[Panning] Strategic Uses of VaR for PC Insurers

8/8/2019 [Panning] Strategic Uses of VaR for PC Insurers

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84

THE STRATEGIC USES OF VALUE AT RISK:LONG-TERM CAPITAL M ANAGEM ENTFOR

PROPERTY

/ CASUALTY

INSURERS

William H. Panning*

ABSTRACT

In cont rast to alternative measures of risk, value at risk (VaR) has imp ortant virtuesint elligibilit y,

comparability, and practicalitythat make it a potentially valuable tool for strategic decision

making and capital management in a wide variety of industries. However, capital-management

decisions in most industriesincluding financial services, such as property/casualty insurance

have time horizons far longer than the one-day horizon that prevails in commercial and invest-

ment banking, where the use of VaRis now concentrated. For VaRt o be usefully applied t o long-

horizon decisions, it m ust address three fundamental problems unique to t hat context: estim ationrisk, adaptive risk modification, and franchise risk. This paper describes each of these problems,

shows how they can be solved, and provides examples applicable to property/ casualty insurance.

The purpose of computing is insight, not numbers.

Richard Hamming

1. INTRODUCTIONOf the twin concepts, risk and return, that comprise

the foundation of modern financial analysis, risk has

long been the more troublesome. Return is straight-

forward both t o understand and, i n most cases, to

measure, despit e t he persist ent efforts of account ant s.

Risk, by contrast, is elusive, and measures of it, such

as standard deviation or beta, fail the critical test of

intelligibili ty: can they be explained in simple t erms

to ones famil y or, for that mat ter, t he board of direc-

t ors? The measures also fail t o answer a simple but

crucial question. If one alternative has a higher stan-

dard deviati on or a higher bet a t han another, just how

much greater should its expected return be to justify

choosing it ? The answer t ypicall y offered, which is

based upon the aggregate choices of others in the

*William H. Panning, Ph.D., is Executive Vice President and Chief

Investment Officer at ARM Financial Group, 515 West Market Street,

Louisville, Kentucky 40202.

marketplace, exhibi t s a Panglossian confidence in col-

lective wisdom and the virtue of conformity. Despite

these disquiet ing inadequacies, the textbook mea-

sures of ri sk retain t heir sway over t he generat ions of

MBAs that now populate the world of business.

Value at risk (VaR) offers an alt ernati ve way of

thi nking about, measuring, and managing ri sk t hathas three important virt ues lacked by other, more tr a-

diti onal measures of risk. First, VaR is int elli gible.

Roughly speaking, VaR for a firm, a project, or a se-

curit y is simply the amount of money that it could lose

under ext remely adverse cir cumst ances. More pre-

cisely, VaR is an estimate of the maximum loss that

could occur under all but a specifi ed percentage of

possibl e scenari os, ordered from best t o worst . Sec-

ond, VaR is comparable. This potenti al loss, in dollars

or percentages, can be compared across alt ernati ves

and bears directly on a classic rule for decision mak-

ing: never risk more than you can afford to lose. That

is, VaR specifies the amount of risk capital a fi rm

needs to undertake a project without threatening its

survival. Thir d, VaR enables us t o answer import ant

quest ions. If one alternat ive has a higher VaR than

anotherand therefore requires more risk capital to

undert ake it then thi s addit ional amount, mult iplied

by t he fi rms marginal cost of capital, indicates the


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THE STRATEGIC USES OF VALUE AT RISK 85

additional return it should require to justify selecting

that alternative.1

These import ant vir tues make VaR a potenti all y im-

portant t ool for strategic decision making i n a wide

range of sit uat ions. During t he past decade, for ex-

ample, VaR has become an indispensable tool in com-mercial and investment banking, where i t is used t o

determine the amount of capital needed to support a

firms overall risk exposures. If the amount of capital

needed exceeds t he amount avail able, t he fir m re-

duces its risk exposures by hedging its current port-

folio of assets and liabilities. The use of VaR for this

purpose is now widespread and has been endorsed by

bank regulators. Moreover, banks also use VaR to at-

tribute capital to the component parts of their busi-

ness, so t hat they can determine t he relative contr i-

buti ons of each t o t heir overall return on capit al. This

int ernal applicati on of VaR i s t herefore an essenti al

t ool in banks conti nual quest t o achieve t he maxi-

mum return from their capital.

Nearl y every fir m faces a simi lar need t o est imate

it s overall need for ri sk capit al and to att ri bute capit al

t o it s component operations or to potential new ones.

Consequent ly, t he evident useful ness of VaR in bank-

ing str ongly suggests it s potential applicabilit y in

other indust ri es as well . However, as with other im-

portant t ools, one must calculate and int erpret VaR

in a manner that is appropriate to the context at

hand. It is t herefore i mport ant t o consider whet her

t here are important differences bet ween banks and

other firms t hat need to be taken into account inadapting VaR for more general application.

What dist inguishes invest ment banks from most

other i nsti tut ions is not the part icular t ype of risks to

which they are exposed, but rather t he t ime hori zon

over which information can be gathered and practical

decisions implemented. In large invest ment banks

VaR is typically calculated for a one-day horizon, and

relevant changes are made to its portfolio of financial

assets and l iabil it ies each day t o ensure t hat VaR re-

mains within specified regulatory and internal limits.

This short ti me horizon and rapid response t ime is

made possible by the abili ty of banks t o uti li ze marketinformation to price their assets and liabilities each

day. In most other i nst it ut ions, by contr ast , assets and

1Here I have described VaRs application to a choiceamong mut ually

exclusive, stand-alone alternatives. By cont rast, what matt ers for a

firm with multiple projects or lines of business is the amount of

capital needed to support all of them taken together. In this more

typical case, what matters in evaluating an alternative is its marginal

impact on the firms overallVaR. However, VaRh as the same virt ues

in this more complex problem as well.

liabilities can be priced only occasionallyquarterly,

yearly, or longerand ri sk exposures must be mea-

sured and managed over these much longer ti me

spans.

A propert y/ casualt y i nsurer provides an excell ent

example of t he long-horizon natur e of some risks andthe relevance of VaR to strategic decisions. The bal-

ance sheet of a propert y/ casualt y insurer consists

principally of the it ems shown in Table 1. On the asset

side of t he balance sheet are it ems t hat can be ana-

lyzed by methods currently used in investment bank-

ing, since t he principal risks are market and credit

risk. The liability side, by contrast, is more challeng-

ing. The two largest items are the unearned premium

reserve and t he combined l oss and loss adjustment ex-

pense reserve (although these are often separated for

reporti ng purposes, I shall here tr eat t hem together

and refer to them as simply the loss reserve). The ec-

onomic value ( in contr ast t o t he report ed value) of

the unearned premium reserve is the discounted fu-

t ure loss and expense payments for fut ure accidents

covered by in-force policies. By contrast, the loss re-

serve is an esti mate of future payments for past

accidents.

For most insurers t he loss reserve is t he single larg-

est li abili ty on i t s balance sheet and i s t he focus of

considerable attention by regulators and financial an-

alysts. The existence of this reserve results from the

delays between the occurrence of an accident and the

submission of a claim, and between the claim report

and a final negotiated settlement. For some types ofri sk both delays can be substanti al. Decades may

elapse, for example, between an insureds exposure to

a t oxic substance and t he eventual appearance of

physical symptoms that tr igger a claim. The ti me lag

between provision of coverage and payment of cl aims

can be abbreviated if an insurer switches from occur-

rence policies, which indemnify t he insured against

loss from accidents t hat occur during t he term of cov-

erage, to claims-made policies, which pert ain to

claims actually submitted during the term of cover-

age. In either case, however, the lag from claim sub-

mission t o fi nal set tl ement is often prolonged by legalwrangling.

