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Market risk for foreign currency options: Basle's simplified model.

In 1988, the Basle (Switzerland) Committee on Banking Supervision of the Bank for International Settlements established minimum capital requirements for financial institutions' credit risk. The Committee's objectives were to strengthen the safety and soundness of the international banking system and to ensure a level playing field among international banks (see Basle, 1988).

One of the regulators' major concerns, one which was not addressed in the 1988 Accord, was market risk, the losses that result from adverse market moves such as changes in interest rates, exchange rates, and equity and commodity prices.(1) Thus, beginning in 1992, Basle began the process of establishing guidelines that required banks to set aside capital to protect against market risk. According to these guidelines, implemented in December 1997, to determine capital requirements, banks must choose either a standardized model proposed by Basle or their own internal model.

Because corporate use of foreign exchange (FX) currency derivatives to manage foreign exchange risk exposure has increased over the years, with banks dominating the over-the-counter (OTC) markets (see Bodnar, Hayt, and Marston, 1995, 1996; and Phillips, 1995), our paper focuses on one of Basle's simplified models, which estimates capital charges for over-the-counter FX options.(2)

In 1993, Basle proposed a framework for measuring market risk, offering financial institutions several standardized models, and invited comments on these models. Banks strongly criticized the standardized models and claimed their sophisticated internal models provided more reliable forecasts of market risk or value at risk (VAR). After receiving these and other public comments, in a 1995 ruling, Basle permitted the use of internal models, but required that local regulators approve these models. However, new backtesting rules for internal models, which penalize banks for inaccurate forecasting of VAR, could encourage banks to reexamine the benefits of standardized models.

In an August 1996 ruling, the Federal Reserve Board (Fed) required all US banks to use internal models for estimating market risk. The ruling requires banks with trading activity that exceeds either 10% of their assets or $1 billion, whichever is lower, to develop their own internal models (see Federal Reserve System press release, August, 1996).

Although the Fed's ruling eliminated the use of standardized models (with a minor exception for estimating specific risk for debt and equity instruments), local regulators in other countries, including Canada, Japan, and the European Union, do permit the use of standardized models. One advantage of these models is that the burden of modeling VAR is borne by regulators, not by banks.

Our paper shows that Basle's current simplified model for foreign currency options does not systematically relate capital requirements to market risk. We offer two alternative models, the Simplified Incremental Model (SIM) and the Simplified Value-at-Risk Model (SVAR). We compare these models to an internal model based on RiskMetrics[TM] (RM), a risk management system developed by J.P. Morgan (1995) to measure and manage market risk. We show that it is possible to develop a standardized model that can be as effective as an internal model.

In Section I, we describe Basle's guidelines for setting capital requirements for foreign currency options and show that charges are not consistently related to VAR. In Section II, we define the SIM and SVAR models. In Section III, we compare the three models to an internal model. Section IV concludes the paper with a discussion of the results.

Capital Requirements for Foreign Currency Options: Basle's Simplified Model

In this section, we explain how capital charges are estimated under Basle's simplified model. When estimating capital charges, banks that use the simplified model can combine the underlying currency with a long put or call or they can consider each position separately. In Subsections A and B, we describe how capital charges are estimated when cash positions and naked options are considered separately. In Subsection C, we demonstrate how charges are estimated when the positions are considered together. Subsection D shows that charges and VAR are not consistently related.

A. Capital Requirements for Underlying Cash Positions

Estimation of capital charges for cash positions in foreign currencies first requires the bank to determine its net position in each currency after the foreign currency is translated into the local currency. Net long (short) positions are then summed across currencies and a capital charge of 8% is levied on the larger of the two positions.

For example, assume a bank is net long $150 million in Japanese yen ([yens]), $300 million in German marks (DM), and is net short $200 million in British pounds ([pounds]), and $125 million in Swiss francs (fr), for a total net long position of $450 million, and a total net short position of $325 million. Basle requires a capital charge of 8% on $450 million, the higher of the two amounts, i.e., $36 million.

A positive feature of this method is that it recognizes that changes in the values of long and short positions in the same currencies offset one another. However, this method also assumes that the correlation across currencies within each long or short group is plus one, which could be unrealistic.

B. Capital Requirements for Long (Naked) Call and Put Options

If the bank holds only long positions in options that are not part of a hedged position, i.e., they are naked, the associated capital charges are the lesser of 8% of the market value of the underlying currency or the market value of the option (see Table 1). Capital charges levied on options are added to the charges on the underlying positions in foreign currencies, if any.

