Electronic copy available at: http://ssrn.com/abstract=1487145

Correspondance Address: Larry Eisenberg, School of Management, New JerseyInstitute of Technology, Newark, NJ 07102, USA. Email: larry@riskengineer.com

! ! VaR,!Probability"of"ruin!and!Their! ! Consequences!for!Normal!Risks

LARRY EISENBERG New Jersey Institute of Technology, Newark, NJ, USA

Electronic copy available at: http://ssrn.com/abstract=1487145

VaR,!Probability"of"ruin!and!TheirConsequences!for!Normal!Risks

A Despite the use of VaR as a means to control risk, using VaR can have theBSTRACT

opposite effect. VaR is used by bank and insurance regulators more than any other risk

measure. A value-at-risk (VaR) constraint on the probability that future firm equity value

will be less than a floor, when the floor is zero, is also a constraint on the probability of ruin.

A manager who maximizes his firm's expected equity value subject to a VaRconstraint, when the firm is in bad financial health, pays a premium for financialinstruments that increase his firm's volatility and does the opposite when the firm isin good financial health. Regulations with VaR or probability-of-ruin constraints mayincrease banks' or insurers' volatility in bad economic conditions. Hence the use ofVaR may increase the instability of the global financial network when the financialsystem is more vulnerable. Basel II regulations, via the Internal Based RatingsApproach, encourage banks with greater systemic risk, the large banks, to use VaRconstraints thereby encouraging the banks to which the global financial system ismore vulnerable to take on greater risk when it is more vulnerable.

K W : EY ORDS Premium switching, Probability of ruin, Risk management, VaRconstraint

J C :EL LASSIFICATION G11, G12, G22, G32

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1. Introduction1.1 Why Should Economists Care About Value-at-Risk?

Three times the Goldman traders who made outsized profits during the subprimemeltdown were, according to a article (Kelly, 2007) required toWall Street Journalunwind their positions by higher management. Said one commentator (Salmon,2007) "what fascinates me is the way in which value-at-risk, a very quick-and-dirtyrisk measure, seems to have been of paramount importance within the highlysophisticated investment bank".

The trading desk that made billions of dollars of profits is part of Goldman'smortgage trading department. More than once the head of the department, accordingto (Kelly, 2007), "was summoned to Loyd Blankfein's (Goldman's Chief ExecutiveOfficer) office to cut down on risk...Goldman's top executives understood thestrategy but were uncompromising about the VaR. They demanded that risk be cutby as much as 50%..."

For regulators and the managers of many financial institutions VaR is a visibleand often reported number. Basel II (Basel Committee on Banking Supervision,2004) mentions VaR more than any other risk measure. At many financial institutionsthe chief risk officer (CRO) gets a daily report on firm-wide VaR as well VaRdisaggregated across the firm's activities. Often the CRO has a direct report to theboard of directors as well as to the chief executive officer (CEO). A special case ofVaR is the probability of ruin: used by insurers and insurance regulators since Condorcet (1784) introduced it.

Solvency II often known as Basel II for insurers, a proposal for insuranceregulation the European Union plans to have in place by 2012, explicitly uses VaR todetermine insurers' capital requirements ( ). Probability of ruinEuropean Union, 2007is an active research topic in the actuarial literature on ruin theory e.g. Gerber andShiu (1998).

The defects of VaR as a risk measure are well known in the finance literature.(For example, it is not subadditive), yet VaR continues to be used by insurers,sophisticated investment banks like Goldman Sachs and banking and insuranceregulators. Perhaps one defense of VaR is that the marginal price of risk with a VaRconstraint, when the constraint is binding, is coherent (Eisenberg, 2008).

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Be it a faulty risk measure used by regulators and practitioners who should knowbetter or a risk measure to be incorporated into a fundamental theory of decisionmaking, the consequences of the widespread use of VaR on a firm's risk-takingbehavior and global financial stability bear investigation. A manager who maximizesfirm equity value subject to a VaR constraint, when his firm is in bad financial health,may pay a premium over the risk-neutral price for instruments that increase thevolatility of firm equity. This is different from the asset substitution problem. With aVaR constraint the firm will pay a discount for these instruments when prospects aregood.

A risk measure promoted by regulators that may encourage firms in trouble totake on additional risk may exacerbate the volatility of the global banking system. Inemerging economies with more volatile markets one might expect deeper downturnsand a greater number of banks with periods of dim prospects. The use of VaR mayexacerbate the volatility of these economies. Whether the use of VaR actually doeshave a destabilizing effect depends on how firms use VaR to make decisions.

This paper builds on the results of (Eisenberg, 2007) in which a CEO maximizeshis firm's expected equity value subject to a VaR constraint. This constraint is aconstraint on the probability that firm's portfolio value (assets minus liabilities) orequity will be less than a "floor". With the floor set at zero, this probability is theprobability of ruin. derives (Eisenberg, 2007) a manager's marginal price of risk (alottery) when he maximizes his firm's expected equity subject to a VaR orprobability-of-ruin constraint. Here, the risk premium of this price is the focus of ourinterest. Unlike standard asset pricing models, this risk premium is dependent on thestate of the firm and its financial health.

In this paper I examine, for lotteries and portfolios that are normal variates, theproperties of the marginal price derived in (Eisenberg, 2007). From these propertieswe can infer when a CEO will increase or reduce the risk of his firm. For example,when risks are normal, a CEO who maximizes his firm's expected equity subject to aVaR constraint will—when his firm is in trouble—increase the firm's variance. Hedoes the opposite when his firm is doing well. He is a "variance switcher".1

1This claim appears inadvertantly without proof in [12].

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Variance switching means that a manager can switch from pricing a financialinstrument at a premium over its risk-neutral price to a discount or vice versa. Forexample, if the instrument and the firm's portfolio have a bivariate normaldistribution with positive correlation and expected firm equity value is greater thanthe floor, then the manager prices the instrument at a discount to the risk-neutralprice. This is the usual case for a firm in good financial health and we call this case"good times". However, if firm expected equity value is less than the floor—"badtimes"—the manager prices the instrument at a premium. This premium switch iswithout any change in the instrument's correlation with the firm's value. Themanager switches from disliking variance to preferring it when the firm goes fromgood times to bad times. The opposite happens if the correlation of the instrumentwith the firm's portfolio is negative. Whatever the correlation, the manager willreduce the firm's risk in good times and do the opposite in bad times.

Good and bad, however, are relative to the floor. The manager, for example,may manage his firm with the goal of getting the firm's credit rating changed from Ato Aa. Suppose credit ratings are based on a floor and a probability: a floor to theequity value at which a "buffer" between the firm's asset value and the asset value atwhich the firm fails and the probability that the firm will maintain this buffer. Themanager seeks to raise his firm's credit rating, but his firm's expected equity is lessthan the minimum buffer level required for the higher credit rating. In this case, eventhough the firm is A-rated firm the manager will pay a premium for financialinstruments that increase the variance of his firm's portfolio. He will increase thefirm's risk (in terms of variance) because by doing so he increases the probability thatthe required buffer for an Aa-rating will be maintained.

For another example of a CEO whose floor may be below the firm's expectedequity value is a CEO who wants to achieve greater growth in the firm's equity thanhe expects with his current portfolio. Because, independent of his ambition, in thissetup the CEO always maximizes the firm's expected equity, more ambitious growthis represented by raising the floor to the VaR (or if you will equity) constraint.Going forward, firm value and the manager of the firm also stand for portfolio valueand the portfolio manager.

