Using value-at-risk to measure financial risk

Duchac, Jonathan

I. INTRODUCTION

Over the past several decades portfolio and risk management techniques have adapted to increasingly complex financial instruments and risk scenarios. The rapid growth in derivative financial instruments and the derivatives losses reported in recent years have intensified concerns over reliably measuring financial instrument risk exposure. The massive derivative losses experienced by Gibson Greetings 20 million), Proctor and Gamble ($157 million), the Orange County Investment Pool ($1.7 billion), and Barings Bank ($1 billion) illustrate the devastating effects that unexpected shifts in market prices can have on portfolio values.(1) To protect against such huge losses from unexpected market shifts, risk managers and managerial accountants have focused on developing reliable techniques for measuring financial instruments risk exposure.

The most basic method for measuring financial instrument risk is to examine notional principal amounts.(2) This type of analysis, however, provides very lime insight into the risks associated with financial instruments because it does not consider market values, the volatility of market prices, or correlations between financial instruments. Duration analysis improves on the notional principal technique by concentrating on changes in market values associated with a single basis point change in yield.(3) While this technique considers current market values and the effects of changes in interest rates, it only considers the effects of a single basis point movement. Duration is also limited to relatively non-complex environments where a company has only long or short positions, and does not accommodate changes in interest rates at different points in the yield curve.

To overcome some of the limitations of notional principal and duration analysis, value-at-risk techniques have been developed. Value-at-risk is the estimated total loss that may be sustained o a financial instrument (or portfolio of instruments) from an adverse market movement, estimated for a given level of confidence over a specified holding period.(4) The output of a single market risk estimate denominated in a simple common denominator, dollars, makes this measure useful for internal reporting and managerial decision making. Managers can use this information for portfolio management and capital allocation decisions. Performance measures such as risk-adjusted returns can also be enhanced by this information. In a risk management context, managers can use value-at-risk models to further examine the effects that catastrophic changes in market values will have on financial instrument portfolios. Value-at-risk estimates are valuable to the external user as well. Moody’s currently uses value-at-risk estimates as part of its volatility rating system for bond mutual funds and money market funds.(5)

The following section provides two simplistic and general examples of how value-at-risk is determined. This illustration should in no way be perceived as a standard. Rather, it is intended to provide a basic illustration of value-at-risk concepts and how the methodology works (i.e., the basic mechanics). In practice, value-at-risk models vary widely and are significantly more complex than those presented in this discussion. The examples are designed solely to provide some initial insights into value-at-risk and how it can be used to evaluate the risks of financial instruments.

II. MEASURING VALUE-AT-RISK

As discussed above, value-at-risk is the estimated maximum potential loss that will be sustained from an unfavorable movement in market prices. This estimate is determined for a given probability and holding period. While a number of financial institutions and consulting firms have developed their own specific techniques and methods for measuring value-at-risk, these methods are based on the same underlying theory and model. This technique can be applied to individual financial instruments or a portfolio of different instruments. The output of the value-at-risk model is a single number representing the risk of the instrument or portfolio.

Value-at-risk is a function of four factors: market value (price), the standard deviation in market value, a confidence interval, and a holding period.(6) For individual financial instruments, the relevant market value(s) is the current market price of the financial instrument. Price data can be generated Internally or obtained from external data sources (e.g., broker quotes). The relevant market value for portfolios of financial instruments, however, is based on the percentage of the portfolios market value taken up by each instrument.(7) Thus, the portfolio’s market value is equal to the sum of the individual instrument’s market value multiplied by the proportion of the total financial instrument portfolio invested in that instrument. This value serves as the starting point for determining the value-at-risk of the individual instrument or portfolio of instruments.

The second component in the value-at-risk calculation is the standard deviation of market price returns for the financial instrument or instruments being evaluated, which measures the volatility of the instrument’s return(s). When a portfolio of financial instruments is being considered, correlation factors for the correlation between the market prices of the various financial instruments held in the portfolio must also be considered. Both the standard deviation and correlation factors are typically based on historical data over a specified period. Similar to market prices, this data can be generated internally or obtained from external data sources.

The confidence interval is the probability (or level of confidence) that the actual maximum loss experienced will not exceed the maximum expected loss (value-at-risk) generated by the model. For example, using a value-at-risk method, a bank estimates that the maximum potential loss on its financial instrument portfolio will be $10,000 for a 99% confidence interval. The 99% confidence interval indicates that the bank is 99% sure that the maximum expected loss on its financial instrument portfolio will not exceed $10,000.

The last factor used in determining the value-at-risk assessment is the holding period of the instrument or portfolio. This is the period over which the value-at-risk is to be measured. For illustration purposes, this paper will focus on a holding period of one day. However, value-at-risk calculations can also focus on longer holding periods. As the holding period increases, it becomes more difficult to estimate the value-at-risk because of the additional complexities that longer time periods have on the model.