In determining the value of its loss reserve, a prop-

ert y/ casualt y insurer must esti mat e future loss pay-

ments bot h on claims that have been report ed but not

yet sett led and on accidents t hat have been neit her

reported nor set tl ed. Because the payments ult imately

made may differ considerably from t he i nsurers cur-

rent esti mat e of their value, t he insurer must have


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86 NORTH AM ERICAN ACTUARIAL JOURNAL, VOLUME 3, NUM BER 2

Table 1

Principal Component s of an Insurers Balance Sheet

Assets Liabilities

Premiums receivable Unearned premium reserve

Reinsurance recoverable Loss and loss adjustm ent expense reserveO th er recei vab les O th er p ayab lesIn vested assets Sur pl us ( net w orth)

suffi cient net wort h, k nown as surplus, t o absorb po-

t ential adverse deviati ons from it s est imated reserve.

In contr ast to investment banks, which can alt er

t heir asset and l iabil it y exposures rat her quickl y, the

li abili ty ri sks of a propert y/ casualt y i nsurer change

slowly. Propert y/ casualt y r isk exposures are diffi cult

to t rade, so that t here i s neit her an active market nor

dail y pricing, except in a very few highly specialized

circumstances. As a consequence, the principal valueof VaR for this industry is strategic rather than tacti-

cal. For i nsurers, i t can provide a means t o det ermine

t heir overall need for capit al. Moreover, i t can assist

them in identifying the relative contributions of their

vari ous lines of business t o their overall capit al re-

quirement, so t hat t hey can adjust their pricing and

business mix to maximize their return on capital. For

regulators, VaR provides a pot enti all y powerful t ool

for ensuri ng t hat t he companies t hey oversee are ad-

equately capit ali zeda role that it is current ly begin-

ning to play in banking as well.

Because of the illiquid nature of insurance liabili-

t ies, t he st rategic decisions just descri bed have t ime

horizons considerably longer than t he one-day horizon

t hat characterizes VaR measures i n banki ng. Ext end-

ing VaR to decisions and alternatives that span these

longer time horizons has the potential to enhance its

usefulness in nonfinancial industr ies as well . But ex-

tending VaR in thi s way requir es solving at least thr ee

import ant problems t hat have received lit tl e att en-

ti on. These problems are esti mati on ri sk, adapti ve ri sk

modification, and franchise risk.

Estimation risk result s from t he fact that VaR is an

estimate of potential loss under an extreme scenario.

This esti mat e is subject t o err or, since it is based upona forecast of what asset or liabil it y values wil l be under

condit ions that differ from current ones. Esti mat ion

error is likely to be quite small for investment banks

calculating overnight VaR for portfolios of highly liq-

uid financial asset s and li abili ti es. Applying VaR t o

other types of assets and liabilities makes estimation

error more important. Moreover, extending the time

int erval over which VaR is calcul at ed magnifi es poten-

ti al changes from current condit ions and t hus mag-

nifi es t he impact of errors in forecasti ng t he response

of assets and liabilities to such changes. Fortunately,

as I shall show, this esti mati on error can it self be

quantified and taken into account in calculating VaR.

This is as it should be, since the esti mat ion error is

an inherent component of overall risk.

Adaptive risk m odification refers t o the fact that

over an extended period of time the assets and liabil-it ies of a fir m do not remain static but are alt ered in

response t o i mplicit or expli cit decisions prompt ed by

changing circumstances. In the extreme scenari os

t hat are the focus of VaR, decision makers do not t yp-

icall y remain passive but take act ions that alt er t heir

perceived r isk exposures. Adapti ve ri sk modifi cat ion i s

nearl y ir relevant for investment banks calculati ng

their overnight VaR, but it can be significant for any

firm calculating VaR for longer periods. I shall show

how adaptive risk modification can affect VaR and ex-

plain how it can be taken into account.

Franchise risk is the potenti al exposure to loss from

asset s and l iabil it ies that are not reflect ed on a fir ms

current balance sheet . The market value of a fi rm

the value of it s outstanding st ock or, for privat e fir ms,

the amount a potenti al buyer would pay to acquir e

it t ypically exceeds t he economic value of t he assets

and li abili ti es included on it s balance sheet . What I

here call franchise value is t his di fference bet ween a

fir ms t otal market value and i t s balance sheet value.

It consists, in principle, of the discounted expected

cash flows from future business. Protecting and en-

hancing this franchise value is a critical objective at

most fir ms. Consequently, if VaR is to become a

longer-term strategic tool, it must take into accountpotential exposures to loss of franchise value. For ex-

ample, the discovery of an error in the Intel Pentium

chip creat ed only small losses for the fir ms balance

sheet but had a potentially devastating impact

ult imately avert edon it s enormous franchise value

from fut ure sales. A VaR calculation t hat fail ed to t ake

this effect into account would be incomplete and se-

ri ously mi sleading. The challenge, t hen, is t o measure


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Table 2

Payoffs and Risk M easures for Four Alt ernat ives

Alternatives

Possible Payoff atProbability

50% 49% 1%Expected

Value

Risk M easures

StandardDeviation

Probabilityof Loss

ExpectedLoss

Worst-Case Loss

A 75 7 5 2,575 100 249 0% 0 0

B 20 220 220 100 120 50 10 20

C 249 50 0 100 149 49 24.5 50

D 104 100 100 100 20 1 1 100

the components of franchise value and include them

in calculati ons of VaR. In thi s case I offer only a part ial

solution, which consists of measuring franchise value

for one type of fi rm and showing how this component

of VaR can be calculated for one type of risk.

I begin by presenti ng a somewhat more t horoughintr oduction to VaR and it s differences from other

measures of risk. I then devote a separate section to

each of the three problems just posed and end with a

brief conclusion. Although these problems and their

solutions are pertinent to any kind of firm, I present

examples appli cable t o propert y/ casualt y i nsurance.

Throughout, I st ri ve to make t he exposit ion widely ac-

cessible, although some aspects of VaR are unavoida-

bly technical.

2. VAR VERSUS ALTERNATIVE

M EASURES OF RISKIn order to distinguish VaR from other risk measures,

it is helpful to consider the specific example shown in

Table 2. The t able shows four alt ernatives, each of

which has mult iple possible payoffs occurri ng wit h t he

different probabili t ies shown. The four alt ernatives

have t he same expect ed value, but dif ferent ri sks.

Standard deviation, t he fir st ri sk measure, is a st a-

t istical esti mat e of vari abili t y around a mean or ex-

pected value, variabili t y that i s often port rayed graph-

ically by a bell -shaped cur ve. For fi nancial assets or

liabilities, this measure is calculated from a time se-

ri es of changes i n their value and appropriately adjusted to produce the annualized standard deviation

of percentage change in value, the standard measure

of retur n volati li t y. Despit e i t s widespread use, vola-

ti lit y of return has t wo import ant deficiencies as a

measure of risk. First, it does not conform to the in-

t uit ive definit ion of ri sk used by most business profes-

sionals. To see why, consider t wo invest ment alt erna-

t ives, one of which (say, one-year Treasury bill s or

T-bil ls) wil l defi nit ely return 5%, while t he second has

a 50-50 chance of retur ning 8% or 10%. Alt hough the

second alt ernative has great er vari abili t y of return

t han t he T-bill , most invest ors would not consider i t

riskier, because its return is always higher. Investors

would t ypicall y consider t he second alternat ive as ri sk-ier only if it had some posit ive probabili ty of retur ning

less t han the T-bill . Their aim is not to achieve cer-

t aint y, but t o avoid loss. Most business professionals

thus consider risk as involving a probability of loss

either absolutely (as in a negative return) or relative

t o some riskless alternati ve (for example, relati ve to a

T-bill). A second difficulty with standard deviation as

a measure of ri sk is t hat it can be highly misleading

when applied to return distr ibutions that differ dra-

matically from a bell-shaped curve. In fact, many of

the new financial instruments developed in the past

decade were specifi call y designed t o creat e such un-

usual return distr ibuti ons, and t heir creati on has led

to the search for improved risk measures.

