Mark D. Flood

Mark 0. Flood is a visiting scholar at the Federal ReserveBank of St. Louis. David H. Kelly provided research assistance,

On the Use of Option PricingModels to Analyze DepositInsurance

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I HE FAILURE rate of banks and thrifts hasexploded over the past decade, making reformof the deposit insurance system a topic of con-siderable interest to regulators, bankers) andeconomists. As illustrated by figure 1, whichshows the total number of failed commercialbanks (excluding thrifts) for each year since thechartering of the Federal Deposit Insurance Cor-poration (FDIC), the annual number of commer-cial hank failures in each of the last severalyears has exceeded its previous peak, attainedduring the Great Depression. The status of thethrift industry is even more gritn, with losses tothe Federal Savings and Loan Insurance Cor-poration (FSL1C) estimated at $160 billion ormore. The primary consequence of these fail-ures for public policy is the enormous losses,especially to the FSLIC, as depositors in thesefailed institutions are reimbursed.

This article considers a particular set of eco-nomic tools used to evaluate deposit insurance.’Option pricing models am’e among the techniquesavailable for analyzing the deposit insurancesystem. These models can he used to assign

specific values to the claims of each of the in-terested parties involved in the deposit insur-ance system — the insurer, financial institutions,and depositors. Such valuations can then heused, for example, to estimate the net value ofthe government’s insurance fund or to deter-tnine a fair price that a bank should pay for itsinsurance. More generally, by comparing in-surance valuations with different model parame-ters, one can investigate the system of incen-tives under a given regulatory schemne, such asthe risk incentives for hank shareholders anddepositot’s under the present system. Finally,comparisons of insurance values and incentivescan be made across various proposed regulatoryschemes. These applications are illustrated be-low with some examples.

‘The usefulness of option pricing models forevaluating deposit insurance is of special in-terest for two reasons. First, the consensusamong the interested parties is that the presentdeposit insurance systetn has contributed to thecurrent crisis. Second, in the context of thisdebate, a number of economists have used the

iThis article does not addness the issue of why we shouldhave deposit insurance. The rationale for the currentsystem of bank regulation and for deposit insurance inparticular is based on two related principles: protection ofthe depositor and the mitigation of contagious bank runs.See FDIC (1984) and U. S. Treasury (1985). Benston and

Kaufman (1988) identify three reasons for bank regulationin addition to the two traditional rationales, namely disrup-tion to communities from localized bank failures, moralhazard induced by deposit insurance and restrictions oncompetition.

Figure 1Bank Failures (Insured andUninsured)Number of Failed Banks200

0

160

modern theory of option pricing to explain theincentives and measure the costs both of thecurrent system of deposit insurance and ofsome suggested alternatives. This paper pre-sents the basic theory of option pricing, explainshow it can be applied to deposit insurance, andanalyzes some of the issues involved in its use.

A L:.Ti.MELI (JPT.HI.) PRICING

This paper presumes no knowledge of optionsor of the various economic models that havebeen used in the academic literature to assignvalues to options. Thus, it begins with a briefdescription of options and some of the majorcontributions to the theory of pricing optionsthat have been made in the past two decades.

Cmhigenf ~

A call option is a legal contract that gives itsowner the right to buy a specified asset at a fix-ed price on a specified date,~Similarly, a put op-tion gives its owner the right to sell a specifiedasset at a fixed price on a specified date. Optioncontracts are usually sold by one party to an-other.’ The person who owns an option con-tract is called the holder of the option. The per-

son who sells an option contract — that is, theperson who will be compelled to perform if theoption holder invokes her right as specified inthe contract — is called the writer of the option.The act of invoking the contract is called eyer-cising the option. The fixed price identified bythe option contract is called the striking price.The date at which the option can he exercisedis called the eypiration date of the option.

120 These legal contracts are probably best knownby the stock options that are bought and soldby brokers in the trading pits of organized op-tions exchanges in Chicago, New York andelsewhere, In addition to options on common

40 stock, there are active markets fom’ options onagricultural commodity futures, foreign curren-

0 cies, stock index portfolios, and government

securities, to name only a few. The definition ofan option, however, does not limit the term tothose contracts actively traded on the floors oforganized financial exchanges. By definition, anoption is any appropriately constructed legalcontract between the writer and the holder,regardless of whether it is ever traded,

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Consider now the value to the holder of anexpiring put option, as illustrated in figure 2,The value of the underlying asset specified bythe contract is given on the horizontal axis,while the value of the option itself is given onthe vertical axis- The point K on the horizontalaxis is the specified striking price for the asset,If the value of the underlying asset is above thestm’iking price on the expiration date, then theput option will not be exercised; anyone whotruly wanted to sell the asset would do sooutright at the going price, rather than usingthe option and receiving the striking price. Inthis case, the option expires worthless, and theoption holder experiences no gain or loss on theexpiration date.

On the other hand, if the value of the asset isbelow the striking price, then the holder willexercise her option and receive the striking

2This definition is a paraphrase of the definition given byCox and Rubinstein (1985), p. 1, It describes a “Euro-pean” option, which is distinguished from an “American”option. An American option gives its owner the right to buyat any time on or before the specified date.

not compel the owner of the contract to do anything.Although they are valuable, nothing in the definition of anoption requires that they be offered for sale; that is, theirvalue does not depend on how they were obtained,

Number of Failed Banks200

160

120

ao

40

1934 39 44 49

source: roic (1989)

54 59 64 69 74 79 84 1988

3They are sold, because options have a non-negative value;because they are a right to buy (or sell) the asset, they do

Figure 2Value of Put Option to Holder

price for the asset. In this case, her net gain onthe expiration date will he (K — A

7), the differ-

ence between the striking price and the currentprice, since she can turn around and replacethe asset immediately, if she wants to, Thus, theexpiration value of the option and the decisionabout whether to exercise are contingent uponthe value of the underlying asset at that time:

(1) State Action Option value

AT < K Exercise P = K — A7A7 K No exercise P = 0.

