ASTIN BULLETIN Volume 32, Number 2 - Casualty Actuarial Society

148 Pages·2015·6.89 MB·English

ISSN 0515-0361
Volume 32, No 2 November 2002
A S T I N
B U L L E T I N
The Journal of the ASTIN and AFIR Sections of the
International Actuarial Association
CONTENTS
EDITORIAL
H. NNAMLHOB
New Math for Life Actuaries 209
:SROTIDE A. SNRIAC
ASTIN Bulletin Online 212
Andrew Cairns
Paul Embrechts ARTICLES
.S WANG
A lasrevinU Framework rof gnicirP laicnaniF and ecnarusnI sksiR 213
:SROTIDE-OC
W. HORLIMANN
Stephen Philbrick lacitylanA Bounds rof Two ksiR-ta-eulaV slanoitcnuF 235
John Ryan .S ,NESSUMSA .F ,MARVA M. LEBASU
Edangian Approximations for Finite-Horizon Ruin Probabilities 267
K.TH. ,SSEH A. ,DLAWEIL K.D. TDIMHCS
LAIROTIDE "DRAOB
An Extension of Panjer's Recursion 283
Alois Gisler D.C.M. ,NOSKCID H.R. SRETAW
The Distribution of the Time to Ruin in the Classical Risk Model 299
Marc Goovaerts
.P OCIREV
Mary Hardy Bonus-Malus Systems: "Lack of Transparency" and Adequacy Measure 315
Ole Hesselager
WORKSHOP
Christian Hipp I.M.E ORIEDROC
Transition Intensities for a Model for Permanent Health Insurance 319
Jean Lemaire
Gary Parker BOOK REVIEW
Jukka Rantala A. SNRIAC
K. Sandmann and EJ. Sch6nbucher (ed~): Advances in Finance and
Axel Reich Stochasticg Essays in Honour of Dieter Sondermann 347
Robert Reitano MISCELLANEOUS
Shann Wang The Actuarial Meetings in Cancun 349
International AFIR Colloquium 2003 350
Invitation to the ASTIN Colloquium 2003 in Berlin/Germany 351
PEETERS
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insurance. Since then a well-established non-life methodology has resulted, which is also applicable to
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Since the foundation of the AFIR section of IAA, i.e. since 1988, ASTIN BULLETIN has opened its
editorial policy to include any papers dealing with financial risk.
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Copyright © 2002 SRETEEP
NEW MATH FOR LIFE ACTUARIES
The fact
The model used in the technique of the Life Actuary is built on
i) probabilities of insured events, e.g. death, survival, disablement, just to
mention the most common examples for such events,
ii) time value of money.
The model has been in use since more than 200 years (1762 Old Equitable)
hence its practicability has been proved. Beyond that it has been the basis for
insurance legislation all over the world. Hence the model is iron cast by law to
be the right one.
The challenge
All models are wrong. Still we all know that they can nevertheless serve their
purpose if they are applied with care. That means, if we add what the prac-
titioners call "experience" and what the academics call "understanding of the
limitations of the model" they become very valuable and useful tools. The
question hence arises by how much a useful model may be wrong. My point is
that for dealing with the time value of money (item ii) above the classical actu-
arial technique is nowadays so far off economic reality that it needs to be
fundamentally revised. Here is a program for revision. The ideas are not new.
One finds them in several scientific papers, some also published in the ASTIN
Bulletin. My aim is to communicate these ideas more widely such that they
will be applied and used.
1) Recognize that Life Assurance is always expressed in units. In private
insurance (social insurance needs another definition of units) these units
can always be thought of as units of a (possibly dynamic) portfolio. Such
units can contain guarantees, one only needs to add the appropriate (Euro-
pean) options to the portfolio. As a special case, if the units are defined as
the unit of the local currency, the portfolio is made up by Zero Coupon
bonds plus options for the contractual guarantees.
2) Understand that in the language of units one unit always remains one unit.