If losses are what mat ter, probabili ty of loss could

be used as a measure of risk. In Table 2, alternative A

has t he highest st andard deviati on of t he four alt er-

nati ves, but t he lowest probabili ty of l oss. What t his

measure ignores, unfortunately, is the magnitude of

potenti al losses. It therefore fails t o dist inguish be-

t ween t wo alternati ves that have different losses, for

example, one small and the other large, but the same

probabili ty of l oss. Note t hat in Table 2 alt ernatives B

and C have nearly identical probabilities of loss, 50%

and 49%, but t hat t he magnit udes of t hose losses dif-fer considerably: 20 in one case, 50 in the other.

Expected loss compensates for thi s defici ency by

taking i nto account both the probabilit ies and t he

magnitudes of potenti al losses. It s value for a given

alternative is obtained by multiplying each potential

loss by its probability and summing these products.

Potential gains are totally ignored. This measure

t herefore dist ingui shes sharpl y bet ween alt ernati ves B


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and C in Table 2 and considers the latter to be far

more risky (i n fact, t he ri skiest of t he four) . Expected

loss is the actuarially fair cost of insuring against loss

from choosing that alt ernati ve, that is, the least

amount one could possibl y pay someone t o compen-

sate for a l oss should i t occur. In investment t erms,expected loss is the minimum cost of a put opti on

with an exercise price of zero. Where the payoff con-

sists of a conti nuous range of possible values wit h

corresponding probabili ti es, expected loss can be

calculated by an appropriate application of the Black-

Scholes (1973) option-pricing formula. For property/

casualty insurers, Doherty and Garven (1986), Cum-

mins ( 1988), and Butsic (1994) have shown how ex-

pected loss to poli cyholders can be calculat ed and

used to determine insurance prici ng, fair premiums

for st at e insurance guarantee funds, and mini mum

capital requirements for insurers.

A final risk measure, one often requested by senior

executives, is the worst-case scenario, defined as the

maximum possible loss. Unlike probabili t y of l oss, t his

criterion finds alternative C in Table 2 to be more

ri sky t han alt ernative B and considers alt ernative D t o

have t he highest ri sk of all . I n another respect, how-

ever, thi s measure is not very discrimi nati ng, for it will

fail t o differenti ate bet ween t wo alt ernat ives t hat have

the same worst-case loss but dramati call y different

probabil it ies of loss. Despit e t his defect , worst -case

loss is a valuable measure for one very important rea-

son. It measures the amount of capital needed to sur-

vive the worst outcome that can occur for t he alt er-nat ive being considered. If prudence dict ates t hat one

never risk more than one can afford to lose, then

worst-case loss informs the decision maker how much

capit al he or she must be prepared t o lose in choosing

a parti cular alt ernative. As wit h expected l oss, thi s

measure can be generali zed t o alt ernat ives wit h a con-

tinuous range of payoffs. In such cases one calculates

not a worst -case l oss (which may be infi nit e) but a

hi ghly improbable loss. The procedure requires that

one rank order t he payoffs from best t o worst . Then

one select s a (very low) probabili ty p of occurrence,

1%, for example, and det ermines t he payoff at t he pt hpercenti le of t he ordered distr ibuti on. If t his payoff is

a loss of 500, t hen one i nfers t hat in selecti ng t his

alternative a capital of 500 is needed to ensure a 99%

probability of survival, or, equivalently, to ensure that

losses greater than 500 will occur with a probability

of 1% or less. I f the dist ri bution of possible outcomes

is normal or lognormal, then the same value can be

directly calculated from the mean and standard devi-

ation of the distribution. Such highly improbable

losses, where t he probabili ty is specified, are also

called VaR, or VaR(1 p), where t he term ( 1 p) is

t he probabil it y of survival. That is, VaR is a measure

of the nearly worst-case outcome.

Generating the Distribution of OutcomesSince a fi rms capit al can absorb r isks from any of it s

component operati ons, VaR analysis is t ypically ap-

plied t o t he fir m as a whole. However, it can equall y

be applied to a project, a line of business, or a partic-

ular asset or li abil it y, depending on ones purposes. In

each case, it is calculated from a distribution of pos-

sible futur e outcomes. Most of the work needed to

calculate VaR consists of generating that distr ibuti on.

Here I outli ne the principal steps in doing so and focus

on calculating VaR for t he value of a fir m whose assets

and liabilities consist of fixed-income securities.

1. Identify and price the components of the firmsvalue. Since the objective of VaR analysis is to es-

timate potential changes in the value of the firm,

the starting point must be an esti mate of the firms

current value. In investment banking, this is done

by determini ng t he market values of t he fi rms as-

sets and liabilities, which consist principally of se-

curit ies that are priced daily. Alt hough many of

t hese securi t ies are not acti vely t raded and t here-

fore do not have pri ces t hat can be directl y ob-

served, t heir pri ces are i nferred fr om t hose of ac-

ti vely t raded securit ies. A simil ar combinati on of

observat ion and inference can be ut il ized in cal-culating the current value of other types of firms.

2. Identify the underlying processes that affect out-

comes. The price of a bond and t he present value

of a loss reserve are determined by the discounted

values of their future cash flows, which are in turn

a function of Treasury spot rates (zero-coupon

bond yields) at different matur it ies. To model po-

tential changes in the value of a bond or a l oss

reserve, one must t herefore model changes in

these underlying Treasury rates.

3. Estimate the parameters of these underlying pro-

cesses. For t he bond, one must est imat e t he vola-

ti lit ies of the spot rates that determine it s price.

Volatility is the standard deviation (or variance) of

changes in each rate. However, changes in spot

rates at different mat urit ies are not independent of

one another. Rat es at different but nearby mat ur-

ities tend to change in tandem. To take this into

account, one must also est imat e t he covariances

among rate changes for different maturities. Since

the result ing vari ance/ covari ance matri x can be

quite large, this process is often simplified in one


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of three ways. One is to use fewer rates and inter-

polate between them as needed. Another is to use

fact or analysis t o reduce t he size and complexit y of

t he vari ance/ covari ance matr ix. A t hird is t o skip

parameter esti mati on entir ely and instead use

bootst rappingsampling changes that have oc-curred in historical datain the following step.

4. Generate the distribution of potential changes in the

underlying processes. For a bond or a loss reserve,

this is t he mult ivariate distr ibution of changes in

each of t he r elevant Treasury r at es, using t he pa-

rameters estimated previously. This step is often

implemented by means of a Monte Carlo

simulation.

5. For each point in this underlying distribution, cal-

culate the effect on the firm. For bonds or loss re-

serves, one must use t his dist ri buti on of spot rate

changes to calculate t he dist ri bution of changes in

t heir value. This is done by discounti ng the relevant

cash fl ows using t he combinati on of spot rates at

each point in the multivariate distr ibution. Doing

this for all points i n the underlying distri bution

produces the distr ibution of overall changes in

value.

6. Select an appropriate value ofp, and calculate VaR.

As explained earl ier, thi s is done by ( a) orderi ng

t he vari ous out comes from best to worst, and (b)

selecting t he outcome such that onlyp%of the out-

comes are worse. If p is, say, 5%, t hen t his outcome

is referred to as the 95%VaR. Its interpretation is

str aightforward: only 1 p or 5%of the time is theoutcome lik ely t o be worse t han t his value.

The preceding outline is extremely general. More

det ail ed descript ions of VaR calculations for portfoli os

of financial assets can be found in Jorion (1997) and

J. P. Morgan (1995).

3. ESTIM ATION RISK

The Nat ure and Consequences of

Estimation Risk

In calculating VaR, as in any statistical modeling ef-fort, t here are numerous sources of error or ri sk that

result from our imperfect understanding of the phe-

nomena being modeled. Two t hat are especially im-

port ant are specificati on risk and esti mati on risk.