For this reason, options are also referred to as“contingent claims” on the underlying assets,

The corresponding net payoffs to the writerof the put option are given in figure 3, Noticethat his payoffs are exactly the inverse of thosefor the option holder, Also note that the payoffat expiration to the writer of an option is neverpositive; at best it is zero. It is for this reasonthat options are sold to the holder, rather thanbeing given away free of charge. ‘t’he price in-itially paid for the option — the option price oroption premium — could be incorporated intothe figures by simply shifting the holder’spayoffs down and the writer’s payoffs up bythe appropriate amount.

The payoffs at expiration to the holder andwriter of a call option are given in figures 4and 5, respectively. The corresponding analysisfor’ call options is precisely analogous to theanalysis just given for put options.

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Having described the value of an option at ex-piration leaves the question of its value prior toexpiration unanswered. Instead of being a sim-ple function of A and K, the value of an optionbefore maturity depends on several additionalfactors. Although a number of bounds had beenplaced on the value of an unexpited option byusing relatively simple arbitrage arguments, animportant advance in the valuation of unexpiredoptions was made by Black and Scholes (j973)4

They obtained an exact equation for the valueof a put option under an unrestrictive set ofassumptions.~Their result has since been dab-orated and generalized by others.’

In their model, the value of an unexpired op-tion depends on five things:

4For an exposition of the arbitrage bounds on option prices,see Merton (1973), or Cox and Rubinstein, ch, 4.

‘Almost alt derivations of option pricing models, includingthat of Black and Scholes, are stated in terms of callrather than put options, As it happens, this distinction islargely irrelevant, because call option valuations are readi-ly converted to put option valuations, and vice-versa, viaan arbitrage relationship known as “put-call parity”. Put-

call parity is an exact relationship for European optionsand an approximate one for American options (see Coxand Rubinstein, pp. 150-52); throughout this paper it istreated as exact. Put-call parity is first presented by Stoll(1969).

‘One such generalization is found in the shaded insert. Fora partial survey of option pricing models, see Cox andRubinstein, ch. 7.

Expiration Value of Option

Figure 3Value of Underlying Asset at Expiration

Value of Put Option to Writer

0 Kvalue of Underlying Asset at Expiration

22

K = the striking priceA = the current asset priceT = the time remaining to expirationa = the volatility of the asset priceB = the risk-free interest rate,

Almost as notable is what the option’s valuedoes not depend on: any characteristic of theholder or the writer! Under their assumptions,Black and Scholes are able to include an optionin a riskless portfolio. Such a portfolio mustearn the risk-free interest rate, and they areable to use this result, along with an assumptionabout the probability distribution of the assetprice, to identify an exact value, P, for a putoption:

P = (K’e—RT).N(X+rnfñ — A-N(X),

where:x = ..J.. [ln(K.e’RT)_ln(A)] — ½a’ff

N() = the —Jstandard normal cumulativeprobability function

ln() = the natural logarithm function

e = the base of the natural logarithm

Although this formula may at first appearcomplicated, a rough intuition can be providedrelatively painlessly.’ First, e_RT is just the pre-sent value discount factor for T periods at in-terest rate B with continuous compounding, sothat K’e~’~Tis the present value of the strikingprice. Keeping in mind that N(X + crti5 is a pro-bability, the first term is the expected presentvalue of the striking price at expiration, giventhat A7 < K, Similarly, the second term, AN(X),is the expected present value of the expiration-day asset price, again given that A7 < K,’ ‘l’hus,the value of the option is the expected presentvalue of its value at expiration, given by condi-tion I above.

Unfortunately, no easy, correct interpretationcan be attached to the specific probabilities,N(X + ø’ñï and N(X), in the two terms. These

Figure 4Value of Call Option to HolderExpiration Value of Option

0

0 K

Value of Underlying Asset at Expiration

Value of Call Option to Writer

probabilities are closely related to the probabili-ty that A7 < K, but they are not quite thesame, because the present value of the strikingprice is known with certainty, whereas the pre-sent value of the asset’s price on the expirationday, AT.eTtT, is not; the current asset price, A,appears instead,

7For example, one might suspect that the holder’s attitudestoward risk or her beliefs about the asset price at expira-tion should influence the option’s value to her. This is notthe case, however. Also note that four of the five factors,at least theoretically, are well-defined and directly obser-vable at the time of valuation. The exception is assetvolatility, which must be estimated from observable fac-tors; see Cox and Rubinstein, pp. 280-87, for an exampleof an estimation technique.

‘A full derivation of the formula is fairly involved and willnot be presented here. Interested readers are referred to

Malliaris (1983) for a mathematically advanced approachor to Cox and Rubinstein, ch. 5, for a longer but lesstechnical derivation.

‘The corresponding expected present values for the casewhen A7 is greater than K are both zero, because then theexpiring option is worthless and will not be exercised;hence, this possibility adds nothing to the current value ofthe option.

Figure 5

0 ICValue of Underlying Asset at Expiration

23

In spite of its complexity, the option pricingequation is still a useful tool. In one sense, theformula can be treated as a black box in whichthe five parameters (K, A, T, a and B) enter atone end, and the value of the put option, P,comes out at the other; a computer spreadsheetor calculator can be programmed to performthe intervening calculations defined by the for-mula, For example, if the current asset value isA = $985, the standard deviation of assetreturns is a = 0,3 percent, the striking price ofthe option is K = $1000, the time to expirationis one year, and the riskless interest rate is &percent per year, then the Black-Scholes equa-tion tells us that the put option is worth $8545,Figure 6 graphs the Black-Scholes value of a putoption for a range of current asset values fromzero to $1500, where the values of the otherfour parameters are the ones just given.

