Abstract

The variety and complexity of portfolio holdings have given rise to the need for additional analyses for purposes of risk management. A framework for risk analysis includes three dimensions: sensitivity analysis, value at risk (VaR), and stress testing. This article describes each dimension and suggests a procedure for achieving a VaR measure. Once individual holdings are analyzed, attention can be directed to portfolio-level analyses and the types of output suitable for monitoring purposes. In combination, this framework can capture the important features of portfolio risk.

Introduction

Risk control in asset management is the ability to manage the uncertainty associated with the investment process. Fundamental to risk control is risk measurement, which can be thought of as quantification of the characteristics of risk.

Early attempts at risk quantification dealt with investments in relatively simple security types. This approach included both fixed and known cash flows, as is the case for Treasury securities and equities described by lognormal return distributions. Risk was characterized by volatility of returns, measured by quantities such as variance, standard deviation, or mean absolute deviation (Markowitz 1952, 1959).

As the concept of risk measurement and risk control evolved over time, additional approaches were introduced. In the case of fixed-income securities, the concept of duration, modified duration, and effective duration became widespread tools for risk management (see, e.g., Fabozzi 1988, Appendix A). For equities, beta coefficients and fundamental betas were introduced to provide additional capability in managing the risk of equity portfolios (Sharpe 1964; Rosenberg and Guy 1976a, 1976b). These analytical paths are indicative of specialization by asset type since the earlier attempts at risk management. Portfolio-oriented measures such as the concept of shortfall risk also have been introduced (Leibowitz and Henriksson 1989).

As the structure of marketable assets has become more complex and as market conditions have exposed the limitations of the traditional measures of risk, a number of recommendations have emerged to address the perceived need for additional risk analysis insight. The Group of Thirty (1993) reviewed the derivative product industry practice and suggested capital at risk as an appropriate risk measure. The Group of Thirty's Derivative Policy Group further described specific parameters for a capital-at-risk analysis.

The complexities of the many risk factors and their interaction call for a multidimensional approach to risk measurement. The nature of complex marketable assets has increased the requirements for the necessary analytical methods. In general, these methods represent a revisit to the early macro perspective in viewing risk from an overall portfolio standpoint.

These methods must deal with the multiplicity of risk sources and their correlations. They must also recognize the asymmetry of the return distribution. Derivative securities, such as options or swap transactions with embedded options, exhibit a skewed price distribution that cannot be adequately analyzed using the traditional risk measures suitable for simpler investments.

The objective of this study is to describe the methods appropriate for quantifying the risk of complex investments that are subject to a variety of risk sources. The overall methodology consists of three functional elements: sensitivity analysis, value at risk (VaR), and stress testing. Each element has its unique contribution to comprehensive risk measurement. Sensitivity analysis provides a basic building block to risk analysis and is a necessary input for hedging activities. VaR provides a useful summary, under prespecified conditions, of the amount at risk, given the risk characteristics of the portfolio. Stress testing complements VaR by providing the results of extreme scenarios of joint risk-factor change.

This chapter discusses ways of measuring and analyzing quantifiable risks, with emphasis on assessment of total portfolio risk and on the tools for risk management. The discussion, for illustrative purposes, focuses on fixed-income portfolios, because they typically contain the largest percentage of derivative securities and transactions. The principles of the analysis, however, apply to portfolios of all asset types.

Risk Sources

An investment portfolio of securities, derivatives, or contracts is exposed to many types of risk, including the following:

• Market risks. The value of a portfolio may change because of changes in market conditions, such as changes in a stock market index or in the level of interest rates.
• Foreign exchange risks. The value of foreign investments may change because of exchange rate movements.
• Option risks. The counterparty may exercise its options at a time that is disadvantageous to the security holder.
• Prepayment risks. Principal payments and other cash flows may be accelerated at a time when prepayments are undesirable.
• Credit risks. The counterparty or issuer of the security may be unable to fulfill its obligations.
• Specific risks. Individual securities may be subject to price changes not explained by changes in the level of the market.
• Liquidity risks. A security may be difficult or impossible to sell or liquidate at its proper value.
• Management and operations risks. The portfolio may be poorly managed or maintained, such as by engaging in disadvantageous transactions or executing trades improperly.
• Administrative risks. A loss in value may result from excessive fees or fraud.
• Regulatory risks. The governing laws and regulations may change, requiring adjustments to a portfolio that can affect its value.
• Event risks. The portfolio may be vulnerable to specific events such as political instability in a country in which the portfolio has an exposure.

