at “cur-rent exit value” for GAAP financial reporting pur-poses. See the article in this issue “An InternationalFinancial Reporting Standards (IFRS) Phase IIDiscussion Paper Primer” by Tara Hansen and MarkFreedman for a more detailed examination of thepreliminary views expressed in the Discussion Paper.Some of the implications of those views may be sur-prising. A few are enumerated below.

Credit StandingAs the Discussion Paper is currently written, theimpact of an insurer’s credit standing will be a factorin the discount rate for its liabilities. If the creditstanding were to decrease, it would cause liabilityvalues to decrease, increasing income and surplus.Conversely, if credit standing were to improve, itwould cause liability values to increase, reducingincome and surplus. It may be difficult to explainthese gains when there is a credit downgrade, gener-ally thought of as a “negative” event, or the losseswhen there is a credit upgrade, which is generallyconsidered a “positive” event.

Earnings VolatilityAnother implication is the potential for earningsvolatility. It may not be surprising that if there isduration, convexity or hedging mismatch betweenthe assets and liabilities there could be earningsvolatility from changes in interest rates or other cap-ital market fluctuations. However, even where assetsand liabilities are well matched, there could bevolatility if some assets are accounted for in theincome statement on a basis other than fair value (ora similar measure). After all, current accounting doesnot require changes in fair values of securities ormost other financial instruments to flow through netincome. And certain invested assets, such as realestate, are not even eligible for a fair value option.

Earnings volatility might also emerge from othersources, even if liabilities are duration matched andwell hedged. All of an insurer’s liabilities wouldreflect its own credit standing. Invested assets, how-ever, typically have diverse credit ratings, not neces-sarily equal to the insurer’s. Thus, if there were anon-parallel shift in credit spreads, the asset and lia-bility values would not move consistently, creatingearnings volatility.

For example, assume an insurer has an AA rating.And assume that AA credit spreads relative to therisk-free rate increase by 10 basis points. Further,assume that the credit spreads on invested assetsincrease by an average of 20 basis points. Even ifassets and liabilities were otherwise well matched andproperly hedged, the non-parallel shift in creditspreads would cause the asset fair values to decreasemore than the liability current exit values, generatinga potentially large loss.

This could occur if there is a perception in the mar-ket that an insurer’s credit standing has changed,even if no change has actually occurred. An examplecould be presented from recent events, as follows.Assume that during the recent sub-prime credit cri-


Potential Implications of IASB’S PreliminaryViews on Insurance Contractsby Leonard Reback

8

sis some insurers were perceived by the market to beexposed to sub-prime credit risk. The credit spreadsfor their own obligations would have increased, evenif in reality they had no such exposure. Thisincreased credit spread, however, would likely havebeen considered consistent with observed marketcash flows, and thus may have been required to bereflected in discounting those insurers’ liabilities.This would have generated a reduction in the cur-rent exit value of an insurer’s liabilities, generatingpotentially large gains, whether or not the insurer’screditworthiness was actually impacted. If thosecredit spreads reversed in subsequent periods, thegains would reverse, producing large losses. Inessence, the market’s view of a company’s credit riskwould likely be deemed “correct” for purposes ofvaluing the liabilities, whether or not that view wasaccurate.

Also, changes in liability experience and assumptionswould create earnings volatility, similar to currentDAC unlocking on FAS 97 contracts. For example,if current estimates of future mortality were tochange, that change would impact the current exitvalue of the liabilities immediately, and wouldimpact net income. However, there would be no cor-responding change in invested assets (although rein-surance assets might be impacted). Depending onthe direction of the change in liability current exitvalue, a large gain or loss could result. These gains orlosses could be greater than the impacts currentlyseen from DAC unlocking on FAS 97 contracts,because DAC unlocking impacts are mitigated bythe amortization ratio, which is typically less than100 percent. With current exit value changes, therewould be no such mitigation. And under current exitvalue these changes would also impact contracts cur-rently accounted for under FAS 60 under US GAAP.