Value-at-Risk Illustration — single financial instrument

To illustrate the basic value-at-risk model, consider an entity that holds only one financial instrument, a treasury bond position. The following information is known about the treasury bond position:

(Table omitted)

The following model assesses the value at risk for a single financial instrument:(8)

VAR = Current Market Value * Confidence * sigma * sq. root of Holding-Period

Using the factors identified above, the calculation of value-at-risk for a 99% confidence interval is:

VAR = $15 Million * 2.33 * .0004 * sq. root of 1 = $13,980

This result indicates that the entity is 99% confident that they will not lose more than $13,980 from this individual treasury bond position. Managers can use this information when assessing their firm’s overall risk exposure and making resource allocation decisions.

Value-at-Risk Illustration — two financial instruments

While the value-at-risk calculation for a single financial instrument illustrated above is relatively straightfoward, the consideration of more than one financial instrument increases the complexity of the calculation. The primary difference in the calculation of value-at-risk for more than one financial instrument is that the covariance of return for all the financial instruments in the portfolio is considered. This component reflects how different instruments’ return move in relation to one another. If the covariance is positive, the returns move in the same direction, while a negative covariance indicates that the returns move in opposite directions.(9)

Consider a firm that has a portfolio with two financial instruments, the treasury bond (TB) discussed above, and a foreign exchange contract (FXC). The market values, volatility measures (standard deviation), confidence intervals, and holding periods for the two instruments are as follows:

(Instruments omitted)

To determine the covariance of the instruments returns and the standard deviation for the two instrument portfolio, the correlation (rho) between the two instruments returns must be determined. In addition, the current market value for the two instrument portfolio (CMV sub rho ) is adjusted to reflect the portion of the portfolio invested in each of the two instruments. The market value of the portfolio, the covariance of the returns, and the volatility (standard deviation) of the portfolio (sigma sub rho ) is calculated as follows:(10)

(Table omitted)

The calculations for the treasury bond — foreign exchange contract portfolio are as follows:

(Equations omitted)

The portfolio’s current market value and standard deviation are then used to calculate the portfolio’s value-at-risk as follows:

VAR=Current Market Value * Confidence * sigma * sq. root of Holding Period

VAR = $12.5 Million * 2.33 * .0008529 * sq. root of 1 = $24,841

This calculation indicates that the firm is 99% confident that it will not loss more than $24,841 of its portfolio holdings during any one day period. Because the market prices of the Treasury bond and foreign exchange contract are partially correlated (rho = .2), the impact of the individual instruments on value-at-risk is not additives.(11)

Using value-at-risk techniques provides managers with a unique and robust mechanism that allows them to compare risks that would otherwise be distinct. The outputs are useful in assessing and evaluating the risks and performance of an entity’s financial instrument portfolio. Maximum risk estimates can also serve as the basis for determining capital reserves necessary to protect against potential financial instrument losses. In addition, managers can use these measures to develop guidelines on the amount of risk the entity is willing to undertake.

III. LIMITATIONS OF VALUE-AT-RISK

The value-at-risk methodology, however, is not without its flaws and limitations. First, value-at-risk estimates are typically based on past performance. While historical performance, volatility, and correlation estimates provide insight into the future, there is no guarantee that future movements of these variables will be consistent with the past.(12) Structural changes in correlations and volatilities can lead to actual financial instrument losses well in excess of those estimated by the value-at-risk model.

Second, value-at-risk does not consider the effects of catastrophic events. Victor Makarov, Managing Director for Risk and Analytics at Chase Manhattan Bank, notes that risk is too complicated to describe with one number.(13) By definition, value-at-risk excludes the effects of changes in market prices that rarely occur. To fully understand the risks associated with a firm’s portfolio of financial instruments, the effect of large, infrequent swings in market values must also be considered. Assessing the impact that such rare shifts in market prices have on a portfolio of financial instruments is accomplished by stress testing the portfolio. Stress testing is a simulation technique that measures the effect that severe price changes can have on financial instrument values. Considering the effects of such rare price changes in conjunction with value at risk estimates provides managers wit a more comprehensive picture of the risks associated with financial instruments than can be obtained by considering value-at-risk estimates alone.

Variations in the application of value-at-risk techniques also make it difficult to interpret the resulting estimates. Because a common value-at-risk model does not exist, the resulting risk estimates may partially be a function of the model employed. In addition, some model inputs are somewhat discretionary and contingent upon the individual manager’s subjective interpretations and estimates. The discretionary nature of these inputs to the estimation process highlights the limitations associated with relying solely on value-at-risk estimates to evaluate the risks associated with a firm’s financial instruments.