C. Capital Requirements for Carved-Out Positions

Under the simplified model, banks can choose to separate options and matching cash positions from the rest of their currency portfolios, i.e., "carve out" their positions to estimate their capital requirements. Banks can carve out long puts with matching long currency positions and long calls with matching short positions in currencies, essentially hedging the cash positions with options. The payoffs of the hedged position mimic those of a naked call and put, respectively, and are referred to in our paper as "call equivalents" or "put equivalents." Capital charges are based on this carved-out position, rather than on the risk of the two individual positions. For combined positions, capital charges are the largest of 8% of the market value of the underlying currency minus the option's intrinsic value or zero (see Table 1). Banks can choose not to carve out if capital charges are lower when estimated separately for the option and underlying currency (see Huckins and Rai, 1996, for details). Also, banks can decide not to carve out if they hold long calls (puts) and are long (short) the underlying currency.

D. Problems with Basle's Simplified Model

Although the simplified model is easy to use because [TABULAR DATA FOR TABLE 1 OMITTED] it bases capital charges either on a position's intrinsic value, its market value, or a combination thereof, it presents three problems. For equal increments in current market values or payoffs, options and option equivalents have different capital charges, capital charges are not monotonic, and absolute levels of capital charges for puts, calls, and their equivalents are not equal.

1. Options and Option Equivalents Have Different Capital Charges

We illustrate the differences in capital charges for calls and call equivalents in Figure 1. We base the figure's graph on the numerical example shown in Table 2. In the example, we assume that a foreign currency (FC) contract with an exercise price of $1.50 has a notional value of FC 100,000. For clarity, we assume the options' time premiums equal zero. (This assumption is relaxed in Section III.)

Comparing Columns 4 and 8 in Table 2 shows that the payoffs of call options and call equivalents differ by a constant amount, $150,000 (X). For the call equivalent, X represents a constant payoff obtained by exercising the in-the-money put embedded in the combined position. The put places a lower bound of $150,000 on the position's payoff, which then becomes insensitive to market risk. As a result, the market risk of calls and call equivalents is the same, given equal incremental payoffs.

Note that in our paper, incremental payoffs equal the difference between a position's market or intrinsic value and its lower bound. Figure 1 shows that when X [less than or equal to] S [less than] X/0.92, charges for long calls increase dollar for dollar with payoffs (slope = 1.0), charges for call equivalents increase by $0.08 (slope = 0.08). When X/1.08 [less than or equal to] S [less than] X, charges for long calls are zero and those for call equivalents increase by $1.08 per dollar change in the spot (Table 2, Columns 7 and 10) even as payoffs remain constant.

Similar problems occur for puts and put equivalents. In put equivalents, the call places a bound of minus X on the cost of covering the short position. If the call is in the money (S [greater than] X), risk from adverse changes in the market is zero. Whenever S [less than] X, incremental payoffs are X minus S because minus X is never exposed to market risk.

As Figure 2 shows, when X [less than] S [less than or equal to] X/0.92, charges for long puts are zero, but because payoffs vary inversely with the spot, those for put equivalents decline by -$0.92. When X/1.08 [less than or equal to] S [less than] X, charges for long puts decrease dollar for dollar with decreasing payoffs, but those for put equivalents increase at a rate of $0.08. Inexplicably, when S [less than or equal to] X/1.08, capital charges for put and put equivalents increase at a rate of $0.08 when payoffs are falling.

One source of the difference in charges for options and option equivalents is Basle's inclusion of counterparty risk when X [less than or equal to] S [less than] X/0.92 for calls and X/1.08 [less than or equal to] S [less than] X for puts.(3) In this range, charges for option equivalents are based on the positions' total, not incremental, payoffs.

To illustrate this for calls, Table 2, Column 6, shows charges based on the total payoffs of call equivalents. Note that when S [greater than or equal to] X, capital charges in Columns 6 and 7 are equal. However, for S [less than] X, capital charges in Column 7 are lower, because counterparty risk is not considered. If Basle's intention is to include counterparty risk, it is not clear why it should be considered for a limited price range.