1.2 Results

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The main contributions of this paper are: 1. For normal risks (a lottery, the firm's asset-liability portfolio), when the firm isin bad financial health a higher correlation of a lottery with the firm's portfoliocorresponds to a higher premium to the firm's price for the lottery over the risk-neutral lottery price. The opposite holds when the firm is in good financial health. 2. The equation for security market line of the CAPM is a special case of themarginal price of risk equation derived in .Eisenberg (2007) 4. At a critical probability that the firm's value will be less then the floor the firm'smanagement switches from averting risk, as measured by the variance of the firm'sequity, to preferring risk. 5. When a firm is in bad times, the firm will price its own portfolio at a premium.Management prefers an increase in scale, that increases risk over leaving the scaleunchanged or reduced. In good times the opposite holds. 6. The firm's forward price for its asset-liability portfolio equals the floor. 7. If regulators respond to a downward shock to the economy by reducing the riskof markets, they will make it more difficult for firm in bad times that have violatedtheir VaR constraints to satisfy them without an equity infusion. 8. Extrapolating the results of the static model investigated here with sequence ofmyopic static models, given an unchanging multivariate distribution for tradedlotteries, management that maximizes the expected value of the firm's portfolio willbecome more risk neutral over time. Of these results perhaps the first is the most important. Basel II encourages thosefirms with the greatest systemic risk, the large banks, to use VaR constraints, andbanks that are subject to VaR constraints are likely take on more risk when they arein trouble thus placing the banking system at greater risk when it is more vulnerable.

1.3 Systemic Risk, Firm Size and Basel II Basel II allows banks to choose between the standardized and internal ratings

based (IRB) approaches. The latter requires an extensive investment in informationtechnology. Due to the cost of implementing the IRB approach large banks are morelikely to choose it (Hakenes and Schnabel, 2006) than small banks. Because defaultby firms with greater liabilities pose a greater systemic threat than default by firmswith less liabilities, Basel II encourages those firms that could pose the greatest

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systemic threat to the financial system to operate with a VaR constraint; and firm'sthat have such a constraint may be taking on more risk when the financial system ismore vulnerable. For a review of the literature through 2006 on the effect of Basel IIon risk-taking of banks by size and on aggregate risk the reader is referred toHakenes and Schnabel (2006). Recent papers on the IRB approach include Kwakand Lee (2007) and Fees et. al. (2008).

In contrast to the partial equilibrium model used here, these papers, for the mostpart, use general equilibrium models, so comparing results is problematic. It isinteresting to note, however, that (Hakenes and Schnabel, 2006) conclude that ifregulators were to apply the IRB approach uniformly across banks, then stability isimproved. The results here on the variance switching suggest otherwise. In thepartial-equilibrium static model used here, any firm whose manager maximizes hisfirm's expected equity subject to a VaR constraint will increase the variance of hisfirm's asset-liability portfolio when his firm is doing poorly. Variance switching mayhave exacerbated the financial crisis of 2007-2009.

1.4 Variance Switching and the Financial Crisis of 2007-20091.4.1 Variance Switching, Margin Calls and Liquidity Spirals The potential procyclical effects of capital requirements and margin calls are wellknown. Regulations that require firms to reduce their positions in response to adverseshocks may be responsible for amplifying those shocks through liquidity spirals andother consequences. There is a substantial literature on determinants of liquidity andmarket instability starting perhaps with Allen and Gale (1994). Studies since thebeginning of the 2007-2009 crisis include Brunnermeier (2009), Brunnermeier andPedersen (2009), and Huang and Wang (2009). There appear to be few studies, ifany, on the procyclical effects of what I call variance switching. However significanta role margin calls or liquidity spirals played in this crisis, variance-switching mayhave also played a role and bears further investigation.

1.4.2 Problems with VaR: Stuffing the Tails vs. Variance Switching Variance switching provides some insight to the contribution of VaR regulations tothe financial crisis of 2007-2009. Among the many ways that VaR regulations mayhave contributed to the financial crisis, it is commonly conjectured that VaR

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regulations helped cause the financial crisis or made the crisis more severe via traders"stuffing the tails", and that regulations using CVaR rather than VaR would preventtail stuffing.

Consider one-year 95% VaR which gives the maximum loss that occurs with aprobability of at least 5% in one year. This VaR is completely insensitive to lossesthat occur with a probability of less than 5% in one year, so a constraint on one-yearVaR at the one-sided 95% for the firm's portfolio is unaffected by positions in atrader's book that have less than a .5% probability of a loss. For example the sale ofan out-of-the-money put option that has less that a 5% chance of expiring in themoney in one year has no effect on one-year 95% VaR. However a drastic enoughdrop in the market price of the put's underlying, can cause a loss orders of magnitudegreater than the revenue from the sale of the put.

Increasing variance when portfolio value is normally distributed is quite differentfrom stuffing the tails. Strictly interpreted, stuffing the tails can be viewed as alteringthe functional form of a distribution. However, one can move large losses to the tailsof a normal distribution and still have a normal distribution.

Consider a portfolio of credit default swaps (CDS) that we model as normallydistributed. By underestimating the correlations to the bonds underlying the CDS,the CDS is estimated as having a low variance. 5% of the area of a standard normaldensity lies below approximately -1.65. Let denote the value of the CDS portfolio!

with non-standard normal parameters , then with .! " ! "! " ! "" ! # $ % % & ' (" )

Suppose the current value of the CDS portfolio is and its true is $100 million.! "

Suppose also that because the correlations of the underlying bonds to the CDS areunderestimated, the estimated is $20 million. Then the estimated 95% VaR is a"̂

loss of 1.65 times $20 million or $33 million, but the actual 95% VaR is $165million. By underestimating one can stuff the tails. That is we assign large losses"

to a high- move when they are in fact a low- move. We stuff the tails by" "

underestimating ."In contrast to stuffing the tails, variance switching has nothing to do with a biased

under reporting of variance. Variance switchers don't under report variance, theychange the actual variance; and sometimes they make it bigger rather than smaller.One can sometimes lower the probability of large losses by increasing variance rather

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than reducing it, that is by making more risky investments do not have a higherexpected value than less risky investments.

Increasing the variance reduces the probability mass in any finite interval, and asthe variance approaches infinity, the probability mass in any finite interval goes tozero. When the portfolio value is normally distributed, the probability that therealized value of the firm's portfolio is between the floor and the mean portfolio valuedecreases as the variance increases. As the variance approaches infinity theprobability that the portfolio will be less than the floor approaches one-half. If thefloor is greater than the mean portfolio value, then increasing the variance reducesthe probability that the portfolio value will be less than the floor. The opposite is thecase if the floor is less than the expected portfolio value. With variance switching,the firm's preference for or aversion to volatility is evident in the risk premium of itsprice for a lottery. The sign of the risk premium of the firm's price for a lotterydepends on the economic prospects of the firm and the correlation of lottery with thefirm's asset-liability portfolio, where the prospects are good if expected portfoliovalue is greater than the floor.

Consider a firm "on its VaR constraint". That is the firm has a binding andfeasible VaR constraint. The firm's "constraint price" for a lottery is the cash tradedfor the lottery that leaves the firm on its constraint after the trade. In a universe ofnormal risks, consider two lotteries and with the same expected payoff but has! * !

a positively correlation with the firm's portfolio and a negative correlation. As*

shown here, in bad times, the firm's constraint price for will be higher than that for!

* ! !. In fact the constraint price for will be at a premium to the risk-neutral price for and the firm's constraint price for will be at a discount to the risk-neutral price for*

*.

1.5 A Sequence of Myopic Two-Date Model and Long-Term Risk Neutrality1.51 Assumptions to a Sequence of Myopic Two-Date Models

We gain a key insight into the dynamic effects of VaR regulations by considering asequence of the dates where the static model considered here is used myopically.Given a stationary multivariate distribution for all traded lotteries, firms will becomerisk neutral. Because the assumptions are restrictive the conclusion that firms will

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become risk neutral is not intended as a prediction. Rather its is cautionary andprovides a benchmark for non-myopic models.

In the static model used here, the firm trades on the first date with the goal ofmaximizing the expected value of its asset-liability portfolio on the second datesubject to a VaR constraint. Consider a daily sequence of models where each modelis the two-date model analyzed here. This implies that the firm has: 1) a rollinghorizon equal to the time between the two dates in the static model and tradesmyopically; 2) VaR regulations require firms to be in compliance at the close ofbusiness; 3) there are no exogenous changes in the parameters of the multivariatedensity.

1.52 Endogenous Changes in Portfolio Variance and Changes in the Firm's RiskPreferences in a Sequence of Myopic Two-Date Models

Consider how changes in the variance of the firm's portfolio due to trades executedby the firm affect subsequent trading decisions. In bad times, the firm seeks toincrease portfolio variance, and in good times the firm does the opposite.

