Preface to Section III: Risk Management Practices
AExpected return
I. A.1.7.4 Downside RAPMs
Psychologically as well as practically, economic agents are often more concerned by the downside risk, or risk of underperformance, than the upside risk, or risk of overperformance. That is true for fund managers who like to compare their performance to a benchmark or a peer group. It is also true for managers and regulators who want to limit the risk of insolvency of firms to very low levels and thus help maintain credit ratings and avoid financial debacles. For instance, investment-grade firms (rated BBB and above) exhibit historical default probabilities of the order of 0.5% over a year, and AA-rated firms probabilities of the order of 0.05% only.
With a concave utility function, the negative utility impact of a loss is greater in absolute terms than the positive utility impact of the opposite gain. And utility theory makes use of the full probability distribution of outcomes. However, some simplified applications of the maximum EU principle, such as the mean–variance criterion leading to the use of the Sharpe ratio, rely on a single, symmetric characteristic of a probability distribution, in this case the variance. Therefore they may not be safe to apply to distributions that are obviously asymmetric. We illustrated this problem in the previous section and suggested an adjustment to the Sharpe ratio to account for skewness and excess kurtosis.
I.A.1.7.4.1 RAROC
Several other approaches have been used to simplify calculations and to circumvent a detailed statement of risk attitude whilst stressing downside risks. Thus, one may impose a downside risk constraint, for example that potential losses should not exceed a given threshold with more than a certain probability. That is what banking regulators have chosen to do.25 Banks are free to take on risks as long as their eligible capital – as defined by the regulator – is sufficient to ensure that their probability of default over a year remains less than a small percentage. The difference between the current value of a business and a specified loss quantile ơ at a specified time horizon T is called the value-at-risk and is more fully described in Part III of the Handbook.
As the risk constraint is translated into a capital requirement, it has a cost that must be taken into account. This is what financial institutions do when they develop their own RAPMs. They look at maximising the ratio of net returns over the corresponding capital requirements. Bankers Trust were the first to publicise the use of a home-grown RAPM when they introduced the risk-adjusted return on capital (RAROC) in the late 1970s. There is now a whole zoo of such measures with minor definitional variations; Matten (2000, pp. 146–149) gives a guided tour. In this Handbook there is an entire chapter devoted to capital allocation and RAPMs. This chapter is so fundamental to risk management that we have placed it as preliminary reading to all chapters in Part III – it is thus numbered Chapter III.0! The generic form of a RAROC is:
RAROC = (Expected return net of costs and expected losses)/(Economic capital), (I.A.1.10)
where the numerator may include an adjustment for risks. For example, the cost of funding may be related to risk. The economic capital is an estimate (internal to the firm) of the amount of capital necessary to cover possible losses up to a certain confidence level chosen by the firm, such as 99.9%, per year. More details are given in Section III.0.2.
A downside RAPM this is closely related to RAROC is the return over VaR (RoVaR), defined as RoVaR = (Expected return net of costs and expected losses)/(VaR), (I.A.1.11)
The difference between (I.A.1.10) and (I.A.1.11) is that whilst economic capital is normally assessed at a very high significance level (such as 0.03% for a firm that targets a AA rating) the VaR in (I.A.1.11) can be assessed at any percentile (in the example below we use 5%).
Returns and risks are usually estimated at a one-year time horizon on the assumption that the RAPM will help rank continuous activities rather than projects with a finite life. Often, financing costs are deducted from the numerator but the risk-free rate of return on the imputed economic
25 See the Basel capital requirements, in particular the Basel II proposals (June 2004).
capital is added back. If the resulting RAPM exceeds the cost of equity capital of the firm, the activity is deemed to add value to the firm. In general, firms will seek to develop activities with high RAPM and curtail activities with low RAPM, especially those where the ratio is below the equity cost of the firm.
But basing business decisions on the RAPM rule may not produce the desired effect of increasing the value of the firm. Turnbull (2000) shows that RAROC measures, whilst taking into account a target probability of default, do not adjust for systematic risk and do not account properly for correlations between activities within a firm. He proposes instead an adjusted net present value rule that recognises marginal economic capital requirements and uses discount rates relevant for the risk of the underlying activity.
It remains the case that RAPMs may be useful in simpler circumstances such as the static portfolio selection problem when particular attention must be given to downside risks (but without setting a probability constraint on low returns).
I.A.1.7.4.2 Sortino Ratio, Omega Index and other Kappa indices
Back in 1959, Markowitz suggested the use of semi-variance rather than variance as a more relevant risk metric for investors worried about the downside. The semi-variance of a risky asset is defined as the expected value of the squared deviation of returns below the expected return. But Markowitz was discouraged by the computational difficulties involved in determining efficient portfolios in a return versus semi-variance framework. This is one of the difficulties with
‘downside’ risk metrics. The other is to ensure that the risk metric is based on a firm theoretical foundation (by establishing some compatibility with an acceptable utility function). As a general rule, downside risk metrics based on quantiles (such as VaR) or extremes are not compatible with utility theory and therefore are likely to lead to dubious results, whereas metrics based on lower partial moments are compatible with some utility functions.