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Not surprisingly, the distribution of assetprices is a crucial factor in determining the ex-act form of the option pricing equation. In theirderivation, Black and Scholes assumed that theprice of the underlying asset progressed ran-domly through time according to geometricBrownian motion, This is the assumption thatleads to the specific normal probability func-tions in their pricing equation.

Brownian motion was first used to describethe random progress of a single moleculethrough a gas from a given starting point.’’ It isa mathematical model of motion that identifiesthe way the particle can move. Three restric-tions are implied by Brownian motion:

1)The path followed must be continuous;”

2)All future movements are independent ofall past movements;12

3)The change in position between time s andtime t is normally distributed with meanequal to zero and a standard deviationequal to

Note that standard deviation is directly pro-portional to the amount of time that has passed.

Figure 6Value of Put Option to HolderOptton Value Prior to Maturity(K1000, 7=1, 0 = ‘3, B

1= .08)

1000

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Figure 7

Thus, the longer one waits, the less certain oneis about the location of the molecule, or, in ourcase, the price of the asset.

Simple Brownian motion is not completelysatisfactory for describing asset prices, how-ever. While a normally distributed random vari-able can take on negative values, an asset pricecannot. Therefore, geometric Brownian motion,a variant, is assumed for the Black-Scholesmodel.” Under geometric Brownian motion, thethird restriction is modified, so that the loga-rithm of the change in position, rather than the

‘°Weare here concerned with only a single dimension ofmotion, for example, the East-West coordinate of themolecule or the price of an asset.

“Although this may be true of molecules, it need not be thecase for asset prices, as is considered in the shaded in-sert on Merton’s lump-diffusion model,

“This implies, for example, that the molecule cannot buildup momentum or that prices do not have a predictable

trend. It does not mean that the future location is indepen-dent of the past location,

“By way of terminology, simple Brownian motion (alsoknown as arithmetic Brownian motion) and geometricBrownian motion are examples of the Wiener process (alsoknown as the Gauss-Wiener process), which, in turn, is aspecial case of the Ito process,

Asset Value

Value of Net Asset Position

0 A=750 1500Asset Value

change in position itself, is normally distributedwith mean zero and standard deviation a’J(t — s).This distributional assumption gives us the spe-cific functional form which appears in the Black-Scholes equation. Thus, this assumption is im-portant: a different distribution would generallyyield a different pricing equation, as illustratedby Merton’s (1976) jump-diffusion option pricingmodel, which is presented in the shaded inserton the opposite page.

The option pricing equation has the paradox-ical property that, although risk (as measuredby volatility in the asset price) is itself a factorin the option’s value, the attitudes toward riskof the holder and the writer (and anyone else)are not. The option’s value is a function of fivevariables, none of which depends on the charac-teristics of the individuals involved. Black andScholes achieved this by showing that the op-tion can be made part of a completely hedged(that is, riskless) portfolio. Any option writerwho offered a risk discount when selling an op-tion would find himself selling many option con-tracts to investors who, in turn, could hedgethe risk completely and pocket the risk discountas an arbitrage profit. It is for this reason thatthe discount rate which appears in the pricingequation is the risk-free rate of interest, and at-titudes toward risk are irrelevant to the valueof the option.

‘Fo see how the hedged portfolio works, con-sider the value of a put option to the holderbefore expiration, depicted in figure 6, and thevalue of the underlying asset purchased for theamount A, depicted in figure 7. The value ofthe net asset investment increases one for oneas the price of the asset increases, and the val-ue of the option decreases, although not in aconstant proportion.

The key to the hedged portfolio is to buy putoptions and underlying assets in the appropriateratio, so that, when the asset price increases,the increase in value of the net asset investmentwill be precisely offset by the decrease in valueof the option position, and vice-versa. This im-plies a riskless total portfolio. Of course, the ap-propriate ratio (called the “hedge ratio” or “op-tion delta”) also changes as the asset pricechanges, because the value of a put option does

not decrease as a constant proportion of theasset value (the put option’s value is representedby a curved line). This implies that the holderof a completely hedged portfolio must con-tinuously adjust the relative proportions of op-tions to assets if the hedged portfolio is to re-main riskless. Black and Scholes presume that atleast some investors are large and sophisticatedenough to do this.

Because the risk of an option can be com-pletely diversified, the risk-free rate is the ap-propriate interest rate to use for discountingthe option’s uncertain payoff at expiration.Nevertheless, the risk (defined as price volatility)of the underlying asset is a factor in the op-tion’s value, because asset risk affects the ex-pected value of the option’s payoff. This is dueto the limited liability nature of the option.Although increasing the volatility of the assetprice increases both the chance of getting avery high expiration-day asset price and thechance of getting a very low expiration-dayasset price, the bad (high price) outcomes allhave a weight of zero in the put option valua-tion, while the good (low price) outcomes havea weight of (K — A7). The volatility of the op-tion’s value also increases with that of the assetprice, but the volatility of the option’s value isirrelevant, because it can be completely hedged.

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The analysis of deposit insurance is a natural,albeit not obvious, extension of option pricingmodels. The connection between the two comesthrough the limited liability property commonto both options and common stock,” This pro-perty implies an “expiration-day” payoff fordeposit insurance that can be modeled as an or-dinary put option. Similarly, other claims on afinancial intermediary’s assets can be modeledas options or combinations of options. Thebenefit is that, given such a model, option pric-ing theory allows us to assign values to each ofthe claims. These values are the key to optionpricing’s usefulness in this context, because theyallow two sorts of comparisons to be made.

First, variations in the parameters of the op-tion pricing equation can be considered.” Suchvariations are of special interest, because, in the

MThis connection was first made by Black and Scholes andfirst applied to deposit insurance by Merton (1977).