Increased portfolio variance in bad times decreases the absolute value of riskpremiums, so increasing portfolio variance moves the firm towards risk-neutralpricing. In the limit as the firm's variance increases to infinity, the firm's pricesbecome risk neutral. In bad times one could infer the firm moves towards risk-neutralpricing but more slowly as it's variance increases. More slowly because the premiumat which the firm it requires to sell a lottery that increases its variance decreases, andthe discount it is willing to buy lotteries that decrease its variance increases.

Once the firm becomes risk neutral if the market rewards higher variance withhigher return, then the firm's variance will increase very fast as it seeks to maximizeexpected value without regard to risk. If the market does not reward increasedvariance with greater expected return, the evolution of the firm's variance andexpectation once the firm becomes risk neutral is ambiguous.

Decreased portfolio variance in good times, increases the absolute value of riskpremiums, so the firm will reduce its risk at an accelerating rate. There are twopossible medium-term outcomes for the composition of the firm's portfolio and for itsattitude towards risk. The firm will convert its portfolio to the risk-free asset

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completely, or it will do partially because the expected value of the firm's portfoliovalue will equal the floor before the portfolio is completely converted.

Which occurs depends on the market price for the firm's portfolio. If the marketprice for the portfolio is less than the floor discounted at the risk-free rate, conversionwill be partial, and the firm will be risk-neutral. If the market price for the portfolioequals the discounted floor, then conversions is complete and the firm is completelyrisk neutral. If the market price of the portfolio is greater than the discounted floor,coversion will be complete and the firm will not be risk neutral.

If the market rewards greater risk with greater expected return, then the firm'sinitial portfolio (before conversion) will be priced by the market lower than aquantity of the risk-free asset with the same expected payoff as the initial portfolio.Hence conversion will be partial and the firm will become completely risk neutral.

1.6 Exogenous Changes1.61 Exogenous Changes to the Portfolio Mean

An exogenous downward shock to the portfolio mean may cause the firm to be inviolation of its VaR constraint. Also the constraint may no longer be feasible withoutexternal equity. If the firm does not issue equity, and the current constraint is notfeasible, then floor or the maximum tolerable probability of reaching the floor willhave to be reset. The firm may have to trade to adjust the distribution of its portfolioto satisfy the new feasible constraint. The model analyzed here says nothing abouthow a VaR constraint would be reset.

A VaR constraint may remain feasible after an exogenous downward shock to theportfolio mean. The firm will have to alter the composition of its portfolio to satisfyits constraint by raising the portfolio mean or reducing (increasing) the portfoliovariance if the firm is in good (bad) times.

An exogenous upward shock to the portfolio mean will cause the firm to becomemore conservative in its risk preferences.

1.62 Exogenous Changes to the Portfolio VarianceBecause there are endogenous changes in the portfolio variance, one time shocks to

the variance should not affect portfolio variance over longer periods.

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1.7 Advice for Regulators1.7.1 Short and Long-Term Consequences

Regulators need to be aware that firms in trouble may switch from seeking toreduce risk as measured by the variance of their asset-liability portfolios to increasingit when the expected payoff of the firm's portfolio is low relative to the floor. BaselII encourages those firms with the greatest systemic risk, the large banks, to use VaRconstraints, and banks that use VaR constraints are likely take on more risk whenthey are in trouble placing the banking system at greater risk when it is morevulnerable. More generally regulators need to consider the potential procyclicaleffects of their regulations of which variance switching is one. Also, there may belonger-term consequences of VaR regulations.

If firms behave in accordance with a sequence of myopic two-date models, thengiven no change in the multivariate distribution of traded lotteries (the weights of thelotteries in the firm's portfolio can change), firms may eventually move towards risk-neutral pricing. While one be must skeptical about modeling financial decisionmaking with a sequence of myopic two-date models, the implications of such asequence are a warning to regulators and can be used as a benchmark prediction fordynamic models.

1.7.2 CVaR vs. VaR and the Uses of VaRVaR can be used in basically three ways: passively, defensively or actively (Jorion,

2001). These are: to monitor risk, to set limits or to allocate risk. An allocating ofrisk can be viewed as the solution to a constrained optimization problem where thelimits are the constraints. If so, one cannot use VaR actively unless one also uses itdefensively. To use VaR defensively, one must measure VaR; so VaR cannot beused defensively unless it is used passively.

Regulations could stipulate passive, defensive (hence also passive), or passive,defensive and active use of VaR or CVaR. CVaR may be superior to VaR forpassive and defensive use. It is not clear that it is superior for active use.

Used to monitor risk CVaR or expected shortfall is commonly viewed as preferableto VaR because one cannot stuff the tails with CVaR. Because the CVaR of a firm'sportfolio is the expected value of the portfolio up to its floor level, CVaR is sensitive

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to even highly improbable events. There is another benefit to using CVaR to monitorrisk.

Not all firms are always in compliance with regulations. The variance switchingresults in this paper and in Eisenberg (2008b) hold for a firm whose constraint isbinding and feasible and is in compliance with its VaR constraint. For firms that arenot in compliance with their constraint, but accurately report their exposures, there isgood news for CVaR relative to VaR. CVaR provides more information about tailrisk for all firms including those that are non-compliant. The fact that on its VaR orCVaR constraint a firm is a variance-switcher may be irrelevant to firms outside oftheir constraint.

To set limits, there are many way that VaR or CVaR can be used. The limits maybe defined in terms of VaR, deltas and gammas, number of positions or notionalexposure. Limits on VaR are VaR constraints, and VaR constraints have implicationsfor the allocation of risk, so this defensive use of VaR cannot be separated from usingVaR actively. More generally, the implications of using VaR to trigger limits definedin terms of some other risk measure depend on the specifics of what the limits are andhow they are set. Used defensively CVaR might be superior for the same reason itmay be superior to monitor risk: CVaR is sensitive to improbable events.

To allocate risk, however, it is not clear the CVaR is superior to VaR. The firm'sconstraint price when the firm has a CVaR constraint will usually be very close to thefirm's constraint price with a VaR constraint (Eisenberg, 2008b). With a CVaRconstraint, the price has two terms. The first term is the constraint price when thefirm has a VaR constraint, and the second term, subtracted from the first, has amagnitude equal to that of the lottery payoff divided by the portfolio floor. Ingeneral any lottery will be of smaller magnitude than the firm's portfolio and its floor,so the first term dominates; and the risk premium to the constraint price with a CVaRconstraint will be close to that with a VaR constraint. Hence the constraint price riskpremium will be about the same under either constraint.

The bad news about switching to regulations that stipulate the defensive (withCVaR limits) or active use of CVaR is that, as with a VaR constraint, when the firmis on its constraint the firm is a variance switcher. The risk allocation decisions of afirm with a VaR constraint should not differ much from its decisions with a CVaR

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constraint. For firms that do not violate their constraints, whether the constraint is aVaR or a CVaR constraint should not matter.

1.7.3 Conservative Strategies

When the management of a firm that has a VaR constraint claims to adopt a moreconservative strategy by seeking a higher credit rating, such a change may imply thatthe firm will, counter to intuition, take on more risk as measured by variance witheither constraint. The reason is that if aiming for a higher credit rating impliesaiming for a greater capital buffer and hence a higher floor, then the firm may beraising the floor to its VaR constraint.

1.8 Advice for ManagersFirst, astute financial managers already know that monitoring risk using numbers

from VaR reports is no safeguard against stuffing the tails. Those who don't alreadyknow should. This means that management most impose constraints on the firm'srisk in addition to the VaR constraints currently required by regulators.

Second, managers should be aware that if: 1) they seek to maximize the value ofthe firm's asset liability portfolio; 2) they model risks as normal; 3) their VaRconstraints are binding and 4) their trades are consistent with their objective; theneven if they have not been aware of the results in this paper, they will trade as if theyhave been explicitly using these results. The implication is that in bad times even ifthey have not violated their VaR constraint, they may be increasing their firm's risk.

Third, as mentioned using a CVaR constraint may cause the same risk-takingbehavior as a VaR constraint. However because CVaR is linear it is a much moreconvenient measure to use than VaR. For this reason, perhaps more than any other,2

CVaR may be preferred to VaR. Fourth, as mentioned above, consider a manager who 1) already is maximizing

expected equity value subject to a VaR constraint; 2) wants to increase the firm'sexpected equity as represented by increasing the floor to the firm's VaR constraint; 3)does not change in the universe of tradable securities (or projects, such as thosenewly available due to technological advances). To achieve his objective this

2Properties of the measure VaR are different from those of the constraint price with a VaRconstraint. The latter is additive, and therefor subadditive. The former is not subadditive.

