Global risk minimization in financial markets

In modern portfolio selection models this goal can be mathematically formulated as finding the global minimum of a risk function (4-7),

, where p i is the positive or negative amount of capital invested in asset i, and s i = sign (p i ) ∈{-1, 1} are binary spin variables; r i is the expected return of asset i such that

; C ik is the covariance between assets i and k; and γ is the margin account requirement which sets the fraction of capital that the investor must deposit in a margin account before buying or selling short assets. With the inverse C -1 of the covariance matrix C the minimum risk distribution p = (p 1 ,…, p n ) becomes

. It is known that finding the absolute risk minimum is computationally equivalent to the ground state problem of the random field Ising model (4,6). This is evident after inserting p into the risk function while neglecting fixed terms that do not depend on spin variables:

, where we introduced an interaction term

, and a random field

. Because covariance between assets can be both positive and negative (see, for example, inset in Fig. 1A), globally minimizing risk means finding a ground state of the random field Ising model with random spin glass interactions, which in general belongs to the class of NP-complete decision problems (8,9) and for which efficient computational algorithms remain unknown. The computational intractability arises from the non-convexity of the cost function R; non-convex problems are much harder to solve computationally than convex optimization problems for which efficient algorithms exist (10). In the context of financial markets, the non-convexity of the spin glass model prevents equilibration into a ground state and is viewed as an inherent source of risk (3,4).

We can now demonstrate that ground states are efficiently accessible in the random field spin glass Ising model provided the margin requirement γ remains below the critical value ( )

, where Δ is the Laplacian matrix defined as

. This upper bound on the margin requirement ensures that there exists a related but convex risk function

, which in matrix form reads ( ) ( )

We note that in the special and simpler case with nonnegative interactions J ik ≥ 0 similar objective functions have been studied in semisupervised machine learning (11). Our prerequisite

, and so nowhere along the path-including its end-the risk in R can be lower than at the beginning. Therefore, in contradiction to the assumption, it is R(s*) ≥ R(s), which proves that s is a global minimum of the Ising model. -, indicating that in larger portfolios efficient risk minimization imposes stricter limitations on margins, which is consistent with the estimated decrease in the critical margin requirement (Fig. 1B, graph EOD1). However, the observed deviation from the expected scaling, with c n α γ and 1.8 α ≈in Fig. 1B, suggests that intermittency effects in price fluctuations may also be important. For portfolio sizes below 100 n ≈ , this downward trend was robust against changes in price sample selection (graph EOD5), and against smoothing of the data (graph EOD1s5).

Based on our result, the efficient minimization of risk may provide a market instrument for curbing volatility if financial products are traded below the critical margin requirement, and if investors and traders rationally optimize their portfolios.

The second condition is both desirable and realistic in today’s highly computerized markets, although it may have been less realistic in the past when computers were not widespread and therefore complex financial decisions were to a lesser degree rational. But the first condition seems to be in conflict with interests of traders and lenders who, in individual contracts, seek to reduce default risk by increasing margins. From a collective market perspective, however, higher margin requirements may have a destabilizing effect through higher transaction costs, which can drive traders from the market place; this may lead to a lower overall liquidity thus making the market more susceptible to volatility (12,13). Hence, in financial markets where minimum margin requirements are regulated a reduction of risk by lowering margins is conceivable. Historically, the possibility of such a regulatory approach is indirectly supported by the fact that both the 1987 and the 1929 financial market crashes were accompanied by an increase in margin requirements which exacerbated liquidity problems and which might have contributed to rapid downfall (14,15).