Abstract
Portfolio analytics have three key dimensions: exposure, performance, and risk. Substantial time and effort have been allocated to the treatment of exposure and performance analysis of a fixed-income portfolio. It is now time to consider the third and final dimension: risk.
Never was anything great achieved without danger. Nicolò Machiavelli
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Notes
- 1.
The exception is the 80 % probability of losing in cards to my father-in-law; this figure is derived from hard-earned experience!
- 2.
The computation for a single game is simply: \(0.5 \cdot \$1 + 0.5 \cdot -\$1 = \$0\).
- 3.
A sequence of coin tosses—each in itself a Bernoulli trial—is described by the binomial distribution. See Casella and Berger [4] for more on the binomial distribution. We suppress the notion of statistical distribution in this section, but will return to it explicitly in the following discussion.
- 4.
The probability of this outcome is very small at \(\frac{1} {1024}\) or 0.1 %.
- 5.
Extreme-value theory is a complete area of study. See Embrechts et al. [8] for more information.
- 6.
This is a familiar feature of the binomial distribution.
- 7.
One can go much further in selecting a risk measure. Artzner et al. [1] posited a set of mathematical properties that each desirable risk measure should possess. Measures possessing all of these properties are termed coherent.
- 8.
With an investment portfolio, the returns also have a time dimension. Empirically, the returns from one time period to the next are generally uncorrelated. Consequently, we ignore this form of dependence, although it may, albeit weakly, actually exist.
- 9.
None of the risk measures in Table 10.9, with the exception of the mean, is linear—that is, you cannot compute them as a simple weighted average of the underlying components. Try it as an exercise!
- 10.
- 11.
This is the central thesis of Taleb [14], which offers a scathing review of risk-management practices.
- 12.
This is termed volatility clustering. Volatility behaviour is a broad area of financial research. See, for example, Engle [9].
- 13.
- 14.
A moment measures the shape of a set of points—they are basically summary measures for a statistical distribution. The first moment (i.e., the mean) basically describes the shape of the set of average observations, while the second moment (very roughly) describes the average squared observations: this is often termed standard deviation or volatility. Third and fourth moments—also called skewness and kurtosis, respectively—examine cubic and quartic transformations of the observations.
- 15.
This is shown by the points in north–west and south–east quadrants of Fig. 10.10.
- 16.
Although we will make extensive use of it, correlation is not the only approach to measuring dependence. See Embrechts et al. [7] for an excellent discussion of the pros and cons of using correlation.
- 17.
There are actually more points in the two southern quadrants because of the negative skew in the simulated distribution. Without skewness, the points would be more equally spread throughout the four quadrants.
- 18.
Recall that ρ is the Greek letter that is typically used by statisticians to designate the correlation coefficient.
- 19.
Since we are running simulations, there is still some noise in these computations. If we performed a very large number of simulations, however, we would expect risk to be an increasing function of correlation.
- 20.
Moreover, this is a very extreme example. It is probably rather difficult to actually construct a robust portfolio with a return that, under all market conditions, differs from the strategic benchmark by a constant amount.
- 21.
Much more will be said in the subsequent chapters on the construction, estimation, and analysis of this covariance matrix.
- 22.
See Hwang and Satchell [10] for a more detailed discussion of this point.
- 23.
This notion is called backtesting and it is discussed in Chap. 12
- 24.
- 25.
This follows from Euler’s theorem for homogeneous functions and is discussed in detail in the next chapter.
- 26.
The ex-post tracking error provides useful information for assessing the historical closeness of the portfolio to its strategic benchmark. As a yardstick to measure permissible deviations, however, it is rather unhelpful. This is because, since it looks only backwards, no remedial action can be taken.
- 27.
Active exposures, as we have seen, can be helpful in this regard.
- 28.
An interesting introduction to the foundations of risk analysis is found in Berstein [2].
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Bolder, D.J. (2015). Introducing Risk. In: Fixed-Income Portfolio Analytics. Springer, Cham. https://doi.org/10.1007/978-3-319-12667-8_10
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