2. Capital Charges Are Not Monotonic

The second problem with the Basle proposal is that [TABULAR DATA FOR TABLE 2 OMITTED] for the same incremental change in payoffs, capital charges for long options and option equivalents do not increase monotonically in every price range. As Table 1 and Figure 1 show, capital charges for options and option equivalents jump at two different points without any economic rationale. For long calls (puts), charges jump from 0 to S-X (X-S) at S = X, and a second time to 0.08S when S [greater than] X/0.92 (S [less than] X/1.08). For call equivalents, capital charges jump from zero to 0.08S-(X-S) at S = X/1.08 and to 0.08S when S = X. Similarly, capital charges for put equivalents jump from zero to 0.08S-(S-X) at S = X/0.92 and to 0.08S when S = X.

3. Absolute Level of Capital Charges Differs between Puts and Calls

A third problem is that the absolute levels of capital required for puts, calls, and their equivalents can differ even when the difference between a position's market value and its lower bound are equal (S-X equals X-S). When X/1.08 [less than or equal to] S [less than or equal to] X/0.92, the current model systematically requires more capital for calls and call equivalents than it does for puts and put equivalents. Thus, the bank is not equally protected against potential losses.

II. Alternatives to Basle's Simplified Model

In this section, we propose two alternatives to Basle's Simplified Model. These two alternative models resolve the inconsistencies described earlier, yet maintain the simplicity of Basle's model. We compare all three models to J.P. Morgan's RiskMetrics[TM], a popular internal model.

A. The Simplified Incremental Model (SIM)

The SIM model ensures that capital charges are consistently related to VAR. As in the Basle model, VAR equals a contract's marked-to-market value, a definition also used by the derivatives market to estimate margin requirements. In the SIM, capital charges equal a fixed proportion ([Alpha]) of a contract's marked-to-market value. The charges are shown in Table 2, Columns 5 and 9, where we assume the time premium equals zero and [Alpha] is fixed at 8%, as is assumed by Basle. Capital charges for in-the-money long calls and call equivalents are equal and proportional to S-X, which is each position's incremental payoff (from its lower bound). When calls are out of the money and the incremental loss is zero, capital charges also equal zero. We can also show that when capital charges for in-the-money puts and put equivalents equal [Alpha](X-S), charges for out-of-the money puts and put equivalents equal zero.

Therefore, capital charges under this model are consistently related to VAR, and charges for options and option equivalents are equal. Ideally, the parameter ([Alpha]) should be determined by individual banks, be a function of a portfolio's risk, and range from zero to one. Thus, the SIM model offers the advantage of simplicity, but its efficacy depends on a bank's choice of [Alpha].

B. The Simplified Value-at-Risk Model (SVAR)

The SVAR model extends the SIM by incorporating both price and volatility risk so that charges reflect the nonlinear characteristics of options, as required by Basle. Since price risk varies with an option's moneyness, we use X/S as a proxy for price risk, where X equals an option's exercise price, and S represents the currency's spot price. As a proxy for volatility risk, we use the standard deviation of the underlying currency ([Sigma]). Daily volatility estimates can be obtained from J.P. Morgan.

In the SVAR model, we link capital charges and contract-specific risk by setting [Alpha] = (X/S)[Sigma]. Defining ct in this manner ensures that capital charges increase with the underlying currency's volatility and decrease as a proportion of contract value as the option's moneyness rises. The results are presented in Table 3 and will be discussed in Section III.

C. J.P. Morgan RiskMetrics[TM] (RM) Internal Model

We estimate an internal model based on a parametric model presented in the 1995 J.P. Morgan RiskMetrics[TM] Technical Document. Internal models are tailored to each institution's portfolio, incorporate a relatively complex measure of VAR, and must satisfy Basle's quantitative and qualitative standards, which are described below. Parametric models make distributional assumptions and require the use of an option-pricing model, such as the Black-Scholes (1973) model. Nonparametric models, which are distribution-free, simulate expected future losses under different scenarios.

We assess the effectiveness of the SIM, SVAR, and Basle models by comparing their charges to those determined by the internal model, which serves as a benchmark for comparison.

The simplified and internal models are differentiated by their definitions of VAR. In internal models, VAR equals a position's expected maximum loss for a given probability over a specified period of time, but in the simplified models, VAR equals a contract's marked-to-market value.(4)

Basle's qualitative standards apply to the procedural aspects of risk management. The quantitative standards pertain to market risk measurement. Since our paper focuses on risk measurement, we discuss only quantitative standards.