“For example, in Black and Scholes’ model, the fiveparameters: K, A, T, o and R would be varied.

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Second, various deposit insurance structurescan he compared. The structure of deposit in-surance is defined here by the number andtype of options pertaining to each of the in-tet-ested parties. Changes in deposit insurancestructure are different from the parameterchanges within an insurance structure, con-sidered in the preceding paragraph. Thus, forexample, the FDIC could use option models toestimate the net increase or decrease in the pre-sent value of the insurance fund caused by aswitch from one structure to another; or itcould examine the change in risk incentives oc-casioned by the same switch. ‘I’hree differentstructures, illustrating some of the issues involv-ed, are presented in the following examples.”

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To see how deposit insurance and options arerelated, consider the following simplified bank-ing scenario. A single banker both owns andruns a bank, a single large depositor providesthe entire liability portfolio of the bank, and asingle insurer, the FDIC, insures deposits andwill liquidate the bank in the event of insolven-cy-.’7 The liability pottfolio consists of a singledeposit due at year-end. Also at year-end, theFDIC examines the bank to determine the valueof assets, which will, in turn, determine wheth-er liquidation occurs. If the hank is economi-cally insolvent, it is closed by the FDIC, whichliquidates the assets at market values and paysoff the depositor in full,” If the bank is eco-nomically solvent, control remains with thebanker, who can either renegotiate the depositor liquidate the bank.

Now considet- the payoffs to the thtee in-terested parties — hanket-, FDIC and depositor— when the year-end audit is performed. Thesepayoffs are illustrated in figures 8-10. Each par-ty’s year-end payoff is plotted as a function ofthe year-end value of the hank’s assets. Notethat the sum of the payoffs to all of the parties(obtained by adding the graphs vertically) equalsthe value of the bank’s assets. These functionsshow how the bank’s assets will be distributedafter the audit is performed. Also note theshape of the payoff functions for the hankerand the FIJIC; in effect, the banker’s portfolioconsists of the hank’s assets, whose value isuncertain before the audit hut known after-war-d, the bank’s deposits, whose value is knownto be L, and a put option with striking price Lwritten by the FDIC.” The FDIC, on the otherhand, has effectively written the put option onthe assets of the bank and sold that option tothe banker for the price of the deposit insur-ance premium.20 The depositor has issued thebank a risk-free loan, which pays off the amountL, including accrued interest.

With this in mind, the usefulness of an optionpricing model to evaFuate deposit insurance be-comes more apparent. An option pricing modelprovides an estimate of the actuarial dollarvalue of deposit insurance, as well as a toolwith which to analyze the economic incentivesthat deposit insurance creates. The depositor,for example, has a portfolio, I), that is worth, atthe beginning of the year, simply the presentvalue of the deposit liability discounted at theriskless rate, Le~T if his year-end payoff, L, is$1000, and the riskless rate, B, is 8 percent,then the value of this portfolio at the beginning

“Full coverage is considered first, because it is the simplestinsurance structure possible, and because it approximatesthe current system of extensive coverage combined withthe FDIC’s tendency to arrange purchase and assumptiontransactions, rather than deposit payouts, for failed banks,

l7The assumptions of a single owner-manager for the bankand a lone depositor are clearly broad abstractions fromreality. The owner-manager assumption allows us to ignoreprincipal-agent incentives; see Barnea, Haugen andSenbet (1985). Similarly, the assumption of a singledepositor effectively precludes the ability of depositors towithdraw their funds individually without forcing an im-mediate closure of the bank. Although these are both im-portant issues, the purpose of the present analysis is to il-lustrate the general principles involved in the application ofoption models to deposit insurance, rather than to model abank in its full complexity.

“The bank is defined as economically insolvent when themarket value of its assets is less than the present value ofits liability to the depositor. This terminology is meant to

contrast with an illiquidity, or legal insolvency, which isbrought on by an inability to meet maturing short-termliabilities with liquid assets. The legal profession has aseparate terminology for these two concepts: the “balancesheet test” is used to determine economic insolvency, andthe “equity test” is used to determine legal insolvency.See Symons and White (1984), pp. 603-16, for an exposi-tion, Since there are no short-term liabilities in thesimplified world here, legal insolvency is not germane.

‘tThe bank’s deposits represent a “short” position, or bor-rowing, for the banker, The net payoff shown in figure 8can be gotten by drawing the individual payoff graphs forthe components of the portfolio and adding these togethervertically as before. This portfolio is also equivalent, viaput-call parity, to a simple call option on the assets of thebank.

20Compare figure 9 with figure 3. The insurance premium isconsidered a sunk cost at the time of the audit and henceis not included in the graphs.

Figures 8-10Payoff to

Banker

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Payoff toFDIC

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Payoff toDepositor

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of the year is $1000-e °‘ = $923.12. The FUIC,on the other hand, has written a put optionwith striking price L — $1000; if, for example,the standard deviation of the bank’s asset re-turns is a = 0.3 percent, and the current valueof the bank’s assets is $9&~5,then the value ofthe FDIC2s pot-tfolio is given 1w the Black-Scholes equation as —P — —$85.45.

Of particular interest are the incentives cre-ated by deposit insurance. Under 100 percentinsurance, the depositor does not care about thevalue of the bank’s assets, since he receives hisdeposit back with interest, regardless of thebank’s condition. ‘The hanker, however, receives

Asset Value - - , , , -

the postttve equtty capttal, tf the bank ts solvent;if the bank is insolvent, the loss is charged tothe FIJIC. This “heads I win, tails you lose” ar-rangement is certainly not peculiar to banks; itapplies to any corporate entity with limitedstockholder liability. In the absence of other in-centives, the banket- will make the corpor-ationas risky as possible.”

