A Volume 30, No. 2

1SSN 0515-0361

N o v e m b e r 2000

EDITORS "

Andrew Cairns

Paul Embrechts

Co-EDITORS:

Stephen Philbrick

Ren6 Schnieper

EDITORIAL BOARD :

Alois Gisler

M a r c Goovaer ts

M a r y Hardy

Ole Hesselager

Chris t ian Hipp

Jacques Janssen

Wil l iam S. Jewel l

J e a n Lemaire

J u k k a Ranta la

Axel Reich

Robert Rei tano

The Journal of" the ASTIN and AFIR Sections of" the International Actuarial Association

CONTENTS

EDITORIAL

INVITED ARTICLE

M. LANE Pricing Risk Transfer Functions 259

ARTICLES

J.P. NIELSEN Super-Efficient Prediction Based on High-Quality Market Information 295

B. SUNDT Multivariate Compound Poisson Distributions and Infinite Divisibility 305

R. GROBEL, R. HERMESMEIER Cornputation of Compound Distributions II: Discretization Errors and Richardson Extrapolation 309

T. MACK Credible Claims Reserve: The Benktander Method 333

WORKSHOP

A.J. MATA Pricing Excess of Loss Reinsurance with Reinstatements 349

M.C. GUTIERREZ-DELGADO, A. KORABINSKI Initial Selection and Cause of Disability for Individual Permanent Health Insurance 369

J.F. WALHIN, J. PARIS The True Claim Amount and Frequency Distribt, tion of a Bonus-Malus Systern 391

J.P. NIELSEN, B.L. SANDQViST Credibility Weighted Hazard Estimation 405

MISCELLANEOUS

Acluarial Vacancies 419

Report on the International AFIR Colloquium 2000 421

2001 AFIR Colloquium Announcement 423

Report on the International ASTIN Colloquium 2000 425

~ E E T E R S

E D I T O R I A L P O L I C Y

,4STIN BULLETIN started in 1958 as a journal providing an outlet for actuarial studies in non-life insurance. Since then a well-established non-life methodology has resulted, which is also applicable to other fields of insurance. For that reason ASTIN BULLETIN has always published papers written from any quantitative point of view - whether actuarial, econometric, engineering, mathematical, statistical, etc. - attacking theoretical and applied problems in any field faced with elements of insurance and risk. Since the foundation of the AFIR section of IAA. i.e. since 1988. AS'ITN BULLETIN has opened its editorial policy to include any papers dealing with financial risk.

We especially welcome papers opening up new areas of interest to the international actuarial profession.

ASTIN BULLEI'IN appears twice a year (May and November), each issue consisting of at least 80 pages.

Details concerning submission of manuscripts are given on the inside back cover.

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Members of AST[N receive ASTIN BUI.Lh2ThV free of charge. As a service of ASTIN to tile newly founded section A FIR of I AA, members of A FIR also receive ASFIN BULLETIN free of charge.

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I N D E X T O V O L U M E S 1-27

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Copyfight ,~ 2000 PEETERS

EDITORIAL C H A N G E S

Following the recent resignation of Henrik Ramlau-Hansen as editor the committee of ASTIN has agreed that Andrew Cairns should be promoted to editor alongside Paul Embrechts. The Editors are delighted that we will also be joined by Stephen Philbrick of the USA as Co-editor in succession to Andrew.

EDITORIAL DATA

Readers of ASTIN Bulletin will know,just from looking at the bulging issues on their bookshelf, that the journal is going from strength to strength. The numbers of papers submitted in 1998 and 1999 were 31 and 42 respectively. The acceptance rate remains steady at about 50%. The aim of the editors is that there should be no backlog of papers awaiting publication and, to date, final versions of accepted papers have gone into the next issue (appearing within 6 months on average).

Processing times for papers submitted during 1998 and 1999 have been analysed with good results. In particular, the mean time between submission of a paper and the communication of a decision after the first set of reports from the two referees is 5.0 months with a standard deviation of 2.1 months. There is no significant difference between accepted and rejected papers. Note that we record time to decision rather than time to publication. Since authors are often requested to make substantial improvements to their contributions before the paper can be published, the time to publication can be substantially longer than the time to decision.

The referees of ASTIN Bulletin, through thorough and conscientious reviews, make this journal possible. Their task is difficult and time consuming with no material reward. On behalf of the ASTIN Bulletin readership, the Editors and the Editorial Board offer their sincere thanks.

ANDREW CAIRNS

ASTIN BUI . I .ETIN. Vol. 30, No. 2, 2000, p. 257

PRICING RISK T R A N S F E R TRANSACTIONS

BY

MORTON N. LANE

Lane Financial LLC, Kenilworth, USA

INTRODUCTION

Should the pricing of reinsurance catastrophes be related to the price of the default risk embedded in corporate bonds?

If not, why not? A risk is a risk is a risk, in whatever market it appears. Shouldn't the risk-

prices in these different markets be comparable? More basically perhaps, how should reinsurance prices and bond prices be set? How does the market currently set them? These questions are central to the inquiry co,atained in this paper.

Avoiding unnecessary suspense, our answers are: Yes, cat prices should be related to credit prices because both risks contain a characteristic trade-off between the frequency of and severity of adverse events. We leave the question of how prices should be set to others and focus on the empirical question of how they have been set by the markets. In the process, we develop a fairly robust pricing mechanism and explore its potential uses in many different contexts.

The 1999 Insurance-Linked Securities (ILS) market (a.k.a., Cat Bond market) provides l, he empirical springboard to the discussion. The ILS market is only 4 years old. As such, it represents a new and unique intersection of reinsurance and financial markets. It provides a wonderful laboratory f o r exploring risk-pricing.

The ILS market, still in its experimental phase, appears to require more generous (cheap) pricing of insurance risk than does the bond market of default risk. So much so that academics have begun to weigh in on the question of why 2 Previously, insurance pricing discussions had been confined to practicing insurance professionals, particularly actuaries 3. For finance professionals, insurance pricing, much less reinsurance pricing, seldom made the index of their financial texts - though even that is beginning to change.

This paper was presented on June 22, 2000, as a keynote speech at the AFIR 2000 Colloquium in Tromso, Norway.

2 See Bantwal & Kunteuther (I) and Froot & Posner (6). 3 See Krcps (8) (9) and Mango (15).

ASTIN BULLETIN. Vol. 30. No. 2. 2000, pp. 259-293

1800 ~c~

II R R I R K G I I R F

1600

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1999 ILS I m p l i e d Spreads over LIBOR

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Note: The scale for the ratings axes is not linear and only part of the range of probabilities is shown.

CCC

PRICING RISK TRANSFER TRANSACTIONS 261

Perhaps academic reticence occurred because prior to the advent of cat bonds, reinsurance markets were opaque. Reinsurance prices were unavailable to the investing community. With insurance securitization, however, the pricing of embedded insurance risks is exposed to, and must be made appetizing for, investors. Prices ought to converge towards other alternative fixed income assets. At the very least, risk-pricing of insurance can now be compared with other investor alternatives. And yet, insurance is different.

Insurance is Different?

Any appraisal of the risks contained in insurance or reinsurance covers must take into account the fact that the statistical distribt.tion of profit and loss outcomes may be severely skewed. Conventional risk measurement (i.e., the standard deviation) deals with random outcomes that are symmetric in nature. Price volatility is usually viewed as symmetric. Event or outcome risk (a characteristic of insurance) is not. How is the asymmetry to be captured? What are the components of event-risk and how do they factor into price? Indeed, how should "'price" be articulated?

There is general agreement that the "price" of an insurance-linked security is the spread over and above (a) the risk-free rate and (b) the loss expected from the insurance. There is also general agreement that this "excess spread" is a function of, and reward for, the risk assumed. But how is that excess spread calculated and evaluated?

The most conventional - and respectable - risk measure is the standard deviation of outcomes. It is analytically convenient for portfolio as well as individual investment evaluation. And yet, its original promoter, Harry Markovitz, preferred the semi-variance to capture asymmetries. Its popularizer William Sharpe has largely confined its application to price-risk, which may be largely viewed as symmetrical in behavior.

In a pioneering work for actuarial literature, Rodney Kreps (Kreps (8) and (9)) tried to relate reinsurance pricing to capital markets. (This was largely before the 1LS market existed.) His rationale was that the deployment of risk- capital in underwriting should be related to the deployment of risk-capital in investments. Indeed, it should. The ILS market gives us the first opportunity to see whether it actually does.

Kreps' analysis led (in general terms) to the conclusion that the "excess spread" should be a fraction of the standard deviation of the outcomes of the cover being reviewed. Furthermore, for a wide range of parameter choices, that fraction would likely be in the order of 20% to 40% of the standard deviation. This is the equivalent of saying that cat bonds should have Sharpe Ratios around 0.3. But, in a remarkable example of the "dog that didn't bark" theory,

t Care must be taken in defining the exact interpretat ion of lhe Sharpe Ratio that particular analysts use. Here it is the ratio o f excess spread to s tandard deviation arising from credit events. It does not include returns on volatility from interest rate risk.

262 MORTON N. LANE

no single private placement memorandum (PPM) of a 1999 ILS transaction provided standard deviation as a risk measure for investor consideration. All PPMs offered frequency risk and expected loss as risk measures for investors.

Antecedents

The antecedents of this paper explored the use of standard deviation in the context of the first ILS in 1996 ("A Year of Structuring Furiously").

As more securitizations emerged, it became clear that standard deviation was not entirely adequate for the insurance task. Our 1997 annual review paper ("Price, Risk, and Ratings for Insurance-Linked Notes") began to explore the use of Conditional Expected Loss (CEL) as a better measure for asymmetric insurance risk embedded in insurance-linked notes. Appealing as it was, it was not robust enough. This led to the joint consideration of two risk measures - the CEL and the probability of first dollar loss (PFL) in our 1998 paper, "Risk Cubes". (In retrospect, our 1997 enthusiasm for the CEL may have arisen because so many of that year's deals were pitched at the same PFL level.) This year's review of the 1999 securitizations continues the two-way (CEL and PFL) risk analysis addressed in 1998.

Empirical Approach

The conditional expected loss (CEL) is known more familiarly as severity of loss. The probability of first dollar loss (PFL) is referred to as frequency of loss. The influence of frequency and severity on risk assessments is not a new idea. Indeed, it is very old. What is fresh, however, is the use of transacted market prices as a mechanism for empirically measuring how the market makes a trade- off between these two components of risk. A "revealed risk-preference" function is derived.