The lower partial moment of order n (n > 0) of a random return R, given a threshold return ƴis defined as:
LPMn(ƴ) = E[max(ƴ – R, 0)n].
For example, the semi-variance is equal to LPM2(µ) where µ = E[R]. A lower partial moment can be used instead of a standard deviation in the denominator of the Sharpe ratio. This yields a family of kappa indices as follows:26
Kn(ƴ) = (µ – ƴ)/ (LPMn(ƴ))1/n. (I.A.1.12)
The second-order kappa index, introduced earlier by Sortino and van der Meer (1991), is called the Sortino ratio. The first-order kappa index is closely related to the omega statistic introduced by Keating and Shadwick (2002). Omega is the ratio of the expected return above thresholdƴ over the expected return below thresholdƴ, that is:
ƙ(ƴ) = E[ max((R – ƴ), 0)]/ E[max(ƴ – R, 0)]. (I.A.1.13)
Kappa indices have the following properties:
x
x x x
They decrease monotonically with the choice of thresholdƴ (they are all equal to zero whenƴ µ).
Their sensitivity toƴ increases with the order, n.
The kappa indices for portfolios with equal mean and variance are also sensitive to skewness (positive) and kurtosis (negative).
The rankings of different assets under a kappa index depend on the choice of both n and ƴ; large values of n and small values of ƴemphasise the effects of skewness and kurtosis.
The definition of the kappa indices could also be extended to include a different return threshold for the calculation of excess return (numerator) and the calculation of lower partial moment (denominator). Currently there is much debate among fund managers and academics about the appropriate choice of parameters for kappa index with many recommending that these indices be calculated over a range of values for the parameters.27
To illustrate the possible merits of downside RAPMs we return to the four gambles in Figure I.A.1.5. All four gambles have the same expected value of 5 and the same standard deviation of 15 and therefore the same Sharpe ratio of 1/3. Do downside RAPMs reveal preferences closer to what many observers feel intuitively, i.e. that A is the best investment followed by B, C and D in that order?
Table I.A.1.2 shows that the answer very much depends on which downside RAPM is used. But this result does not indicate which RAPM is most appropriate in which circumstance. For example, it is not clear which RAPM is the best metric for comparing the performance of hedge funds.
26 So-called by their inventors, Paul Kaplan and J. Knowles (2003).
27 We note that if the threshold for excess returns is taken as the risk-free rate, the kappa index will produce preference rankings congruent with the use of a piecewise power near utility function of the form:-li
u(x) = x for x ƴ,
u(x) = x – c (x – ƴ)n for x ƴ.
Table I.A.1.2: Comparison of four investments according to four downside RAPMs
Omega is defined in (I.A.1.13) with a threshold of 0;
I.A.1.8 Summary
We have surveyed the main risk preference criteria for making rational decisions under uncertainty. We now summarise the choice of criteria to be applied in specific circumstances and describe their likely effects compared to less formal approaches.
It may be helpful to visualise decisions under uncertainty in three dimensions:
i. Size of risks: Are they large or small compared to net worth?
ii. Portfolio effects: Are there dependencies among the risks we are taking, and in particular dependencies with existing risks (status quo) and external opportunities (securities markets), or can the new risks be considered independently of background risks?
iii. Absolute levels or relative ranking: Is the problem to identify a class of efficient portfolios or do we have to decide on quantities, such as the size of an investment or level of insurance?
For the more complex cases involving large risks correlated with existing uncertainties (internal or external) and where decisions are about absolute levels of commitments (capital or other resources), there is no alternative to a full application of utility theory. A firm should have a corporate risk policy expressing risk preference in terms of a utility curve on its economic value (market value of equity or some more easily measurable proxy). This utility curve may be reduced to one or two parameters such as a risk-tolerance coefficient and the sensitivity of this coefficient to changes in value. The main effect of using a formal expression for risk attitude will be to improve the quality of communications and analyses concerning choices under uncertainty.
It is likely that more projects deemed too risky at a departmental level will be recognised as
A sensible lower moment thresholdƴ is therefore the return level beyond which the investor is risk neutral. Below x = ƴ the coefficient of risk tolerance is proportional to ƴ– x; a strange, endogenous utility function, but better than no utility representation at all.
valuable and passed on to higher decisional levels. It is also likely that the firm will benefit from greater consistency in risk taking across divisions and over time.
Where decisions are not so critical or complex, analytical shortcuts are possible. But one should remember that where decisions bear on absolute levels of risk, a value criterion must be used.
