A New Risk Measure
Risk vs. Return
This section is for individual investors/mutual fund managers who are interested in long term investments, in particular those interested in having managed portfolios that beat the market returns in the long run (typically > 6 years). For such investments, short term glitches such as crashes do not matter, since they average out in the long run.
In this section we introduce a novel Risk measure called Tsallis-relative entropy (TRE) (also referred to as q-relative entropy). Before discussing that, let us see what is the risk measure that is currently used by investors to assess risk vs. return.
In constructing portfolios that beat market returns, the commonly used risk measure is 'beta' defined in the Capital Asset Pricing Model (CAPM). The CAPM, which is based on efficient market hypothesis (see emh for details) was tested by Black, Jensen and Scholes in 1972. These tests and several other studies, including our own, show that the portfolios managed for long periods (~16-18 years), in general, show the behavior of increasing excess returns with increasing risk. However, for shorter portfolio periods (~6-9 years), the risk return behavior is not always consistent. Sometimes, the portfolio returns actually decrease with increased risk. The following figure shows the average (averaged over 6 years) six-month relative risk-excess return patterns, obtained using beta as the risk measure. The vertical axis shows excess returns in percentage. 2001-2007 includes the .com bubble period; 2007-2013 includes the 2008 crash period.
Note that during the .com bubble period, high risk stocks actually perform more poorly than the low risk stocks. During the 2007-2013 interval, the performance is flat, i.e, on the average, the yield from higher risk stocks is about the same as those from lower risk stocks. During 2013-2019 (coming out of recession), the higher risk portfolio yields higher returns than the lower risk portfolio. This inconsistent behavior leaves to question as to what kind of risk one needs to take to get reasonable returns.
A New Risk Measure
Based on non-extensive statistical mechanics, we have introduced a novel risk measure called 'Tsallis-relative entropy' ( TRE ) (also referred to as q-relative entropy) which gives a consistent behavior of increased returns with increased risk even for periods as short as 6 years. The following figure shows the relative risk - excess return patterns obtained using TRE as the risk measure.
Note that during all intervals (including the .com bubble and 2007 crash periods), this risk measure shows a consistent pattern of increasing return with increasing risk.
TRE Portfolio Performance
In case of managed portfolio returns, what one is interested in is the cumulative returns (dividends and capital gains re-invested) for a certain period (typically more than 6 years). How do portfolios with TRE as the risk measure perform? For details click Portfolio Performance.
A tutorial describing the steps involved in creating and managing a portfolio using TRE as the risk measure is given in Portfolio Management.
In the above discussions, the assumption is that the stock market distributions are symmetric. Tests on past 12 - 15 years of data shhow that this may not be true. We have modified the risk estimation to take into account this asymmetry. For details see the post Asymmetry in Finanancial Markets.
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