This risk preference function will be familiar to economists. It is similar to the Cobb-Douglas Production function. This workhorse of economics illustrated to generations of students how a trade-off is made between capital and labor to optimize production. Here a trade-off is made between frequency and severity to best explain preferred risk return positions (i.e., transacted prices).

What adds wind to our analytical sail is that the empirical insights derived from the 1LS market may be useful in other apparently unrelated fixed income markets. Default risk (a.k.a., credit risk) is asymmetric whether it emanates from simple corporate bonds, CBOs or leveraged loans.

Here is the pricing that should bear some relation to the ILS market. In a "David tells Goliath what to do" exercise, this paper presents a revealed

risk preference that emanates from the 1999 ILS pricing and tries to predict bond prices. The results are tantalizing. Although not entirely scientific, the results certainly motivate further research. The bond market is, of course, a bigger and more volatile market than the ILS market. Revealed risk preferences from bonds should dictate I LS prices not the other way around.

PRICING RISK TRANSFER TRANSACTIONS 263

That is the prospect that is held out by our approach - that empirical studies of both markets may lead to revealing a "universal risk-pricing grid".

Outline

These ideas are presented in three parts. The first reviews the ILS securities issued in 1999. (Note that Lane & Beckwith (10) put the securities in the context of current trends.) Readers more interested in the concepts may conveniently skip to the next section. Part Two fits a risk preference function to the 1999 data. It also develops some of the uses to which such a fit might be put. Finally, in Part Three the implications of the risk preference approach is explored in the framework of corporate bonds. We conclude with some opinions about the way the rating agencies can use this approach to improve their rating categories of insurance-linked (and perhaps other) securities.

Some caveats are in order. First, the risks described herein may be considered to be non-hedgeable, or more precisely, locally non-hedgeable because they are event-driven. They are not, therefore, susceptible to well-known options pricing models (e.g., Black Scholes, et al.). Neither is there an attempt to establish equilibrium pricing or portfolio effects - all of which we consider to be important - but beyond our present scope. The attempt is simply to suggest a form of, and a mechanism for, measuring the market's utility function for risk. In any such exercise, it is relative pricing that is examined. Absolute risk-return trade-offs are for other parts of the market to decide upon.

P A R T I: REVIEWING THE INSURANCE SECURITIZATIONS OF 1999

The (re)insurance securitizations of 1999 are listed in Table I. Approximately $2.0 billion of insurance risk was transferred to the capital markets in approximately a dozen transactions. The word "approximately" is used to signal the fact that full details may n o t be available about known deals, and not all deals may be known. (After all, the market is a private placement market.)

Table l details for each transaction (and any tranches), the Special Purpose Reinsurer (the name by which the deals are often known), the ceding company, lead underwriters, amount, ratings given, date of issue and maturity, together with various financial statistics. Certain of the financial statistics have been obtained directly from the PPM. Two statistics, "Expected Excess Return" (EER) and "Conditional Expected Loss" (CEL) are derived numbers.

In certain transactions, the term to maturity is different from the term for which the investor is on (insurance) risk. This was true of the Kelvin transaction. The senior Kelvin tranche could not go on risk until the second year of the three- year term to maturity. The noteholder was receiving coupons for all three years. In order to compare and contrast reinsurance-equivalent prices, Table 1 adjusts prices to reflect equivalent annual risk periods.

Another adjustment involved the conversion of LIBOR - by definition, based on 360-days accounting - to an actual 365-day count. This affected nearly all of the deals. The LIBOR component was also extracted from fixed coupon

T A B L E IA

1999 ~LS GENERAl. STATISTICS

S P V Cedent Lead Underwriters Amount S&P 3food),'~ DCR Fitch 3/99-3]00 Maturity ~Waturlty Erposure (US $) Rating Raling Rating Rating Is,~ue Date " Term Term

Analyzed Securities

Mosaic ~'~ USF&G Goldman Sachs 24.3 - BB Mar-99 Feb-O0 12 12

Mosaic 2B USF&G E.W. Blanch 20.0 - - B Mar-99 Feb-00 12 12

Halyard Re Sorcma Memll Lynch 17.0 - B B - Apt-99 Apr 02 36 36

Domestic Re Kemper Aon 80.0 BB+ Ba2 - Apr-99 Apr-02 37 37

Concenlric Re Oriental Land Goldman Sachs 100.0 BB+ Bal BB+ May 99 May-04 60 60

Juno Re Gerhng Goldman Sachs 80.0 BB - BB+ Jun-99 Jun-02 36 36

Residential Re USAA Goldman Sdchs/Lehman 200.0 BB Ba2 - Jun-99 Jun-00 12 12 Bros./Merrill Lynch

Kelvin Ist Event Koch Goldman Sachs 21 6 - B - Oct-99 Feb-03 39.9 ~6

Kelvin 2nd Event Koch Goldman Sachs 23.0 BB - B B B - BB+ Oct-99 Feb-03 39.9 24

Gold Eagle A Am Re Am Re/ML 50.0 Baa3 B B B - Oct-99 Apr-01 17 17

Gold Eagle B Am Re Am Re/ML 126.8 Ba2 BB Oct-99 Apr-01 17 17

Namazu Re Gerling Aon I00.0 BB BB Nov-99 Dec 04 60 60

Atlas Re A SCOR Goldman Sach';/ 70 0 BBB+ BBB BBB Mar-00 Apr-03 36 36

Alias Re B SCOR ~.tarsh McLennan 30 0 BBB - B B B - B B B - Mar-00 Apt-03 36 36

Atlas Re C SCOR " " " 100 0 B - B B - MarCO Apr-03 36 36

Seismic Lid Lehman Re Swiss Re CM/ 145.5 BB+ Ba2 - Mar-OO Dec-01 22 22 Lehman

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T A B L E I A ( c o n t i n u e d )

Amount S&P ~tood "~ DCR Fitch 3]99,-3[00 3tatarit " Maturity Expo.~ure SP V Cedent Lead Underwriters ( U S $) Rating Rating Rating Rating Issue Date Term Term

Other Notable ILS Securities - Not part o f Pricing Analysis

Units:

Mosate (Units)

Domestic Re (Shares)

Gold Eagle (Units)

Seismic Ltd.

Option"

Circle Maihama

CLOCS

Credit:

SECTRS A

SECTRS B

SECTRS C

USF&G Goldman Sachs 1.4 - AAA Mar-99 Feb-00 12 12

Kemper Aon 20 0 - Apr-99 Apr-02 37 37

Am Re Salomon SB 5.5 _ - Oct-99 Apr~:ll 17 17

Lehman Re Swiss Re CM/ Lehman 4.5 - - - Mar-00 Dec~)l 22 22

Oriental l.and Goldman Sachs I00 0 A - A - May-99 hlay-04 60

Jan-00 Dec-02 36 ReAC Sv.'lss Re Cap Mkts 75.0

Gerling GKS Goldman Sachs 245.5 AA Aa2 AA - Apr-99 Apr-02 36 36

Gerhng GKS Goldman Sachs 127.5 A A2 A - apr-99 apr-02 36 36

Gerling GKS Goldman Sachs 82 0 BBB+ Baa2 BBB - Apr-99 apt-02 36 36

Other related ILS market transactions of note'

WINRS Enron Mernll Lynch 105.0 Sep-99

Surety ResidenSea Ltd Centre Solutions 280.0 O¢t-99

SWAP Not Disclo,c.ed Marsh McLennan 50.0 S¢p-99

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Note&" - The table displays securities and/or their tranches that were issued between March 1099 and March 2000. - Upper panel shov, s 16 deals and/or their tranches that are anal)7~ed in this paper Lower panel records related transactions - The exposure term of the Kelvin (Koch) transaction ~s less than the maturity of the notes Traded weather seasons do no cover the ,,,,'hole )'ear. Accordingly. an adjustment is made to the spread to make il

comparable to a 365-day e',posure year.

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1999 11.8 FINANCIAL STATISTICS

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SPV Cedent Lead Underwritern Spread Adjusted Spread Expected Probabilit , Probability

Premium Premium Long o f I~t $ Long o f Exhaust to LIBOR (bpdn) (Annual) (Annual) (Annual) (Annual)

Expected Exees~ Return (Annual)

Conditional Expected Lost

Anal3,2ed Securities

Mosaic 2A

Mosaic 2B

Halyard Re

Domestic Re

Concenlnc Re

Juno Re

Restdential Re

Kelvin Ist Event

Kelvin 2nd Event

Gold Eagle A

Gold Eagle B

Namazu Re

Atlas Re A

Atlas Re B

Atlas Re C

Seismic Lid.

USF&G

USF&G

Sorema

Kemper

Oriental Land

Gerling

USAA

Koch

Koch

Am Re

Am Re

Gerhng

SCO R

SCOR

SCOR

Lehman Re

Goldman Sachs 400 4 08% 0.42% 0.0115 0.0042 3.64% 36.52%

E.W. Blanch 825 8 36% 2 84% 00525 0 1150 5.52% 54.10%

Merrill Lynch 450 4.56% 0 63% 0.0084 0.0045 3.93% 75.00%

Aon 369 3.74% 0 50% 0 0058 0.0044 3.24% 86.21%

Goldman Sachs 310 3.14% 0.42% 0.0064 2.72% 65.63%

Goldman Sachs 420 4.26% 0.45% 0.0060 0.0033 3 81% 75.00%

Goldman Sachs/ Lehman 366 3.71% 0 44% 0.0078 0.0026 3.27% 57.89% Bros./ Merrill Lynch

Goldman Sachs 1570 10.97% 4.45% 0 1210 0.0050 6.52% 36.78%

Goldman Sachs 870 4.82% 0.30% 0.0158 0.0007 4.52% 19.23%

Am Re/M L 295 2 99% 0.17% 0.0017 0 0017 2.82% 100.00%

Am Re/ML 540 3.48% 0.63% 0.0078 0.0049 4.85% 80.77%

Aon 450 4.56% 0.75% 0.0100 0.0032 3.81% 75.00%

Goldman Sachs/ 270 2.74% 0. I 1% 0.0019 0.0005 2.63% 57.89%

Marsh McLennan 370 3 75% 0.23% 0.0029 0 0019 3.52% 79 31%

" " " 1400 14.19% 3.24% 0.0547 0 0190 10.95% 59.23%

Swiss Re CM] Lehman 450 4 56% 0.73% 0.0113 0 0047 3.63% 64.60%

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T A B L E 1B (cont inued)