IV. SUMMARY

As discussed above, measuring and assessing the impact of market risk on financial instruments is a major concern for risk managers and accountants. Value-at-risk techniques generate an easy-to-understand estimate of these risks that mangers can use to measure performance and allocate resources. It is critical to understand, however, that value-at-risk estimates are only one factor that should be examined when evaluating the effects of market risk on financial instrument values. Stress testing should also be performed t6 determine the effect of large, highly unlikely market shifts on the portfolio’s value. To be most effective, value-at-risk estimates should not be considered in isolation, but rather as part of a comprehensive risk management and analysis plan.

PRELIMINARY: PLEASE DO NOT QUOTE WITHOUT AUTHOR’S PERMISSION

* Jonathan Duhac, is Assistant Professor of Accounting, Calloway School of Business and Accountancy, Box 7285 Reynolda Station, Wake Forest University, Wintson-Salem, NC 27109 Phone: (910) 759-4458, Fax: (910) 759-6133, email: duchacje(at)wfu.edu, February 10, 1996

1 Duchac, J.E., and J. Wilkerson, “Evaluating and Controlling Derivatives Operational Risk: Working Paper. Wake Forest University (December 1995).

2 “Measuring Value-at-Risk: Derivatives Week, September 26, 1994, p.6.

3 “Measuring Value at Risk,” Derivatives Week, September 26, 1994, p.6.

4 Sullivan, R.P., “Value-at-Risk vs. Capital-at-Risk,” Presentation in Wake Forest University’s Value-at-Risk: Financial Risk Management Tools and Methods, November 6, 1995.

5 “Moody’s Volatility Ratings Examine Derivatives Risk,” Derivatives Week, May 8, 1995.

6 Sullivan, R.P., “Value-at-Risk vs. Capital-at-Risk,” Presentation in Wake Forest University’s Value-at-Risk Risk: Financial Risk Management Tools and Methods, November 6, 1995.

7 Sullivan, R.P., “Value-at-Risk vs. Capital-at-Risk,” Presentation in Wake Forest University’s Value-at-Risk: Financial Risk Management Tools and Methods, November 6, 1995.

8 Sullivan, R.P., “Value-at-Risk vs. Capital-at-Risk,” Presentation in Wake Forest University’s Value-at-Risk: Financial Risk Management Tools and Methods, November 6, 1995.

9 Copeland, T.E., and J.F. Weston, Financial Theory and Corporate Policy, Addison-Wesley, New York, New York, 1988.

10 Sullivan, R.P., “Value-at-Risk vs. Capital-at-Risk,” Presentation in Wake Forest University’s Value-at-Risk Financial Risk Management Tools and Methods, November 6, 1995.

11 The value-at-risk for the foreign exchange contract in isolation is $17,475.

12. Kersey, Craig, “Emperor’s Clothes,” Banker, November 1994, p. 17.

13 Heap, P., “Insider Derivatives: Studying Effect of Market Catastrophes Helps Measure Risk, Analyst Says,” The Bond Buyer, August 1, 1995, p.30.

REFERENCES

Copeland, T.E., and J.F. Weston, Financial Theory and Corporate Policy, Addison-Wesley, New York, New York, 1988.

Duchac, J.E., and J. Wilkerson, “Evaluating and Controlling Derivatives Operational Risk,” Working Paper: Wake Forest University (December 1995).

Heap, P., “Insider Derivatives: Studying Effect of Market Catastrophes Help Measure Risk, Analyst Says,” The Bond Buyer, August 1, 1995, p.30.

Kersey, Craig, “Emperor’s Clothes,” Banker, November 1994, p.17.

Longerstaey, J., “Riskmetrics,” Presentation in Wake Forest University’s Value-at-Risk: Financial Risk Management Tools and Methods, November 6, 1995.

McHenry, J., “What the World Thinks About Derivatives,” The Journal of Bank Cost Management and Accounting, Vol. 8, No. 1 , p.5.

“Measuring Value-at-Risk,” Derivatives Week, September 26, 1994, p.6.

“Moody’s Volatility Rating Examine Derivatives Risk,” Derivatives Week, May 8, 1995.

Smithson, C.W., C.W. Smith, and D.S. Wilford, Managing Financial Risk: A Guide to Derivative Products, Financial Engineering, and Value Maximization, Irwin Professional Publishing: New York, New York, 1994.

Sullivan, R.P., “Value-at-Risk vs. Capital-at-Risk,” Presentation in Wake Forest University’s Value-at-Risk: Financial Risk Management Tools and Methods, November 6, 1995.

Copyright National Association for Bank Cost & Management Accounting 1996

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