Article Detail

A Risk Measure for S-Shaped Assets and Prediction of Investment Performance

Journal 34: Cass-Capco Institute Paper Series on Risk

Qi Tang, Haidar Haidar, Bernard Minsky, Rishi Thapar

In this paper, we study the option valuation of S-shaped as-sets. S-shaped assets are frequently encountered in technological developments, grant funding of research projects, and to a degree, hedge funds and stop-loss controlled trend-following investment vehicles. We conclude that the quantity σ2/μ (variance of return/expected return) replaced the traditional variance risk measure σ2 in the Black-Scholes option valuation formula. We further study the interesting property of σ2/μ in predicting the turning point of performance of a portfolio of hedge funds in the early months of 2008 (and indeed, for earlier historical turning points).

In technology development, it is well known that the S-curve descriptor is widely used in assessing the maturity of technology projects [Nieto et al. (1998)]. Our study is about the valuation of assets whose price changes follow the pattern of an S-shaped asset. That is, the asset prices either remain unchanged or go up at each time interval of observation.

In the equity investment world today, there are mainly two kinds of funds run by professionals:

  • Mutual funds – these funds seek “relative return” [Harper (2003)], and their performance is measured against certain benchmarks, such as S&P 500, MSCI world index, or other indices. So if the index returned -5 percent and the fund returned -2 percent for the period, the fund is regarded as an excellent fund as it beats the market by 3 percent.
  • Hedge funds – these funds seek “geared, absolute return,” and their performance is usually measured against 0 (or in some cases, against the risk-free bank deposit interest rate, which is much harder to achieve). Hence, at every report time window, the fund is expected to have a positive return (or in some cases, better than depositing the capital in the bank) compared to the end of the previous period. Consequently, in the case of hedge funds, negative returns are regarded as highly undesirable.

In real life, when we read the hedge fund monthly returns report, occasional monthly negative returns do appear. In the normal times, the negative returns are very rare, but during financial crises, we notice that the number of negative returns is more pronounced. Hence, we anticipate that if we model hedge funds using the concept of S-shaped assets during normal times, we may have interesting findings when we approach financial crises.

It has been widely noticed that the bond market feels financial crises be-fore they happen. We are out to demonstrate, in this paper, using the three most recent financial crises as examples, that hedge funds feels financial crises before they take place with some clear measurable indicators.

In particular to this topic, we notice that in his report to the U.S. House of Representatives, Lo systematically analyzed the existing methods of studying systematic risks in the financial market, he postulated that the hedge funds should be regarded as a group, and suggested that researchers should use some systematic approaches to look at the risks that the financial markets are facing [Lo (2008)]. In particular, he quoted some of the network approaches in studying systematic risks in the hedge fund world. Taking the idea of systematic risks, we look at the hedge fund world from a different angle, but follow a similar approach: we use differently weighted portfolios of a large set of hedge funds and calculate the risk measure derived by us, to conclude that when approaching a financial crisis the hedge fund world as a whole begins to deviate from the traditional “higher risk implies higher return” behavior. The “stress” in the pre-crisis financial market implies that “higher risks lead to lower returns.” This observation is not obvious when using a particular fund or index as observation reference, it can only be observed through a systematic approach as we explain below in detail.

In this paper, we follow the standard binomial formulation to establish the value of call/put options for S-shaped assets. Under appropriate assumptions, we further derive the approximate Black-Scholes (BS) European option pricing formula. It is interesting to note that in our situation, the involvement of the variance of the asset returns in the BS formulae is replaced by σ2/μ (variance of asset returns)/(expected return). And from this, we further reveal how hedge funds (as a group) anticipate the arrival of financial crises.