SP V Cedent Lead Under writers

Spread Adjusted Spread Expected Probability Probability Expected Exees~ Conditional Premium Premium Loss of Ist $ Lo~s of Exhaust Return (Annual) Expected Loss

to LIBOR (bpds) (Annual) (Annual) (Annual) (Annual)

Other Notable ILS Securities - Not Part of Pricing Analysis

Units

Mosaic (Umts) USF&G

Domestic Re (Shares)Kemper

Gold Eagle (Umts) Am Re

Seismic Ltd Lehman Re

Option'

Circle Malhama Oriental Land

CLOCS ReAC

Credtt:

SECTRS A Gerling GKS

SECTRS B Gerling GKS

SECTRS C Gerling GKS

Goldman Sachs 190

Aon

Salomon SB 850

Swiss Re CM] Lehman

Other related ILS market transactions of note:

WINRS Enron Merrdl Lynch

Surety ResidenSea Ltd Centre Solutions

SWAP Unknown Marsh McLennan

75

E+45

E+85

E+I70

0 0060 0.0082 0.0058 0.968

0.0017 0.0017 0.0017 1.00fl

0.0113 0.0113 0.0113 1.000

0.0064

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NA NA NA

NA NA NA

NA NA NA

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Notes" - The table displays securities and/or their tranches that were issued between March 1999 and March 2000 - Upper panel shows 16 deals and/or their tranches that are analyTed in this paper. Lower panel records related transactions. - Shaded columns show the data that is used m that subsequent price analysis. - All deals are converted to a 365-day year. LIBOR convention uses a 360-day year, but CAT risk ts a 365-day activity. The adjusted spreads displayed are comparable to reinsurance pricing. - The exposure term of the Kelvin (Koch) transaction is less than the maturity of the notes. Traded weather seasons do not cover the whole year Accordingly. an adjustment is made to the spread to

make it comparable to a 365-day exposure year. - The Kelvin (Koch) transaction was issued as a fixed-income instrument An adjustment is made to provide an equivalent floating rate basis. - Expected Excess Return is defined as Adjusted Spread Premium less Expected Loss - Conditional Expected Loss is defined as Expected Loss divided by the Probability of First Dollar Loss

t-O

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268 MORTON N. LANE

k n o w n to

traditional itself. This future.

Finally,

deals (i.e., Kelvin), to isolate the risk-price as opposed to the financing-price. (When deals are quoted on a floating rate basis (e.g., LIBOR plus a spread), that separation has already taken place.) The adjusted spread is now comparable to reinsurance quotations.

1999 was a decidedly active and experimental year. Particularly notable was the range and inventiveness of the deals brought to market. Deals covering earthquake and wind perils were repeated from the previous year (e.g., Mosaic, Residential Re) and new deals were completed that combined or extended these risks (e.g., Halyard, Domestic, Concentric, Juno, Gold Eagle, Namazu, Atlas, Seismic). European wind and Midwest quake were added to the more familiar exposure regions. More importantly, two entirely new risk classes were securitized: weather (via Kelvin) and trade credit (via SECTRS). One company (Gerling) isstied three securities all difl'erent - making it second to USAA and Reliance who have both issued similar securities four times.

Several structural innovations stand out. Domestic Re presented the market with the first use of a domestic SPV (via

INEX). It is said that investor acceptance was thereby expanded. Certain classes of investor were not previously able to purchase ILS because of the offshore nature of the SPV.

Kelvin also stands out structurally. Not only did this security contain a new risk class (a portfolio of weather risks from U.S. cities) but the tranching was also unique. The so-called second event tranche was activated if and only if the first event had been previously attached - even if that first event tranche was not necessarily exhausted. Furthermore, the second event cover could only be brought on-risk at certain pre-specified dates. Once on risk, it would only attach after the first event exhausted. In the end, the nature of this tranching, combined with a new risk class, may have proved to be too complicated. Originally targeted as a $200 million issue, the offering was closed after $54 million.

Gold Eagle was based on a portfolio of equally sized "Industry Loss Warranty"-type covers. Each individual cover exhausted in full the moment it attached. Several such individual covers needed to attach to exhaust the junior tranche. The limit of the senior tranche was, however, set equal to the size of the individual covers. By design, it exhausted immediately when it was attached (i.e., C E L = 100%).

Seismic is also worthy of comment. Lehman Re bought index cover from Seismic Re using the PCS catastrophe index for California. Lehman Re was also

have underwritten part of the California Earthquake Authority's reinsurance placement. Lehman Re thereby created a basis risk for may be a harbinger of the way the ILS market will develop in the

by way of innovation, Concentric Re and Circle Maihama stand out. Concentric was not an issue from an insurer or reinsurer, but from the insured itself (Oriental Land). One potential consequence of insurance securitization is that the insured will bypass the insurance industry and go directly to the capital markets. This was the first concrete evidence of such disintermediation. The principle business of Oriental Land is Tokyo Disneyland. A sizeable earthquake

PRICING RISK T R A N S F E R T R A N S A C T I O N S 269

anywhere in and around central Tokyo would affect Disneyland's business. Upon the occurrence of a specified earthquake, Concentric would immediately pay Oriental to compensate for business loss. The exact payment was based on a synthetically constructed scale (i.e., an index payment).

A sister part of Oriental Land's securitization was Circle Maihama. This was a standby facility. It allowed for Oriental Land debt issuance, and could be contingently activated if and only if Concentric was attached. The contingent debt provided Oriental Land with working capital.

The debt is to be issued on prearranged terms that will not change subsequent to an earthquake.

PART I1: COMPARING 1999 PRICES

The gross price of a set of I LS securities issued at par may be expressed as the coupon accruing to the investor. As already observed, however, this is part financing-risk (LIBOR) and part insurance-risk (the spread over LIBOR). As the footnotes in Table I make clear, these spreads have to be adjusted to equiwdent risk periods and day counts in order to make appropriate comparisons i. The true gross price of the insurance-risk is therefore, the "'adjusted spread over LIBOR".

This adjusted spread can be broken down further into two parts. The first part compensates the investor for his expected losses (EL). The second part compensates the investor for assunaing the risk of the investment. It is the alnount the investor requires to commit his risk capital. Financial markets refer to this second spread as "expected excess return" (EER). This is what will be referred to as the price of a security - the net price, if you will.

In a riskless, perfect market, there would be no "expected excess return". However, these markets are not riskless and are demonstrably far from perfect. The risks taken are not obviously hedgeable and the investor needs a return to compensate for risks taken. The EER represents expected profit on the transaction over and above his financing return and expected losses. For the investor (the risk taker), the bigger the risk, the bigger the required expected profit. Even perfect-market financial theorists acknowledgc that risk-adjusted returns will be higher for larger risks.

Reinsurance underwriters see the same thing but through a different lens. Underwriting premiums (pure prices - since there is no prefunding in traditional reinsurance) are viewed as consisting of expected losses plus a " load". Expected losses are defined as the same for both markets (presuming the same data and analyses). The " load" is therefore the insurance analogue of the "expected excess return". It is the price we seek to examine.

For the 1999 transactions, it varies considerably (See Table 2). The lowest "price" is in the 250+ basis points range per annum (e.g., Concentric Re and the

Credit quality is not an issue in these securities because they are fully funded.

270 M O R T O N N. LANE

senior t ranche o f At la s Re). The highest "p r i c e " is 1095 basis po in ts (for the j u n i o r t ranche o f At las Re). Evident ly , there is qui te a range o f riskiness.

TABLE 2

EXPECTED EXCESS REURNS (EER) VS FREQUENCY (PFL) AND SEVERITY {CEL) OF 1999 TRANSACTIONS

Transaction EER PFL CEL

Mosaic 2A 0.0364* 0.0115 0.3652 Mosaic 2 B 0.0552 0.0525 0.5410 Halyard Re 0.0393 0.0084 0.7500 Domestic Re 0.0324 0.0058 0.862 I Concentric Re 0.0272 0.0064 0.6563 Juno Re 0.0381 0.0060 0.7500 Residential Re 0.0327 0.0076 0.5789 Kelvin I st Event 0.0652 0.1210 0.3678 Kelvin 2nd Event 0.0452 0.0156 0.1923 Gold Eagle A 0.0282 0.0017 1.0000 Gold Eagle B 0.0485 0.0078 0.8077 Namazu Re 0.0381 0.0100 0.7500 Atlas Re A 0.0263 0.0019 0.5789 Atlas Re B 0.0352 0.0029 0.7931 Atlas Re C 0.1095 0.0547 0.5923 Seismic Lid. 0.0383 0.0113 0.6460

* The units tire expressed here as decimal points of par. Thus, the Mosaic 2A tranche investor expects to make a profit of 3.64% (0.0364) for taking a risk that has a 1.15% (0.0115) chance of happening. If a loss happens, it is expected that 36.52% (0.3652) of principal will be lost.

H o w are these relat ive prices de te rmined? How does the marke t adjus t for risk? Ill what follows, we are p ropos ing fresh answers to these quest ions . Hopefu l ly this will add new insight to an a l r eady rich deba te on the subject o f r isk-pricing.

We also acknowledge enter ing this inquiry with cer ta in prejudices (hopeful ly based on ra t iona l observa t ion) . First , we believe that one way to cap tu re the a symmet r i c na ture o f loss d i s t r ibu t ion is to measure the " c ond i t i ona l expected loss" ( e E L ) from cer ta in key threshold poin ts (like the a t t a c h m e n t poin t o f the layer). Second, a most impor t an t risk measure is the p robab i l i t y that the cover will a t t ach and that pr incipal and interest will be impai red . Refer to this as the p robab i l i t y o f first do l l a r loss (PFL) .

The intui t ive appea l s o f e E L and P F L are numerous . The chance o f exper ienc ing a loss is an obv ious concern to any investor. It is also the stuff o f ra t ing agency evalua t ions . A l t h o u g h there is some deba te between agencies a b o u t the exact in te rp re ta t ion o f a ra t ing in the contex t o f insurance securit ies (Lane (12)), there is no deba te abou t the in te rp re ta t ion for a s imple c o r p o r a t e

PRICING RISK TRANSFER TRANSACTIONS 27 [

rating. It represents a ranking of the probability of default. Since that involves interest and/or principal, the rating is another name for a security's PFL.