The preference order it induces should still correspond to the preference order induced by a
‘reasonable’ utility function. For instance, the certain equivalent of a risky opportunity, measured by the expected value of its outcomes less a fraction of the incremental variance it creates, is a five-parameter measure that encapsulates the five essential ingredients: expected value and standard deviation of the new opportunity, standard deviation of existing risks, correlation between the new and the old risk, and trade-off between risk and return. It is the certain equivalent corresponding to an exponential utility function and jointly normally distributed risks;
it is still a good approximation of a certain equivalent when the risk is not too large nor too skewed.
When the risks to be compared are not normally distributed, a value criterion based on symmetrical metrics such as the mean and variance may be misleading. Corrections can be made to include higher moments of the distributions or to focus on the more critical outcomes, for example the downside risks. But with powerful computers, one may doubt the wisdom of applying analytical approximations if it is at the expense of implying odd utility functions.
Where the problem at hand is not to decide on an allocation of resources but rather to compare performance, another simplification is possible: only the mildest assumptions about risk attitude, such as stochastic dominance, may be sufficient to compare performance. This is the classical portfolio selection problem where the issue is to identify an ideal portfolio or a family of so-called efficient portfolios. It is not a decision problem, only a first step towards making an investment decision. Under restrictive conditions, a ratio of excess return over the risk-free rate divided by the chosen risk metric can be justified as a relative performance criterion.
The Sharpe ratio is the grandfather of performance ratios and is still the most widely used. The risk metric in the denominator is the standard deviation of return. It is good for comparing the performance of assets with similarly distributed (two-parameter) returns, for instance some traded securities, but it does not apply to the comparison of activities that are not scalable, do not have similarly distributed returns, or have finite life cycles. The latter cases are not amenable to
the two-step approach: (i) identification of efficient portfolios through a relative performance measure and (ii) optimal investment in an efficient portfolio.28
Alas, risk-adjusted performance measures– an elaboration on the concept of Sharpe ratio usually embedding a constraint on solvency risk and therefore relying on a lower quantile risk metric – have been applied to problems that only look superficially like portfolio selection of tradable securities. One should check whether applications of RAPMs are congruent with the widely claimed goal of maximising shareholder value; often they are not, especially if they do not account for systematic risks.
Finally, it may be worth noting that day-to-day risk management is mainly devoted to estimating expected losses rather than risks (see Chapter III.A.1). Many problems can be solved by an analysis of costs and benefits without resorting to a risk analysis. It should be possible to assume risk neutrality for all decisions where the range of consequences does not exceed a very small fraction of the net worth of a firm (say, 0.01% of capital for a bank, which would be €1 million for a €10 billion bank). The bar has been set lower than this in some financial institutions. The consequence is that too much management attention is given to minor risks, to the detriment of more significant risks. The management challenge is to identify the large risks and analyse them soundly.
28 They may also necessitate a separate statement of time preference, as opposed to a simple discounting of cash flows at a certain cost of financing or target rate. The use of a higher discount rate to take into account the riskiness of future returns can be defended only in the case where returns are i.i.d. and a first-moment–second-moment criterion applies, because in those cases first and second moments are homogeneous in time. Otherwise it is misleading.
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Appendix I.A.1.A: Terminology
Everyday words carry many connotations, some of which may be unhelpful in the context of a specific theory. On the other hand, some nuances between similar words may be irrelevant for the purpose of a formal theory. But everyday words we must use lest we fall into technical jargon. When discussing decisions under uncertainty, no matter how serious these may be, we use the same formal language as when discussing games of chance. We do not mean to imply that life is a game in any moral sense or that there is no difference between investment and speculation in the business world. We are not addressing these issues and we do not want to disconcert or offend the reader with the use of apparently frivolous words. Hence we wish to clarify our use of the following terms.
Utility: The utility theory we discuss here has only historical connections with the now dated concept of economic utility and the rule of diminishing marginal utility, which applies to deterministic situations. It is also only remotely connected with the socio-philosophical school of utilitarianism. But utility is the word that has been used historically and was deemed good enough to be retained by von Neumann, Morgenstern, Savage and others. Therefore we keep it. We use many other words with a specific intent:
uncertainty, risk, risk attitude, decisions, preferences, rationality, outcomes, gambles, lotteries, probabilities, etc. Many are defined in the text, some key terms are reviewed and compared here.
Outcome: A possible consequence of an action. An action may, of course, lead to an indefinite chain of outcomes and subsequent actions that would be infeasible to explore comprehensively. We focus our attention on outcomes that are material for the purpose of deciding between alternative courses of action.
Alternative words: event, consequence, result, reward, prize, profit, loss, penalty.
Gamble: A finite collection of exclusive and exhaustive outcomes that may result from a decision. Each outcome in a gamble has a certain probability of occurrence in the mind of a decision maker, and these
Gamble: A finite collection of exclusive and exhaustive outcomes that may result from a decision. Each outcome in a gamble has a certain probability of occurrence in the mind of a decision maker, and these