Gain or Loss at IssueGains or losses at issue are another possibly surprisingimpact. Under current GAAP, gains generally do notoccur upon issuance of a contract. Losses upon issueare rare, and are generally limited to situations whereacquisition costs are not recoverable, or where thepresent value of expected benefits and expenses underthe contract exceed the present value of premiums.

Under the IASB preliminary view, gains at issuewould be permitted (although the IASB indicates

that they expect this to be rare), and losses may occureven if expected premiums are adequate to coverexpected benefits and expenses. This may happen ifthe premiums are adequate to cover benefits andexpenses, but not adequate to cover benefits andexpenses plus a risk margin consistent with that ofother market participants.

Another situation where a loss may occur at issue iswhen an insurer has a particularly efficient expensestructure and builds the resulting low expenses intoits pricing, generating a low premium relative to therest of the market. Because the market level ofexpenses is higher than that of the insurer, the insur-er may need to assume those higher expenses whendetermining its liability value. And where thoseexpenses are higher than what was assumed in deter-mining the premium, there may be a resulting loss atissue. Of course, if the insurer’s own expected lowexpenses do materialize, the insurer will realize high-er profits as those expenses emerge.

Furthermore, the IASB preliminary view restricts therecognition of future premiums on a contract, unlessone of three conditions is met:

1. The insurer can compel premium payment;2. Including the premiums and associated benefits

increases the liability value; or3. The insured must pay the premium to retain

guaranteed insurability.

This would seem to preclude recognition of futurepremiums on most universal life contracts in excessof the minimum amount needed to cover contractu-al charges. It would also seem to preclude recogni-tion of any future premiums on most variable annu-ity contracts that are classified as insurance contracts.Any such excess premiums would be considered anunrecognized customer relationship intangible.

This provision appears to make losses on issue ofthese contracts likely. After all, premiums in excess ofthe bare minimum to keep the contract in force aregenerally assumed in pricing the contract. Thus,those premiums typically include elements to recov-er acquisition costs. If those premiums cannot berecognized, a loss at issue may result.


>> Potential Implications of IASB’S ...

continued on page 10>>

Leonard Reback FSA,MAAA, is vice presidentand actuary,Metropolitan LifeInsurance Co. inBridgewater, N.J. He can be contacted at 908.253.1172 or [email protected]


>> Potential Implications of IASB’S ...

Similar concerns may apply to lapse-supported prod-ucts. The beneficial policyholder behavior, in thiscase lapsation, may not be recognizable under thepreliminary views. Therefore, even likely future laps-es may need to be excluded from the liability valua-tion. This could also generate losses at issue.

A mechanism which might have the opposite effectis the IASB’s preliminary view on the restriction ofrecognition of dividends (on participating contracts)and interest credits in excess of minimum guarantees(on UL contracts) to those which the insurer has alegal or constructive obligation to pay. It is notentirely clear how strong the expectation for pay-ment of the dividend or interest credit would need tobe in order to be recognized. If dividends and inter-est credits that were assumed in pricing cannot berecognized in the liability valuation, this could gen-erate a gain at issue.

Business CombinationsAssets and liabilities acquired in a business combina-tion are required to be initially measured at fair valueunder FASB Statement No. 141, and under IFRS 3for entities following IASB standards. As of theissuance of the Discussion Paper, the IASB had notidentified any significant differences between currentexit value as described in the Discussion Paper andfair value. If that remains the case, then current exitvalue, as described in the Discussion Paper, maybecome the valuation basis for insurance contractsacquired in business combinations—a developmentthat could have interesting implications.

For example, if a company is acquired in a competi-tive bidding process, the fair value of the acquiredassets net of the current exit value of the acquired lia-bilities may well exceed the purchase price. After all,all the other companies that participated in the bid-ding process would have required receiving moreassets from the selling company. But in situationswhere an entire company is acquired, the resultingdifference could be considered a goodwill asset.