What is peculiar to banks undet- 100 percentfiat-rate deposit insut-ance is the absence ofsuch other incentives for the depositor and thehanker to limit risk. Normally, creditors impose

Asset Value a risk premium on corporations, based on theriskiness of the firm’s assets.22 By definition,flat-rate insurance implies that the FIJIC chargesno risk premium. Similarly, the depositorscharge no risk premium, because they are fullyinsured. The result, in our simplified model, isthat the hanker has an unmitigated incentive toincrease the riskiness of the bank’s assets, whilethe FDIC has the invet-se incentive to reduce thebank’s risk-taking.2’ The risk incentive impliedby extensive, flat-rate deposit insurance is theimpetus for most of the current proposals fordeposit insurance reform.24 In analyzing both

Asset Value the current system and proposed reforms, manyauthors have used option pricing models.”

2’Recall that the value of an option to the holder increaseswith the volatility of the underlying asset. For the Blackand Scholes model, risk is defined as the standard devia-tion of the logarithm of the asset’s value,

22This is the function of bond rating services, such asMoody’s and Standard and Poor’s, See Barnea, Haugen,and Senbet, especially pp. 33-35, for an exposition of therisk-incentive problem.

23For this reason, risk-taking is restricted by extensiveregulation of commercial bank activities, In practice, thebanker’s incentive may also be mitigated by the potentialloss of a valuable bank charter or by nonpecuniary factors,for example, the potential loss of a bank manager’s pro-fessional reputation in the event of a failure. These factorsare beyond the scope of the option model,

‘4fleform proposals include: risk-based insurance premia,risk-based capital requirements, larger capital re-quirements, reduced insurance coverage, depositor co-insurance, subordinated debt requirements, increasedsupervision and more stringent asset regulation. SeeWhite (1989) for a survey of current proposals.

2’See, for example, Merton (1977, 1978), McCulloch (1981,1985), Sinkey and Miles (1982), Pyle (1983, 1984, 1986),Brumbaugh and Hemel (1984), Marcus (1984), Marcus andShaked (1984), Ronn and Verma (1986, 1987), Thomson(1987), Furlong and Keeley (1987), Pennacchi (1987a,1987b), Osterberg and Thomson (1988), Flannery (1989a,1989b), and Allen and Saunders (1990).

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We can now extend the options model toother arrangements for deposit insut-ance, toevaluate their relative impacts. Two illustrativecases will be pm’esented: a coverage ceiling clauseand a deductible clause. A significant character-istic of both these cases is that they impose apottion of the hank’s asset risk on the depos-itor. The FDIC benefits ditectly from such pro-visions, because they shift some potential lossesdirectly to the depositors. In addition, imposi-tion of a possible loss on the depositor mitigatesthe risk-incentive problem that exists under 100percent, flat-rate deposit insurance. In general,the depositor will monitor the bank more close-ly and will require a higher interest rate tocompensate for the possibility of default.

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Although 100 pet-cent covet-age is often treatedas the status quo de facto of federal deposit in-surance, coverage extends legally only to thefirst $100,000 per depositor per institution.” Aniaximum coverage limit is a form of co-insur-ance, a technique used by insurers to reducethe moral hazard problem (the tendency of in-surance to alter the behavior of the insured).Other basic forms of co-insut-ance are the deduct-ible and fixed proportional sharing of losses.’~

‘l’he applicability of the maximum coveragelimit considered here is comphcated by theFUIC’s current closure protocol. Bank closuresby the FDIC can take one of two forms: pur-chase and assumption or deposit payout. Undera purchase and assumption closure, healthyassets and a/I deposits are transferred to an-other healthy bank, with the FIJIC absorbingthe problem assets and any net loss.” This sortof ttansaction is best modelled by the 100 per-cent coverage considered above.

Under a deposit payout closure, the FDICitself takes all of the bank’s assets amid liabilitiesinto receivership. It then sells the assets andpays the depositors up to the maximum cover-age limit plus any excess of asset sales over in-surance claims, distributed on a pro rata basis.As a result, this method is best modelled by thedeductible considered below. ‘I’he upshot is thatthe payoffs under the FIJIC’s maximum cover-age limit do not conform to the familiar (from,say, automobile or health insurance) maximumcoverage arrangement illustrated here.

Consider a maximumn coverage limit of NIdollars for the depositor (where NI < L), illus-trated in figures 11—13. Under this arrange-inent, the depositor receives the full depositamount, L, in the event of any insolvency orshortfall, up to the amount, NI, of the coveragelimit. Thus, the depositor’s portfolio contains thedeposit amount, L, and he has written a put op-tion on the bank’s assets with striking price (L— M). ‘This put option is the result of the max-imum coverage limit. ‘The FDIC holds the putoption with striking price (L — NI), but has writ-ten a second put option on the bank’s assetswith striking price L. As before, this put option(with striking price L) is held by the hanker,who also owns the assets and owes the amountI, to the depositot-.

Because of the put option written by thedepositor and held by the FDIC, the depositornow shares in the risk of the hank’s assets. Hisdeposit is now worth less, and he will discountthe promised payoff more steeply. Extendingthe example given above for the case of 100percent covet-age, the depositor’s portfolio, whichcontained only the riskless deposit, wot-th $932.12when discounted, is now augmented by the putoption ~-vritten with a striking price of (L — M).If, for example, the coverage limit is set at NI =

$100, so that the striking price is $900, thenwith a = 0.3 percent, B = 8 percent and A —

$985 as before, the I3lack-Scholes value of thisput option to the depositor is —P = — 847.96,and the total value of his portfolio is D =

26The original limit was $2500 under the Banking Act of1933; see FDIC (1984), pp. 44, 69. The impact of thecoverage ceiling is limited by the availability of brokereddeposits and by the tendency of the insurer to arrangepurchase and assumption solutions to bank failures.