Briefly, Basle's quantitative standards for options require banks to estimate VAR on a daily basis, using a 99th percentile, one-tailed confidence interval, i.e., a loss of a given magnitude should occur only 1% of the time. VAR estimates must also be based on a ten-day holding period, which must be approximated from at least one year of data. Data sets must be updated at least once every three months.(5)

Banks are also required to recognize the nonlinear characteristics of option contracts. Specifically, models must incorporate an option's price and volatility risk by examining the underlying currency's risk characteristics.

In our paper, we use the Garman-Kohlhagen (1983) model, which is a modified version of the Black-Scholes (1973) model, to estimate FX option prices. According to the Garman-Kohlhagen model, all FX option's value depends on the price (S) and volatility ([[Sigma].sup.2]) of the underlying currency, the option's exercise price (X), its maturity (T-t), and the difference in the foreign ([r.sub.f]) and domestic ([r.sub.d]) risk-free interest rates ([r.sub.d]-[r.sub.r]). Specifically, a call option's price equals:

[TABULAR DATA FOR TABLE 3 OMITTED]

c = [[[Se.sup.-r].sub.f].sup.(T-t)]N([d.sub.1])-[[[Xe.sup.-r].sub.d].sup.(T-t)]N([d.sub.2]) (1)

where

[d.sub.1] = ln(S/X) + [([r.sub.d] - [r.sub.f]) + 0.5[[Sigma].sup.2]](T-t)/[Sigma][-square root of T-t] (2)

and

[d.sub.2] = [d.sub.1] - [Sigma][-square root of T-t] (3)

Although the expected change in an option's value can be expressed as a function of all five parameters, Basle specifically requires that price and volatility risk

(1) be considered. A RiskMetrics' model that captures both risks shows that the change in an option's value (dV) equals:

(2) dV = [Delta]dS + 1/2[Gamma]d[S.sup.2] + [Lambda][Sigma] (4)

where

dS = the expected change in the underlying currency's price

d[Sigma] = the expected change in the underlying currency's volatility

[Delta] = [Delta]V/[Delta]S

[Gamma] = [[Delta].sup.2]V/[Delta][S.sup.2]

[Lambda] = [Delta]V/[Delta][Sigma]

The parameters [Delta], [Gamma], and [Lambda] are derived from Equation (1). Specifically, d[Sigma] is set to 0.01 and

dS = [[Sigma]/[(250).sup.0.5]]S (5)

[Delta] = [[[e.sup.-r].sub.f].sup.(T-t)]N([d.sub.1]) (6)

[Gamma] = [N[prime]([d.sub.1])

[[[e.sup.-r].sub.f].sup.(T-t)]]/[S[Sigma][(T-t).sup.0.5]] (7)

[Lambda] = [[[e.sup.-r].sub.f].sup.(T-t)] N[prime]([d.sub.1])S[(T-t).sup.0.5] (8)

III. Comparison of the Basle, SIM, SVAR, and RM Models

We first compare the Basle and SIM models to the RM model by expanding the example from Section I. The new example appears in Table 3.

In the example, the British pound ([pounds]) replaces the general FC contract. The currency option we consider has a strike price of $1.50 and a notional value of [pounds]100,000. The value of the underlying currency ranges from $1.30 to $1.70.

We add to the example estimates of volatility, domestic and foreign interest rates, and time to expiration, resulting in a positive time premium. Time to maturity is held constant at 0.25 years. We approximate interest rates by the one-year Eurosterling rate and the one- year Eurodollar rate, obtained from the Financial Times on January 7, 1998 ([r.sub.f] = 7.3%, [r.sub.d] = 5.7%). We obtained 60 days of volatility estimates (from October 10, 1997 through January 2, 1998) for the British pound from J.P. Morgan's RiskMetrics[TM] regulatory set. J.P. Morgan bases these estimates on rolling ten-day holding periods, updates them daily, and multiplies them by 2.33.

For our example, we obtain a 60-day average volatility of 0.38 by dividing the daily estimates by 2.33 and averaging them. All Garman-Kohlhagen (1983) call prices and corresponding RM model charges appear in Table 3, Columns 4 and 8, respectively, and in Figure 3, together with the charges for the Basle, SIM, and RM models.(6) Basle's charges are for long calls only. Basle's charges for call equivalents remained unchanged from Table 2.