Insurers refer to the PFL as frequency. If the annual probability of getting a loss is 1%, then the interpretation is that the "frequency" is once every 100 years. In Table 2, PFLs range from 0.19% (Atlas senior tranche) to 12.1% (Kelvin junior tranche).

The CEL is the severity of a loss. If a loss happens (the conditional aspect), how big do you expect it to be'? Obviously, the larger the potential hit, the higher (in some sense) is the riskiness of the security. More abstractly, the flatter the loss curve, the higher will be the CEL. In the extreme, i fa loss is total, the CEL " expressed as a percent - of exposure - will be 100%. This is the case with the Gold Eagle senior tranche in the 1999 offerings. Contrast this with the Kelvin junior piece, where, i fa hit happens, it is expected to result in only a 19% loss of principal. Given equal chances of being hit (i.e., the same PFLs), a rational investor would charge a much higher price for Gold Eagle than Kelvin.

Conversely, equivalent severity investments (i.e., equal CEL) would rank investment premiums by rating (i.e., PFL).

In short, there is a trade-off between the frequency and severity that must enter the risk-pricing framework. The precise empirical trade-off is presented in Table 2, which contains an extract of the essential financial statistics from Table I B.

The hypothesis in examining 1999 prices is that the risk-price is a function of frequency and severity of loss.

EER = Function (PFL, CEL)

Further, we are emboldened by early analysis of the 1998 prices to suggest a functional form that is familiar to readers of old economic texts as a general form of the Cobb-Douglas Production Function vis-~i-vis

EER = "),(PFL) ~' x (CEL) ~

Before proceeding to examine this and other possible relationships, we observe the special case where "7 = c~ = /3 = I. Then by definition

EER = PFL x CEL = Expected Loss.

In other words, if we found that "7 = c~ = /3 = I, that would validate the old fashioned " load" rule where the gross premium would be (in this case) twice the expected loss.

This observation also exposes the rule for determining the CEL of a security. All of the 1999 securities PPMs contain statistics on PFL and EL numbers. To extract the CEL, simply divide the EL by the PFL.

CEL = EL 4- PFL.

PFL is a number between 0 and I. The CEL will be a number between 0 and I, where I represents a 100% loss. Note also that we have chosen to express PFL and EL in terms of annual probabilities and annual expected loss. The

272 MORTON N. LANE

probability of first dollar loss over the term to maturity of the investment could also be used, as long as it were consistently associated with total expected loss over the same term.

The intuitive appeal of the Cobb-Douglas form should not blind us to the possibility of other functional forms. The "Risk Cubes" paper provided extensive analysis of the linear form

EER = 7 + cvPFL + f i C E L

with and without a specified intercept c. A respondent, Richard Phillips I (Georgia State University), examined that data and suggested that our fit could be improved by a Taylor 's expansion, at least to quadratic form vis-fi-vis EER = -yl + oel PFL + f i lCEL + 72 + cv2PFL2 + J32CEL2. He further urged us to use the form where the intercept was included as part of the regression. As he observed, without the intercept, interpretations of R2 become problematic. For completeness, six regression fits of the data are conducted. The statistical results are disphtyed below in Table 3.

Professor Phillip's assessment of lhe data is available on the web sile, LaneFinancialLLC.com.

PRICING RISK TRANSFER TRANSACTIONS

TABLE 3

REGRESSION SUMMARY STATISTICS

1999 DATA SET

273

Model I Model 2 Model 3 Model 4 Model 5 Model 6

Dependant Linear Quadratic Log Linear

No Intercept With Intercept No Intercept With Intercept No lutercept Hqth Intercept

Intercept - 0.0325 - 0.016 I

Std Err -0 .02 0.02

(t-slat) 1.90 0.74

PFL 0.6114 0.4404 1.7889 1.7557

Std Err 0.14 0.15 0.38 0.39

(t-slat) 4.52 2.87 4.69 4.48

CEL 0.0456 0.0032 0.0519 0.0029 Std Err 0.01 0.02 3.14 0.07

(t-stat) 6.19 0.14 1.60 0.04

PFL 2 - - -11.4199 - I 1.3106

Std Err 0.03 3.20

(t-slat) -3 .64 -3 .54

CEL 2 - - -0.0248 0.01 I 0

Std Err 0.04 0.06

(t-stat) -0.65 0.18

LN(PFL) . . . .

Std Err

(t-stat)

LN(CEL) . . . .

Std Err

(t-star)

R 2 0.70 0.44 0.80 0.74

R 2 Adjstd 0.23 0.35 0.66 0.65

F Statistic 13.60 5.03 11.80 7.91

p-value 0.002437 0.024059 0.000684 0.002950

-0.5887

0.28

-2 .12

O.5955 0.4946

0.03 0.05

2O.93 9.2 I

0.7777 0.5741

0.21 0.21

3.72 2.79

2.77 0.86

-0 .16 0.84

35.93 43.64

0.000033 0.000001

T h e r e g r e s s i o n s s h o w t h a t the fit is i m p r o v e d by a c o , n p l e t e f o r m o f the C o b b -

D o u g l a s e q u a t i o n vis-f i -vis E R R = 7 ( P F L ) " x ( C E L ) ~ a n d t he bes t fit ex i s t s

w h e r e 7 = 5 5 % , ~ = 4 9 % , a n d /3 = 5 7 % . ( N o t e t h a t "7 is the a n t i l o g o f the e s t i m a t e - 0 . 5 8 8 7 (i.e., / n ( - 0 . 5 8 8 7 ) = 0 .55. )

All p a r a m e t e r s a re s t a t i s t i c a l l y s ign i f ica , l t . A d j u s t e d R2 is a h e a l t h y 8 4 % a n d

the p v a l u e is e x t r e m e l y low - i n d i c a t i n g a n a p p r o p r i a t e n e s s o f the f i t ted m o d e l .

T h e f i t ted e q u a t i o q

E E R --- 0 . 5 5 ( P F L ) ° 4 9 5 ( C E L ) 0"574

274 MORTON N. LANE

is different f rom the best fit o f the 1998 ILS transaction. In that year, evidently a higher /3 coefficient (a round 2.2) indicated that greater pricing emphasis was being placed on severity o f loss. However , the data set was smaller - a less good fit - and less well documented . We focus on the 1999 model.

One compet i tor to risk-pricing used extensively by investment bankers is the multiple o f expected loss. In particular, it is used extensively to compare with bond pricing. The fitted form does not negate that approach. It simply refines it. In effect, the "mul t ip le" is a function o f both PFL and C E L 1

V i s u a l i z a t i o n

The fitted number can only tell part of the story. It is impor tan t to visualize exactly what is going on in this revealed preference function. Il lustration is provided in two ways. First, imagine an underwriter pricing an excess-of-loss cover for four different tmderlying loss distributions - each cover incepting at the same cumulat ive probabil i ty o f a t t a c h r n e n t - using the fitted model to calculate the " load" . The picture is shown in Figure 2 and the numerical results are shown in Table 4. Clearly the effect o f skewness o f the various loss distributions is captured well in C E L numbers. More important ly, our fitted function provides load and total premiunas that accord well with intuitive assumptions.

TABLE 4

RISK LOADS ON THE SAME COVER WHEN APPLIED TO DIFFERENT UNDERLYING DISTRIBUTIONS

Prob of I s' $ Conditional Expected Expected Spread over Loss Expected Loss Loss Excess Return LIBOR

COVER" 5XSS PFL CEL EL EER SPREAD DISTRIBUTIONS Frequenc), Severity Burning Cost Load Premium

Normal 0.050 23.8% 1.2% 5.5% 672 Log Normal 0.050 43.8"/0 2.2% 7.8% 1004 Gamma 0.050 62.7% 3.1% 9.7% 128 I Discrete 0.050 71.9% 3.6% 10.4% 1403

Mathematical Equivalent of this statement: Risk Premium: = 7(PFL) a * (CEL) a Expected loss: = PFL * CEL .'.Full Premium: = (PFL) • (CEL) + 7(PFL) ~' • (CEL) '~

PFL[CEL + 7(PFL) ''-I • (CEL) a-I) EL(I +7(PFL) "-~ • (CEL) fl-']

Full premium is indeed a multiple of Expected Loss. But the exact multiple will vary by CEL and PFL of the deal being considered.

0.030

0.025

0.020

0.015

0.010

0.005

0 .000

I Discmle

i Norm al

iLoa Normal

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I 2 3

PROBABIL IT IES

i~iili :; i~ ̧ ii~ii~i~ill

4 5 6 7 8 9 10 11

LOSS OUTCOMES

12

$6

$5

S4

$3

$2

$1

~ So 13 14 P A Y O U T S ($M)

( o r RECOVERIES)

Z

> Z

-]

> Z o~ > -]

z

FIGURE 2: Comparative distributions incepting with the same (5%) PFL. (Together with a 5 XS 5 Cover, i.e. a Call Spread).

276 M O R T O N N. LANE

The second visualization is shown in panels A, B, C and D of Figure 3 - the process of fitting the cover - from coarse to smooth. Figure 3A shows the 1999 transacted prices as points of individual pyramids in three dimensional space (EER, PFL, CEL). Figure 3B shows the function

EER = 0.55(PFL) ° 495(CEL)°574 draped as it were over the price points. Certain securities stand out, as will be illustrated in the section on cheapness and dearness. Figure 3C shows the surface itself, and Figure 3D shows the trade-off between frequency and severity implicit in the modeled market.

Extrapolation

We can use the fitted parameters to suggest or predict the " load" or "risk premium" that should be attached to proposed securities. An interesting example is a weather bond proposed during 1999. What price should it have had if it was priced consistently with the other 1999 securities? "WINRS'" was the 1999 security proposed by Merrill Lynch that was floated as a concept but withdrawn before issue. Its statistical specifications are listed in Table t. They are E L = 0 . 0 1 2 7 and P F L = 0 . 0 4 7 . By deduction, CEL = (0 .0127-0 .047 ) - -0 .2702 . W I N R S was a weather bond like Kelvin, and its CEL is low (27%) as was Kelvin's at 19%. The cedent for W I N R S was Enron who evidently did not like the market reaction and/or its market price. What should that price have been?

If it were to be priced consistent with the other 1999 securities, it would have been priced as follows:

Expected Loss: = 0.0127 Net Price: EER

Full Price:

Adjusted for Day Count: And converted to basis points Final Price:

= 0.55x(0.047) (°4946) (0.2702) (05741)

= 0.0577 = EL + EER = 0.0704 = 0.0695

= LIBOR + 695 bps

(Note that this assumes a coincidence of the risk period with the term of the note.) Perhaps the market was demanding too big of a concession to this number, or perhaps Enron estimated it was more expensive than other alternatives available to them and withdrew.