‘fl’here are other possibilities. For example, deposit in-surance in the United Kingdom involves fixed proportionalsharing combined with a coverage ceiling [see Llewetlyn

in the United States was to have sharing in staggered pro-portions Isee FDIC (1984), p. 441. Analyses of co-insurancetend to focus on proportional sharing arrangements. SeeBoyd and Rotnick (1989), and Benston and Kaufman(1988), ch, 3.

“Defining a “healthy” asset is a difficult chore. The task isgenerally accomplished by individual evaluation of assets,rather than the application of a generic rule,

(1986). p. 20], and a temporary deposit insurance program

Figures 11-13Payoff toBanker

0

Payoff toFOIC

$932.12 — $47.96 = $884.16. In other words,given deposit insurance with a $100 coveragelimit, $884.16 is the amount deposited at thebeginning of the year in exchange for a prom-ised year-end payoff of $1000.

In general, we can use an option pricingmodel to solve algebraically for a measure of

the risk premium that the depositor wouldcharge under this risk-sharing arrangement. Onthe one hand, we can think of the deposit as ariskless deposit combined with a put option. Onthe other hand, we can think of it as a singlerisky promise of repayment from the banker, tobe discounted at some risk-adjusted interestrate, r, so that the value of the deposit is:

D = L-e rT We can equate these two inter-pretations thus:

D = L.e_~<T— P(L — M,A,T,a,B)

= L.etT

where P(s) is the value of the put option beforeexpiration, as defined, for example, by theBlack-Scholes model, and r is the risk-adjusteddiscount rate implied by the presence of thecoverage limit. Given the other variables, wecan rearrange this to find the risk premium:

(2) (r — It) = --iln[ e~T~~~1P(.)1- B.

T L -~

Applying this to the numerical example, the

stated risk-adjusted interest yield on the deposit is:

r = — In[e-’°’~’t— $47.96 I$1000.00

= — ln(0.87515) = 13.3%,

implying a risk premium, r — H, of 5.3 percent.

In practical terms, such an estimate of themagnitude of the risk premium implied by agiven coverage ceiling might be useful in calibra-ting the degree of market discipline in a reformof the insurance system. If bank risk-taking is tobe curbed by limiting deposit insurance, forcingriskier banks to pay higher risk premia as somehave suggested, then the insurance limitationsmust be such that the risk premium implied by

Asset Value equation 2 is large enough to make bankers altertheir behavior!’

The magnitude of the risk premium might alsoserve as a readily observable vital sign, registeringthe financial health of the bank’s assets, andaiding the regulator in scheduling audits. Thispresumes that depositors have some advantageover regulators in assessing the bank’s risk bet-ween audits.” Such applications, how-

“For some examples of market discipline proposals, seeBoyd and Rolnick (1989), Gilbert (1990), Gorton and San-tomero (1988, 1989), or Thomson (1987).

‘°Notethat uninsured or co-insured depositors might find itmore practical to engage in capital rationing, i. e,, limiting

the amount lent to a bank based on that bank’s risk, inaddition to pricing that risk,

L Asset Value

0

L-M I. Asset Value

L-M L

ever, are subject to some limitations which at-cillustrated by the next example.

Another form of co-insut-ance is a deductible.‘The case of a deductible on insurance coverageintroduces a twist to the problem. Now thedepositor’s portfolio effectively consists of twoput options, one written and one held, in addi-tion to the promnised repayment of the depositwith interest. This case is of special interest,because it applies to a deposit payout closure asconsidered above and because it can also he ap-plied to subordinated debt, which is the objectof a recent debate on sources of mnarket disci-pline of bank risk-taking. In both cases, thepayoffs to one of the hank’s creditors can hemodeled as a pair of put options with differentstriking prices.”

In this example, the depositor is promised thereturn of his deposit amount, ivith acci-ued in-terest, for a total of L dollars. Becamtse of thedeductible ptovision, however, this promisedtepayment is not certain; in the event of thebank’s insolvency, the depositor will he the firstto share in the shortfall, For year-end assetlevels helow L, the shortfall is deducted from thedepositor’s payoff until the deductible amount, U,is exhausted. Any shortfall beyond that is ah-sotbed by the FUIC. Thus, the depositor effec-tivelv holds the deposit amount L and has writ-ten a put option with striking price L, that isheld by the hanker; in addition, he holds a putoption with striking price (L — U), which iswritten by the FDIC. These payoffs are illus-trated in figures 14-t6.

The deductible provides a cushion for theFUIC, which, in the preceding examples, hadwritten a put option with striking price Lrather than (L — U). The yeat--end payoffs fotthe hanker are the same as before. The real dif-ference applies to the depositor’s incentives andthe resulting impact on the price he charges thebanker for the deposit. Although he alwaysprefers a higher asset value, as before, his at-titude toward the riskiness of the bank’s assetsis now ambiguous, because he is long one putoption and short another, with two diffet-entstriking prices. Volatility in the hank’s asset

Figures 14-16Payoff toBanker

0

Payoff toFDIC

0

(L-U)

Payoff toDepositor

I.

L-U

0

returns increases the value of the long positionand decreases the value of the short position.

As Black and Cox (1976) point out, the net im-pact of these countervailing forces will dependon the current asset value relative to the strik-ing prices. Specifically, there is an inflection

“The relevant creditors in each case are the depositor andthe subordinated debt-holder, respectively. For an analysisof option models in the case of subordinated debt, seeBlack and Cox (1976) and Gorton and Santomero (1988,1989). For an analysis of subordinated debt and bank

regulation, see Gilbert (1990). The approach here is atodds with that of Ronn and Verma (1986), who calculatethe value for a single put on total debt and then scale thatvalue down by the proportion of insured to total liabilities,

L Asset Value

Asset Value

L-U L Asset Value

point equal to the discounted geometric meanof the two striking prices.” For an asset valueabove the inflection point, which includes allcases in which the bank is solvent (i. e., A >

L.e~T), the effect of the short position out-weighs that of the long position, and the de-positor will prefer less risk. Conversely, whenthe current market value of assets falls belowthe inflection point, the long position outweighsthe short, and the depositor would prefer ariskier asset portfolio, given the low asset value.‘Thus, a decrease in the “risk premium” chargedby the depositot no longer necessarily impliesthat the bank’s assets are less risky; for exam-ple, such a decrease could instead be the resultof an increase in the current asset value and anincrease in the volatility of those assets.