If only a block of insurance contracts is acquired,however, rather than an entire company, there wouldbe no goodwill. Under the preliminary viewsexpressed in the Discussion Paper, any differencebetween the purchase price and the net of the fair

value of acquired assets over current exit value ofacquired liabilities would be considered a loss (orgain, if applicable) at the time of the transaction.

SummaryThe preliminary views expressed in the IASBDiscussion Paper on insurance contracts may gener-ate some surprising and, perhaps, disturbing results.Because future accounting standards for entities cov-ered by the IASB and by FASB may be impacted bythe results of this project, interested actuaries shouldfollow these developments closely and make them-selves heard in the process.

10

$

T he recent FASB Statement 157—Fair ValueMeasurement—describes, among other modi-fications to the current fair value methodology,addition of risk margins for the risks other than cap-ital market risks. In particular, for investment guar-antees on variable annuities, GMxB’s (e.g.,Guaranteed Minimum Accumulation Benefit—GMAB, Guaranteed Minimum WithdrawalBenefit—GMWB), which are subject to fair valueaccounting (FAS 133), one major risk category ispolicyholder behavior risks.

This paper describes a methodology to calculate riskmargins attributable to these types of risks. Whileexamples will focus on policyholder behavior risk forGMxB’s, the approach may be extended to a broad-er range of non-market risks and other insuranceproducts.

For purposes of the risk-neutral liability value calcu-lation (a current, widely used FAS 133 method), cer-tain assumptions are made about future policyhold-er behavior. Let’s call this set of assumptions thebaseline. The policyholder behavior risks arise froma chance that corresponding assumptions will be dif-ferent from the baseline assumptions and willadversely impact the value of guarantees. Such riskscreate an uncertainty about future liability cashflowsand clearly affect the liability transfer price, or “exitvalue,” described by FAS 157.

Systematic vs. Idiosyncratic RisksOne question is what type of risks should be consid-ered: a systematic deviation from the baseline or arandom noise with the mean being the baselineassumption.

In the capital markets world, CAPM assigns com-pensation in form of higher risk premium only fornon-diversifiable or systematic risks. Systematic risksare the risks simultaneously affecting the entire cap-ital markets. Idiosyncratic risks are risks specific toindividual assets.

To extend this approach to policyholder behaviorrisks, an example of systematic risk would be the riskof a large number of policyholders simultaneouslyswitching to a lower lapse “regime.” A random noisetype of a risk, where lapse experience each period fluc-tuates around the mean assumed in pricing for eachpolicy/cohort, is an example of an idiosyncratic risk.

In terms of its impact on the fair value of liabilities,the systematic behavior risk of this nature may resultin more severe outcomes than a random fluctuationtype of risk. Note that a simple application of theCAPM principles suggests that diversifiable risksshould be excluded from the margin calculation.Random noise types of risks seem to be of suchnature.

While it’s not suggested to exclude idiosyncraticbehavior risks from the margin calculation entirely,the focus here is going to be on systematic risks.

Wang TransformOne possible methodology for the calculation of riskmargins is a technique called Wang transform (Wang2000, Wang, et al, 1997). For an arbitrary insurancerisk X, where X is, for example, a distribution ofinsurance losses, Wang transform describes a distor-tion to the cumulative density function (CDF) F(X).The distorted CDF F*(X) then can be used to deter-mine price of risk, where the premium equals theexpected value of X.

In a context of GMxB-specific risks, e.g., behav-ior risks, each observation of the variable X canbe viewed as a market-consistent value of liabili-ties corresponding to a certain state, representedby a set of policyholder behavior assumptions.Thus, a range of possible sets of policyholderassumptions will translate into a distribution ofliability values, where each of them will corre-spond to a market-consistent (risk-neutral) valueunder a given set of behavior assumptions. In par-ticular, the value corresponding to the baseline setof assumptions is the baseline value of liabilities,as determined using the current FAS 133methodology.