The total risk of a portfolio is the potential decline in its market value. Measuring this risk requires quantification of possible market value changes, under probable as well as extreme circumstances, resulting from the individual risk sources and their interplay.

A quantitative measure of the contribution of the various types of risk to the market value change is possible for only some of them, including market, option, prepayment, foreign exchange, credit, and specific risks. Additional risk components, notably liquidity, management, administrative, regulatory, and specific event risks, must be determined by other means, including judgmental procedures, and incorporated into the total risk assessment.

An investor may be able to hedge or otherwise compensate for some of these risks; for example, the portfolio's interest rate risk can be easily counterbalanced by short positions in interest rate futures contracts, and foreign exchange risk can be eliminated by forward currency hedges. The specific risk of the portfolio may be diversified away by the investor's other holdings. Proper risk management, therefore, requires measuring the exposures to the sources of risk in such a way that they can be reduced or eliminated.

Risk Exposures

An essential basis to risk measurement and management is determining the security and portfolio exposures to risk factors. Risk factors are market characteristics whose change affects the value of a given security or contract. For fixed-income derivatives, the principal risk factors are as follows (for each of the currencies involved):

• Interest rate level, the overall level of the term structure whose shifts will affect the portfolio value
• Rates of benchmark maturities, the specific maturity points along the term structure by which changes in interest rates can be measured
• Spreads over government rates (such as various corporate bond quality sectors, swap spreads, or mortgage-backed security spreads), the yield premiums attributable to specific nongovernment securities
• Volatility of rates, measures of interest rate variability for various maturities and forward horizons
• Exchange rates, measures of currency exchange rates

Denote the values of these risk factors by . If P is the value of a security, then the change in the security value resulting from the change in the risk factors can, in the first approximation, be given as

1

The quantities in this equation are the exposures, or sensitivities, of the security to each of the risk factors. They measure the percentage change in the value of the security resulting from a unit change in the value of the factors.

A well-known example is the exposure to changes in the level of interest rates, which is the security duration. Another example is the exposure to changes in volatility (relevant especially for options), sometimes referred to as vega. In equities, the exposure of a stock to a stock market index move is proportional to its beta.

If we postulate a linear relationship between changes in the value of the factors and the percentage price change represented by Eq. (1), then the exposures to the factors are defined by the partial derivatives as

2

If F_i is the interest rate level, Eq. (2) is the familiar definition of duration. It generalizes in the same form (apart from the choice of sign in Eqs. (1) and (2), which is strictly a matter of convention) to other risk factors as well. Care needs to be taken that the duration and all other exposures are correctly measured on an options-adjusted basis. If so, the price sensitivities will have already taken into account any embedded options affecting price changes.

For several reasons, except as a first-order approximation, Eqs. (1) and (2) are not a satisfactory representation for the price change of a security. First, the price change is not a linear function of the factor change, particularly for derivatives. Second, the changes in the factors are not instantaneous, so a change resulting from the passage of time needs to be incorporated. Third, the market move may not explain fully the change in the value of a security. Fourth, it is more appropriate to characterize the dollar change rather than the percentage value change because derivatives such as swaps and other contracts often start with a low or even zero value.

Assume that the change in the market value of a security over an interval is governed by the equation

3

where are changes in the value of each risk factor, and Y is the risk specific to each security. The quantities are then the linear and quadratic exposures of the security value to the factors. They are analogous to the dollar duration and dollar convexity measures of interest rate exposure. So that the nonlinear price response is properly approximated, however, should be measured for a finite factor change rather than the infinitesimal one given by Eq. (2). The exposures are estimated as

4

and

5

where and are the prices of the security calculated under the assumption that the risk factor F_i changed by the amount of and , respectively. Some considerations (related to the theory of Hermite integration) suggest that should be taken specifically to equal

6

where is the volatility of over the interval . In Eqs. (4), (5), and (6), the exposures characterize the global response curve of the security price rather than the local behavior captured by durations and convexities.

The quantity A in Eq. (3) is equal to

7

where μ is the expected return, .

This representation of price behavior facilitates risk analysis and measurement. The linear risk exposures and the quadratic risk exposures combine in the portfolio as simple sums of those exposures for the individual securities. Thus, if is the linear exposure of the k-th security to the i-th risk factor (and similarly for ), then

and

would be the risk exposures for the portfolio.

A risk-management process may then consist of a conscientious program of keeping all the portfolio risk exposures close to zero,

to eliminate an undesirable dependence on market factors. This approach is equivalent to hedging against all sources of market risk. The specific risks, , which combine by the formula

can only be reduced by diversification.