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manager must increase the firm's risk . While it may seemas measured by varianceobvious when a manager seeks greater growth he may take on more risk it is lessintuitive that he may take on more risk when he become "more conservative" byseeking a higher credit rating via raising his floor.

1.9 Structure of the PaperSection 2 discusses variance-switching behavior, and reviews the literature on this

topic. A manager who is a premium switcher may switch between paying a premiumover the risk-neutral price of a lottery (the residual risk of a hedge, policy, financialsecurity or contract) and paying a discount without a change in the correlation anylottery with any other lottery.

Section 3 compares and contrasts variance switching with asset substitution.Section 4 .reviews results from Eisenberg (2007) used hereSection 5 discusses properties of the risk premium to the firm's constraint price for

a lottery.Section 6 shows that a firm will seek to increase the size of its portfolio rather than

doing nothing when it is bad times.Section 7 discusses how a firm that is in violation of a feasible VaR constraint can

change its portfolio to satisfy the constraint. Section 8 concludes. Proposition proofs are in the Appendix where denotes the end of a proposition!

proof. Otherwise it denotes the end of a definition or proposition statement.

2. Variance Switching Standard asset pricing models such as the Capital Asset Pricing Model (CAPM)

(Litner; 1965a, 1965b; Mossin, 1966; Treynor, 1961) or Arbitrage Pricing Theory(APT) (Ross, 1976) make no inferences made about the state of a firm and its pricefor risk. VaR regulations are, intentionally, are about the state of a firm. Though notmotivated to influence a firm's price for a lottery, VaR regulations have consequencesfor how a firm prices risk. The price is a reservation price, and it is specific to thefirm. Given market prices the firm may trade a lottery or it may not.

In standard asset pricing models, the risk premium for a lottery cannot change signunless its correlation with the market or with a factor changes sign. However with a

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VaR constraint the firm's constraint price for a lottery can have a risk premium thatchanges sign without any change in the lottery's correlation an underlying factor oranother lottery. A change in the firm's financial health changes the risk premium ofits price for a lottery. A small change in firm's financial health can cause the firm'sprice for a lottery to change sign. When a portfolio has a high probability of beating a benchmark, decreasing thevariance raises the probability of beating the benchmark and increasing the variancehas the opposite effect. However when the portfolio has a low probability of beatingthe benchmark, these effects are switched. If the benchmark is the amount owed tocreditors or a level that represents the liquidation drawdown for a hedge fund, then ifthe manager does not beat the benchmark the firm goes bankrupt or the fund isliquidated. Survival of the firm or fund, and perhaps the manager's employment andfuture employment opportunities, may depend on beating a benchmark. A rough analogy with biology is the flight or fight response of a cornered animal.When successful flight is probable it may be a low-risk strategy, but when it is not,the strategy to fight, if the outcome is more uncertain - the animal may win, may bebetter than an almost certain (and hence low risk) death. Variance switching, though not so named, is not new to the economics and financeliterature. Basak and Shapiro (2001) embed risk management objectives in a utilitymaximization framework to obtain a result similar to one of the results here. UsingCRRA preferences and lognormal prices, they show that portfolio managers with aVaR constraint tend to have larger losses in unfavorable conditions relative tomanagers who do not have a VaR constraint. Regarding a notion fundamental in microeconomics, profit maximization, Radner(1995) argues against the hypothesis put forth by Friedman (1953). Radner arguesfirms that maximize profits have the greatest chance for survival, and he posits thatthe fraction of profit-maximizing firms that survive is negligible. One of his resultsis that firm with little wealth will behave as if they are risk lovers, and firms withgreat wealth will behave as if they are more risk averse. More recently , introduce a theory of the firm thatJarrow and Purnanandam (2004)has implications for capital structure, capital budgeting, and liquidity. In their modelthere are three states of the world: no default, default, and insolvency. Before a firmbecomes insolvent, it must default. Default occurs when the firm violates some

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covenants with its debt holders. Such a model implies that the firm is risk-averse aslong as it is not at the default state. Once in default, but not yet insolvent, the firmswitches to risk-preferring from risk averting behavior (Jarrow, 2004). There are alsoempirical studies of mutual fund managers this suggest these managers engage invariance switching. Some mutual fund tournament studies conclude that losing managers (those withlower than average returns partway through the year) have more volatile funds andwinning managers the opposite. (1996) confirm this hypothesis inBrown et. al.examination of monthly data on growth-oriented mutual funds. Researchconsistent with this conclusion includes , ( andBronars (1987) McLaughlin, 1988)Ehrenberg and Bognanno (1990) Chen and Pennacchi. (2001) obtain similar resultsto (1996) but for tracking error to a benchmark index rather thanBrown et. al.variance. Variance switching, not surprisingly, is also related to the asset substitution (Blackand Scholes, 1973; Jensen and Meckling, 1976) or risk shifting problem. 3. Variance Switching Compared to the Asset Substitution Problem Economic intuition suggests that a company in good financial health should bemore conservative than one close to bankruptcy. Shareholders have much more tolose from bankruptcy if their firm is doing well and their equity is worth a lot than ifthe firm is close to failure and their equity is nearly worthless. Readers familiar with the asset substitution problem will recall that a far-out-of-the-money option has a much greater percentage increase in value from increasingvolatility than a deep-in-the-money option. In the context of the asset substitutionproblem the out-of-the-money option is the equity of a nearly bankrupt firm and thedeep-in-the-money option is the equity of a firm that has little chance of failure.Hence shareholders benefit more from a more aggressive or risky strategy when thefirm is doing poorly. With both variance switching and asset substitution, equity holders benefit moreby increasing risk when the firm is in trouble than when it is not. There is however akey difference between the variance switching and asset substitution. With varianceswitching sometimes shareholders benefit from increasing risk and sometimes bydecreasing it because their firm stays in compliance with its VaR constraint. Their

16

firm is not shut down by regulators, nor is their equity diluted due to issuance of newshares. In the asset substitution problem shareholders always benefit from increasingrisk. The usual statement of the risk shifting problem describes a conflict between bondholders and equity holders as justified by put-call parity and the commonly statedassumptions to the Black-Scholes option pricing model (Black and Scholes, 1973):complete markets, geometric brownian motion asset returns, and assets with alognormal distribution. The risk shifting problem is can also be stated witharithmetic brownian asset returns, a modified option pricing model and assets with anormal distribution.

As described in standard elementary finance textbooks, when the firm has onlyzero-coupon bonds outstanding, all with the equal priority and the same maturitydate, then equity can be represented as a call option on the assets of the firm; and byput-call parity the firm's debt can be represented as default-free debt minus a putoption on the assets of the firm. So if a change in the asset distribution increases thevalue of the call without increasing asset value, equity holders benefit at the expenseof the bond holders.

Because a standard option is monotonic increasing in the standard deviation of"

the returns of its underlying, according to the asset substitution problem equityholders will always benefit by an increase in . Hence equity holders will always"

seek to make the firm more risky.However the equity holders' gains are at the expense of the bond holders.

Intuitively one can see why a standard option is monotonic increasing in under the"

Black-Scholes assumptions. Based on these assumptions calls and puts are theexpected payoffs of these options using risk-neutral probabilities (the expectationusing actuarial probabilities and discounted at the risk-free rate). Increasing moves probability mass to the tails for a normal variate Because the"

value of a call option is the discounted expectation of underlying values greater thanthe option's strike price, moving mass to the tails increases the value of the option forboth types of variates by increasing the probabilities of more extreme values. By put-call parity an increase in the call's value due to an increase in must be matched an"

identical increase in the put's value. Increasing has another effect and this effect is"

important for variance switching.

17

For the normal distribution increasing lowers the probability mass over any"

finite interval. Consider the special case of a probability-of-ruin constraint. In thevariance switching problem the probability that the firm's bonds pay their face valueat maturity is the focus rather than the bonds' expected payoff. In bad timesincreasing lowers this probability and in bad times increasing has the opposite" "

effect. In the asset substitution problem independent of the level of the option strikerelative to expected asset value increasing always reduces debt value."