Wang transform as applied to non-market risks ofGMxB formulaically is as follows:

where X – distribution of fair value of liability result-ing from stochastic nature of non-market risks;


>> Risk Margins to the Non-Market Risks …

Risk Margins to the Non-Market Risks underFAS 157: Suggested Approachby Vadim Zinkovsky

continued on page 12>>

2

for example, a distribution of insurance losses, Wang transform describes a distortion tothe cumulative density function (CDF) F(X). The distorted CDF F*(X) then can be used to determine price of risk, where the premium equals the expected value of X.

In a context of GMxB-specific risks, e.g., behavior risks, each observation of the variable X can be viewed as a market-consistent value of liabilities corresponding to a certain state, represented by a set of policyholder behavior assumptions. Thus, a range ofpossible sets of policyholder assumptions will translate into a distribution of liabilityvalues, where each of them will correspond to a market-consistent (risk-neutral) value under a given set of behavior assumptions. In particular, the value corresponding to the baseline set of assumptions is the baseline value of liabilities, as determined using the current FAS 133 methodology.

Wang transform as applied to non-market risks of GMxB formulaically is as follows:

where X – distribution of fair value of liability resulting from stochastic nature of non-market risks; F(x) – original “loss” cumulative density function. X’s are ranked from the lowest loss(best result) to the highest,�� cumulative distribution function for standard normal;�

-1� inverse of �;

�— market price of risk, a parameter.F*(X) — transformed, or “pricing” distribution.Expected value under the transformed distribution E(X) equals the new price adjusted bythe risk margin.

For normally distributed risks, Wang transform applies a concept of Sharpe ratio fromthe capital markets world. However, it can also extend the concept to the skewed distributions.

Basically, the distribution F(x) describes the real world set of probabilities attributable toa range of possible outcomes, while F*(x) describes risk-adjusted probability distribution for the range of outcomes.

Essentially, Wang transform makes the probability of severe outcomes higher byreducing their implied percentile. This bears similarity with the risk-neutral valuation technique for the capital market instruments. Indeed, Wang transform replicates the results of risk-neutral pricing, Black-Scholes formula in particular, under a set ofadditional conditions, such as market completeness, availability of the risk-free asset,etc. Results of CAPM is another special case of the Wang transform.

Wang transform enables us to calculate a margin to the fair value of liabilities using notions of price of risk, usually denoted by � (lambda), and a measure of risk which isdetermined by the entire distribution. Possible methods to estimate lambda will be discussed later in the article.

[ ]��= � ))(()(* 1 XFXF

Vadim Zinkovsky, FSA, is vice president,Metropolitan LifeInsurance Co. in JerseyCity, N.J. He can be contacted at201.761.4603 [email protected]

F(x)—original “loss” cumulative density function.X’s are ranked from the lowest loss (best result) to thehighest,

—cumulative distribution function for standardnormal;

inverse of ;—market price of risk, a parameter.

F*(X)—transformed, or “pricing” distribution. Expected value under the transformed distributionE(X) equals the new price adjusted by the risk margin.

For normally distributed risks, Wang transformapplies a concept of Sharpe ratio from the capitalmarkets world. However, it can also extend the con-cept to the skewed distributions.

Basically, the distribution F(x) describes the real worldset of probabilities attributable to a range of possibleoutcomes, while F*(x) describes risk-adjusted proba-bility distribution for the range of outcomes.

Essentially, Wang transform makes the probability ofsevere outcomes higher by reducing their impliedpercentile. This bears similarity with the risk-neutralvaluation technique for the capital market instru-ments. Indeed, Wang transform replicates the resultsof risk-neutral pricing, Black-Scholes formula in par-ticular, under a set of additional conditions, such asmarket completeness, availability of the risk-freeasset, etc. Results of CAPM is another special case ofthe Wang transform.

Wang transform enables us to calculate a margin tothe fair value of liabilities using notions of price ofrisk, usually denoted by (lambda), and a measureof risk which is determined by the entire distribu-tion. Possible methods to estimate lambda will bediscussed later in the article.