Under such circumstances, it is a reasonabletaxonomic question whether the interest ratemarkup over the riskless rate should be called arisk premium at all. The current value of thedepositor’s claim and the implicit risk premiumcan be calculated as before:

D = L.e~T_ P(L,A,T,a,R) + P(L—U,A,T,o,R)

value of the deposit resembles more and morethe staggered year-end payoff function of figure16. In fact, if the year-end payoff of figure 16 isscaled down by the risk-free present value dis-count factor, ~ it becomes identical to theextreme case of figure 17 where a = 0. Figure17 also illustrates graphically the Black and Coxargument that for some (higher) asset levels, de-positors will charge a risk premium, while forother (lower) levels, they will offer a risk discount.

It is also clear from the picture, however, thatthe asset level has a much more significant effecton the value of the claim than does the volatili-ty.’4 All of this suggests that bankers, depositorsand pollcymakers should give considerable care toan appropriate definition of risk in this context,and that similar care should be given to designinga practical measure of that risk. Risk defined asvolatility in bank asset returns and measured bythe risk premium charged on equity, subor-dinated debt or uninsured deposits may not beapt for the tasks to which it has been applied.

= L.e’~TSOM1~CAVE ATE

(r - B) =

— I lnfeRT_ ~ IP(L,°) - P(L — U,~)11 — H,T ( Lt JJ

but the risk premiumn so defined is a measureof the expected difference between cash pro-mised, L, and cash ultimately received. It is apoor measure of the volatility of the returns onthe bank’s assets, because the expected dif-ference between cash promised and cash re-ceived depends on several factors and is nolonger a simple direct relation of the volatilityof assets.

Figure 17 graphs the value of the depositor’sclaim for a range of asset levels arid volatili-ties.” In interpreting this graph, note the con-nection between it and figure 16. In particular,as the volatility, a, goes to zero in figure 17, the

The preceding analysis has ifiustrated some usesof option pricing models in evaluating deposit in-surance. There are some limitations, however, onthe use of options models in this context. Most ofthese limitations derive from the assumptions thatform the basis for the option pricing equation,and the extent to which these assumptions arevalid for the case at hand.

Perhaps the most basic problem is the ques-tion, who truly holds the option.” Until now, ithas been presumed, based on the end-of-periodpayoffs, that deposit insurance represents a putoption written by the FDIC and held by thebanker. In fact, however, the FDIC decideswhether a bank is insolvent, and, more impor-tantly, whether to close a bank that is alreadyinsolvent (or one that is not quite insolvent).” Inthe face of a large-scale bank failure or run,

“The discount rate used to calculate this inflection point in-cludes a risk premium. Since all other discounting in thiscontext is at the riskless rate, this means that the inflec-tion point can fall below the present value of the lower dis-count rate, (L — i;)-e’1, if the risk premium, o’/2. is largeenough. For the same reason, the inflection point isalways smaller than the solvency point, L.e~r,

~ riskiess rate was set at 8 percent, time to maturitywas one year, promised repayment was $1000, and thedeductible amount was $200. Similar graphs with othermaturities and deductible amounts reveal no surprises.

‘~Thisfact is noted by Pyle (1983), p. IS.“This issue is addressed by Brumbaugh and Hemel (1984)

and Allen and Saunders (1990).“Recall that the definition of insolvency used in this paper

ignores the possibility that the bank might be deemed in-solvent on the basis of its current ratio — i. e., its inabilityto meet maturing liabilities with liquid assets. This problemrelates to the maturity structure of the bank’s portfolio,which will be considered briefly below.

L-e —RI =923.12

(L-U)-e ~ =738.49

0

short-term political considerations may over-whelm any prior prescriptions on closure poli-cy. The FSLIC’s actions in the thrift crisis in-dicate that this is not idle speculation; numerousthrifts were left open long after their insolvencyhad been discovered. Conversely, as Benstonand Kaufman (1988) suggest, the FDIC couldclose solvent banks, if they have come closeenough to insolvency. lf the insuret- were tofollow scm’upulously a well-defined rule one wayor the other, there would he little at issue, sincethe striking price and year-end payoffs couldthen be easily adjusted within the context of thecurrent model. As matters stand, however,bankers and depositors effectively face a ran-dom striking price, because the insurer decidesif the option will he exercised. As a practical

matter, it is difficult to envision how such awell-defined closure rule might be im-plemented.’~

A related issue is the measurement of bankasset values. The option pricing model pre-sented above presumes that the current assetprice can be readily observed. For stock op-tions, this is an uncontentious assumption,because stock prices can be observed on thefloor of the exchange or in the over-the-countermarket. For an option on the assets of the bank,ho%vever, the relevant pt-ice is not readily obser-vable. Indeed, one of the primary functions ofbank ct-edit analysis is to assign values to assetsfor which there is no active market. Similarly,

‘7A rule is well-defined if it leaves no doubt about the cir-cumstances which imply closure, with no room for FOICdiscretion. Note that defining a closure rule, in general, is

not sufficient for our purposes; the closure rule must bedefined so that the values of the resultant claims conformto the values given by the option pricing equation.

Figure 17Value of Depositor’s Claim

z

0

-3

a .6

.96 400

1200

738.49 A

the insurer must invest significant effort, in theform of an audit, to determine the year-endasset value.