4. Preliminaries4.1 VaR Constraint (Telser,A VaR constraint, otherwise known as a Telser safety-first constraint1955-56) is given by the inequality

+,-./% 0 1 # % 0 +2 23/ 1; (1)4

where the firm's value (payoff) realized at Date 1, the floor to firm's value% 5 52

at Date 1, and the manager's maximum tolerable probability and is the+ 5 34

cumulative probability distribution function that is absolutely continuous. A specialcase of a VaR constraint is a probability-of-ruin constraint where the asset-liabilityfloor 2 is zero.3

From Eisenberg (2007) a manager's reservation bid or offer for a lottery (a contractor project) given that the his objective is to maximize the expected value of his firm'sasset-liability portfolio subject to a VaR constraint is an expectation: theunconditional expectation of the lottery or its conditional expectation given that thefirm or portfolio equals a "floor" level chosen by the manager. Consider a firm with a binding VaR constraint that trades a lottery forward for cash(the cash and the lottery are exchanged at Date 1), and is still on its constraint (theconstraint is feasible and binding) after the trade. For example, a firm buys forward #units of an insurance policy ( ) with a non-negative payoff . is the payoff to# 6 ( * %

the firm's risky portfolio prior to the sale of the policy, and 2 "7 5 % $ * $ 7! "# #

is the payoff to the portfolio with units of added to plus a cash payment per# * % 7

3In the simple case where liabilities due on the second date are not random, as with zero-coupon bonds that all mature on date 1, a VaR constraint with an asset floor equal to the facevalue of the zero-coupon bonds is also a probability-of-ruin constraint.

18

# units of the lottery. If the firm is on its VaR constraint prior to trading , then*

+,-./% 0 1 # +2 4. If at the initial date the constraint is still feasible and bindingwith equality after contracting forward to receive on date 1 and # units of cash ,* 7

7 8 (, then the firm remains on its constraint after the trade

+,-./ 0 1 # +,-./% $ 0 1 # + 92 "7 * $ 7! "# #2 2 4 (2)

One can think of # and as control variables that can be adjusted so that7

+,-./ 0 1 # +2 "7! "# 2 4. The firm is the buyer and has a forward cash outflow or a forward negative cashflow . T Note that if7 8 ( he average forward price per unit of the lottery is . : 7

#

instead of buying the policy the firm sells the lottery, then # 8 (. The forward cashflow per unit of the policy and the forward price do not change, but since ,#" 7 8 (

the firm's forward cash flow for selling | | units of the policy is .# #7 6 (

In contrast to the average forward price, ;7;# the marginal forward cash flow per

unit of the lottery is*

;7

;## : 9

+,-. 2 "7

+,-. 2 "7

<<

#(

<<7

#(

##

#

/ 0 1

/ 0 1

! "! "#

#

2

2

## (3)

4.2 Marginal price of risk with a VaR constraint Suppose the firm trades is the marginal forward cash flow; *# units of lottery . ;7;#per unit of the lottery that leaves a firm on its constraint. equals the negative of* ;7

;#

the conditional expectation of the lottery's payoff given the payoff to the firm's totalportfolio equals the floor (floor event). Let *("= denote the spot marginal constraintprice for *

;7# >? * @ #

;A %

#: B9C * * #:,

("= ("=$ 2 (4)

;7;# is the cash flow per unit of the lottery is opposite in sign to the conditional*

expectation because the firm has a cash outflow when it buys a lottery with a positivepayoff. The marginal spot price per unit of on the constraint is equal to the*

forward price on the constraint discounted at the riskless rate .* *("= ("=D * # DE ( E:):

19

We need to make a distinction between the firm's constraint price and its bid andoffer prices. When the VaR constraint is binding, the firm's bid price for a portfolioof lotteries is not, in general, equal to its offer price. The manager's goal is twofold:(a) to maximize the firm's expected value and (b) to satisfy his VaR constraint. The manager's first goal implies the firm's forward bid price for a lottery cannot begreater than the lottery's expectation, and the second goal implies, the firm's forwardbid price cannot be more than the lottery's conditional expectation. Hence the firm'sreservation forward bid price is the minimum of these two expectations. By similarreasoning, the firm's offer price is the maximum of the expectations:

forward bid (5) non-binding VaR constraint binding VaR constraint#

>*FG'H>*"$ >? @%* # BI2

forward offer non-binding VaR constraintbinding VaR constraint#

>*FJ!H>*"$ >? @%* # BI2

Note that if the forward market bid is lower than the firm's forward offer and theforward market offer is greater than the firm's forward market bid, then the firm doesnot trade . For the rest of the paper, unless explicitly stated otherwise, constraint*

price will mean the marginal forward constraint price with the understanding thatwhen the constraint is binding the bid or offer must equal the constraint price. An analog to equation (5) holds for the price of a portfolio of lotteries. Let ! K %'

denote the vector of payoffs for all the lotteries that trade. The firm's portfolio priorto the trade has a payoff at date 1 denoted by lottery% # ! L ! the dot product of the payoff vector an initial portfolio vector When the firm trades a! with .! K %' portfolio of lotteries firm's prior portfolio, " ! ! "! ! , it adds to the , to end upL %

with the portfolio . From t! "L $ %! [0] he forward cash flow that leaves the;7

probability of breaching the floor unchanged is

;7 # : >? @! !("= ("=L A % # B" !! .# 2 (6)

Again, the signed cash flow is the negative dot product of the lotteries' conditionalexpectations given that the portfolio equals the floor event with . To determine"!

the price per unit of this portfolio define # to be a vector of 1's with the samedimension as . Also define to be , then the forward! "1 unit of the portfolio ! "

" #!!L

constraint price for one unit of is given by equation (6) with substituted for"! "" #

!!L

20

"! and the spot constraint price is . Alternatively to write the forwardDE:)!("= L "

" #!!L

price as denote a portfolio of lotteries the firm trades as : 5 " " 999";7; ) M '# " ! "& & &

where and there are tradable securities. Then we can also write equation" L # # ) '

(6) as

;7# : >? @ >? @

;L A L L % # B % # B L

#! !("= ("=" " " " .# #! !2 2 (7)

4.3 Constrained Maximization Problem From Eisenberg (2007) for distributions that include the multivariate normal giventhe assumptions: (a) everywhere differentiable market supply-demand curves thatgive the market price of a lottery as a function of the quantity of that lottery the firmtrades and (b) the manager's choice set of portfolios is compact, then there is a!

global maximum to the manager's constrained optimization problem. !4 The firm'sbid and ask reservation prices for any marginal trade of a lottery or lotteries includingat the optimum must satisfy equation (5) or its analog.!4

5. Risk Premium with Normal Risks From the risk premium of the firm's constraint price we can infer the riskpreferences of the firm's management. If the premium is positive for a lottery thathas a positive correlation with the firm's portfolio, zero for a lottery with zerocorrelation and negative when the correlation is negative; then the firm's managementvalues a lottery that will increase the variance of the firm's portfolio more than onethat does not. The main results to this paper are that for normal risks, given thestandard deviation of the firm's portfolio, the lottery and their correlation the riskpremium to the firm's constraint price can switch between negative and positive.

5.1 Vega of the Risk Premium During usual business conditions, the floor is below the portfolio's expectedvalue. For lotteries that are positively correlated with firm value the risk premium isnegative. When conditions are less favorable, the premium switches sign. In badconditions the manager pays a premium for lotteries that increase firm variance. Ingood conditions, he does the opposite. This section discusses the changing sign of

21

the risk premium. First we determine the risk premium to the constraint price withnormal risks by obtaining the constraint price.

5.2 Normal Constraint PriceThe following proposition gives the marginal constraint price for a lottery when*

the portfolio is composed of the firm's portfolio prior to trading plus 2 "7 % *! "# #

units of plus cash .* 7

Proposition 1 Normal Pricing on the VaR Constraint: If the joint distribution forthe payoff of the portfolio and the lottery is a bivariate normal distribution,N %" *! "then the firm's marginal reservation price for a lottery with payoff is given by*

* # "7 "7("=;7

;# : C ? $ * B # : C ? *2 / : 2 1 $ * B / 1: ::

#' (:, :,$ $ ! " ! "! "# #2 . 8

' ( (#" # "

"5 *2 / : 2 1 *2 5 / 1

: /*1 $ *%

/2 1! " ! " ! "! " ! " ! "! "# # #

#" 7 " 7 " 7

"72 ; .