Specification risk results from the use of an inappro-

priate model that gives wrong or misleading answers.

For example, volat il it y est imat es are often based upon

an assumed normal distr ibut ion of r ate changes. By

contr ast, t he actual distri bution exhibits a higher cen-

tral peak and fatter tails than a normal distribution

and is more precisely modeled by a t-distribution or

by a mixture of normal distr ibutions wit h different vol-

ati lit ies. Simil arly, changes in securit y prices are

someti mes represented as li near functions of ratechanges, alt hough the tr ue r elati onship is t ypicall y

nonlinear. As in the two instances just cited, specifi-

cat ion error can be deliberate, as a means of speeding

up calculations, and the magnitude of it s effect can

be esti mat ed. More commonly, however, it is due t o

fundamental ignor ance. For example, most VaR mod-

els are i ncomplet e: the financial market ri sks incor-

porated in such models comprise only a port ion

alt hough cert ainly a major port ionof the overall

ri sks t o which invest ment banks are exposed. Losses

due to rogue t raders are one prominent risk not

included in VaR calculati ons. Consequentl y, banks

t hat use VaR t ypicall y compensate for thi s by using

some multiple of VaR to estimate their capital needs.

Even in the absence of specification error, estima-

tion risk ari ses from the fact that t he esti mated par-

ameters of our models will differ from t he tr ue values

of t hose paramet ers. This t ype of error ari ses from t he

fact t hat the data employed to esti mat e parameters

are only a samplea pot enti all y unrepresentati ve

samplefr om a much larger set of possible inst ances.

The following is a graphical example of est imation

ri sk and it s consequences. Suppose t hat t he t rue r e-

lationship between two variables, Y and X, is Y 2X,

and t hat we have 100 observat ions of each, but t hatour measures of Y are subject to random error. What

we observe will not be the tr ue relati onship, but rather

a scatter of points like those shown in Figure 1.

If we now use li near regression to est imate the re-

lationship Y a bX e, where e represents the

random measurement err or, we wil l t ypically not ob-

tain the true values a 0 and b 2, but values that

depend upon our part icul ar sample. The great er t he

random measurement err or in Y, the greater the po-

t ential deviat ion of our est imat es of a and b from their

true values.

Figure 2 shows the various relationships estimatedfrom 12 different simul at ed samples of 100 observa-

tions. Although the range of X in the data was from

5 t o 5, the range of theX axis in Figure 2 has been

increased to better display the differences in the es-

t imates obtained. For 100 different esti mat es from

different samples, t he average est imat ed values of a

and b were nearl y i denti cal to their t rue values, but


7/22


Figure 1

Illustrative Sampled Data

-20

-10

0

10

20

30

-6 -4 -2 0 2 4 6

X

Y

Figure 2

Relationships Est imat ed from Dif ferent Samples

-50

-40

-30

-20

-10

0

10

20

30

40

50

-20 -15 -10 -5 0 5 10 15 20

X

Y

the estimates obtained from any single sample could

vary rather widely, despit e an R-squared of close t o 0.6in nearly every case.

Especiall y import ant in Figure 2 is t he fact t hat t he

different est imat ed r elat ionships agree closely at t he

mean sample values of Y and X and diverge away from

that point. As one moves farther and fart her away

from the average value of the Xs we have observed,

the greater is t he potential for our esti mate of t he

corresponding value of Y t o deviat e considerably from

its true value. Estimation risk is simply this risk that

the true value of Y will differ from our estimate of it

based on inference or extrapolati on from sampled

data.Est imation ri sk can affect VaR calculati ons in at

least two ways. One is in esti mati ng the effect of

changes in market condit ions on t he values of a fir ms

assets or liabilities. Among invested assets, mortgage-

backed securi t ies provide an especially dr amatic ex-ample. The cash flows from such securities depend in

part upon the procli vit y of homeowners t o pay off their

mortgages prior to maturity, especially when interest

rates fall below t he rate they are currently paying.

Consequently, most algorit hms for esti mati ng the

prices of such securities rely upon a statistical model

for forecasti ng prepayments for different types of

mort gages under different rate scenarios. Such fore-

cast s are subject t o est imat ion r isk, which can become

part icul arl y severe under t he extreme scenari os that

are the focus of VaR calculations. However, although

est imation ri sk can be quanti fied and should be in-corporated into these pricing models, it typically is

not. Instead, the forecasts are treated as determinis-

ti c. This omission can have a seri ous impact on t he

est imated pri ce and r isk of coll at eralized mortgage


8/22


obli gati ons (CMOs), which are securit ies that gi ve t he

owner the ri ght to cert ain classes of cash flows from

a pool of mortgages. In particular, CMOs whose cash

flows are distant in ti me have values t hat are ex-

t remely sensit ive to est imation error in t he prepay-

ment models t hat are employed. It appears li kely,then, that the risks of such CMOs are drastically un-

derest imated because of the current practi ce of ig-

noring esti mati on ri sk. For some corroborating evi-

dence see Sparks and Sung (1995).

A second important way i n which esti mati on r isk

can affect VaR is in t he est imation of paramet ers such

as the volati li ty of int erest rates or of stock market

indices, or t he correlati ons among different t ypes of

ri sk. In est imat ing t hese paramet ers, VaR models t yp-

ically use daily data from the past. But the volati lit y

esti mat es t hat are obtained depend upon the length

of t he data seri es employed. Using dat a fr om t he past

several years will produce different volati lit y esti -

matesand, consequently, different VaRsthan a

longer or shorter data series will produce.

The i mpact of esti mati on error or risk on i nvest-

ment decisions is clearl y demonstr at ed by Bawa,

Brown, and Klein (1979) , who use hist orical data con-

cerning the risks and returns of various securities to

ident ify efficient port folios. They show that, in con-

t rast t o r esult s obt ained when esti mati on ri sk is ig-

nored, incorporating estimation risk results in port-

foli os t hat are more conservat ivethat i s, less skewed

t oward assets that have unusuall y high returns but r el-

atively short track records.Estimation risk is especially pertinent for VaR cal-

culations that involve non-market risks, that is, risks

unrelated to security prices. In such cases the data

employed are often l imi ted by avail abili ty rather t han

by choice. Moreover, t here may be a subst antial lag

bet ween the occurr ence of events t hat t ri gger changes

in value and t he t ime at which t hose changes in value

become fully known. During this interval a fi rm ex-

posed to such ri sks faces t he problem of using t he

sample of data available from the past to estimate fu-

t ure changes in i ts value. Esti mat ion ri sk r esult s from

t he fact t hat t his sample may be unrepresentati ve andt herefore misleading.

The potential impact of esti mati on ri sk on VaR is

magnified by the ti me horizon over which VaR i s cal-

culat ed. When t he horizon is long, the potenti al range

of changes from current market condit ions is much

greater. This, in turn, increases the potential impact

of errors in esti mati ng how the prices of securit ieswill

be affected by such large changes. This can be clearly

seen in Figure 2, which shows t hat t he divergence be-

tween the esti mated and the tr ue relationship be-

t ween Y and X incr eases as one moves away from t heir

mean values in the data used t o est imate t he relati on-

ship. This form of estimation risk is further com-

pounded by errors in esti mati ng market volati li ty. Thelonger t he ti me horizon over which VaR is calcul at ed,

the greater the potential impact of this type of error

as well.

