# Downside risk

Downside risk is the financial risk associated with losses. That is, it is the risk of the actual return being below the expected return, or the uncertainty about the magnitude of that difference.[1][2]

Risk measures typically quantify the downside risk, whereas the standard deviation (an example of a deviation risk measure) measures both the upside and downside risk. Specifically, downside risk can be measured either with downside beta or by measuring lower semi-deviation.[3]:3 The statistic below-target semi-deviation or simply target semi-deviation (TSV) has become the industry standard.[4]

## History

Downside risk was first modeled by Roy (1952), who assumed that an investor’s goal was to minimize his/her risk. This mean-semivariance, or downside risk, model is also known as “safety-first” technique, and only looks at the lower standard deviations of expected returns which are the potential losses.[3]:6 This is about the same time Harry Markowitz was developing mean-variance theory. Even Markowitz, himself, stated that "semi-variance is the more plausible measure of risk" than his mean-variance theory.[5] Later in 1970, several focus groups were performed where executives from eight industries were asked about their definition of risk resulting in semi-variance being a better indicator than ordinary variance.[6] Then, through a theoretical analysis of capital market values, Hogan and Warren[7] demonstrated that 'the fundamental structure of the "capital-asset pricing model is retained when standard semideviation is substituted for standard deviation to measure portfolio risk."' This shows that the CAPM can be modified by incorporating downside beta, which measures downside risk, in place of regular beta to correctly reflect what people perceive as risk.[8] Since the early 1980s, when Dr. Frank Sortino developed formal definition of downside risk as a better measure of investment risk than standard deviation, downside risk has become the industry standard for risk management.

## Downside risk vs. capital asset pricing model

It is important to distinguish between downside and upside risk because security distributions are non-normal and non-symmetrical.[9][10][11] This is in contrast to what the capital asset pricing model (CAPM) assumes: that security distributions are symmetrical, and thus that downside and upside betas for an asset are the same. Since investment returns tend to have a non-normal distribution, however, there in fact tend to be different probabilities for losses than for gains. The probability of losses is reflected in the downside risk of an investment, or the lower portion of the distribution of returns.[8] The CAPM, however, includes both halves of a distribution in its calculation of risk. Because of this it has been argued that it is crucial to not simply rely upon the CAPM, but rather to distinguish between the downside risk, which is the risk concerning the extent of losses, and upside risk, or risk concerning the extent of gains. Studies indicate that "around two-thirds of the time standard beta would underestimate the downside risk."[3]:11

## Examples

${\displaystyle SD(X)=\left(\mathbb {E} [(X-\mathbb {E} [X])^{2}1_{\{X\leq \mathbb {E} [X]\}}]\right)^{\frac {1}{2}}}$
where ${\displaystyle 1_{\{X\leq \mathbb {E} [X]\}}}$ is an indicator function, i.e. ${\displaystyle 1_{\{X\leq \mathbb {E} [X]\}}={\begin{cases}1&{\text{if }}X\leq \mathbb {E} [X]\\0&{\text{else}}\end{cases}}}$
• Below target semi-deviation for target ${\displaystyle t}$ defined by
${\displaystyle TSV(X,t)=\left(\mathbb {E} [(X-t)^{2}1_{\{X\leq t\}}]\right)^{\frac {1}{2}}}$.

## References

1. McNeil, Alexander J.; Frey, Rüdiger; Embrechts, Paul (2005). Quantitative risk management: concepts, techniques and tools. Princeton University Press. pp. 2–3. ISBN 978-0-691-12255-7.
2. Horcher, Karen A. (2005). Essentials of financial risk management. John Wiley and Sons. pp. 1–3. ISBN 978-0-471-70616-8.
3. James Chong; Yanbo Jin; Michael Phillips (April 29, 2013). "The Entrepreneur's Cost of Capital: Incorporating Downside Risk in the Buildup Method" (PDF). Retrieved 25 June 2013.
4. Nawrocki, David (Fall 1999). "A Brief History of Downside Risk Measures" (PDF). The Journal of Investing. 8 (3): 9–25. CiteSeerX 10.1.1.22.262. doi:10.3905/joi.1999.319365. Retrieved 27 February 2015.
5. Markowitz, H. (1991). Portfolio selection: Efficient diversification of investment (2e). Malden, MA: Blackwell Publishers Inc.
6. Mao, J.C.T. (1970). "Survey of capital budgeting: Theory and practice". Journal of Finance. 25 (2): 349–360. doi:10.1111/j.1540-6261.1970.tb00513.x.
7. Hogan, W.W.; Warren, J.M. (1974). "Toward the development of an equilibrium capital-market model based on semivariance". Journal of Financial and Quantitative Analysis. 9 (1): 1–11. doi:10.2307/2329964.
8. Chong, James; Phillips, Michael (2012). "Measuring risk for cost of capital: The downside beta approach" (PDF). Journal of Corporate Treasury Management. 4 (4): 346–347. Retrieved 1 July 2013.
9. Mandelbrot, B (1963). "The variation of certain speculative prices". Journal of Business. 36 (4): 394–419. doi:10.1086/294632.
10. Bekaert, G.; Erb, C.; Harvey, C.; Viskanta, T. (1998). "Distributional characteristics of emerging market returns, and asset allocation" (PDF). Journal of Portfolio Management. 24: 102–16. doi:10.3905/jpm.24.2.102. Retrieved 27 February 2015.
11. Estrada, J. (2001). "Empirical distributions of stock returns: European securities markets, 1990-95". European Journal of Finance. 7: 1–21. CiteSeerX 10.1.1.200.4265. doi:10.1080/13518470121786.