Case 1: Normal DistributionConsider two normal loss distributions, Distribution1 and Distribution 2, with same means equal 1 andstandard deviations equal 0.5 and 1, respectively.

If the price was determined as an expectation of loss-es, the two risks would’ve had the same price equalto the mean.

However, Distribution 2 is clearly more risky and, asa compensation for risk, should command a higherprice.

This is consistent with the CAPM efficient frontiertheory where a riskier asset would have higherexpected return. Since both distributions are normaland have same means, standard deviation can beused as a measure of risk here, similarly with theCAPM approach. An assumption of risks normalityis also an underlying assumption of the CAPM.

Now, apply Wang transform to the original loss dis-tribution in order to get the pricing distribution, i.e.,the one that can be used to get price as expectedvalue of losses.

For normal distribution, F(X) with mean and stan-dard deviation and , the transformed distribu-tion F*(X) is also normal, however with the parame-ters and .

Thus, the price as an expectation of the transformeddistribution is expressed simply as:

Where – is the price of the margin, calculatedas expected value of the transformed distribution.

See the graph below showing the transform for thetwo normal distributions:

Assume that the state with the unshocked policy-holder behavior assumptions corresponds to themean result of the loss distribution: E[X] = BaselineRisk-Neutral value, the risk-neutral value correspon-ding to the baseline set of policyholder assumptions.

So the margin for additional non-market risk equalsjust ,

Value with margin = Risk-Neutral Value + .

4

Pricing

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

-3 -2 -1 0 1 2 3 4Losses

Normal: Low sigmaNormal: High sigmaPricing: Low sigmaPricing: High sigma

Lambda x Sigma1

Lambda x Sigma2

Assume that the state with the unshocked policyholder behavior assumptionscorresponds to the mean result of the loss distribution: E[X] = Baseline Risk-Neutral value, the risk-neutral value corresponding to the baseline set of policyholder assumptions.

So the margin for additional non-market risk equals just � �[X],

Value with margin = Risk-Neutral Value + �� [X].

Thus, risk margin is proportional to the standard deviation of a normally distributed risk.Under the normality assumption, a distribution with higher standard deviation will havehigher risk margin, which in this example is Distribution 2.

This result is analogous to the CAPM/Sharpe ratio conclusion which assigns a higher compensation to a risk with higher standard deviation of return, that is, standard deviation is a measure of risk.

Case 2: Long-tailed distribution – LognormalThe normal distribution, being a two-tailed symmetrical distribution with a range ofoutcomes from -� to +�, is not a very realistic one for modeling behavioral risks. If onethinks of a range of policyholder behavior outcomes, e.g., rider utilization scenarios, it’sreasonable to assume an extremely efficient behavior will increase the value of liabilitiesquite significantly. Intuitively, perfectly efficient behavior scenarios should be lowprobability events, perhaps with lower probabilities than normal distribution implies. Suchscenarios will define the right tail of the liability value distribution due to variability in policyholder behavior.

On the other hand, the left tail of the distribution will be impacted by extremely inefficientbehavior scenario types. An example of perfectly inefficient behavior is for example an

4

Pricing

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

-3 -2 -1 0 1 2 3 4Losses

Normal: Low sigmaNormal: High sigmaPricing: Low sigmaPricing: High sigma

Lambda x Sigma1

Lambda x Sigma2

Assume that the state with the unshocked policyholder behavior assumptionscorresponds to the mean result of the loss distribution: E[X] = Baseline Risk-Neutral value, the risk-neutral value corresponding to the baseline set of policyholder assumptions.

So the margin for additional non-market risk equals just � �[X],

Value with margin = Risk-Neutral Value + � � [X].

Thus, risk margin is proportional to the standard deviation of a normally distributed risk.Under the normality assumption, a distribution with higher standard deviation will havehigher risk margin, which in this example is Distribution 2.

