Abstract
The true market portfolio of the CAPM on the efficient frontier (located at the tangent point of the line extending from the riskless rate) is not observable in the real world. Sharpe received the Nobel Prize in Economics for develo** the CAPM and proposed that portfolio performance can be compared by computing their excess returns divided by the standard deviation of returns (or total risk). This so-called Sharpe ratio is one of the most widely used measures of portfolio efficiency. However, it does not take into account beta risk in the CAPM. In this regard, Gibbons et al. (Econometrica 57:1121–1152, 1989) (GRS) proposed a test of whether any particular portfolio is ex ante mean-variance efficient in the context of the market model version of the CAPM. More specifically, they modified the Hotelling \(T^2\) test to take into account whether a portfolio lies on the efficient frontier. While the Sharpe ratio and GRS test are prominent in academic studies on portfolio performance, practitioners have developed other metrics to evaluate portfolio performance. One such popular measure is drawdowns, which is tantamount to value at risk (VaR). If a portfolio exhibits high periodic drawdowns, it is necessary to carefully evaluate the potential gains in the long run that may serve to counterbalance this risk.
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Notes
- 1.
A simple example illustrates the bias. If you start with $200 and lose 20% in the first period, you have \((1 - 0.20)\times \$200 = \$160\) to invest in the second period. If the gain in this second period is 30%, the end value becomes \((1 + 0.30)\times \$160 = \$208\), such that the 2-period return of your investment is \((208-200) / 200 = 4\%\), i.e., \((1 - 0.20) \times (1 + 0.30) - 1\). The additive return of 10% (\(= -0.20 + 0.30\)) in the arithmetic mean formula is clearly too high and therefore yields an upward biased mean return equal to 5% (\(=10\%/2\)).
- 2.
Thus, in the simple example in footnote 1, the correct mean return is \([(1 - 0.2)\times (1 + 0.3)]^{1/2} - 1 = 1.98\%\) (rather than the arithmetic mean of \((-0.2 + 0.3) / 2 = 5\%\)). The 2-period return is then \((1 + 0.0198)^2 - 1 = 4\%\), which matches the correct 2-period return computed above.
- 3.
This value is the Neper or Napierian number after the Scottish mathematician John Napier. The term e is also known as Euler’s number after the Swiss mathematician Leonard Euler.
- 4.
- 5.
Fischer and Wermers (2013) give an excellent discussion of IRR.
- 6.
- 7.
See Chapter 21 in Ferson (2019).
- 8.
A popular downside risk measure is the semivariance:
$$\begin{aligned} \textit{sv} = \frac{1}{T}\sum _{t = 1}^T\min (0, R_t - \bar{R})^2,\qquad \qquad \qquad {(6.12)} \end{aligned}$$where \(\bar{R}\) is the sample mean of the returns.
- 9.
See Goetzmann et al. (2007) for an extreme example.
- 10.
- 11.
The alpha in the ZCAPM can be estimated by computing the average of the residual error terms in the time-series EM regression model.
- 12.
- 13.
For technical details, see Mina and **ao (2001).
- 14.
This methodology is described in Zumbach (2007).
- 15.
Data source: CRSP and https://finance.yahoo.com.
- 16.
For details on generic weighting schemes, see Appendix B in Zumbach (2007).
- 17.
The 95% confidence interval for the 5% VaR here is \((0.046\%, 0.054\%)\) which implies that, for the normal distribution RM1994 1-day 5% VaRs, short positions in the S&P 500 index and CRSP market index as well as long position in portfolio G significantly overstate the realized VaR threshold. That is, for example, a long position in the G portfolio with realized exceedance of \(3.58\%\) indicates that, for a $100 million investment, the realized 1-day 5% VaR would have been $3.58 million rather than $5 million stated by the VaR. However, these findings are not losses incurred, but rather the levels of the realized VaR values for different positions (long/short) of the portfolios.
- 18.
Sometimes drawdown is defined in terms of log returns:
$$r_t = - \max _{s \le t}\,\log (P_s / P_t) = \min _{ s\le t}\, \log (P_t / P_s),$$where \(\log\) is the natural logarithm. Also, drawdown is defined in monetary units at times.
- 19.
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Kolari, J.W., Liu, W., Pynnönen, S. (2023). Portfolio Performance Measures. In: Professional Investment Portfolio Management. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-031-48169-7_6
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