Incorporating Estimation Risk in VaR

A propert y/ casualty insurers loss reserve poses two

import ant st at istical problems. The first is est imating

the expected value of future loss payments. The sec-

ond is estimating the distribution of possible devia-

t ions fr om thi s expected value and the consequent

amount of surplus t he fi rm needs t o absorb adverse

deviations wit h a high probabilit y of remaining sol-vent. The first of these problems has been studied in-

t ensively by actuaries. The second, by cont rast , has

received far less att enti on. I n practi ce, surplus r e-

quirements are stil l widely det ermi ned by two t radi-

tional rules of thumb. One focuses on the ratio of in-

surance premiums to surplus, the other on the ratio

of reserves to surplus. Both are inadequate to the

t ask. The fir st r eall y focuses on t he ri sks posed by wri t-

ing new business, rather than the risks posed by loss

reserves. The second is clearl y pert inent but is typi-

call y based upon histor ical industr y averages r at her

than upon a rigorous quantit ati ve foundati on. Bothare i ncomplet e measures, since t he surplus needed by

an insurer depends upon it s t otal ri sk exposures, of

which adverse reserve development is onl y one com-

ponent. Here I shall focus on the twin problems of

estimating the expected value of an insurers loss re-

serve and det ermi ning t he distr ibuti on of potenti al de-

viations from this expected value. Solutions to these

t wo problems are essenti al if VaR is to become a useful

str at egic tool for propert y/ casualt y i nsurers. More-

over, a correct soluti on t o both problems requires at-

tention to the problem of estimation risk.

Two caveats are in order. First, adverse loss reserve

development is only one of the numerous ri sks t hat

affect an insurers overall VaR and need for surpl us.

Consequently, t he analysis presented here is int ended

to be illustrative rather than complete. Second, loss

payment dat a often exhibit complexit ies not pr esent

in the data used here. Although the model I present

in Table 3 can be elaborated t o t ake such complexiti es

into account, addressing these issues here would de-

tract from the principal point of this section: the fact


9/22


Table 3

Loss Development Triangle

AccidentYear

Development Year

0 1 2 3 4 5 6 7 8 9

0 1,291 791 548 423 334 304 224 230 214 2101 1,492 881 617 471 390 282 263 241 2312 1,640 944 685 546 379 320 295 2583 1,818 1,049 799 522 409 364 2974 1,992 1,171 774 566 482 3765 2,056 1,133 814 644 4766 2,004 1,149 886 6097 2,023 1,264 8248 2,131 1,2039 1,998

Source: Adapted from Corporate 10K.

that long-term VaR must take estimation risk into ac-

count, and i n a way t hat is syst emat ic rather t han ad

hoc.The fir st step in calculating VaR for a l oss reserve

is t o det ermine t he expected futur e cash flows for past

accidents. The dat a used by propert y/ casualt y insurers

and regulators to do this are shown in Table 3, here

for a large firm that writes long-tail business, that is,

lines of business in which loss payments extend over

a l ong period of t ime. The dat a shown here have been

adapted from the firms 10K in two ways. First, all

numerical values have been multiplied by a constant,

to conceal the fir ms identit y. Second, although t he

original data were cumulati ve, from l eft to right, I

have here shown the decumulated values. The use ofdecumulated values is essential for two reasons. First,

they are t he actual payments that occurred in each

peri od and therefore const it ute the correct basis for

estimated expected cash flows in each future period.

Second, I will model these payments as consisting of

two parts: an expected or predictable part, and a ran-

dom deviati on or error. It is these deviati ons or errors

that will enable us to ultimately calculate the distri-

bution of overall deviati ons from t he expected value

of the reserve. The use of cumulative values, which is

common, conflates errors for different peri ods and

makes them difficult to model correctly.

Each row in Table 3 represents a part icular accident

year, defined as t he year in which an accident oc-

curred that was covered by the firms policies. Here I

have r eplaced t he actual calendar year numbers wit h

the digits from zero t o nine. Each column in the table

represents a particular development year, defined as

t he number of years subsequent t o t he accident year

in which payment s were made. A payment occurr ing

in development year zero was in fact made in t he same

calendar year as t he one i n which the accident oc-

curred; payment s i n development year one occurred

in the calendar year following the accident; and soon. The entry Pij in row i and column j of the table is

the total amount paid by the firm for losses incurred

in accident year i but paid in development year j, that

is, j years l ater. Not e t hat t he payments on any diago-

nal of the table were made in the same calendar year

i j.

The firms loss reserve is an estimate of the total

payments it will make in future years for accidents

t hat have already occurr ed. These payment s consist of

the unknown entries in the blank port ion of t he table,

as well as those occurring to the right of the table, in

development years ten and higher.Actuaries have developed a vari ety of methods for

estimating loss reserves. Unfortunately, the methods

most commonly in use are il l-suit ed for det ermini ng

VaR. There are t wo i mport ant reasons for t his. First,

these met hods are essenti all y ad hoc, in that t hey lack

a basis in an underlying model of the data. As a con-

sequence, t hey produce paramet ers wit hout reference

t o a well -defined st at isti cal measure of goodness of fi t

to the data. Second, because of this they provide no

stati sti cal basis for esti mat ing t he magnit udes and

probabili t ies of possible deviati ons from t he esti mat es

t hey produce. The procedure employed here, which

remedies both t hese defects, i s based upon work by

Taylor (1987).

The model used here to represent the data in Table

3 is

P a f( i) f( j) e ,ij ij

where a is an estimate of P00, f(i) is a function that

represents changes in the volume of losses over acci-

dent years, f( j) is a function that represents the pat-

t ern of payouts over development years, and e is an


10/22


11/22


Table 6

Deviations of Actual from Fitt ed Loss Payment Data

AccidentYear

Development Year

0 1 2 3 4 5 6 7 8 9

0 3.5% 4.7% 2.5% 2.5% 2.9% 11.0% 7.0% 4.1% 2.0% 2.5%1 0.3 4.5 1.2 2.7 7.7 6.9 1.4 1.9 1.22 0.7 6.7 0.2 8.4 4.1 3.4 1.1 4.13 3.5 3.7 8.1 3.5 4.2 1.9 5.74 6.7 1.4 1.1 1.4 6.4 0.85 5.4 6.4 0.4 7.1 0.66 0.0 7.8 5.1 1.57 0.4 0.4 3.58 5.0 4.49 0.3

logged actual and fi t t ed values, t he difference bet ween

t he average (l ogged) values in each row and t he av-

erage of t he corresponding values in t he t op row. Sys-

t emat ic development year errors can li kewise be spot-t ed by applying t he same procedure t o t he columns of

the logged actual and fitted data. These comparisons

are shown in Figures 3 and 4. Both confir m a close fit

to the data and the apparent absence of specification

error in the model.

Finally, the distri bution of errors, fit ted here by

means of a procedure call ed kernel densit y esti ma-

tion2 (Silverman 1986), displays symmetry and a rea-

sonable approximati on t o a normal distr ibuti on, as

shown in Figure 5.

The results obtained so far now enable us to obtain

the expected future loss payments that comprise thisinsurers loss r eserve. These forecast values are sub-

ject t o t hree sources of error:

1. Specification error: syst emat ic dif ferences bet ween

the model and t he dat a on which i t is based, dif-

ferences t hat wil l produce a biased forecast . The

extensive analysis of errors and compari sons be-

t ween actual and fi t t ed values were designed t o en-

sure that this source of error has been minimized.

2. Random error: random deviati ons between actual

and fit t ed or forecast payments. These error s are

represented by the error term in the fitted equa-

tion and can be anticipated to occur in the future

2Kernel density estim ation, a procedure for producing a relatively

smooth distribution from N data points, consists of the following

steps: (1) Create Nnormal distributions, each with a mean equal to

one of the data values, and a standard deviation S. (2) Calculate the

overall distribution by averaging the N individual distribut ions.