This result is analogous to the CAPM/Sharpe ratio conclusion which assigns a higher compensation to a risk with higher standard deviation of return, that is, standard deviation is a measure of risk.

Case 2: Long-tailed distribution – LognormalThe normal distribution, being a two-tailed symmetrical distribution with a range ofoutcomes from -� to +�, is not a very realistic one for modeling behavioral risks. If onethinks of a range of policyholder behavior outcomes, e.g., rider utilization scenarios, it’sreasonable to assume an extremely efficient behavior will increase the value of liabilitiesquite significantly. Intuitively, perfectly efficient behavior scenarios should be lowprobability events, perhaps with lower probabilities than normal distribution implies. Suchscenarios will define the right tail of the liability value distribution due to variability in policyholder behavior.

On the other hand, the left tail of the distribution will be impacted by extremely inefficientbehavior scenario types. An example of perfectly inefficient behavior is for example an

3

Case 1: Normal Distribution.Consider two normal loss distributions, Distribution 1 and Distribution 2, with samemeans equal 1 and standard deviations equal 0.5 and 1, respectively.

Distribution of LossesNormal

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

-3 -2 -1 0 1 2 3 4Loss

Normal: Low sigmaNormal: High sigma

If the price was determined as an expectation of losses the two risks would’ve had the same price equal to the mean.

However, Distribution 2 is clearly more risky and, as a compensation for risk, should command higher price.

This is consistent with the CAPM efficient frontier theory where a riskier asset would have higher expected return. Since both distributions are normal and have same means,standard deviation can be used as a measure of risk here, similarly with the CAPMapproach. An assumption of risks normality is also an underlying assumption of the CAPM.

Now, apply Wang transform to the original loss distribution in order to get the pricing distribution, i.e., the one that can be used to get price as expected value of losses.For normal distribution, F(X) with mean and standard deviation 7 and �, the transformed distribution F*(X) is also normal, however with the parameters 7+�� and �.

Thus, the price as an expectation of the transformed distribution is expressed simply as:

E*[X] = E[X] + � �[X],

Where E*[X] – is the price of the margin, calculated as expected value of the transformed distribution.

See the graph below showing the transform for the two normal distributions:

3



0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

-3 -2 -1 0 1 2 3 4Loss







E*[X] = E[X] + � �[X],



3



0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

-3 -2 -1 0 1 2 3 4Loss







E*[X] = E[X] + � �[X],



3



0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

-3 -2 -1 0 1 2 3 4Loss







E*[X] = E[X] + � �[X],



3



0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

-3 -2 -1 0 1 2 3 4Loss







E*[X] = E[X] + � �[X],



3



0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

-3 -2 -1 0 1 2 3 4Loss







E*[X] = E[X] + � �[X],



2











��= � ))(()(* 1 XFXF

2











��= � ))(()(* 1 XFXF

2











��= � ))(()(* 1 XFXF

2











��= � ))(()(* 1 XFXF

2











��= � ))(()(* 1 XFXF



12

Thus, risk margin is proportional to the standarddeviation of a normally distributed risk. Under thenormality assumption, a distribution with higherstandard deviation will have higher risk margin,which in this example is Distribution 2.

This result is analogous to the CAPM/Sharpe ratioconclusion which assigns a higher compensation to arisk with higher standard deviation of return, that is,standard deviation is a measure of risk.

Case 2: Long-tailed distribution –LognormalThe normal distribution, being a two-tailed sym-metrical distribution with a range of outcomes from

to , is not a very realistic one for modelingbehavioral risks. If one thinks of a range of policy-holder behavior outcomes, e.g., rider utilization sce-narios, it’s reasonable to assume an extremely effi-cient behavior will increase the value of liabilitiesquite significantly. Intuitively, perfectly efficientbehavior scenarios should be low probability events,perhaps with lower probabilities than normal distri-bution implies. Such scenarios will define the righttail of the liability value distribution due to variabil-ity in policyholder behavior.