For every spot price, Basle's charges are highest and SIM charges are lowest. Basle's charges are equal to the contract's value until its value exceeds 0.08S. In contrast, RM model charges are always less than the contract's value (Column 5), but increase steadily as contract values rise. RM model charges are high relative to contract value when the option is out of the money, and decline proportionately as contract values rise, ranging from 70% of value (S = $1.30) to 33% of value (S = $1.70). The decreasing proportion reflects the reduction in the volatility of option returns as the option moves deeper into the money.

Although Basle's charges also decrease as a proportion of contract value when charges equal 0.08S, the charges themselves are no longer linked to the option's price and volatility risk. The magnitude of Basle's charges and the sizable difference between Basle's charges and RM model charges suggest that Basle-model users set aside too much capital for market risk.

SIM model charges always equal 8% of the contract's value regardless of the option's price and volatility risk, and are significantly lower than RM model charges. Thus, setting [Alpha] equal to a constant in the SIM might not be an effective method of estimating capital charges, nor does it appear that Basle's parameter of 8% is economically justified.

Results are similar for put options and option equivalents. Figure 4 shows charges estimated under the Basle, SIM, and RM models. Again. Basle's charges are substantially higher than those obtained from the other two models, even in the price ranges where charges vary inversely with VAR. SIM model charges are the lowest.

The model comparisons suggest that SIM is only effective if [Alpha] can vary with a position's price and volatility risk. Charges should rise with the contract's value but decrease proportionately as the option's moneyness increases. The SVAR model alleviates these problems.

Figure 3 and Table 3, Column 9, show the charges estimated under SVAR. In the example, [Alpha] = (X/S)[Sigma], where [Sigma] = 0.38, the volatility estimate obtained from J.P. Morgan. For out-of-the-money options, SVAR model charges lie between SIM and RM model charges, with SIM charges substantially lower than the others. Differences in RM and SVAR charges decline as the option's moneyness increases. SVAR charges range from approximately 44% (S = 1.30) to 34% (S = 1.70). Redefining ct appears to increase the SVAR model's usefulness, because its charges then most closely approximate those of the RM model.

Figure 4 shows the similarity in put and call results. SVAR model charges are closer to RM model charges than are those from any other model. SVAR model charges are higher than RM model charges for out-of-the-money puts, but the differences decline as the option's moneyness increases.

Although linking [Alpha] to price and volatility risk improves the SIM model, it is not clear how robust the measure is to changes in the determinants of option prices. Marshall and Siegel (1997) estimate VAR using a parametric model in which [Delta] is held constant across all currency rates (RiskMetrics' models do not use a constant [Delta]) and note that VAR estimates for FX options are sensitive to underlying parameter estimates for both parametric and nonparametric models. Therefore, we re-estimate RM and SVAR model charges and compare them over a range of values for volatility and for time to maturity. The results, which are not shown here, indicate that RM model charges are significantly higher than SVAR model charges when volatility is low, and that the differences diminish as volatility rises. On the other hand, SVAR model charges increase relative to RM model charges as time to maturity increases. In all cases, SVAR model estimates are closer to RM model estimates than are those from the SIM, which continue to be significantly lower.

IV. Conclusions

This paper shows that Basle's simplified model for estimating capital charges for FX options suffers from three problems. For equal changes in incremental payoffs, capital charges for options and option equivalents are not equal, capital charges are not monotonic, and the absolute level of capital charges differs for puts and calls.

We propose two alternatives to Basle's simplified model, the Simplified Incremental Model (SIM), and the Simplified Value-at-Risk (SVAR) model.

The SIM modifies Basle's model by basing capital charges on a constant proportion of a contract's marked-to-market value. This model alleviates the problems presented in the Basle model by systematically relating capital charges and VAR.

The SVAR model links price and volatility risk to capital charges, using easily obtained parameters. The Basle, SIM, and SVAR models are compared to an internal model, which is based on J.P. Morgan's RiskMetrics[TM] parametric model.

We show that charges under the Basle model are substantially higher than those determined under the RM model. SIM charges are substantially lower than the RM or Basle model charges. In contrast, SVAR model charges approximate internal model charges better than do the charges from either the Basle model or the SIM.

One advantage of a standardized model is that its use shifts the burden of modeling VAR to regulators. This is particularly important if banjos are penalized for inaccurate VAR forecasts due to new rules on backtesting. We conclude that it is possible to construct a standardized model that is as effective as an internal model, thus offering some banks the opportunity to benefit from standardized models.

