Instability of Portfolio Optimization under Coherent Risk Measures

1. The portfolio optimization problem The constraint on the expected return is omitted for simplicity. The feasibility of the optimization problem • Proposition 1. Let {xt} be a finite sample. If the risk estimator is coherent and there is an apparent arbitrage opportunity on this sample then there is no optimal portfolio. • Sketch of proof. (For details, see Kondor and Varga-Haszonits, 2008) • By contradiction, let us assume that u is an apparent arbitrage and w is optimal. • Let a be a positive number and consider the portfolio v = w + au. Clearly, v´1 = 1. • Using the axioms of coherence, it is easy to show that the estimated risk of v is lower than that of w, and the greater a, the lower the risk. Therefore, w cannot be optimal. □ • In other words, coherent measures of risk are designed in such a way that in the presence of an arbitrage opportunity they encourage investors to take as large a position in the arbitrage portfolio as possible. At a first glance this might even seem to be a desirable feature. • However, a finite sample estimator of a coherent risk measure may become unbounded from below even if there is no real arbitrage opportunity. Proposition 1 makes it clear that a coherent risk measure may lead the investor to take a very large position in a risky portfolio, because it seems to be risk free on the given sample. Feasibility and sample size • Intuitively, the occurrence of apparent arbitrage opportunities is a probabilistic event and it depends on the given sample. Moreover, the longer the time series used for estimation, the smaller the probability that an apparent arbitrage shows up where there is no real arbitrage. We can quantify this probability for Gaussian returns, based on (Kondor et al. 2007). • Proposition 2. The portfolio optimization problem under the Maximal Loss measure is unfeasible on the sample {xt} if, and only if there is a seeming arbitrage opportunity on that sample. • For standard normal returns we can also prove the following corollaries: • The probability that there is no false arbitrage opportunity is • The probability of unfeasibility under a coherent risk estimator is at least 1 – p. • In the limit where N ∞ while N / T is fixed, a sample size of T = 2N or longer is needed to ensure the feasibility of the portfolio optimization problem under a coherent risk estimator (depending on the particular risk measure). Notation Notation Description N Portfolio size (the number of assets available on the market). X The N-dimensional random vector representing asset returns. r(Xi) The (real) risk of asset i (i = 1, 2, …, N). w, v, u, … N-dimensional portfolio vectors. r(w´X) The (real) risk of portfolio w. (w´ denotes the transpose of vector w.) T Sample size (the number of historical observations) xtor xit The realized value of the return vector X or the individual return Xi over the observation period t (i = 1, 2, …, N, t = 1, 2, …, T). Further Remarks {xt} or {xit} A historical sample of the return vector X or the individual return Xi. • The existence of an apparent arbitrage is sufficient for the unfeasibility of the portfolio optimization problem, but not necessary. • This discussion can easily be generalized to risk reward optimization. • The instability outlined in this presentation goes well beyond the scope of coherent risk measures: it can be proved that all the monotonous and/or subadditive measures are affected. • The short selling (or other) restrictions do not solve the instability problem, they only hide it. The estimated risk of asset i based on the sample {xit}. The estimated risk of portfolio w. 1 The N-dimensional vector with each component equal to unity. Coherent Risk Estimators • Estimators of coherent risk measures are not necessarily coherent themselves. Therefore, if our chosen risk measure r is coherent, we may want to require explicitly that its estimator also satisfies the axioms of coherence. Monotonicity for all t = 1,…,Tw´xt ≥ v´xt implies Subadditivity Positive Homogeneity for any a > 0 Translation Invariance for any a • The historical estimator of Expected Shortfall, for instance, is coherent in this sense. Arbitrage • As we shall see, the feasibility of this optimization problem is closely related to the concept of arbitrage. However, based on a given sample {xt} we cannot tell in general whether a portfolio is an arbitrage or not. This leads to the concepts of real and apparent arbitrage: Real arbitrage • u´1 = 0 (u is self-financing) • u´X > 0 (u admits no loss) Apparent arbitrage • u´1 = 0, (u is self-financing) • for each t = 1, 2, …,Tu´xt > 0 (no loss occurs on the given sample) • It is clear that a real arbitrage will also be an apparent arbitrage on all possible samples. On the other hand, it is important to emphasize that for any self-financing portfolio with the potential of losses, there is a chance that on a finite sample it only realizes positive returns, giving the false impression that the probability of shortfall is zero. Instability of Portfolio Optimization under Coherent Risk Measures Imre Kondor123 and István Varga-Haszonits241Collegium Budapest; 2Department of Physics of Complex Systems, Eötvös Loránd University;3Parmenides Foundation, Munich; 4Morgan Stanley Hungary Analytics, Budapest This work has been supported by the National Office for Research and Technology under grant No. KCKHA005 Motivation The instability of portfolio optimization Adequate decision making is not possible without sufficient amount of data. In particular, to optimize large portfolios, we need long return time series. It is an important question how much data is needed to estimate the optimum. The answer depends on the risk measure to be minimized: That is, Maximal Loss and Expected Shortfall (as well as VaR) are especially sensitive to measurement noise. Since these risk measures (including parametric VaR) are coherent (Artzner et al. 1997, 1999), it is important to understand whether this instability is common to all coherent risk measures or just an idiosyncratic feature of ML, ES and VaR. Notation and important concepts Summary Earlier studies showed that in a finite sample estimation framework the feasibility of portfolio optimization under certain coherent risk measures (such as Maximal Loss, Expected Shortfall, and parametric Value at Risk) is not automatically guaranteed. The existence of the solution is the property of the sample used for optimization, therefore it is a random event. We demonstrated that this instability is not a specific feature of the investigated models, but a general property of coherent measures of risk. More specifically, we showed that whenever there seems to be an arbitrage opportunity on a given finite sample, there is no optimal portfolio, because the estimated risk can always be decreased by increasing the size of our position in the apparentarbitrage portfolio. This property of the coherent risk measures can lead investors to the false conclusion that they can achieve a very high return at virtually no risk. This is potentially very dangerous because what seems to be a good business on the short term may turn out to be a very bad one on the longer term. The longer the time series used to estimate the risk, the smaller the probability that a false arbitrage occurs on the sample. For standard normal stock returns and large portfolios, the sample size must be larger than a given threshold to ensure the feasibility of the portfolio optimization problem. This threshold depends on the particular risk measure, but for any coherent measure it is at least the double of the portfolio size. References • Artzner, Ph., Delbaen, F., Eber, J.M., and Heath, D.: Thinking coherently, Risk 10, 68-71 (1997) • Artzner, Ph., Delbaen, F., Eber, J.M., and Heath, D.: Coherent Measures of Risk, Mathematical Finance 9, 203-228 (1999) • Kondor, I. and Pafka, S.: Noisy Covariance Matrices and Portfolio Optimization II, Physica A319C, 487-494 (2003) • Kondor, I., Pafka, S., and Nagy, G.: Noise sensitivity of portfolio selection under various risk measures, Journal of Banking and Finance 31, 1545-1573 (2007) • Kondor, I. and Varga-Haszonits, I.: Instability of portfolio optimization under coherent risk measures, submitted to Quantitative Finance (2008). • Varga-Haszonits, I. and Kondor, I.: The instability of downside risk measures, Journal of Statistical Mechanics 12, P12007 (2008)