The overall variability of the portfolio or security value can be calculated from its risk exposures, using the formula

8

which is a consequence of the value change Eq. (3). Here, the σ_ij are the covariances in the changes of the i-th and j-th risk factors; that is,

To the extent possible, the variances and covariances should be obtained from current pricing of derivatives whose values depend on these variances (the implicit volatilities). For instance, quotes are available in the swap market for interest rate volatilities, calculated from market prices of swaptions. These volatilities reflect the market's estimate of the prospective, rather than past, interest rate variability. Only when such implicit volatilities are not available for a given risk factor should a historical variability be used. In that case, care should be taken that the historical period is long enough to cover most market conditions and cycles.

The calculations of the price variability of a portfolio, its sectors, and the individual securities can be an accurate picture of the structure of the risks. In addition to the total price variabilities, risk may also be broken down by source. One possible presentation is

These risks are defined as follows:

Although the risks are measured by standard deviation, and standard deviations do not add, the component risks do add up to the total risk. This summation is accomplished by calculating the risk increment each component adds to the previous subtotal. This method makes the decomposition dependent on the order in which the components are listed, but it also makes the components meaningful: The 3 percent derivative risk, for example, means that the fund's derivative holdings add 3 percentage points to the 12 percent price variability attributable to market factors.

Value at Risk

The capital at risk, also called the value at risk (VaR, not to be confused with variance, Var, of the previous section), is a single, highly useful number for the purposes of risk assessment. It is defined as the decline in the portfolio market value that can be expected within a given time interval (such as two weeks) with a probability not exceeding a given number (such as 1 percent). Mathematically, if

9

then VaR is equal to the value at risk at the probability level α.

To calculate the VaR, it is necessary to determine the probability distribution of the portfolio value change. This distribution can be derived from Eq. (3).

Assume that the factor changes have a jointly normal distribution with zero mean and a covariance matrix . Then the first three moments of are given by Eqs. (7), (8), and (10), where

10

Knowing the three moments, the probability distribution of can be approximated and the VaR calculated. There are theoretical reasons to use the gamma distribution as a proxy for that distribution. The resulting formula for the value at risk is then very simple:

11

where σ is the standard deviation of the value of the portfolio or security, obtained as the square root of the variance given in Eq. (8), and γ is the skewness of the distribution,

12

calculated using Eqs. (8) and (10). The ordinate is obtained from Table 28.1 (corresponding to the gamma distribution). Table 28.1 extends only to the values because that is the highest magnitude attainable for the skewness of the quadratic form in Eq. (3).

Note that the value 2.33 in Table 28.1 corresponding to is the 1 percent point of the normal distribution. In other words, if the portfolio value change can be represented by the symmetric normal distribution, the VaR at the 1 percent probability will be

For most derivative securities and portfolios, however, the probability distribution is highly skewed one way or the other, and the normal ordinates do not apply. The numbers in Table 28.1 represent the proper ordinate values.

In fact, the ratio of the ordinate in Table 28.1 corresponding to the skewness of the portfolio to the normal ordinate provides the increase (or reduction) of the VaR attributable to the portfolio composition. Thus, if the portfolio has a negative skewness of (such as a portfolio of callable bonds), the VaR is percent higher than it would be if the portfolio returns were symmetric. A portfolio with positive skewness of (for instance, holding bonds with puts) will require only percent as much capital to cover the VaR as a portfolio with normally distributed returns.

Eq. (11) does not include the expected return μ, because the mean is of a lower order of magnitude (namely ) than the standard deviation σ (which is of the order ) and can be neglected.

The VaR can be calculated for individual securities, portfolio sectors, and the total portfolio, as well as by sources of risk. This approach can lead to a useful breakdown, such as that presented in Table 28.2. The numbers in Table 28.2 do not necessarily add up, either down or across. The reason is that the VaR resulting from, say, interest rate risk may come from rising interest rates for one security (as for most bonds) and declining interest rates for another (such as an income-only security or a short position in futures). The reason the numbers do not add up across the sources of risk is that events of a given probability (say, 1 percent) do not add up: An interest rate change that can happen with 1 percent likelihood when considered alone is not the same as if it would happen together with, say, an exchange rate movement.

Note: The portfolio is assumed to be composed of several sectors (industries, etc.). Sector A amounts are given for illustrative purposes only; the dash represents amounts for the other sectors, which are not made explicit.