Although the example presented in this paper is for a single currency option, the results suggest that a well-designed, simplified model can be as effective as an internal model when more than one currency is involved. Thus, it can be worthwhile for the banking industry and the Fed to re-examine their rejection of standardized models.

We thank Linda Allen, two anonymous referees, and participants at the Financial Management Association International's 1996 Annual Meeting for their comments.

1 Two other risks not covered in the 1988 Accord are interest-rate risk and payments-system risk.

2 Basle expects the model to be used primarily by banks that are option buyers. They have also proposed two intermediate models, the delta-plus and scenario models, to be used by banks that write options and/or hold complex option portfolios. See Basle (1993, 1995, and 1996) for a description of these models for determining capital charges.

3 Counterparty risk is considered under the 1988 Basle Accord in which banks are required to set aside capital for off-balance-sheet and derivative instruments (see Basle, 1988). As of August 1995, limited netting of derivative contracts is also allowed (see Saunders, 1996).

4 For different views on the definition and measurement of VAR, see Dimson and Marsh, 1995; Hendricks, 1996; Mahoney, 1996; Marshall and Siegel, 1997; and Simons, 1996.

5 See Federal Reserve System press release, August 1996, for a detailed description of these quantitative standards.

6 For Table 3, we re-estimate the simplified model and SIM charges to include a time premium. Inclusion of a time premium increases the charges, but has no impact on the earlier analysis.

References

Basle Committee on Banking Supervision, 1988, "International Convergence of Capital Measurement and Capital Standards," Basle, Switzerland, July.

Basle Committee on Banking Supervision, 1993, "The Supervisory Treatment of Market Risk Consultative Proposal," Basle, Switzerland, April.

Basle Committee on Banking Supervision, 1995, "Planned Supplement to the Capital Accord to Incorporate Market Risk," Basle, Switzerland, April.

Basle Committee on Banking Supervision, 1996, "Amendment to the Capital Accord to Incorporate Market Risks," Basle, Switzerland, January.

Black, F. and M. Scholes, 1973, "The Pricing of Options and Corporate Liabilities," Journal of Political Economy (May/June), 637-654.

Bodnar, G.M., G.S. Hayt and R.C. Marston, 1996, "1995 Wharton Survey of Derivatives Usage by US NonFinancial Firms," Financial Management (Winter), 113-133.

Bodnar, G.M., G.S. Hayt, R.C. Marston, and C. W. Smithson, 1995, "Wharton Survey of Derivatives Usage by US Non-Financial Firms," Financial Management (Summer), 104-114.

Dimson, E. and P. Marsh, 1995, "Capital Requirements for Securities Firms," Journal of Finance 50, 821-850.

Federal Reserve System, 1996, "Risk-based Capital Standards: Market Risk," Press Release, Washington, DC (August).

Garman, M.B. and S.W. Kohlhagen, 1983, "Foreign Currency Option Values," Journal of International Money and Finance 2, 231-237.

Hendricks, D., 1996, "Evaluation of Value at Risk Models Using Historical Data," Federal Reserve Bank of New York Economic Review (April), 39-70.

Huckins, N.W. and A. Rai, 1996, "Estimation of Market Risk for Foreign Currency Options: A Modification of Basle's Simplified Model," Hofstra University Working Paper 98-01.

J.P. Morgan Company, 1995, RiskMetrics[TM] Technical Document, 3rd ed., New York, NY.

Mahoney, J.M., 1996, "Empirical-based Versus Model-based Approaches to Value-at-Risk: An Examination of Foreign Exchange and Global Equity Portfolios," Federal Reserve Bank of New York Working Paper, February.

Marshall, C. and M. Siegel, 1997, "Value at Risk: Implementing a Risk Measurement Standard," Journal of Derivatives (Spring), 91-111.

Phillips, A.L., 1995, "1995 Derivatives Practices and Instruments Survey," Financial Management (Summer), 115-125.

Saunders, A., 1996, Financial Institutions Management, 2nd ed., Chicago, IL, Irwin.

Simons, K., 1996, "Value at Risk - New Approaches to Risk Management," in Federal Reserve Bank of Boston New England Economic Review (September/October), 4-13.

Nancy White Huckins is an Assistant Professor of Finance and Anoop Rai is an Associate Professor of Finance at Hofstra University.
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