I came across this really interesting paper on how to define and measure risk. You can find it here, but I also summarized it below, and wrote down some excerpts which I thought stood out. I bolded some text for emphasis and stuff in [ ] brackets is some notes I took while reading.

“Holton (2004) proposes that a definition of risk has to take into account two essential components of observed phenomena:exposure and uncertainty. Moreover, all the admissible tools available to an investor to cope with risk can model only the risk that is perceived.”

“Attempts to quantify risk have led to the notion of a risk measure. A risk measure is a functional thatassigns a numerical value to a random variable which is interpreted as a loss. Since risk is subjective because it is related to an investor’s perception of exposure and uncertainty, risk measures are strongly related to utility functions.”

“In portfolio theory, a risk measure has always been valuedprincipally because of its capacity of ordering investor preferences.”

“…minimizing the probability of being below a benchmark is equivalent to maximizing an expected state dependent utility function (see Castagnoli and LiCalzi (1996, 1999)).”

[Multiple objectives and multiple benchmarks make risk a multi-dminesional phenomenon]

“… risk is an asymmetric concept related to downside outcomes, and any realistic way of measuring risk should consider upside and downside potential outcomes differently. Furthermore, a measure of uncertainty is not necessarily adequate in measuring risk.Thestandard deviation considers both positive and negative deviations from the mean asa potential risk.Thus, in this case,out-performance relative to the mean is penalized just as much as under-performance.”

“Expected tail loss (ETL), an example of a coherent risk measure, is also know as theConditional Variance at Risk (CVaR), if we assume a continuous security returns distribution. ETL can be interpreted as the average loss beyond VaR.”

[Alternatively, if a returns distribution is estimated with 95% confidence (including the right/positive tail), the CVaR is the mean of the remaining 5% in the left/negative tail.]

“Clearly, if the degree of uncertainty changes over time, the risk too has to changed over time. In this case, the investment return process is not stationary; that is, we cannot assume that returns maintain their distribution unvaried in the course of time.”

“Under the assumption of stationary and independent realizations, the oldest observations have the same influence on our decisions as the most recent ones. Is this assumption realistic? Recent studies on investment return processes have shown that historical realizations are not independent and exhibit autoregressive behavior. Consequently, we observe theclustering of volatility effect; that is, each observation influences subsequent ones.”

[**Cointegeration is when two price series display a consistent spread across time.** It is different from correlation as, in correlation, the direction of movements may be the same, but the magnitude may vary. With cointegeration present, if the magnitude of price movements changes the spread, the spread will show mean-reversion. Price series may be both stochastic and show cointegeration. A pair of price series showing cointegration is called a stationary pair]

“The ex-ante analysis clearly indicates that the minimum variance portfolios (portfolios 1 and 3) present a lower dispersion (standard deviation) and a higher risk of big losses (VaR and ETL) than portfolios that maximize the R-ratio given by (12) (respectively portfolios 2 and 4). Thus the ex-ante analysis suggests thatthe more conservative minimum variance portfolios (portfolios 1 and 3) not always take into account the possibility of big losses.”

“In particular, on 5/31/2004 the final wealth of the three different strategies based on R-, Sharpe and STARR ratios is respectively 1.76, 1.07, and 0.91. Therefore, as we expected, we obtain that the strategy based on the maximization of the STARR ratio provides the most conservative behavior while thestrategy based on the R-ratio permits to increase the final wealth much more than the others.”

“…most investors perceive a low probability of a large loss to be far more risky than a high probability of a small loss.Therefore, investors perceive risk to be non-linear.”