The inherent inaccessibility of bank asset valueshas two implications for option models. First, itis no longer possible for an option holder toconstruct the appropriately hedged portfoliodescribed in the Black and Scholes derivation,because the hedge ratio depends on the value ofthe underlying asset. This casts doubt upon theappropriateness of the riskless rate in discoun-ting the expected end-of-period payoffs.’8 Se-cond, the current asset value is important, be-cause it partly determines the probabilities forthe various end-of-period payoffs. Ignorance ofthe current asset value adds another layer ofuncertainty, and this additional uncertaintysignificantly affects the value of the option.”

A closely related issue is the measurement ofasset risk. The option pricing models presentedhere use the variance of the asset’s returns as ameasure of risk.~°Producing an accurate assess-ment of the variance is problematic, even forstock options, because the volatility that mattersis the variance of the process over the futurelife of the option. For bank assets, the measure-ment problem is compounded, because evenpast values are generally unavailable. Pyle (1983)and Flannery (1 989b) consider some of the im-plications of this problem in using the optionmodels to price deposit insurance.

Just as asset values and the volatility of re-turns are not obsem’vable directly, there is themore general moot question of which stochasticreturns-generating process should be incor-porated in the option pricing model. As we’veseen above, the difference in the assumed re-turns process between Black and Scholes’smodel and Merton’s model resulted in asubstantially different pricing equation.Although the choice of an appropriate returnsprocess for modeling a bank’s assets is beyondthe scope of this paper, it is sufficient to note

“One might resort to the argument that the riskless rate isappropriate if the asset has no idiosyncratic (that is, firm-specific) risk, as noted in the context of Merton’s (1976)pricing model (see the shaded insert), but such anassumption is not particularly credible for bank asset port-folios, prima fade.

“See Pyle (1983) for an analysis of asset value uncertainty.Figlewski (1989) and Sabbel (1989) consider the im-possibility of hedging, along with many other difficulties inthe application of stock option pricing models,

40Merton’s (1976) approach uses that variance together withthe parameters of the jump distributions.

that this choice is a salient factor in the option’svalue, because it determines the probability ofeach of the possible year-end payoffs.

The empirical evidence to date testifies to thesensitivity of the results to the specificationemployed. Marcus and Shaked (1984) use thebasic Black-Scholes model, adjusted for divi-dends, and find that federal deposit insuranceis currently substantially overpriced relative tothe “actuarially fair” estimates provided by theiroption model.” They note, however, that“McCulloch’s (1981, 1983) estimates of insurancevalues derived from the Paretian-stable distribu-tion greatly exceed” their own.4’ Pennacchi(1987b) uses a more complicated model whichincludes the degree of regulatory control wield-ed by the insurer. He finds that deposit insur-ance may be either overpriced or underpriced,depending on the level of regulation assumed.McCulloch’s (1985) study assumes non-normal,Paretian-stable asset returns and non-stationaryrandom interest rates. He finds that insurancevalues are highly sensitive to the level andvolatility of interest rates. Ronn and Verma(1986), however, using a variant of Merton’s(1977) model, conclude that neither random in-terest rates nor non-stationary equity returnssignificantly affect the insurance valuations. Inbrief, the empirical evidence suggests that awide range of insurance valuations can bereached by varying the returns process em-ployed in the model.

Finally, it has been assumed in the precedingexamples that there is a single deposit that doesnot mature until the end of the year. In fact, ofcourse, banks maintain many deposit accountswith a wide range of maturities, starting withthe instant maturity of demand deposits. This issignificant, because it gives many depositorsanother type of insurance. A depositor who canwithdraw his funds from a failing bank soonerthan the FDIC can close it has 100 percent in-surance, regardless of the balance in the ac-count or the insurance scheme in effect.” The

41For an exposition of the dividend adjustment to the Black-Scholes model, see Merton (1973).

“Marcus and Shaked (1984), p. 449. Their reference toMcCulloch 0983) was a working paper, later published asMcCulloch (1985).

~‘lt need only be the case that the bank funds the depositoutflow somehow, for example, through a fire sale of itsassets, Such a run on Continental Illinois by institutionaldepositors prompted the FDIC to extend 100 percent in-surance to uninsured depositors.

result is that the simple application of an optionpricing model does not provide an accurateevaluation of such deposits, because it ignorescertain relevant strategies.

The usefulness of option models in the studyof deposit insurance results from two importantcharacteristics. First, these models distill thehost of economic factors involved down to ahandful of relevant parametet’s whose interac-tion is well-defined by the option-pricing equa-tion. Second, they are able to evaluate depositinsurance claims under a wide variety of in-surance structures. Although only three suchstructures were elaborated here, they can begeneralized to other applications. Thus, optionmodels provide a unified context for analyzingincentives within an insurance structure, as wellas for comparing alternative insurance schemes.

Unfortunately, option pricing models, likemost economic models, are an imperfect toolwhen directly applied to the complexities of thereal world. Beyond certain fundamental qualita-tive results, there are theoretical and empiricalreasons to believe that the insurance valuationsgiven by any particular option pricing modelwill be incorrect, highly sensitive to changes intheir specification, or both. As a result, the ab-solute dollar magnitudes provided by optionsmodels of the value of deposit insurance aresuspect. The contradictory empirical evidenceon fair pricing is indicative of this problem.

In defense of these models, however, there isno reason to believe that option models are anyworse in this regard than any alternativeeconomic model. Indeed, there is some reasonto believe that, although the absolute magnitudeof the valuations provided by option modelsmay be unstable, the rankings they provide fora sample of banks are not.44 Similarly, inac-curacies in determining the scale of insurancevalues do not deny the ability of option modelsto identify the direction of incentives or the im-pact of marginal changes in the structure ofdeposit insurance. Therefore, used judiciously,option pricing models can be an effective ana-lytical tool in the study of deposit insurance.

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