This has the great advantage that we can use the classical actuarial formu-
lae at interest rate zero. This is the first discovery: The technical interest
rate which good actuaries have always handled with suspicion has disap-
peared! The second discovery is even more convenient: The present value
(in local currency) defined as the value of a corresponding (called replicat-
ing) portfolio can always (in principle) be read in todays newspaper! (at
least if it has a reasonable financial section).
ASTIN BULLETIN. Vol. 32, No. 2, 2002, pp. 209-211
210 SNAH NNAMLH3(B
OT the non initiated the above may sound too simple and possibly more risky than
how Life Assurance typically operates now. Let me address these two points.
Simplicity
In this new approach the actuary still needs to construct life tables, multiple
decrement tables, probabilities for disablement etc. The only difference in his
work is the fact that he can apply all his actuarial techniques calculating at
interest rate zero. The more difficult part is the construction (in thought) of the
portfolio which replicates the assured unit. Maybe some of us know already
how to do that. If you don't, ask a Financial Engineer, but insist that he com-
municates to you exactly how this portfolio needs to be constructed and man-
aged. He must in principle come up with the exact instructions how the Invest-
ment Department (if they wanted) could invest in the replicating portfolio.
Hence -- in all fairness -- the suggested new treatment of the time value
of money is indeed conceptually simple to understand, but the practical
implementation has its intricacies. So, nobody should fear to lose his or her
job, if the new ideas should be applied in practice.
Riskiness
The financial risk (which is the most important for any Life Assurer) derives in
the new approach from the fact that in reality Life Assurers often do not invest
in the replicating portfolio. They may act so in the hope to make more profit
by investing differently. If you replace "replicating portfolio" by the tradition-
ally named "matching assets strategy", the same remark applies for the classi-
cal actuarial model. If however with the new approach the Life Insurer invests
exactly in the replicating portfolio then the Financial Risk is Zero. The classi-
cal actuarial model -- even under a matching assets strategy -- can never
achieve this since it does not account for contractual guarantees (most impor-
tant in practice the guarantee of the technical interest rate). It is exactly this
fact, which urges for a change in our cherished traditional actuarial model.
How could the change happen?
)1 My appeal is first to my colleagues teaching actuarial science: Change the
basis of your teaching along the principles outlined above. For the young
generation of actuaries and financial engineers the new approach should
become the obvious one. Therefore my title
NEW MATH
which could also be read
NUMAT namely NUMeraire based
Actuarial
Teaching
(Numeraire = Value of unit)
NEW MATH FOR LIFE ACTUARIES 211
By the way, the time needed to explain the "numeraire" you can gain by
omitting "compound interest".
2) To the actuarial consultants it should be obvious how to use the proposed
approach for calculating Embedded Value. This would then also provide a
common basis for such calculations hence lead to identical results -- inde-
pendent of who of us performs this calculation.
3) It is not nay intention to make any proposals regarding legal and/or account-
ing rules. They rely on a process of consensus hence changes will always
take time. Still from a risk management point of view one has to notice
that the numeraire which has value 1 at time zero and value 1( + i) n at each
time n is -- theoretically not replicable.
HANS BIHLMANN
PS My aim was to communicate ideas. Some fine points have been omitted
on purpose -- e.g. how to deal with surrender values. Such guarantees
having the character of an American option can also be dealt with by
the approach as outlined in my editorial, but need a technically different
treatment than those guarantees which can be replicated by European
options.

A UNIVERSAL FRAMEWORK FOR PRICING
FINANCIAL AND INSURANCE RISKS
BY
SHAUN S. WANG
ABSTRACT
This paper presents a universal framework for pricing financial and insurance
risks. Examples are given for pricing contingent payoffs, where the underlying
asset or liability can be either traded or not traded. The paper also outlines an
application of the framework to prescribe capital allocations within insurance
companies, and to determine fair values of insurance liabilities.
INTRODUCTION
Currently there is a pressing need for a universal framework for the deter-
mination of the fair value of financial and insurance risks. In the insurance
industry, this need is evident in the Society of Actuaries' "Symposium on Fair
Value of Liabilities", and in the Casualty Actuarial Society's "Risk Premium
Project" and "Task Force on Fair Valuing P/C Insurance Liabilities".