Cheapness vs. Dearness

One of the by-products of a model that coherently ties together all risk premiums of a set of securities is that their relative cheapness and dearness can be compared. That is illustrated for model 6 m Figure 4. The "when issued"

P R I C I N G RISK T R A N S F E R T R A N S A C T I O N S 277

199g EX(

(Indi~

EER o r

Risk Load

PFL, Rating or Frequency Note: The scale for the ratings axes k not linear and only part of the range of probabilities is shown.

100%

Figure 3B

199 . . . . . . . . . . . . . . PRE

EER o r

Risk Load

PFL, Rating or Frequency Note: The scale for the ratings axes is not linear and only part of the range of probabilities is shown.

t 00%

FIGURE 3: Transacted prices.

2 7 8 MORTON N. LANE

Figure 3C

19oo D , - U , - ^ I ' - n ~lC:k,_ I-- . . . . . . . ~ . . . . .

P R

EER o r

Risk Load

ccc-

PFL, Rating or Frequency Note: The scale for the rat ings axes is no t l inear and only par t of the range of probabil i t ies is shown.

Figure 3D 1999 R E V E A L E D R I S K P R E F E R E N C E

I N D I F F E R E N C E C U R V E S

(Vert ical V iew)

: : : ~ ;

0% 25% 50%

CEL or

Severity

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75%

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e N Z

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FIGURE 3 (continued): Transacted prices.

1 2 0 0

1000

800

600

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z

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400

I l l 2 0 0 . . . . . . . . . . . . . . . . . . . . . . . . .

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:? ~ . . , ~ ............................................ ., . . . . . . . . x

line ~ : ~ ;;~;~, ~{ ~ Expens,ve Securities below the ~L~,ii~;~ ~i ~ ;~;~i

~, ~,0 , ~ ~ ,, ! ~ . . . . . . -600 ~ : ~l~? ~; ~;~ ~ o . . . . . . . . . . . . . .~.-~.~ ....... ,.o

FIGURE 4: Fitted prices and implied cheapness and dearness on ILS offerings (When issued during 3/99-3/00).

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,-.]

> Z rj~

,.-]

>. 'Z

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280 MORTON N. LANE

market price is plotted against the price as implied by the model. As one would expect from the R2, there is an appealing similarity in the curves. Also shown (in yellow bars) is the difference between the two lines. If the difference is positive (market price is greater than implied), the security is said to be "cheap". Conversely, securities where the implied is greater than the market price, are said to be expensive and are shown below the line. Cheap securities provide a balance of benefits to the investor; expensive securities tire better for the issuer.

Most securities appear to be fairly priced. However, three securities stand out: (a) the junior tranche on the Mosaic Deal; (b) both Kelvin tranches, and (c) the Atlas tranches. The Atlas offering was most recently offered. Its entire offering appeared to be cheap. In the Kelvin transaction, the junior tranche was very expensive; the senior tranche was quite cheap.

Consider the Kelvin senior piece. It could not go on risk until the second of its three-year term and might not go on risk until the third year. Meanwhile, investors received coupons for all three years. Evidently, the investors did not appreciate or understand this calculation and did not adjust their prices upwards. They acquired the deal cheaply. On the other hand, the junior investors appeared to have paid too much. Given the relative tranche sizes, investors who bought both tranches would have been nearer the market price though still on the expensive side.

The junior Mosaic 2 tranche was also expensive. It was issued in March of 1999, and notably, it was not renewed in March of 2000 (which would have been its third renewal). The isstier apparently chose to withdraw from the market rather than pay a price comparable to other then available instruments. Consider that, if renewed on the same terms, a March 2000 investor would have been confronted with this choice

Junior Mosaic 3 Atlas C

PFL 0.0525 0.0547

CEL 0.541 0.592

EER 0.0552 0.1095

There is not much of a choice. Atlas was a "slam-dunk" better. Mosaic chose not to pay and did not renew.

Bidding: Weak vs. Aggressive

The preceding analysis was on a "priced as when-issued" basis. It would be more intellectually appealing to have till the deals priced on the same day. This would avoid shifts in the market pricing that might have occurred between the issues becoming buried in the analysis. Fortunately, one market participant (Goldman Sachs) provides a consistent weekly bid and offer sheet allowing contempora- neous comparisons to be made. Unfortunately, such comparisons are not easy

PRICING RISK TRANSFER TRANSACTIONS 281

with insurance-linked notes. Exposure to insurance risk does not unfold at an even daily rate. Full comparison needs seasonal adjustment - something beyond the scope of this paper.

Nevertheless, an attempt has been made to compare the bids, to see where Goldman's demand lies. One seasonal adjustment is made: Kelvin (junior) is left out of the contemporaneous fit. By March 17, 2000, Kelvin already had a good experience (a warmer than expected winter) under its belt. Probabilities of loss and expected loss would have to be adjusted accordingly. Other securities where no known events have occurred are assumed to have the original risk statistics still extant. Notice that this includes old outstanding issues (Parametric Re, Pacific Re) that were not part of the Class of 1999.)

Table 5 displays Goldman's prices. Figure 5 illustrates its bidding preferences. Goldman likes, or has a demand for, earthquake deals (Pacific, Parametric, Domestic, Concentric, Seismic). Indeed it appears to accumulate and offer portfolios of such deals. It is aggressive in bidding for Kelvin's first event (because of good weather experience), and it also bids for the (originally) cheap Kelvin second event. However, it has weaker bids on the wind portfolio deals of its competitors (Gold Eagle and Atlas). Remember that Atlas came slightly cheap at issue. Goldman bids back but appears to want to avoid the junior tranches (Atlas C, Gold Eagle B). Perhaps it is full on these names.

P A R T I11: EXTENSIONS - A UNIVERSAL PRICING GRID?

The insurance-linked securities market is new. Its dozen securitizations during 1999 may be too small to support extensive statistical analysis and reliable results. And yet, from our rudimentary analysis, we have found a model that seems to provide a good, if not perfect, explanation of the prices of a wide range of differently risked insurance security mulches. Furthermore, it is a model that builds on two widely understood characterizations of risk - frequency and severity. The question that is intriguing is this: "Would the fitted parameters from the ILS market bear any relevance to other risky capital market instruments which can be characterized by frequency and severity? If not, why not?"

We first examine the case of corporate bonds.

Corporate Bonds

A senior, unsecured, AAA rated corporate credit is said by the rating agencies to have an annual probability of default of 1.5 basis points (i.e., probability of 0.00015). Triple-B securities are expected to default once every 588 years (i.e., probability of 0.0017). Their PFLs are respectively 0.00015 and 0.0017 and are shown with other bond ratings in Table 6.

TABLE 5

GOLDMAN SACHS'S ILS QUOTE SHEET MARCH 17. 2000

tO O0 tO

Class SPV Rating Risk t rpe Price Bid]Ark Size Spread Rid/Ask Maturity YLD @ Isxue

U.S Earthquake Domestic Ba2/BB Mid-West Quake 100.9/101.12 5.0MM/- L+315/310 4-30-02 L+369

Seismic Re Notes Ba2/BB California Quake 99.58/100.07 2.0MM/- L+475/445 l-I-02 L+450

Juno Re BB/BB+ East Coast Hurricane 100.43/100.73 5.0MM/ L+395/380 6-26-02 L+420

Japan Earthquake Concentric BaI/BB+/BB+ Tokyo Quake 99.90/100.15 5 0MM/- L+309/304 5-13-04 L+310

Namazu Re BB/BB Tokyo and Eastern Tokal Quake 98.49/9949 2 0MM/3.0MM L+491/464 12-2-04 L+450

Parametric Re Ba2/BB Tokyo Quake 102.09/102.75 5.0MM/- L+388/375 I 1-19-07 L+430

Weather Kelvin Ist Event B- O.S. Weather 101.13/101.59 5.0MM/5.0MM Fixed 2-14-03 15.70%

Kelvin 2nd Event BBB-/BB+/BB+ U.S Weather 98.76/- 5.0MM/- Fixed 2-14-03 8.70%

Typhoon Pacific Re Ba3/BB- Japan Typhoon, Drop Down 100.00/t00 59 5.0MM/- L+370/348 5-31-03 L+370

Atlas Class A (a) BBB+/BBB/BBB European wind, U.S. quake, L+270/261 4-4-03 L+270

Atlas Cla~ B (a) BBB-/BBB-/BBB- European wind, U.S. quake. L+370/36i 4-4-03 L+370

Atlas Class C (a) B/B-/B- European wind. O S. quake. L+1412]1388 4-4-03 L+I400

Gold Eagle A Baa3/BBB- U.S. East & Gulf Coast Hurricane: 100.00/100.48 2.0MM/0 365M L+295/247 Apr-01 L+295 Mid-West and CA quake

Gold Eagle B Ba2/BB U.S. East & Gulf Coast Hurricane, 99.75/100.24 2 0MM/- L+564/515 Apt-01 L+540 Mid-West and CA quake

OTHER ILS QUOTED, BUT NOT CONSIDERED CURRENT IN PRICE ANALYSIS

Grcle Maihama A/A Tokyo Quake Contingent Capital 99.27/- 5 0MM/- L+ 95/- 5-13-04 L+75

U.S. Hurricane Residential Re Ba2/BB East & Gulf Coast Hurricane 100.57/100.67 5.0MM/- L+I00/50 6-1-00 L+366

Portfolios of Risk Japan quake 100.00/100.23 5.0MM/3MM

Japan quake 100.00/100 25 5 0MM/-

Japan quake 99.711100.25 2.0M M/2.5MM

Parametric Re Units Baa3/BBB- Tokyo Quake 101.62/- 5.0MM/- L+185/- 11-19-07 L+206

©

0 Z

Z

Z rn

1200

1000

,.Q

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I,U m," r,

800

600

400

200

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284 M O R T O N N. L A N E

T A B L E 6

PRICING CORPORATE BONDS FROM ILS'S REVEALED RISK PREFERENCE FUNCTION*

,4 A,4 A A A BBB BB B CCC

PFL PFL PFL PFL PFL PFL PFL

CEL 0.57 0.00015 0.0004 0.00075 0.0017 0.0075 0.02 0.08 (Assume constant CEL)

EL = In bps 0.9 2.3 4.3 9.7 42.8 EER = Inbps 51.6 83.9 114.4 171.5 357.4 E L + E E R = In bps 52.5 86.1 118.7 181.2 400.2

Implied Spread over LIBOR 52 86 119 181 400 Actual Bond Spreads** 67 92 120 183 350 Difference - 15 - 6 - 1 - 2 50

Risk Multiple EER/EL 60 37 27 18 8 $ per $1CEL EER/CEL $0.91 $1.47 $2.01 $3.01 $6.27

114.0 456.0 580.6 1152.6 694.6 1608.6

695 1609 580 1147 I 15 462

5 3 $10.19 $20.22

* Uses the following parameters: Gammtl 0.5551; Alpha 0.4946; Beta 0.5741. ** Lehman Bros 10/19/99.