(3) Use one of several criteria (such as maximum likelihood) to de-

termine an appropriate value of S. When S is tiny, the result will

consist of N spikes. As Sis increased, the number of spikes will di-

minish until the overall distribution ultim ately becomes nearly flat.

as well. Their magnitude is estimated by the stan-

dard deviation of the fitted model. Note, however,

that this model applies to the l ogarit hms of the

dat a. Consequently, actual err ors wil l i n fact be dis-tributed lognormally for each payment.

3. Estimation error: differences between t he t rue val-

ues of t he model parameters and t he esti mat ed val-

ues obtained from the data. This source of error is

reflected by t he fact t hat each paramet er has a

st andard error t hat represents the probabili ty dis-

tribution of this actual value around its estimated

value.

All t hree sources of error affect t he values of fore-

cast fut ure payments. Because t he independent vari -

ableshere consisting of powers of i and jare

known for each future payment, we can calculate thelogged forecast payments using t he paramet ers est i-

mated from the available data. However, the expected

unlogged payment is not the antilog of the expected

logged payment. Instead, it is exp(Eij S ij2 / 2), where

Eij is the expected logged value of payment Pij and S ijis the standard deviation of the forecast error of that

logged value. Forecast error reflects both random er-

ror and estimation error.

The variance of the forecast error for the individual

(l ogged) futur e payments i s t he main diagonal of t he

vari ance/ covari ance mat ri x of forecast err ors. This

matrix,V

, is equal to se2

[I X

f (XX

)1Xf] , where se

is the standard error of the regression equation, X is

the mat ri x of independent vari ables used in the re-

gression ( which incl udes a leading column of ones for

estimating the intercept term), Xf is the matr ix of i n-

dependent vari ables for t he unknown l ogged payments

to be forecast, and I is the identity matrix, which has

ones on its main diagonal and zeros elsewhere.

The r esult ing (unlogged) forecast futur e loss pay-

ments thr ough development year nine are shown in


12/22


Figure 3

Actual and Estimated Log Accident Year Payment Ratios

0.00

0.10

0.20

0.30

0.40

0.50

0 1 2 3 4 5 6 7 8 9

Accident Year

Actual

Estimated

Figure 4

Actual and Estimated Log Development Year Payment Ratios

-2.0

-1.5

-1.0

-0.5

0.0

0 1 2 3 4 5 6 7 8 9

Development Year

Actual

Estimated

Figure 5

Fitted Error Distribution

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

6.0%

7.0%

-15% -10% -5% 0% 5% 10% 15%

Fitted

Normal


13/22


Table 7

Estimated Future Loss Payments

AccidentYear

Development Year

0 1 2 3 4 5 6 7 8 9

0

1 2282 256 2503 290 276 2694 334 308 293 2865 396 349 322 306 2986 487 408 360 331 315 3077 627 494 413 365 336 319 3118 852 625 493 412 364 335 319 3119 1,236 837 615 484 405 357 329 313 305

Table 8

St andard Deviation of Est imat ed Future Loss Payment s

AccidentYear

Development Year

0 1 2 3 4 5 6 7 8 9

01 6.0%2 5.2% 6.13 5.0% 5.2 6.24 5.0% 5.0 5.3 6.35 4.9% 5.0 5.1 5.3 6.36 4.9% 5.0 5.0 5.1 5.4 6.37 5.0% 5.0 5.1 5.2 5.2 5.4 6.38 5.2% 5.2 5.3 5.3 5.4 5.4 5.6 6.49 5.5% 5.6 5.7 5.8 5.8 5.9 5.9 6.0 6.7

Table 7, and t heir st andard err ors ( as percent ages)

are shown in Table 8.

It is evident from t he forecast payment s in Table 7

that additional payments are likely to occur in devel-

opment years t en and above. Here t hese payment s

were forecast by assuming t hat t he proport ional de-crease in value from one development year t o t he next

is i dent ical t o t he decrease fr om development years

eight to nine in the fi tt ed values in Table 5. For ex-

ample, in the first row of Table 5, the forecast values

for development years nine and eight are 205 and 210.

The ratio 205/ 210 0.975. Consequently, the fore-

cast values for all development years greater than ten

are E[ Pi9] 0.975(J9).

This procedure makes it possibl e t o develop an over-

all development year payout pat t ern, which consist s of

t he percent age of t otal accident year l osses t hat are

paid out in each development year. This payout pat -

tern for the company analyzed here is shown in Fig-

ure 6.

We have now obtained est imated loss payments

E[ Pij] for all fut ure years. From t hese esti mates we can

calculate the est imated undiscounted loss reserve,

which is E[ Pij] for all j i. We can also discount

each such cash flow at an appropriat e rate t o obtain

a discounted value for the reserve.The procedure j ust descri bed solves t he fi rst prob-

lem: establishing the expected value of the insurers

loss reserve. What remains is the second problem: de-

termini ng t he distr ibuti on of possible deviati ons from

thi s expected value. This i s an essenti al step in cal-

culating VaR for the loss reserve or for the firm as a

whole. The key in both cases is V, t he vari ance/ covar-

iance matri x of forecast err ors. I nspecti on of thi s ma-

tr ix shows t hat forecast err ors are i n fact correlated

wit h one another. That is, one cannot assume t hat t he

forecast err ors for different futur e payments are in-

dependent of one anot her. Fortunately, the int erde-

pendencies among for ecast errors can be t aken i nt o

account by means of Mont e Carl o simulation. Here I

used @RISK to generate 1,000 scenarios, in each of


14/22


Figure 6

Development Year Payout Pattern

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

55 57 59 61 63 65 67 69

Total Future Payments (000)

Figure 7

Distr ibution of Total Nominal Payment s

0.0%

0.5%1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

55 57 59 61 63 65 67 69


which a r andom error was selected for each forecast

futur e payment from a mul t ivari at e dist ri bution t hat

reflected the variances and covariances of forecast er-

rors. For each scenari o I calculated t wo values: t he

sum of the future payments, and their discountedvalue at a 6%annual rate of int erest. The distr ibutions

of t he nominal and discounted tot als are shown in Fig-

ures 7 and 8.

Summary statistics for the two distr ibutions are

shown i n Table 9. The fi rst column shows t he mean of

the two distributions, and the second column shows

the first percentil e for each distr ibution. The differ-

ence bet ween thi s second number and the fir st , shown

in the third column, is the 99th percentile VaR in

each case. This is the amount of surplus the firm

should have in or der t o be 99%confident of being able

t o successfully fund it s loss reserve li abili t ies. Thefi nal

column in Table 9 shows the ratio of the expected

payments to thi s surplus. These ratios are quite highrelative to industry averages and widely used rules of

thumb. However, a port ion of this difference is no

doubt att ributable to the fact that other sources of

risk not examined here require additional surplus.

The objective of this example has been to demon-

strate how estimation risk can be incorporated into

t he calculation of VaR. I t is not int ended t o provide a

complet e analysis of the ri sks and capit al require-

ments posed by an insurers l oss reserve. It t akes no


15/22


Figure 8

Distribution of Total Discounted Payments

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

6.0%

33 34 35 36 37 38 39


Table 9

Summary of Monte Carlo Simulation Results

Simulat ion M eanFirst

Percentile DifferenceReserve to

Surplus Ratio

Nominal 63,120 66,560 3,440 18.3

Discounted 35,863 37,600 1,737 20.6

account, for example, of potential changes in the

value of t he discounted reserve result ing from changes

in int erest rates. This r isk has been extensively dis-

cussed elsewhere (see, for example, Panning 1995)

and is highly correlated wit h concomit ant changes in

t he mark et value of t he i nsurers invest ed asset s.

The overall significance of estimation risk for VaR

estimates depends upon the length of the time hori-

zon, as noted earl ier. For the part icular case of lossreserves, i t also depends upon both t he l oss payment

pattern and the magnitude of random error to which

individual payments are subject. Short-tail li nes of

business, in which most losses are paid within a rela-

ti vely few years, are subject t o less est imation ri sk

than are long-tail lines. This is clearly seen in Figure

9, which shows t he magnit ude of for ecast err or by de-

velopment year. It is clear t hat f orecast error begins

t o grow exponent iall y beyond year nine. Since forecast

error consists of random error and estimation error,

and since the former is roughly constant, the upward

trajectory of the graph is due almost solely to in-creases in the magnitude of estimation risk.