In the financial services industry, this pressing need is evidenced by the
recent Basel Accords on regulatory risk management that require fair value,
analogous to market prices, to be applied to all assets or liabilities, whether
traded or not, on or off the balance sheet. In light of all these current events,
this paper addresses a very timely subject.
The paper is comprised of three parts, summarized as follows:
Part One: The Framework introduces a new transform and correlation mea-
sure that extends CAPM to pricing all kinds of assets and liabilities, having any
type of probability distribution, whether traded or underwritten, in finance or
insurance. This transform is just as easily applied to contingent payoffs that
are co-monotone with their underlying assets or liabilities.
In its simplest form, the new transform relies on a parameter called the
"market price of risk", extending a familiar concept in CAPM to risks with
non-normal distributions. The "market price of risk" can either be applied to,
1 SCOR, One Pierce Place, Itasca, IL 60143, E-mail: [email protected]
NITSA ,NITELLUB .loV ,23 .oN ,2 ,2002 .pp 432-312
214 NUAHS .S GNAW
or implied from, a distribution, in order to arrive at a "risk-adjusted price" for
the underlying risk in question. The "market price of risk" increases continuously
with duration, and is consistent at each horizon date between an underlying
and its co-monotone contingent payoff.
When the return for an underlying asset has a normal distribution, the new
transform replicates the CAPM price for that underlying asset, and recovers
the Black-Scholes price for options on that underlying asset.
Part Two: Examples of Pricing Contingent Payoffs illustrates the application
of the new framework to pricing call options on traded stocks, and to pricing
weather derivatives.
Part Three: Capital Allocation & Fair Values of Liabilities illustrates the appli-
cation of the new framework to insurance company capital allocations, and
to the determination of fair values of insurance liabilities. In particular, it
addresses a challenging issue concerning the long-term duration of liabilities.
Also, the framework is equally applicable to primary insurance business and
excess-of-loss reinsurance when calculating fair values of liabilities.
TRAP ONE. THE KROWEMARF
Capital Asset Pricing Model
CAPM is a set of predictions concerning equilibrium expected returns on assets.
Classic CAPM assumes that all investors have the same one-period horizon,
and asset returns have multivariate normal distributions. For a fixed time
horizon, let ~R and RM be the rate-of-return for asset i and the market portfo-
lio M, respectively. Classic CAPM asserts that
ERh.l= r + fli{ERM-r},
where r is the risk-free rate-of-return and
Cov Ri , MR
~i-- 2
G M
is the beta of asset i.
Assuming that asset returns are normally distributed and the time horizon is
one period (e.g., one year), a key concept in financial economics is the market
price of risk:
i2 _ ERi- r
71 i
In asset portfolio management, this is also called the Sharpe Ratio, after Wil-
liam Sharpe.
A FRAMEWORK FOR PRICING FINANCIAL AND INSURANCE RISKS 215
In terms of market price of risk, CAPM can be restated as follows:
~i El~.l-r CovRi,RM ERM-F
= i7¢ : i~¢ MT~ (7 M -- Pi, M" ~M~t
where Pi, M is the linear correlation coefficient between Ri and R M. In other
words, the market price of risk for asset i is directly proportional to the cor-
relation coefficient between asset i and the market portfolio M.
CAPM provides powerful insight regarding the risk-return relationship,
where only systematic risk deserves an extra risk premium in an efficient market.
However, CAPM and the concept of "market price of risk" were developed
under the assumption of multivariate normal distributions for asset returns.
CAPM has serious limitations when applied to insurance pricing when loss
distributions are not normally distributed. In the absence of an active market
for insurance liabilities, the underwriting beta by line of business has been dif-
ficult to estimate.
Option Pricing Theory
Besides CAPM, another major financial pricing paradigm is modern option
pricing theory, first developed by Fischer Black and Myron Scholes (1973).