On the other hand, the left tail of the distributionwill be impacted by extremely inefficient behaviorscenario types. An example of perfectly inefficientbehavior is for example an assumption of zero riderutilization rate. But under this assumption the liabil-ity value will be naturally capped by present value offees with the minus sign. Thus, a distribution with alonger tail for higher losses and with outcomes limitedby the lowest (best result: e.g., no claims) value shouldbe more realistic.

One such distribution with some well-behaved prop-erties is lognormal. Here is how it compares with thenormal distribution:

The distributions shown on the graph have samemean and standard deviations. However, the lognor-mal has “fatter” right tail for high losses. We’ll seebelow that Wang transform assigns higher risk mar-gin to the lognormal distribution in this example.

For the lognormal distribution the transformedCDF per Wang transform is also distributed lognor-mally. If the original F(X) is lognormal with param-eters and , the transformed distribution F*(X) isalso lognormal with parameters and .

The mean of the lognormal distribution equals:.

So the liability value with the margin, calculated asthe mean of the transformed distribution, equals:

.

Again, assuming that the expected value under theoriginal, “real world” loss distribution corresponds tothe baseline liability value, E[X] = Baseline Risk-Neutral value, we’ll get the following result:

Value with margin = Risk-Neutral Value*e .

And

Here is how transformed distributions look:

Assuming =0.3,expected values of transformed “pricing” distribu-tions are:1.15 – normal,1.152 – lognormal

6

For the lognormal distribution the transformed CDF per Wang transform is alsodistributed lognormally. If the original F(X) is lognormal with parameters < and �, thetransformed distribution F*(X) is also lognormal with parameters



14

Thus, lognormally distributed losses resulted in aslightly higher price.

Note that both mean and standard deviation of theoriginal loss distributions were the same.

Margin adjustment to the price is as follows:Normal: New Value = Old price + Lognormal (-.11, 0.47): New Value =Old Price x Exp

As mentioned before, Wang transform increases prob-abilities of severe outcomes.

As an example, consider the 99.5th percentile of ouroriginal lognormal distribution L( -.11, 0.47) whichequals 3.02. Under the original distribution, proba-bility of an outcome in excess of 3.02 is 0.5 percent.The Wang transform changed the original lognormalto the lognormal with parameters L( 0.03, 0.47).The transformed distribution assigns higher proba-bility to the outcomes greater than 3.02, equal 1.14percent.

Recommendation for GMxB’s RiskMargin CalculationIn practice, modeling non-market risks stochastical-ly is a difficult task, both computationally and interms of underlying assumptions. Take as a specialcase policyholder behavior. Multiple parametersneed to be estimated to describe stochastic processesfor lapse, partial withdrawal, rider utilization, can-cellation, optional step-ups, etc. Usually, there is verylittle historical information available for such typesof calibration. If each of these risks is modeled sto-chastically, even in its simplest form, at least an esti-mate for the standard deviation of each risk shouldbe needed. The resulting liability value distributionis likely to be of some arbitrary shape other than awell-behaved distribution. However, there is no rea-son to believe such a distribution will be any betterin terms of its predictive power.

At the same time, a set of “worst case” type of behav-ior assumptions may be defined as part of sensitivityanalysis by pricing actuaries and the liability valueestimated.

Fitting DistributionsMoreover, there may be a certain probabilityassigned to a state described by a set of shocked pol-icyholder behavior assumptions. For example, suchestimates may be needed for economic capital calcu-lation purposes.

In any case, making an assumption of the probabili-ty of such “shocked behavior” would define a pointon the tail of the liability value distribution due to an

uncertain nature of policyholder behavior. Alongwith the assumption that the baseline set of behaviorassumptions corresponds to the mean value of theliability distribution, this defines two points on theliability CDF.

For two-parameter distributions, such as normal orlognormal, defining two points on the probabilitydistribution curve is sufficient to find the distribu-tion parameters.