Various studies show that the recovery or salvage rate for defaulted senior, unsecured corpora te debt is a round 43% of par. It varies from period to period, but 43% represents a good long-term average. The condit ional expected loss (CEL) o f senior unsecured debt is therefore 57%.

Given the P F L and C E L o f a set o f bonds, what price would be implied by the fitted model? The answers are displayed in Table 6. To stick with the BBB example, if the P F L = 0.0017 and the C E L = 0.57, the annual expected loss on BBB's is 9.7 bps. The implied risk premium would be

EEE = 0.55 × [(0.0017) (0"4946)] × [(0.057) (0"5741)]

= 171.5bps.

Therefore, the implied price o f BBBs should be 9.5 + 171.5 = 181 bps. The actual market spread is displayed in Table 7. It is 183 bps. This was

quoted by Lehman Brothers in the middle o f the 12-month period under exam (10/19/99). The extreme closeness o f the BBB compar i son is a for tunate coincidence. Notwi ths tanding, it lends support to the idea that a risk premium is a risk premium is a risk premium - in whatever market it appears.

Figure 6 shows the coincidence o f pricing for all other corpora tes ratings. Diagrammatical ly , Figure 6 displays a tantalizing similarity between actual and ILS-implied bond prices. Closer examinat ion o f the Figure 6 (or Table 7) shows that the ILS market demands a higher premium than the bond market for the lower rated tranches. Last year it demanded a higher premium on senior tranches. Perhaps there is a novelty value to the new, more prolific, junior tranches in the ILS market.

P R I C I N G RISK T R A N S F E R T R A N S A C T I O N S 285

T A B L E 7

CURRENT SPREADS FOR INTERMEDIATE* DEBT

Yield ( TW) Spread

USTR 5.98 - Aaa/AAA 6.65 67 Aa/AA 6.9 92 A/A 7.18 120 Baa/BBB 7.81 183 Ba/BB 9.48 350 B/B II .78 580 Ccc/CCC 17.45 1147 Cc/D 22.6 1662 144-A IG 8.02 204

Source: Lehman Brothers 10/19/99. * Approximately 5 years maturity.

A complete and rigorous comparison of prices is beyond the scope of this paper. Strictly done, bond yi'elds must be adjusted for maturity, liquidity, call provisions, conversions, etc. But the spot comparison above motivates further research, as do other implications outlined below.

The spot comparison contained in Figure 6 (Table 7) is one where the CEL is held constant and a comparison is made between different ratings. How about deals where the rating is held constant and the severity differs? Does the model predict pricing consistent with experience in these areas?

Two areas where these questions arise are: comparison between different industrial sectors and comparison between seniority of claims.

Implications by Sector

Table 8 shows the differing severity of loss experienced in four different industrial sectors and the predicted spread of BBB credit in each sector based on ILS pricing. Market spreads by sector are surprisingly difficult to obtain with any consistency. Indeed, the severity measures themselves are rarely given. These are from the DLJ study of 1997. Utility defaults typically result in smaller absolute losses to bond holders than do defaults in finance companies (CEL's of 39% vs. CEL's of 61%). For equivalently rated BBB credits, therefore, the ILS market suggests that utility bonds should trade 44 bps more expensive than bonds from the finance sector (145 bps vs. 189). Other industrial sectors should display similar discounts to utilities (manufacturing 19 bps for example). While exact market comparisons are difficult, a recent spot check of BBB securities by sector showed Finance-Utility spreads of 48 bps and Manufacturing-Utility spreads of 28 bps.

1800

1600

1400

1999 ILS Implied Spreads over L IBO~

Actual Corporate Spread over LIBOR i

L e h m a n Ind icat ions (10/19/99)

1200

D.

1000 ¢,.. , I

r t 8oo <

ILl n,, D.

600

400

©

© Z

> z

200

"+ 7.5 17

AA A BBB BB B

R A T I N G FIGURE 6: Implied vs actual corporate bond yields (Spread over LIBOR).

Note: The scale for the ratings axes is not linear and only part of the range of probabilities is shown.

CCC

PRICING RISK TRANSFER TRANSACTIONS

TABLE 8

SEVERITY OF LOSS BY INDUSTRY

287

Industrl" CEL * Implied

( BBB rated) ( Approx ) Spread

Utilities 39% 145

Energy 41% 149

Manufactur ing 48% 164

Consumer Durables 63% 192

Finance 61% 189

* Source: DLJ Default Study.

Implications by Seniority

Similar comparisons can be made between the seniority of claim of bondholders. In distress situations, different claimants for the remaining assets rank ahead of others. Senior secured debt is first, junior debt is last. The amount of loss each class experiences is therefore the average severity of loss or CEL (see Table 9). Given a set of equivalently rated BB tranches, the 1LS pricing model would predict a spread of 138 bps between senior secured debt and junior subordinated debt (351 bps vs. 479 bps).

TABLE 9

SEVERITY OF LOSS BY SENIORI'FY

Seniori o, C E L * hnplied ( BB rated) ( Approx ) Spread

Senior Secured 46% 35 I

Senior Unsecured 57% 400

Senior Subordinated 64% 430

Junior Subordinated 76% 479

* Source: DLJ Default Study.

In practice, such comparisons are difficult to make because not all debt of different seniority is rated the same. And, not all industrial sectors issue in the same rating classes. Market spreads therefore might be wider or narrower than predicted here. Our objective, however, is not to predict spread. Rather, it is to illustrate the virtues of a revealed preference function that utilizes both frequency and severity as components of risk pricing. It appears to be consistent with the way the bond market discriminates between credits. One other example may drive the point home: leveraged loans and general corporates.

288 MORTON N. LANE

Implications for Leveraged Loans

The "leveraged loan" market is increasingly a traded market where market prices are obtainable. Leveraged loans are loans to sub-investment grade borrowers that nevertheless have senior secured claims in default. Their severity of loss is reckoned to be very low (CEL _< 20%). Compared to an equivalently risked corporate, the ILS revealed risk preference function would predict an 83 bps differential between leveraged loans and corporates vis-fi-vis:

Leveraged Loans Corporate (BBB rated) CBBB rated)

PFL 0.0017 0.0017 CEL 0.20 0.57

Risk Premium 0.94 171.5 Adjusted Spread 96 bps 179 bps

This is somewhat consistent with market observation. Another discrepancy that exists in credit markets is the price differential

between equivalently rated corporates and CBO or CLO tranches. The explanation to this spread we believe also emanates from CEL differentials, though detailed examination is beyond us here.

In this section we have demonstrated that the fitted revealed risk preference fi'om the ILS market has plausible implications for other fixed income markets. However, in no way do we assert the superiority of the ILS market. To the contrary, the bond market is bigger, deeper, more liquid, and longer lived as a traded market. It should dictate risk preferences to the ILS market not the other way around. But as far as we know, no one has tried to explain prices in either market in quite the way done here. To our mind, it calls for an empirical study of corporate bonds to gauge the risk-return trade-offs implicit in the credit markets.

Two Way Rating System - A Suggestion

We could not leave this discussion of risk-pricing without a call to rating agencies to enhance their categorizations. They do a magnificent job of grading all sorts of credits by the likelihood that the credit will fail to honor its obligations - to default. However, this "frequency estimate" is only one dimension of risk assessment and therefore of risk pricing. Severity is as important as frequency. Why not rate credits on both dimensions - a two-way system?

PRICING RISK T R A N S F E R T R A N S A C T I O N S

T A B L E 10

RISK PRICING BY TWO-WAY RATING (USING FI'FrE[)* ILS PRICES)

289

PFL PFL PFL PFL PFL PFL PFL

0.00015 0.0004 0.00075 0.0017 0.0075 0.02 0.08

EXPECTED LOSS(EL) AAA AA A BBB BB B CCC

CEL 20% I 0.3 0.8 1.5 3.4 15.0 40.0 160.0 OR 40% I I 0.6 1.6 3.0 6.8 30.0 80.0 320.0

SEVERITY 60% III 0.9 2.4 4.5 10.2 45.0 120.0 480.0 OF 80% IV 1.2 3.2 6.0 13.6 60.0 160.0 640.0 LOSS 100% V 1.5 4.0 7.5 17.0 75.0 2 0 0 . 0 800.0

EXPECTED EXCESS RETURN (EER)* or "'LOAD" AAA AA A BBB BB B CCC

CEL 20% I 28.3 46.0 62.7 94.0 195 .9 3 1 8 . 3 631.8 O R 40% II 42. I 68.4 93.4 140 .0 291 .7 4 7 3 . 8 940.5 SEVERITY 60% Ill 53.2 86.4 117 .9 176 .7 368.1 598 .0 1187.0 OF 80% IV 62.7 101.9 139 .0 2 0 8 . 4 434 .2 705 .3 1400.2 LOSS 100% V 71.3 115.8 158 .0 236 .9 4 9 3 . 6 801 .7 1591.5

SPREAD OVER LIBOR (EL+EER) AAA AA A BBB BB B CCC

CEL 20% I 28.6 46.8 64.2 97.4 210 .9 3 5 8 . 3 791.8 OR 40% II 42.7 70.0 96.4 146 .8 321 .7 553 .8 1260,5 SEVERITY 60% 111 54.1 88.8 122 .4 186.9 413.1 718 .0 1667.0 OF 80% IV 63.9 105.1 145 .0 222 .0 4 9 4 . 2 865 .3 2040.2 LOSS 100% V 72.8 119.8 165 .5 2 5 3 . 9 568.6 1001.7 2391.5

Data presented m annual basis points. * Uses the following parameters: Gamma 0.5551; Alpha 0.4946; Beta 0.5741.