M%

M9

risk premium' 5

continuous risk-free interest rate, 5

$ 5 time between date and date ( )

2 5 floor 7 5 forward payment contracted at date 0 and received at date 1 in exchange for payoff of units of at date 1# *

the portfolio payoff at date 1 forward2 5 % $ * $ 7"! "#" 7 #

payment contracted at date 0 and received at date 1 expected value of O 5 O" 2: O # *" %" ! "#" 7

units of and cash # #%% 5 * 7portfolio prior to adding units of and cash if % 5 * 7 # )portfolio prior to adding # #%

covariance of and "! " ! "*P 5 * P" P # %" 2 #" 7

)! " ! "*P 5 * " P # %" 2correlation of and P #" 7

"! " ! "O 5 2standard deviation of O" O # *" %" #" 7

(! "! "*2 5#" 7 "

"

! "! "! "! "*2

2

#

#

"7

"7M

22

!

5.3 Price of a Small Normal LotteryNext we obtain the marginal constraint price when the firm trades a "small" lottery.

Whatever definition is used for 1 unit of and 1 unit of , and are* % * %#" # "! " ! "%

invariant to them. unit of the lottery there is a standardGiven a definition for onedeviation to that unit . Given there is a standard deviation to the" # "! " ! "* %%

diversified portfolio prior to the inclusion of the lottery. Whatever arbitrarydefinition for these units is used and are invariant to them. To# " # "! " ! "* %%

simplify notation let Let will say a lottery relative to the portfolio# *%*%# ) 5 #"

# "! "! "%

is small when . Define one unit of . Given ,* "# ( * *! "lim#Q(

%* # # " " # #! "! " ! "A " % " * # ( # ( 7 *. Note that when the cash exchanged for is

also . Taking the limit of from proposition 1 as both and go to zero we get( 7;7;# #

the result of the next proposition.

Proposition 2 Small Normal Contract: The spot constraint price for a small lottery* is

* # C : C ? *% / : % 1 $ * B9;7

;: :

("=:,/=:R1 :,/=:R1

Q( = (10)lim* #

(! " 2

!

To get the bid ask price from equation (5) when the VaR constraint is binding

bid (11) # $ C *% / : 21 $ * *% / : % 1 8 (

: : :

C * *% / : % 1 S (: :

:,

:,

$

$( (

(! " ! "! "2 2

2

offer (12)# $ C * *% / : % 1 0 (: :

C *% / : % 1 $ * *% / : % 1 6 (9: : :

:,

:,

$

$(

( (! "! " ! " 2

2 2

The sign of the risk premium depends on the beta of a lottery with the firm'sportfolio and the firm's financial health as in this definition.

Definition 1 Bad, Neutral and Good Times : A firm is in bad (good) times whenthe expected value of its asset liability portfolio is less (greater) than the floor When9

23

they are equal the firm is in neutral times. Note that this definition holds for firms inviolation of their VaR constraints as well as those in compliance.

2 : % 9:6 (# (8 (

%&'

bad times neutral times good times

(13)

!

Proposition 3 Variance Switching with Normal Risks When the firm is in badtimes it has a preference for risk, and when it is good times it has an aversion to risk as summarized in table 1. The firm's economic condition (good, neutral or badtimes) and the sign of the correlation of the lottery with the firm's portfolio determinethe risk premium of the firm's constraint price.

[TABLE 1]!

5.6 Critical Probability Behavior Switching Note that is a standard normal variate. With denoting the inverse%:% :)

_

"! "% T

cumulative distribution function for the standard normal, let is the+ 5 T4 %:%:( )_

"! "%probability that the firm's floor is breached when the firm is on its constraint, then

T + # 92 : %::) 4! " _

(14)"! "%

Suppose the firm's cash and non-stochastic obligations fix the floor, then the mean ofthe firm's portfolio determines the firm's variance switching behavior. One can ask atwhat value for the portfolio mean does the firm switch between preferring andaverting risk, i.e. at what value for the mean is the firm risk neutral. Alternatively ifthe firm is considering whether to issue equity or debt with non-stochastic payments,one can ask at what floor, given the portfolio mean, is the firm risk neutral.

Definition 2 Critical Probability and Floor $% &%

24

a) Given the multivariate distribution of the firm's portfolio and all lotteries, the%

constraint price risk premium for all lotteries is for , the critical floor.' ( % # 2U

That is, using the constraint price from equation (4), for any lottery , * % # 2U solves' # >? * @ #% 2U B : * # (: .b) The critical probability is the value of the cumulative distribution of the+U 3

firm's portfolio evaluated at the critical floor . That is 2 2U U3 + #:)U! " 9

!

If 3 is a non-standard cumulative normal, then normalizing to get the standard c.d.f.3 + # T + #:) :)

U U! " ! "2U:%:

% becomes . 2U

"! " Note that switches sign atT +:) 4! "+ # +4 4) )

M M, and the floor equals the mean. As increases from below to greater than ,

the firm switches from being variance averse to preferring variance.

Proposition 4 Critical Values for Good or Bad Times with Normal Risksa) .2U # % + #: and U

)M

b) On a VaR constraint with parameters and , the+ + 6 + V V4 4U2 2 2" 6 #U %:

firm is in bad times and is positive (negative) for lotteries that increase (decrease)'

the firm's risk. For , the inequalities are reversed.+ 8 +4U

!

To better understand variance switching, we examine in detail the risk premium tothe firm's constraint price when risks are normal.

5.7 Risk Premium ExamplesBecause the risk premium to the firm's constraint price is our main interest we

examine the premium for normal risks in more detail.5.7.1 Sign of the Risk Premium for # W (

Using proposition 1 suppose in (8) that the insurance company has sold insurance* ; *, and is selling more: . has a non-negative payoff for the insured. By# #, 8 (

selling of , the insurer has a liability, a short position in a non-negative; *# unitsvalued asset. insurer The gets positive cash inflow for increasing its short position in* ;7 6 (, , so the left-hand side is negative.

25

What is the sign of the right-hand side? Under usual conditions, business is sound,and the expected value of the asset-liability portfolio is2 %

: :! "# #" 7 # $ * $ 7:

above the floornormal conditions X ( 6 2 : 2

:! "#" 7 . (15)

Next examine . For a short position in , .#" ) " " #"M M! " ! " ! " ! " ! "* $ *% * % * * 8 (

Suppose first that is uncorrelated with any asset, held short or long, in , that* 2

) ) " " #"! " ! " ! " ! " ! "*% *% * % # ( *# (. In this case . Multiply (15) by and get aM

positive number. Multiply again by and add . The right-hand side: C : C *::, :,$ $

is negative as it should be in this case. 5.7.2 Zero Correlation Risk Premium5.7.2 1 and. # )# ( *% # ( ! " When # ), , the price of the lottery reduces to the present value of the! "*% # (

expected payoff.

;7

;# : C *:

#:,$ . (16)

5.4.2 2 . 2 : 2 # (_ ! "#" 7

When the expected level of the diversified portfolio is the floor level, then the riskcharge is zero.

5.4.3 Sign of the Risk Premium with )! "*% # ("# W 0 The risk charge is the part of the price that is not due to the expected loss of thelottery * (:. This charge in the case of correlation and before time discounting is

,GYZ U[J,NC # : 9/ : 2 1 *

:

/2 1

2 ! " ! "! "#

#

" 7

" 7

#"

"

M

M (17)

Going forward we will use the small contract price.

5.8 SML of the CAPM

26

When the firm's portfolio is the market portfolio and the floor is the marketportfolio compounded at the risk-free rate the small-contact price equation, equation( ),10 expressed in return format is the s SML) of theecurity-market line ( equationCAPM (Litner; 1965a, 1965b; Mossin, 1966; Treynor, 1961). Hence the SML is aspecial case of . This is a claim only about the SML equation. Theequation 10( )CAPM is a general equilibrium model, so it cannot be a special case of a partialequilibrium model.