The significance of esti mati on ri sk also i ncreases

wit h the magnit ude of random error. The random

component of forecast error vari es across diff erent

li nes of business and across different insurers. When

random error is small , as i n thi s example, esti mat ion

ri sk is also relati vely small . But as random error in-

creases (such as across dif ferent li nes of business), so

does the magnitude of estimation risk. Nonetheless,

as this example has demonstrated, the magnitude ofestimation risk can be quantified and included in the

determination of VaR.

4. ADAPTIVE RISK M ODIFICATIONIn calculating VaR for an extended time horizon, one

cannot safely assume t hat t he assets and li abil it ies of

a firm remain static. Even under the most benign sce-

narios, changes will occur because of normal business

operations. Under the extreme scenarios that are the

part icular focus of VaR calculations, however, dra-

matic changes wil l almost cert ainl y occur as decision


16/22


17/22

100 NORTH AM ERICAN ACTUARIAL JOURNAL, VOLUM E 3, NUM BER 2

Figure 10

Momentum Response Curve

0%

20%

40%

60%

80%

100%

-100% -50% 0% 50% 100%

Change in Stock Price Since Inception

PercentageofPortfolio

InvestedinStock

Figure 11

Return Distr ibutions for Alternative Str ategies

0%

2%

4%

6%

8%

10%

12%

14%

16%

80 90 100 110 120 130 140 150 160

Ending Value of Portfolio

Probability

MomentumStrategy

Buy & Hold

Buy Low,Sell High

shown. At the midpoint of the response curve, the re-

sponse t o st ock price changes is approximately b/4.That is, for the momentum manager, a 1%change in

the stock price will induce him or her to increase the

st ock percentage of his port folio by 2.5%, relati ve t o

the proporti on held the previous day. Note t hat a por-

tion of this increase will have occurred naturally due

to the increase in the price of the stock.

For each yearlong scenario, consisting of 250 daily

price changes, the ending values for the t hree port -

folio managers will typicall y be di fferent. For 1,000

such scenarios, the distribution of year-end portfolio

values for t he t hree port folios is shown in Figure 11.

The distr ibution for the buy-and-hold port folio islognormal. Relati ve to it , the momentum str ategy

clearly has a longer upside tail and a lower downside

ri sk, and t he buy-low-sell -high strategy has just t he op-

posit e characterist ics.

Furt her insight int o t he consequences of t he t wo

active strategies can be gained from Figures 12 and13, which show the relationship between the ending

value of each portfolio and the ending price of the

st ock for each of the 1,000 scenarios. The st raight l ine

in both graphs shows the ending value of a portfolio

consisting only of stock. The results for the momen-

t um st rategy closely r esemble t hose obt ained by own-

ing the stock and buying a put opti on on the stock

wit h an exercise price of about 90. By contrast , t he

result s for t he buy-low-sell -high st rategy resemble t he

results obtained from owning the stock and selling a

call opti on on the stock wit h an exercise price of

about 110.As one would expect from t hese graphs, t he mo-

mentum str ategy has the lowest VaR of the three port -

foli os, and t he buy-low-sell-high st rategy has t he high-

est . The result s for one set of 1,000 scenari os are


18/22

THE STRATEGIC USES OF VALUE AT RISK 10 1

Figure 12

Results of M omentum Str ategy

2001801601401201008060

60

80

100

120

140

160

180

200

EndingValue

ofPort

folio

Ending Stock Price

Figure 13

Result s of Buy-Low -Sell-High St rat egy

2001801601401201008060

60

80

100

120

140

160

180

200

EndingValue

ofPort

folio

Ending Stock Price

summari zed i n Table 10, i n which VaR is expressed asa percentage of the initial portfolio value.

These results show that, when VaR is calculated for

a l ong-term horizon, one cannot simply examine t he

initial composition of a portfolio of assets or liabili-

ties. Instead, one must include in the VaR calculation

the likely response of asset or l iabili ty managers to the

events included in t he scenari os that serve as t he basis

for the VaR calculation. While this may seem difficult

or even impossibl e, in fact it presents a valuable op-

portunit y, since different st rategies have quit e differ-

ent implicati ons for the capital needed by the firm.

The real challenge is to make the strategies that de-cision makers follow explicit rather than implicit, to

systemati cally examine their implicati ons for the firm,

and then t o ensure t hat these st rategies are i n fact

followed. Doing this can have substantial benefits for

the firm. For example, it is unlikely that most invest-ment officers reali ze that foll owing a buy-low-sell -high

st rategy requires a considerably higher commit ment

of the firm s capital t han following a momentum str at-

egy, quite apart from differences i n the average r e-

t urns of the t wo str at egies.

This conclusion applies not only to i nvestment port-

folios, but to any port folio of asset s and/ or l iabil it ies.

For example, many fi rms foll ow an impli cit momen-

tum strategy with the various lines of business that

t hey manage, a st rategy that consists of invest ing t o

expand a business when its profitability is high, and

contr acti ng or sell ing it when it s profit abili t y declines.Using such a str at egy creat es what are call ed r eal

opti ons, asymmetr ic return profil es similar to the


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Table 10

VaR for Three Portfolio Strategies

Percent ile Mo men tum Buy and Hold Buy Low Sell Hig h

95% 5.3% 6.6% 8.4%

99 6.2 11.8 18.799.9 7.1 14.5 24.7

one produced by t he momentum st rategy just pre-

sented. This subject is t reat ed in depth by Dixit and

Pindyck (1994) and Trigeorgis (1996).

5. FRANCHISE RISKThe object ive of most str ategic decisions is to increase

the value of the firm. VaR is potentially useful in that

context because i t focuses on the marginal impli ca-

ti ons for risk capit al of t he different alt ernatives underconsiderati on. But if VaR is t o become a useful st ra-

tegic tool, i t must be able to take into account the

vari ous ri sks t o which t he value of t he firm is exposed.

This, in turn, implies that VaR must be based upon a

correct initial estimate of the firms total value.

In t he case of a propert y/ casualty i nsurer, an obvi-

ous procedure for esti mati ng the value of the firm

would be t o ascert ain t he market value of t he fir ms

assets and subtr act from t hat t he est imat ed present

value of the firms liabili ti es, suitably adjusted for

their risk. This is in fact a common practice in esti-

mating and managing the sensitivity of an insurerssurplus t o changes i n int erest rates. Unfort unately,

thi s procedure is highly misleading, for it ignores that

portion of the firms value that cannot be ascertained

from its balance sheet.

The market value or purchase price of an insurer is

typicall y higher t han t he amount obtained by econom-

icall y valuing it s balance sheet . The difference consists

of what I here call franchise value: t he pr esent value

of the firms expected cash flows from business it will

writ e i n the future. Most of an insurers cli ents t end

to renew their policies, since there are significant

practical cost s t o swit ching t o another fi rm. Moreover,

insurers creat e i ncenti ves for their clients t o renew.

As a consequence, a typical insurer can expect to re-

tain around 6090% of its existing clients from one

year to the next. Even if it att racts no new clients, an

insurer can ant icipate signifi cant futur e profit s from

retent ions alone. Unfort unately, accounti ng st andards

do not permi t t his franchise value t o be recognized

unless a fi rm is i n fact purchased, i n which case t he

excess of the purchase price over book value is tr eated

as an asset and i nserted ont o t he balance sheet of t he

acquirer.

This fact has t wo import ant implicati ons for the

strategic use of VaR. First, franchise value is too im-

port ant to be ignored, since enhancing t hat value i s

an important objecti ve of many strategic decisions.