To illustrate how this might work, consider Table 10. It shows a matrix classification of risks and a current estimate of pricing in each class. The vertical

classification is the convent iona l rat ing analysis using, in this case, S&P classifications. The horizontal strata is simply a division of risks in to diffe,ent CEL classes, labeled I to V for each 20% interval.

Thus, a I I /AA risk would have a loss severity of between 20% to 40% and a likelihood of default of 0.0004 per a n n u m . A deep ou t -of - the-money weather call comes to mind. It would be priced at approximate ly LIBOR + 70 bps. A IV/ B security would imply loses a round 60% to 80% and a likelihood of occurrence of 2%. Priced consistent with the I LS market , it should c omma nd LI BOR + 865. Interestingly, equity which could be viewed as having a CCC chance of a lmost total loss should carry a spread of close to 25% - very similar to what is ant ic ipated for equity over and above the risk-free rate.

1500-

SPREAD OVER LIBOR (in basis points)

(EL+EER)

1000

50(

SEVERITY ( or CE L)

20%

CCC

CONVENTIONAL RATING (FREQUENCY)

(or PFL)

FIGURE 7: Risk pricing by two-way rating categorization.

PRICING RISK TRANSFER TRANSACTIONS 291

More exploration of two-way rating is not appropriate here. Suffice it to say that we believe such a system would be a boon to investors and a welcome new product for the agencies themselves.

Concluding Remarks

This paper has done three things. First, it has presented a record of insurance securitizations that occurred between March 1999 and March 2000.

Second, it has fitted a function to transacted prices that goes a long way to explaining those 1999 prices in terms of the frequency and severity of loss. The fitted function is a "utility function" or a "revealed risk preference" function that is well ordered, and satisfies the basic requirements of such ft, nctions. We have illustrated the usefulness for comparing relative cheapness and dearness of ILS prices and for discriminating between weak versus aggressive bidding patterns.

Thirdly and finally, the paper has sought to explore the implications of this risk-pricing in the context of corporate bonds. Both corporate credit risk and insurance risk is locally non-hedgeable and might be expected to carry similar risk premiums. The exploration is persuasive enough to motivate a large empirical study of risk premium in the bond market. Joint estimation of risk preferences should then lead to superior insights for insurance risk managers and portfolio investment managers alike.

Beyond the base line observations, we think that the admission of the existence of a universal pricing grid or utility function can lead to some important insights in other contexts. Consider VAR (Value-at-Risk) as a risk management tool. Always controversial, VAR sets a (PFL) level as a mechanism to control risk taking. However, the 1998 credit crunch accompanied by the collapse of Long Term Capital Management have caused some to want to also control the conditional loss beyond the VAR attachment point. It is referred to in the literature as controlling "conditional tail loss", but essentially it is about controlling severity once an acceptable frequency of loss has been agreed upon. Anyone doubting that equal VAR points can have differing "conditional tail loss" ,night flip back to Figure 2, Table 4 to refresh their ideas on how underwriters view such risks.

Another context where frequency and severity questions are increasingly being asked is in the performance of hedge funds. One analyst, Leah Modigliani (16) at Morgan Stanley insists that performance returns should be adjusted for (among other things) frequency of draw downs (losses) and the severity of such losses. It seems that she is asking whether the ex-post performance of funds is consistent with market trade-offs between these two risk measures. Our ex-ante trade-otis estimate might well contribute to that analysis.

Whatever the financial context, the use of revealed frequency and severity trade-offs exposes some fascinating new vistas and some intriguing questions. For example, can the individual frequency/severity trade-off parameters be used for the whole investment portfolio? Can they be used by the reinsurer to gauge

292 MORTON N. LANE

where to take leverage in his book - in reinsurance or investments? Can the shift ing spreads in the bond market be ascribed to revised probabi l i ty estimates or to changing risk preferences? How can the two be dist inguished?

In summat ion , th inking abou t ILS securities has led to th inking abou t credit risks. This cross fertilization exercise has, we believe, many impor tan t implicat ions for further future examinat ion .

REFERENCES

I. BANTWAL, V.J. and KUNREUTHER, H.C. (2000) A Cat Bond Premium Puzzle? In The Journal of PSl|:Chology and Filzancial Markets, Vol. I, No. 1, 76-91.

2. BLUME, M.E., DONALD, B.K. and SA A.P. (1991) Returns and the Volatility of" Low Grade Bonds. Journal ~" Finance 44: 909-922.

3. CANABARRO. E.. FINKEMEIER. M., ANDERSON, R.R. and BEND[MER,XD. F. Analyzing Insurance-Linked Securities," The Journal oJRis'k Finance, Vohune I, No. 2, 49-75.

4. DEROsA-FARAG, S., BLAU, J., MATOUSEK, P, CHANDRA, 1., JAGG], H. and REBEG A. (1998) Default Rates in the High Yiehl Market." An Examination Within the Context of Overall Market Rirk. Donaldson, Lufkin 7 Jenrette Securities Corporation,

5. ELTON, E., and MaRTin J.G. (2000) Explaining the Rate Spread on Corporate Bonds forthcoming..Iournal of bTnance.

6. FROOT, K.A. and POSNER, S. (2000) Issues in the Pricb~g of Catastrophe Ri~k. Guy Carpenter Marsh & McLennan Securities White Papcrs. May 2000. < http://www.guycarp. com/publications/white/whitep.html >

7. KEALHOI:E.R, S., KWOK, S. and WENG, W. (1998) Uses and Abuses of Bond Default Rates. KMV. LLC. hulustrv htsider. Technical Papers. 3 March 1998.

8. KRF.PS, R.E. (1990) Reinsurer Risk Loads Ji'om Margimd Surphts Requirements. PCAS LXXVll, 1990.

9. KREPS, R.E. (1998) Investment-Equivalent Reinsurance Pricing. 1998 PCAS Proceedi,gs. Also available from Guy Carpenter Instrat Publications, 14 February 2000.

10. LANE, M.N. and BECKWlTH, R.G. (2000) Trends in the Insurance-Linked Securities Market. Lane Financial L.L.C. Trade Notes, May 31, 2000. < http://www.lanefinancialllc. con] >

11. LANE, M.N. ~.nd MOVCHAN, O.Y. (1999) Risk Cubes or Price, Risk and Ratings (Part II). Sedgwick Lane Financial LLC Trade Notes, March 15, 1999. < http://www.lanetinancialllc. corn >

12. LANE, M.N. (1998) Price, Risk and Ratings for Insurance-Linked Notes: Evaluating Their Position in Your Portfolio. Derivatives Quarterly, Spring [998. Based on a presentation made before the conference, "'Rethinking Insurance Regulation 1998" sponsored by the Competitive Enterprise Institute in Washington, DC on April 13, 1998.

13. LANE, M.N. (1997) A Year of Structuring Furiously: Promises, Promises... Sedgwick Lane Financial LLC Trade Notes, January 31, 1997. Also published in Eaergy hzsura,ce Review. Spring 1997. < hltp://www.lanefinancialllc.com >

14. LITZENBERGER, R.H., BEAC, LEHOLE, D.R. and REYNOLDS, C.E. (1996) Assessing Catastrophe Reinsurance-linked Securities as a New Asset Class. Journal of Portfolio Management (December): 76-86.

15. MANGO, D. (1999) Risk Load and the Default Rate of Surplus. In Casuahv Actuarial Socieo, 1999 Discu.~sion Papers on Securitization Risk. < http://www.casact.org/pubs/dpp/ dpp99/index.htm >

PRICING RISK TRANSFER TRANSACTIONS 293

16. MODIGI.IANI, L. (1997) Investment Strategy: Are Hedge Funds Worth the Risk? In Morgan Stanley U.S. Im,estme, t Research (December 12, 1997).

17. VAN DE CASTLE, K. and K EISMAN, D.(1999) Recovering Your Money: I nsighls Into Losses From Defaults. In Sta,dard & Poor~v Credit H"eek, June 16, 1999, 29-34.

M O R T O N LANE Lane Fi , ancial L L C

321 Meh'ose A venue Kenilworth IL 60043-1136 USA

S U P E R - E F F I C I E N T PREDICTION BASED ON H I G H - Q U A L I T Y M A R K E R I N F O R M A T I O N

BY

JENS PERCH NIELSEN

Codan. Gammel Kongevej 60, 1799 Copenhagen, Demnark

ABSTRACT

Nielsen (1999) showed the surprising fact that a nonparametric one-dimensional hazard as a function of time can be estimated x/~-consistently if a high quality marker is observed. In this paper we show that the hazard relevant for predicting remaining duration time, given the current status of a high quality marker, can be estimated v~-consistently if a Markov type property holds for the high quality marker.

K E Y W O R D S AND PHRASES

Counting Process, Hazard, Kernel, Marker, Nonparametr ic estimation, Predic- tion, Survival analysis, Asset-Liability management, Datamining, High fre- quency data.

1. INTRODUCTION

Prediction of future events is important in many fields (e.g. biostatistics, actuarial science, finance and economics). For example in prevalent data studies, one often wishes to be able to make predictions based on the available information. Ii1 this paper we consider a general way of modelling a marker process and a hazard such that good prediction power remains. We do not have any parametric assumptions of the marker process or the hazard, but we do assume that the marker process obeys a Markov property and that the marker process is of high quality as defined in Nielsen (1999) that estimates the traditional deterministic hazard rate as a function of time. This estimation is improved through the knowledge of a high quality marker and the resulting procedure is more efficient than traditional approaches to nonparametric hazard estimation that do not use this extra information, see Nielsen (1998) for an overview of some of these more traditional kernel hazard estimators. The present paper is concerned about the estimation of the future hazard given the

ASTIN BULLETIN, Vol. 30. No, 2. 2000. pp. 295-303

296 JENS PERCH NIELSEN

current state of a continuous marker. The marker therefore enters as an integrated part of the model and the stochastic Future hazard that we aim at estimating. Therefore the concept of high quality markers seems to even more important in the current study than in Nielsen (1999).

A lot of actuarial work is about modelling, in which the important thing is to make the right approximation to the problem at hand. Statistical testing is not always appropriate since the actuary knows very well that his model is not true. The actuary has another consideration - how can I make a very general model that is easy to understand and that gives me appropriate p,'edictions for the future'? The modelling in this paper can be considered as a procedure following this actuarial tradition. Though testing procedures of the validation of the model indeed need to be developed, the most important Fact is that here is a rather general model that gives good prediction power. Our postt, late of good predicting power is based on the fact that our non-parametrically based estimator of the future hazard given current marker conditions is estimated v~-consistently. This is in general not possible without parametric assumptions, the surprising fact that it holds in our situation is a result of the complicated interaction between the model assumptions of the marker process and the hazard function. We illustrate the applicability of the model in § 5 by considering possible applications in so diverse fields as asset-liability management, datamining, biostatistics and the analysis of high frequency financial data.