Proposition 5 CAPM's SML as a Special Case: Let the firm's floor equal themarket price for the market portfolio, and the firm's portfolio be the market&F(

portfolio times a scale factor then the formula for the firm's spot constraint price&"

for a lottery , stated as a return is the equation for the SML for the expected return*

of *9!

Figure 1 below graphs the constraint price risk premium as correlation varies from-1 to 1 and expected equity value (One of the problems with using the normal torepresent equity prices that it allows for negative values.) varies from -100 to 100with the equity floor equal to 0. Hence the constraint is on the probability of ruin.Figures 2 and 3 graph the bid and offer risk premiums respectively.

Given the distribution for the firm's portfolio there is a 1-1 correspondence betweenthe floor and the cumulative probability evaluated at the floor.

[FIGURE 1]

[FIGURE ]M

[FIGURE ]\

6.4 The Firm's Constraint Price for its own EquityWe can apply the constraint pricing formula not only to some lottery , but also to*

the firm's portfolio.

27

Proposition 6 Portfolio Constraint Price and Return: The forward constraint pricefor the firm's portfolio equals the floor % 9( 2

!

6. Homogenous Risk ManagementIt is easy to show that in bad times, the firm's constraint price for its own portfolio isat a premium. Because the firm's portfolio has a positive correlation with itself,adding quantity increases the variance of the firm's portfolio. By contrast, fromproposition 1 the firm's constraint price for a lottery with negative or zero correlationwith the firm's portfolio will be respectively at a discount or risk-neutral. Hence inbad times, management will prefer increasing risk via a homogeneous expansion ofthe portfolio to leaving the portfolio and its risk unchanged. The opposite is true ingood times.The following proposition says that when the firm is in trouble the firm prefers toincreasing its scale to doing nothing or reducing its scale. By increasing its scale thefirm increases the variance of its portfolio, so it is also increasing its risk. Theopposite holds in good times. The firm prefers contracting to doing nothing orincreasing its scale.

Proposition 7 :Homogenous Risk Expansion (Contraction) in Bad (Good) TimesIn bad times the firm's constraint price for its own portfolio is at a premium. Becausethe firm's portfolio correlation with itself is positive, the firm pays a premium toincrease risk via homogeneous expansion.!

7. How to Satisfy a Feasible Constraint that is ViolatedExogenous shocks to the portfolio mean can cause the firm to be in violation of its

VaR constraint. If the constraint is infeasible without diluting shares by issuing moreequity, then the firm must issue new equity or reset the parameters to the VaRconstraint the maximum tolerable probability of breaching the floorthe floor or 2

+4. The model examined here says nothing about how the constraint would be reset.However if it is not reset, the firm must trade to alter the portfolio mean or variance

28

! "! "" . The following proposition explains how they must changed to satisfy theconstraint.

Curiously if regulators, the treasury of the Fed respond to a downward shock to theeconomy by attempting to reduce the risk of assets it will make it more difficult forfirms in trouble (bad times) to satisfy regulatory constraints.

Proposition 8 How to Satisfy a Feasible Constraint that is Violated (a) When the firm is in violation of a feasible constraint because of a downward shiftin the portfolio mean it can trade to increase the mean or lower (increase) theportfolio variance in good (bad) times.(b) Suppose a downward shock to the portfolio mean occurs with an upward shock inthe portfolio variance. If the portfolio mean cannot be increased, then in good timesthe portfolio variance will have to be changed more than in bad times.!

8. SummaryThe goal of the paper has been to determine when a firm increases or reduces risk

with a VaR constraint; to provide some insight as to how a firm may use VaR and todetermine the consequences for the volatility of a firm. VaR is the most often riskmeasure mentioned in Basel II and Solvency II (known as Basell II for insurers), yetlittle is know about how firms use VaR. This study investigates the risk premium ofthe marginal price of a lottery (contract, security or residual risk of a hedgedinstrument) when a manager maximizes his firm's expected equity value subject to aVaR constraint.

One result is "variance switching". A manager can "swing for the fences" whenthe probability of that the firm's equity value will breach the floor is greater than acritical level. This is evident when he pays a premium for a lottery that increases thefirm's equity variance when expected equity value is low. How low depends on thebivariate distribution for a lottery (contract or financial asset) and the firm's portfolio.Clearly regulations, meant to make the financial system more stable, that mayexacerbate volatility in bad economic conditions deserve greater scrutiny.Specifically Basell II encourages large banks, via the IRB approach, to use VaRconstraints. Hence regulations meant to make the financial system more stable

29

encourage the banks that present the greatest systemic risk to the financial system tobecome more risk when the system is most vulnerable.

This study is at the level of the firm. Embedding the setup used here in networkmodel such as Eisenberg and Noe (2001) might bear investigation.

AppendixProposition 1 Normal Pricing on the VaR ConstraintProof: For the normal distribution, a change in affects the two parameters of the#

distribution:

;+,-./2 "7 0

;

! "#

#

2 1 (18)

# $ 9<+,-./2 "7 0 < /2 1 <+,-./2 "7 0 <2 "7

< /2 "7 1 < <2 "7 <:

:! " ! " ! " ! "! " ! "# " # # #

" # # # #

2 21 1M

M

Adding a small change in the amount of cash required to offset this change gives;7

( # ;<+,-./2 "7 0 < /2 "7 1

< /2 "7 1 <

! " ! "! "# " #

" # ##

2 1M

M

$ ; $ ;7<+,-./2 "7 0 <2 "7 <2 "7

<2 "7 < <7:

: :! " ! " ! "! " * +# # #

# ##

2 1. (19)

* can be thought of as the cash received by the customer who bought a policy fromthe insurance company, so The insurance company is short, so . If the* S (9 8 (#

insurance company sells an additional amount, . The marginal price .; 8 ( 6 (# ;7;

#

Going forward we suppress ! "#" 7

;7

;# :

$

#

<+,-./20 < /21 <+,-./20< /21 < <<2

:<2:

<+,-./20<2:

<2:

<7

2 2

2

1 1

1

" # #"

M

M

. (20)

# : $< /21 <2 <2

< < <7

: :, -. /* +

<+,-./20< /21

<+,-./20<2:

M:)2

2

1

1

"M "

# #. (21)

30

The first factor in the first term on the left-hand side of (19) is the partial of a non-standard cumulative normal with respect to :"M/21

<+,-./2 0 <

< /21 < /21# '/2A 2" /211 ; 2

:2 1

" "" !

M M:

M0 ! "]

2

(22)

# F/2A2" /211 ; 2 A< )

< /21 M /21

:" '"

" !M M

:

M1 0 ! "]

2

(23)

F/2A2" /211 5 M /21 '/2A 2" /2119 / 1: :

" '" "M MM1 24

Changing variables to standardize, let . Let and( (/%1 5 # / 1 #2:2 2:2: :

2 /21" "! " ( % _

'/ A (" )1( be the standardized normal.

<+,-./2/= 1 0 < )

< /21 < /21# F/ A (" )1 /21;

M /21

2 1

" " '"( " ! (

M M M:1 0 ! "]

(

(25)

# '/ A (" )1 ; 9<

< /21"( ! (

M:

0 ! "]

(

(26)

We get a partial of the cumulative standardized normal. With + "5 /21M

T/ " 211 5 '/ A (" )1 ; A " 21 : 21: : :

( (/ / # /+ ( ! ( + +0 ! ":

:

]

(

2)M (27)

(28)<T/ 1 <T <

< < <# 9

(

(

(" +

+ +

(29)<T < : 21

< < M# '/ A (" )1 A # : A

:

((

(

++

/2 :\M (30)

This gives<T/ 1 : 21

< M# : '/ A (" )1 "

:(

(" /+

++

2 :\M (31)

and<T/ 1 <T <

<2 < <2: :# # : '/ A (" )1 9(

(

((

" ++:)

M (32)

Get

;7 : 2 < /21 <2 <2

; M /21 < < <7# : $

: : :

# " # #

"2 32 34 52M

M:)

. (33)

31

Also:< /21 <

< <# / * 1 $ M *% $ / % 1

"

# ##" ## " # "

MM

% %M (34)4 5! " ! " ! "

(35)# M /*1 $ *% 94 5! "#" # "M%

Substituting for obtain (4)."M/21

!