Some actuarial researchers have noted that the payoff functions of a Euro-
pean call option and a stop-loss reinsurance contract are similar, and have pro-
posed an "option-pricing" approach to pricing insurance risks. Unfortunately,
the Black-Scholes formula only applies to lognormal distributions of market
returns, whereas actuaries work with a large array of distributional forms.
Furthermore, there are subtle differences between option pricing and actu-
arial pricing (see Mildenhall, 2000). One way to better appreciate the differ-
ence between "financial asset pricing" and "insurance pricing", is to recognize
the difference in types of data available for pricing.
Options pricing is performed in a world of Q-measure (using risk-adjusted
probabilities), where the available data consists of observed market prices for
related financial assets. On the other hand, actuarial pricing is conducted in
a world of P-measure (using objective probabilities), where the available data
consists of projected losses, whose amounts and likelihood need to be con-
verted to a "fair value" price (see Panjer et al, 1998).
Because of this difference, the price of an option is determined from the
minimal cost of setting up a hedging portfolio, whereas the price of insurance
is based on the actuarial present value of costs, plus an additional risk pre-
mium for correlation risk, parameter uncertainty and cost of capital.
A Universal Pricing Method
Consider a financial asset or liability over a time horizon 0, 7. Let X = Xr
denote its future value at time t = T, with a cumulative distribution function
216 NUAHS .S GNAW
(cdf) F(x)=Pr{X< x}. In Wang (2000), the author proposed a universal pricing
method based on the following transform:
F * )x( : *~-l ))x(F( + ,2 (I)
erehw )1( si eht standard normal evitalumuc .noitubirtsid ehT yek retemarap 2
si dellac eht tekram ecirp of ,ksir gnitcelfer eht level of citametsys risk. ehT
mrofsnart (i) si won better known sa eht ,gnaW mrofsnart among laicnanif
sreenigne dna risk .sreganam ehT gnaW mrofsnart saw partly deripsni yb eht
krow of lareves tnenimorp lairautca ,srehcraeser gnidulcni Gary retneV ,1991(
)8991 dna treboR cistuB .)9991(
roF a nevig tessa X htiw cdf ,)x(F eht gnaW mrofsnart lliw ecudorp a -ksir"
"detsujda cdf .)x(*F ehT mean eulav under ,)x(*F detoned yb ,X*E lliw enifed
a detsujda-ksir "fair "eulav of X ta time ,T hcihw nac eb further detnuocsid to
emit ,orez gnisu eht eerf-ksir tseretni .etar
ehT gnaW mrofsnart si ylriaf easy to yllaciremun .etupmoc Many erawtfos
segakcap evah both )1( dna )1( I- sa ni-tliub .snoitcnuf In tfosorciM ,lecxE )y()1(
nac eb detaulave yb )y(TSIDSMRON dna )z(1-)1( nac eb detaulave yb -MRON
SINV(z).
One fortunate property of the Wang transform is that normal and lognor-
mal distributions are preserved:
• If F has a Normal(g, o )2 distribution, F* is also a normal distribution with
*t.I = ~ - ~,o and o* = o.
• If F has a lognormal(g, o )2 distribution such that In(X) - Normal (la, o2), F*
is another lognormal distribution with g* = ~t- ~o and o* = o.
Stock prices are often modeled by lognormal distributions, which implies that
stock returns are modeled by normal distributions. Equivalent results can be
obtained by applying the Wang transform either to the stock price distribution,
or, to the stock return distribution.
Consider an asset i on a one-period time horizon. Assume that the return ~R
for asset i has a normal distribution with a standard deviation of oi. Applying
the Wang transform to the distribution of Ri we get a risk-adjusted rate-of-
return:
E* /~. : ERi-2a .i
In a competitive market, the risk-adjusted return for all assets should be equal
to the risk-free rate, r. Therefore we can infer that L = (ER~ -r)loi, which is
exactly the same as the market price of risk in classic CAPM. With ~ being
the market price of risk for an asset, the Wang transform replicates the clas-
sic CAPM.
Unified Treatment of Assets & Liabilities
A liability with loss variable X can be viewed as a negative asset with gain Y=
-X, and vice versa. Mathematically, if a liability has a market price of risk ~,