Second, the components of franchise value, and their

exposures t o various t ypes of r isk, should be modeled

and included in VaR in the same fashion as compo-

nents t hat are in fact recognized by accounti ng st an-dards. In this section I show how this can be done for

a propert y/ casualt y insurer with respect to one source

of risk. A more detailed treatment of this problem is

presented in Panning (1995).

In the example presented here I focus exclusively on

franchise value and the potential changes in its value

due t o changes in int erest rates. I have simul ated an

insurer that currently writes policies with a premium

of 100 and loss payments totaling 100 with a payout

patt ern t hat decreases graduall y from 25% to 2% in

development years zero t hrough ten. It s retent ion rate

is 90%, and t he curr ent int erest rate i s 6%.3

All othercomplexities, such as expenses, taxes, and changes in

loss ratios wit h subsequent renewals, are ignored f or

purposes of illustration. The present value of its prof-

its from the firms retentions (I ignore potential sales

to new clients) is approximately 83.6.

If interest rates change but all other conditions in-

cluding premiums remain constant, the present value

of these retentionsthe firms franchise valuewill

change accordingly. Figure 14 shows the effect of dif-

ferent rate changes on the franchise value. Note that

franchise value incr eases when rates ri se: behavior

that is opposit e from that of a bond. This is due t othe fact that the duration (interest rate sensitivity) of

future premiums is lower than the duration of future

loss payments (which occur later in ti me t han pre-

mium inflows).

3This is not the firms cost of capital. The fact that a firm must pro-

vide a reasonable rate of return to its suppliers of capital has strong

implications for the profit margins it must generate, but virtually no

bearing on the appropriate interest rate for discounting its future

cash flows.


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Figure 14

Eff ect of Int erest Rat e Changes on Franchise Value

W hen Premiums Remain Constant

-40%

-30%

-20%

-10%

0%

10%

20%30%

40%

-3% -1% 1% 3% 5%

Change in Interest Rates

PercentChangein

FranchiseValue

Figure 15

Dist ribut ion of Changes in Franchise Value

W hen Premiums Remain Constant

0%

1%

2%

3%

4%

5%

-30% -20% -10% 0% 10% 20% 30%

Percent Change in Franchise Value

Probability

Since i nt erest rates do in fact change, I have sim-ulated the distribution of changes in franchise value

t hat result s when t he volati li ty (standard deviati on of

int erest rate changes, as a percentage of the ini t ial

rate) is 15%. This dist ri bution, shown in Figure 15, i s

reverse l ognormal, as one would expect from Figure

14. The 99% VaR for franchise value i s 21%of it s ini-

tial value.

Now let us add one small but important and re-

alisticcomplexity to the simulation. Let us assume

t hat , when int erest rates rise or fall , at least some

firms in the insurance marketplace alter their prices

(premiums) to gain or retain market share. If the fir m

we are simulati ng maint ains it s current price, then its

retention rate will decline, since at least some of it s

clients will be att racted to the competi tors lower

prices. On the other hand, if it matches its competi-t ors price cut s, t hen it s franchise value will be lower

than would otherwise be the case.

In eit her case t he overall effect is nearl y t he same:

lower cash flows from r etenti ons. The result s for t his

more complex case are shown in Figure 16. Note that

t he effect of i nt erest rate changes on fr anchise value

is now reversed: an i ncrease i n int erest rates now re-

duces the firms franchise value.

The distri bution of possible changes i n franchise

value for t his sit uat ion is shown in Figure 17. The 99%

VaR is again 21%of the ini ti al franchise value, but thi s

now occurs under condit ions t hat are t he opposit e of

those found in the preceding simpler case.


21/22


Figure 16

Eff ect of Int erest Rate Changes on Franchise Value

W hen Premiums Vary Inversely Wit h Int erest Rates

-40%

-20%

0%

20%

40%

-3% -1% 1% 3% 5%

Change in Interest Rates

PercentChangein

FranchiseValue

Figure 17

Dist ribut ion of Changes in Franchise Value

W hen Premiums Vary Inversely wit h Interest Rates

0%

1%

2%

3%

4%

5%

-30% -20% -10% 0% 10% 20% 30%

Probability

Percent Change in Franchise Value

What makes t his simpli fied analysis of franchisevalue part icularly significant is t he fact that the i n-

surer can subst anti all y r educe t he VaR for it s fran-

chise value by altering the duration of its invested as-

sets. In the first case, in which premiums remain

constant, the insurer can reduce VaR by extending its

asset durati on. In the second case, in which the in-

surer alt ers premiums or loses business t o competi-

tors, i t can reduce VaR by shortening t he durati on of

it s invest ments. In eit her case, by making t his change

the insurer creates a sit uati on i n which the change in

its franchise value produced by a change in interest

rates is exactl y offset by an equal and opposit e change

in the value of its invested assets. However, by acting

appropriately to achieve such a reduction in franchise

risk, t he i nsurer will appear to be t aking addit ional

ri sk, since fr anchise value i s not included on i ts bal-ance sheet . (For a more det ail ed example and dem-

onstration of these points see Panning 1995.)

This example has several important impli cati ons.

First, when applying VaR to the firm as a whole, it is

important to include in t he analysis all components of

the firms value, including those not recognized by

current accounting standards. Second, in calcul ating

a fi rms VaR for an extended horizon, the analysis

should t ake int o account r isk factors that are not nor-

mally included in short-term VaR calculations, factors

such as t he effect of competi ti on. This is, i n fact, a

generalizat ion of the point presented i n t he above

analysis of adapti ve ri sk modifi cati on. When VaR is

calculated for long-t erm horizons, it is essenti al to

include in the analysis the likely actions of relevant


22/22


decision makers, including cust omers and competi -

t ors as well as t he managers of the fir m.

6. CONCLUSION

The concept t hat underl ies the calculati on of VaR isin fact quite old. VaR i s in principle nothing more

than what statisticians call a one-sided confidence in-

t erval. What is new i s t he avail abili ty of high-speed

computers and hi stori cal data needed t o implement

t hese calculat ions for large and complex port folios of

securit ies. These same resources also make i t possibl e

to calculate VaR for assets and liabilities other than

securi t ies, for fi rms other t han invest ment banks, and

for decisions that are strategic rather than t acti cal.

But VaR will be useful in these other contexts only if

it successfully takes int o account the longer t ime ho-

rizons that prevail in them.

As we have seen, t ime horizons longer t han t hose

prevalent in investment banking pose three problems

for VaR analysis. Estimation riskthe risk of inaccu-

rately forecasting the response of assets or liabilities

t o changing condit ionscan i n fact be quanti fied and

incorporated into VaR by using appropriate statistical

methods. Adapti ve r isk modifi cat ion t he responses

of decision makers to changes in their circum-

stancescan li kewise be incorporated by correctly

modeli ng t hem and incl uding t hese responses in each

of t he scenari os from which VaR is calculated. Finall y,

franchise r isk is appropriately incorporated by r efus-

ing to take the firms balance sheet at face value andby again carefully modeli ng and including pert inent

strategic decisions in the analysis.

The arguments, examples, and solutions presented

here by no means exhaust t he challenges t hat must

be addressed in extending the scope of VaR to other

indust ri es and to decisions t hat are str at egic rather

t han t acti cal. Nonetheless, I beli eve t hey provide a

useful starti ng point for making greater use of the

considerable virtues of VaR.

ACKNOW LEDGM ENTSI am indebted to the Society of Actuaries for its sup-

port and encouragement, and t o my former coll eagues

at Wil li s Corr oon and two anonymous referees for

helpful comments and criticisms.

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Discussions on this paper can be submitted until Oc-

tober 1, 1999. The author reserves the right to reply to

any discussion. Please see the Submission Guidelines

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