Proposition 2 Small Normal Contract:Proof: Let consider small . Substituting for in equation (16)* # * * #! " ! "5 #"

# "! "! "*%%

get for the forward constraint price (36)

lim lim# #, ,7Q( 7Q(

% %

% %M M MM

%

;7 / : 21/ * % $ *% 1

;# $ *

:

?/ % 1 $ M % *% $ / % 1:

#

* # " # " # "

# " * # # " ) * # # " .

]2 ! " ! " ! " ! "! " ! " ! " ! " ! " ! "! "

Get the spot constraint price discounting at the risk-free rate (37)

# : C $ ** / : % $ * $ 7 1/ $ *% 1

% ? $ M *% $ )

: ::lim

#,7Q(

:,/=:R1

%

%M2 3! " ! " ! " ! "! " ! " ! " ! " .

]" # # * # )

# " * # ) * #

2

If we define 1 unit of the firm's portfolio so 1, then the spot constraint price is#% #

given by equation (10).!

Proposition 3 Variance Switching with Normal Risks:Proof: The risk premium for a lottery when risks are normal is

' # (! "*% / : % 1:2 (38)

by equation (10). , and the lottery if purchased increases the( )! " ! "*% 6 ( ^ *% 6 (

standard deviation of the firm's portfolio . Similarly ," ( )! " ! " ! "% *% 8 ( ^ *% 8 (

and the lottery reduces . In bad times , so " )! " ! "% : % 6 ( *% 6 ( ^:2 ' 6 (. Alsoin bad times . In good times the inequalities are reversed.)! "*% 8 ( ^ ' 8 (

Obtain Table 1.!

32

Proposition 4 Critical Values with Normal Risks:Proof: a) From equation (38) for any (! "*% , definition 2

' # + +( ) "! " ! " ! "*% / : % 1 # *% * T # ( T # (:2 :) :)( ) ( )U U' (39)

' +U # % 9)

M:and (40)2U #

b) By (a) and that the normal cumulative distribution is monotonic strictly increasing.!

Proposition 5 CAPM's SML as a Special Case of the Constraint Price Equation:Proof: Let

$ 5 time between date 1 and date 2 ' 5 the number of traded lotteries

F 5F( market price of the market portfolio (The superscript denotes the

market's price for the market portfolio in contrast to the firm'sconstraint price for the market portfolio.)

( (( ('5 Kthe market portfolio where %

) (( ('5 Kthe firm's portfolio where %

firm's spot constraint price for ; ( in general is not equal toO 5 O *"F" % O( (

the market price for O D O 5 # O O # *"F" %! " O

O( gross return on ;

, O 5 O # D O : ) # OA O # *"F" %! " ! "net return on net return on continuous risk-free rate, 5

D 5 # CE,gross risk-free rate

"! " ! "! "D O 5 D O A O # *"F" %standard deviation of "! " ! "! ", O 5 , O A O # *"F" %standard deviation of D

:O 5 D O A O # *"F" %! " ! "expectation of

,: O 5 , O A O # *"F" %! " ! "expectation of covariance of and "! "! " ! "D O D P # O PA O # *"F" %A P # *"F" %

covariance of and "! " ! " ! "! " ! ", O , P # , O , P A O # *"F" %A P # *"F" %

33

)! "! " ! "D O D P # A O # *"F" %A P # *"F" %"" "

! "! " ! "! " ! "! " ! "D O D PD O D P

Let: 1) be the market price for the market portfolio ; 2) the firm's floorFF( ((

2 # F # where is a scale factor and 3) the firm's portfolio & & &DE (F( () ( . Then

the SML of the CAPM for the expected return is true if and only if the constraint*:

price is true. Hence the SML formula is a special case of the constraint priceformula. Equation 10 Making( ) is the formula for the firm's spot constraint price. substitutions into this equation we will get the SML equation.

From proposition 6, the firm's spot constraint price for its own portfolio equals%( the floor discounted at the risk-free rate Hence Substituting2 2D E X % # D! " (

:)E .

for 2 the firm's constraint price for its own portfolio is) (( (# &

% # D( (:)E 2 # & &F 9F

( That is the firm's constraint price for its portfolio ( equals

the market price for .&((

The beta of the prices of and in terms of the beta of their net returns is* %

( (! " ! "! " ! "*% # , * , %4 5*

%9

(

((41)

Substitute forD E C *% % % D: ::) :, , * , %

, % E! " ! " ! "for , and $ ""( )( )*

%(

(

! "! " ! "! "! "M ( D % %( (for for2

into equation for t( ), the equation he firm's spot constraint price for the lottery .10 *

Next substitute . Divide by & &F % FU U( (( for to get

* # *( (D E /D E : F 1 $ * 9, * , %

, %: ::)

M! " ! " ! "2 3! "! " ! "! "! "4 5"

"D (42)

Divide by *(

) #*

D E /D E : % 1 $ 9, * , % *

, %: :

:)M

! " ! " ! "2 3! "! " ! "! "! "4 5"

"D

((43)

Subtract from both sides. Multiply both sides by -1, substitute for and* *: :

* *( (D:

*! "substitute for , D: * * : )

:! " ! "

, D: * D E / F :D E 1 9, * , %

, %:! " ! " ! " ! "2 3! "! " ! "! "! "# :)

M4 5"

"(44)

34

!

Proposition 6 Constraint Price and Return for the Firm's Portfolio:Proof: Denote the spot constraint price for the firm's portfolio by . With The spot%(

price per share is

% # %()

9D

DE

:)E>? % @ # % %B # (45)

!

Proposition 7 Homogenous Risk Expansion (Contraction) in Bad (Good) Times:Proof: By assumption 2 6 %:

% # C ? $ % B # C ? / : % 1 $ % B: : :(

:, :,$ $' (! "%% 2 (46)

# C ? / : % 1 $ % B # C 6 C %: : ::, :, :,$ $ $2 2 (47)

' # C / : % 1 6 (::,$ 2 . (48)

!

Proposition 8 How to Satisfy a Feasible Constraint that is Violated Proof:(a) Given the parameters to the firm's distribution and +4 2 the set of values for theparameters of the portfolio distribution ! " ! "( )! " ! "" T # + " is given by , so 2:%:

%"! " 4

must satisfy

2 # T + "% $ %: "! " :) 4! " (49)

where is constant. Note that from proposition 4T +:) 4! "YGN' + : # YGN' T + # YGN' :

)

M4 5 6 7! " ! "4 :) 4 2 %: . (50)

Hence if the sign is negative, a downward exogenous shift in the portfolio mean,)% 8 (: must be compensated for by trading so that

, , ,) M M% $ % $ %: : "! "T + ":) 4! " (51)

35

where is not,M indicates shifts from trading. If the previous portfolio mean %:

achieved , then to satisfy the constraint, ,) M% $ % 8 (: :

YGN' % # %:! "! ",M" YGN' : 9! "2 (52)

Hence in good times the firm reduces portfolio variance, and in bad times it increasesportfolio variance.(b) By (a).!

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40

Economic Conditions ) ) )! " ! " ! "*" % *" % *" %6 ( # ( 8 ($ ( :( ( (: ( $

222

: % 6 (:

: % # (:

: % 8 (:

(bad) (neutral) (good)

Table 1: The constraint price premium for a lottery asdetermined by the correlation between the lottery and the firm's

portfolio and the floor minus expected portfolio value

41

Normal constraint price risk premia with * + + + + +! ! # $ $! ) ! ) &+&,-

. *#+ #//+ +&, &+ - #//+ #//,- -

# #+ * + - #0#, *9\" 9)" M(" 0 and

Figure 1

42

Normal bid price risk premia with * + + + + +! ! # $ $! ) ! ) &+&,-

. *#+ #//+ +&, &+ - #//+ #//,- -

# #+ * + - #0#, *9\" 9)" M(" 0 and

Figure 2

43

Normal offer price risk premia with * + + + + +! ! # $ $! ) ! ) &+&,-

. *#+ #//+ +&, &+ - #//+ #//,- -

# #+ * + - #0#, *9\" 9)" M(" 0 and

Figure 3