To get started, consider n individuals with survival times TI,..., T,, and marker processes {Zl(s) ,s t + to}.

We make the stability assumption that h.~,to(t) does not depend on to and write h,,(t) = h,,to(t). This assumption is realistic if the marker process is a Markov process and of high quality. A marker is of high quality if the hazard at any given time only depends on the marker, see Nielsen (1999). Under the stability assumption and the asst, mption that the marker is of high quality, we show that h.~.(t) can be estimated v/~-consistently. In the more precise model formulation below, we allow for filtered data includi,ag prevalent data. In §2 we formulate the two models, the simple time model and the marker-only model. In §3 the estimators are defined and in §4 we state the pointwise asymptotic theory of the v,5~-consistent estimator of the hazard as a function of time. We use standard counting process theory to Formulate our model and consider n individuals i = 1, ..., n with N, ('') counting observed failures For the i ' th individt,al in the time interval [0, T]. We assume that N ('') = (NI"),...,N,I '')) is an n-dimensional counting process with respect to the increasing, right conti,mot, s filtra- tion b e, = cr(N(s), Z(s), Y(s); s < t ) , I E [0, T] ,,,here Y(")= (YI ''), ..., Y,I '')) is the n-dimensional exposure process, x~here ~") is a predictable process taking wflues in {0, 1}, indicating (by the value 1) when the i'th individual is t, nder risk.

( 7 (n) 7 (n)'~ We assume that we get some further marker information Z OO = ~ l ,..., '-,, J,

SUPER-EFFICIENT PREDICTION BASEl) ON HIGH-QUALITY MARKER INFORMATION 297

where ZI '') is a d-dimensional , predictable C A D L A G marke r process and that we are in the high quality marker case, which Nielsen (1999), defined in the following

(. ('0 ..., AI~')) based on the increased way: the s tochast ic intensity process A(") = ~A~ , filtration 5 c, = o-(N(s), Z(s) , Y(s); s < t) depends only on the value o f the marke r . in the following sense

g"/,)--

Apar t f rom smoothness assu, 'nptions the functional form of the marke r -on ly hazard, e~, is assumed to be unknown. We fu r the rmore assume that the marke r process have the Markov property that

G,,,,o(Z) = P r { Z i ( ; + to) _< z I ze ( ;o ) = y , Ti > t + to}

is independent o f to and use the nota t ion

F,..t(z) = F,,.,,,o(z ).

Let ./j,,, (z) be the density o f F~,.,(z) with respect to the d-dimensional Lebesgue measure. We also assume the following condi t ion of tmbiasedfilterffTg:

F,,,t(z) = Pr{Zi(t + to) < z [ Zi(to) = y, Yi(to) = ri(t + to) = 1}

Let also

F,(z) = Pr{Zi(t) _< z I Y,(0) = I}

and let fr(z) be the density o f Ft(z) and

Hy( l ) - Pr{Yi(t + to)= 11 z;(;o)=y, ~',(,o)= l}. We assume that the marke r Z,(s) has suppor t on some compac t set A/" and that the densities~,,t and the hazard c~ are uniformly bounded away from zero and infinity.

We asst, me that E{Yi(s)} = H(s) , where H(o) is cont inuous . The marke r Zi(s) is only observed for those s such that Yi(s) = I. Let

-* s { Z,.(s) when Y i ( s ) = I

Z; (.) = - oo when Yi(s) = O.

We call Z 7 the observed marke r process. We assume that the s tochast ic processes (NI, Z{, Y,,), ..., (N,,, Z,~, Y,,) are lid for the ,; individuals. Under the " above model assumpt ions a more precise definition of the M C F H is

= e [o , {z ; ( ; + ;0)} I z;(r0) = y, ~',(;0) = v;(; + ;0) = 1] : f h.v( t) ct(g)f v,t(z)dz.

In the following we will show that under the above assumpt ion hy can be es t imated v57-consistently.

298 JENS PERCH NIELSEN

2. DEFINITION OF THE ESTIMATOR

The following estimator of a was introduced and analyzed in Nielsen & Linton (1995) and it was employed by Fusaro et al. (1993) to estimate the risk of Aids given current marker status based on the San Fransisco Mens' Health Study. Let ffi(z) = n- ' ~_#i f T Kb{z -- Zk(s)} Yk(.v)(£" and

n -l ~/~.#, fo' Kh{Z -- Zk(s)}dNk(s) &(~) = a~(~_) '

where K is a second order kernel of d dimensions. Both ffi and ~i are leave-one- out according to the definition given in Nielsen. Linton & Bickel (1998).

We estinaate hy(t) by the empirical version of J'o,(z)f,v(z)dz:

g,(t) = ~ ' ' : ' "l°f & {z ' ( t + s)} r,(t + s) Y , ( s ) & { y - Z,(s)}ds ~','=, fo T Yi(t + s) Yi(s)Kb{y - Zi(s)}ds

Remark. Full knowledge of the marker process is only possible if the marker changes deterministically between observed time points. In many examples, including using CD4 cell counts or other surrogate markers to predict onset of Aids of HIV+ infected individuals, the marker is only observed at discrete timepoints. Therefore the methodology of this paper requires an interpolation and extrapolation technique. Interpolation is done between two observed points in time and extrapolation is performed from a point in time, where no succeeding observation exists within a suitable time, see Fusaro et al. (1993) and Nielsen (1999) for more comments on this issue.

Let

and

Let also

3. PROPERTIES OF THE ESTIMATOR

&, t(z) = .fo T f ,,r(z)Hy(t)[s(y)H(s) / {,~o TJi~(z)H(u)dtt }d&

/o gy(t) = H~,(t)A(y)H(s)ds

.~,.(t) = n -I Yi(t + s) Y~(s)Kh{y - Zi(s)}ds. i= I

~0 7" Xi = c~i{Zi(t + s)} Yi(t + s) Y i ( s )Kb{y- Zi(s)}ds - h v(t)'~.,,(t).

SUPER-EFFICIENT PREDICTION BASED ON HIGH-QUALITY M A R K E R INFORMATION 299

The error of estimation can be written as

g( , ) - (,)}-',,-, x, i = I

n L'/" + { g ' ( / ) } - ' " - ' ,=~l • py , t {Z i ( s l } dM i ( s )

+ {£ (0)- ' {R, (,) + ,%(0},

where

p,.,,,,(z) = , , - ' ~ kg:i

Rl(t) = . - I ~ i= 1

n

R~(,) = , - ' ~ i = I

o r K h { Z , . ( t + s ) - =}Kb{y - Zk(~)} r , ( .~) / .~{Z~( t + s ) } &

o"['k,,,,{ z,(,,,) } - r,.,,,,{ z,(.~) } ] da4~(,~),

L T[(~; _ ~,){z,(~ + ~)}] v,(, + 4 r , ( a ) K ~ { y - Z~(.~)}d.,, where c:; corresponds to ~i, but with Ni replaced by its compensator A,:

,

300 JENS PERCH NIELSI-"N

inserting consistent versions of the unknown quantities and then use these estimators to estimate the empirical variance of the sum of the identically distributed stochastic variables.

Theorem I. [Pointwise Convergeneel Assume that oe is twice contimwusly dif/'erentiable and that nb d+l ~ oo and n b 4 ~ O. Assume finally that the kernel K sati.~es the Lipschitz condition given in Theorem 2 in Nielsen & Linton (1995) in each coordinate. Then

where

and

/ A .1 n-{h~,(t) - h.,:(t)} ~ N(O o~?,)

-~ 2 0 "2),,, = {g.,,(t)} -(Cyl..,.., + o'~_ r,, + 2cr~.2.,,.,).

j.jo ,T -) = v ( x , ) , = = E(X, W,) crT,Z,y,~

4. EXTENSIONS

The v@Consistency of the predicting hazard estimator will remain even if the hazard model is extended to some semiparametric model

= o

where m0 is some parametric model specification and LI")(t) is some marker process that can contain continuous as well as descrete markers. A traditional actuarial finite state space Markov model can therefore be included in the model. The estimation of the parameter 0 can be performed by a traditional semiparametric analysis, see Nielsen et al. (1998) for an example of a semiparametric hazard estimation technique beyond the traditional cox regression. Also the Markov assumption of the marker process can be relaxed. One can allow for a parametric trend, for example a linear trend and estimate and include the parameters without violating the v'Z-consistency of the predicting hazard estimator.

5. EXAMPLES

In this section we specify four examples where the above methodology seems to provide a helpful methodology for prediction. The examples are taken from asset-liability management, datamming, biostatistics and an area of recent interest, namely high frequency transaction data in financial studies.

SUPER-EFFICIENT PREDICTION BASEl) ON HIGH-QUALITY MARKER INFORMATION 301

5.1. Asset-Liability management: Predicting the hazard of the remaining time to prepayment of mortgage bonds

Danish prepaid bonds can be a difficulty in asset-liability management of Danish insurance cornpanies, where up to 60% of the invested capital is in this type of bonds. In particular it can be hard to say something precise regarding the expected duration of the bonds. Here we think of duration of bonds in the original sense of Macauley (1938) and of asset-liability management in the sense indicated by Redington (1952). The duration of prepaid bonds is difficult because the payment times are stochastic variables. People might choose to prepay their loan and they might not. One good marker of whether people actually choose to prepay their mortgage is if the market interest rate is considerable below the nominal interest rate. Therefore the difference between the market interest rate and the nominal interest rate is a good candidate for a high quality rnarker. It is reasonable to assume that this marker obeys the needed Markov property and that the hazard of prepayment can be described from the marker information alone. The procedure of this paper therefore seems valuable when calculating the predicting hazard of prepayment bonds and hereby their expected remaing life time or their duration.

5.2. I)atamining: Predicting the hazard of the remaining time a given customer will remain in an insurance company or predicting the remaing time before the next claim

Let us assume that we through datamining have come tip with some relevant marker for expected customer loyalty. If this marker is assumed to obey the Markov property of this paper, then the method of this paper can be used to give a precise estimate of the expected custo