Assume, as an example, that the “shocked behavior”liability value corresponds to the 99.5th percentile.We’ll show below how the mean and the 99.5th per-centile, can be used to uniquely identify parametersof the normal or lognormal distributions Ì and Û.

Normalizing DistributionReal life distributions of liability market values mayspan over a wide range of outcomes. However, thelognormal distribution we’ve considered so far, withparameters =-0.11 and =0.47, denote it L(-0.11,0.47), has its domain between [0; ], as any lognor-mal distribution. In order to map the results of a reallife distribution to the domain of lognormal distri-bution a one-to-one relationship needs to bedefined. A simple linear transform will describe sucha relationship:

(1)

Where,

– real life outcome variable,– lognormal variable, L(-0.11, 0.47).

A and B – constants to solve for.

Two equations linking the two “observed” values ofthe liability value, baseline and 99.5th percentile totheir mapped values to the lognormal L(-.11, .47)will uniquely identify constants A and B.

Here are the main steps of the derivation of parame-ters A and B. The mean of the lognormal distribu-tion:

Mean

The 99.5th percentile for L(-.11, 0.47) equals 3.02 orapproximately four standard deviations away fromthe mean, and variance (standard deviation squared)of the lognormal distribution equals:

Variance

Since the real life distribution is just a linear trans-form of the original distribution, the same relation-ship between the mean and 99.5th percentile musthold: four standard deviations apart.

8

Normalizing DistributionReal life distributions of liability market values may span over a wide range of outcomes.However, the lognormal distribution we’ve considered so far, with parameters C=-0.11and �=0.47, denote it L(-0.11, 0.47), has its domain between [0; �], as any lognormal distribution. In order to map the results of a real life distribution to the domain oflognormal distribution a one-to-one relationship needs to be defined. A simple linear transform will describe such a relationship:

Xreal = A+B*X, (1)

Where,

Xreal – real life outcome variable,X – lognormal variable, L(-0.11, 0.47).A and B – constants to solve for.Two equations linking the two “observed” values of the liability value, baseline and 99.5th percentile to their mapped values to the lognormal L(-.11, .47) will uniquelyidentify constants A and B.Here are the main steps of the derivation of parameters A and B. The mean of the lognormal distribution:

2

2��+

= eMean

The 99.5th percentile for L(-.11, 0.47) equals 3.02 or approximately four standard deviations away from the mean, and variance (standard deviation squared) of the lognormal distribution equals:

)1(222�=

+ �� eeVariance

Since the real life distribution is just a linear transform of the original distribution the same relationship between the mean and 99.5th percentile must hold: four standard deviations apart.

Therefore, standard deviation of the original distribution equals:

Sigma = (99.5th percentile - Mean)/4.

Next, constant B equals 2 x Sigma.

Next, recall the result of the Wang transform for the lognormal distribution.Price with the margin equals:

��

��

eiceOrigeinicewithM *PrargPr 22

==++

Going back to the real life distribution X using formula (1) and remembering that Mean=Original Price =1:

Margin = A + B*PricewithMargin – (A+ B*LMean)= B*Lognormal margin

}1{**2arg �= �� einM (2)

8


Xreal = A+B*X, (1)

Where,


2

2��+

= eMean


)1(222�=

+ �� eeVariance






��

��


==++



}1{**2arg �= �� einM (2)

8


Xreal = A+B*X, (1)

Where,


2

2��+

= eMean


)1(222�=

+ �� eeVariance






��

��


==++



}1{**2arg �= �� einM (2)

8


Xreal = A+B*X, (1)

Where,


2

2��+

= eMean


)1(222�=

+ �� eeVariance

Since the real life distribution is just a linear transform of the original distribution the same relationship between the mean and 99.5th percentile must hold: four standard deviations a

Documents

The Financial Reporter - SOA...The Financial Reporter Published by the Life Insurance Company Financial Reporting Section of the Society of Actuaries 475 N. Martingale Road, Suite