An alternative to Table 28.2 would be to measure the component values at risk incrementally. This method would mean that the VaR attributable to derivative risks, for example, would be calculated as the difference between the VaR obtained when considering jointly interest rate risk, other market risks, and derivative risks and the VaR obtained when considering interest rate and other market risks alone. Such measurement depends on the order in which the sources of risk are taken, which seems somewhat arbitrary. Moreover, when applied down the table for securities and sectors, this method also presumes that the risks of, say, Security 2 are measured on top of those of Security 1, which may make sense in some situations (such as a futures position as a hedge on top of bond holdings), but makes less sense in others.

Stress Testing

Although VaR provides a useful assessment of potential losses from various sources of risk and their interplay, it should be complemented by a series of stress tests. A stress test consists of specifying a scenario of extreme and unfavorable market conditions occurring over a specific time interval and then evaluating the portfolio gains or losses under such scenarios. This approach is useful for a number of reasons: It allows for consideration of path-dependent events such as cash flows on collateralized mortgage obligations; it does not rely on a specific form of the value-response curve, such as the quadratic form in Eq. (3); it appeals to intuition by showing the situations under which a loss can occur, which is lost in VaR alone; and last but not least, it is required or recommended by the various oversight agencies and auditors.

Table 28.3 shows a possible stress test output table. Scenarios 3 and 4 represent the US dollar interest rate term structure steepening or flattening, which may affect long/short rate basis swaps. Scenario 9 is a combination of an exchange rate change and foreign interest rate change, which may affect currency swaps and the like.

Stress tests are less systematic and somewhat ad hoc compared with VaR. Their usefulness is in an analysis of the portfolio response to market-condition changes that are more extreme or persistent than those likely to occur in a short time interval. Over large and protracted market movements, the value of a security or portfolio may show a response curve that is not well represented by a quadratic form such as Eq. (3). An example is provided in Figure 28.1, which shows the price response to interest rate movements for a callable bond.

The VaR calculations for this security would be based on an interest rate move possible with 1 percent probability over a time interval, such as two weeks, which may be some 50 basis points. Within that range, the price-response curve is adequately described by the assumptions of the VaR calculation. The VaR is therefore a proper measure of instantaneous, or current, portfolio riskiness, which is all that would be necessary if all securities in the portfolio were perfectly liquid and if the portfolio risk were managed on a continuous-time basis. Because this assumption is often unrealistic, it is advisable to measure portfolio value in response to extreme stress tests.

Conclusions

The risk in complex portfolios can be quantified. The market characteristics that affect the value of a security or portfolio are called risk factors. Risk factors that affect fixed-income derivatives include interest rate level, benchmark maturity rates, spread over government rates, volatility of rates, and exchange rates. A quadratic approximation may be used to quantify the risk exposure to those risk factors, and then a standard deviation may be calculated. The risk-summary number, value at risk (or capital at risk), is defined as the dollar amount that the total loss might exceed within a certain time period with a certain probability. The VaR may be calculated by a gamma distribution approximation. Although VaR gives a summary risk number, it does not tell the source or direction of the risk. To see the possible loss under extreme or least favorable market conditions, a series of stress tests must be performed. The VaR and the result of a comprehensive stress test give a better risk picture than either of them alone.

Risk measurement of fixed-income investments is an involved process. The result of using three methods—sensitivity analysis, value at risk, and stress testing—is an ability to evaluate complex return outcomes. Each of these techniques has an important role; in combination, they represent the comprehensive risk measurement necessary for portfolios with complex structures and interrelationships.

References

1. Fabozzi, F.J. (1988). Fixed Income Mathematics. Chicago: Probus Publishing.
2. Group of Thirty. (1993). Derivatives: Practices and Principles. Washington, DC: Group of Thirty (July).
3. Leibowitz, M., and R.D. Henriksson. (1989). “Portfolio Optimization with Shortfall Constraints: A Confidence-Limit Approach to Managing Downside Risk.” Financial Analysts Journal, 45 (2) (March/April), 34–41.
4. Markowitz, H.M. (1952). “Portfolio Selection.” Journal of Finance, 7 (1) (March), 77–91.
5. —. (1959). Portfolio Selection. New Haven, CT: Yale University Press.
6. Rosenberg, B., and J. Guy. (1976a). “Beta and Investment Fundamentals.” Financial Analysts Journal, 32 (3) (May/June), 60–72.
7. —. (1976b). “Beta and Investment Fundamentals—II.” Financial Analysts Journal, 32 (4) (July/August), 62–71.
8. Sharpe, W.F. (1964). “Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk.” Journal of Finance, 19 (3) (September), 425–442.