Abstract
Strategic asset allocation leads to the identification of a number of efficient portfolios in the risk-return space. This common feature brings the topic of portfolio selection to the fore, since it is obvious that when presented with a variety of long-term investment options which cannot be ranked (being all optimal), we face a problem of choice and selection. This chapter illustrates the methods and instruments for portfolio selection available to institutional investors for a more aware identification of the “optimal portfolio”, taking into consideration management objectives and constraints. These tools and/or methodologies have been formulated in contributions mainly related to the areas of risk budgeting and financial planning that appeared after Markowitz’s Modern Portfolio Theory. A detailed analysis will clarify that the tools and/or methodologies being considered aim: to enhance our knowledge of the nature and intensity of the diversification offered by the portfolios in the opportunity set; to gain a more profound and/or alternative assessment of the risk of optimal portfolios; to verify, at least in probabilistic terms, the extent to which optimal portfolios satisfy the need to “protect” a financial result or defend against possible losses; to analyse possible future scenarios of cumulative wealth associated with investment in efficient portfolios.
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Notes
- 1.
Choueifaty and Coignard (2008).
- 2.
- 3.
We have considered risk decomposition from the point of view of portfolio volatility, because—when using the mean-variance approach—this is the risk measure which is associated by definition with each efficient portfolio. It can also, however, refer to other risk measures. Yet note that one condition must be respected: the metric chosen must be a first order homogenous function (e.g. standard deviation) or, equivalently, a linear homogenous function. Only by meeting this condition Euler’s Theorem can be applied, and a similar function translated into the sum of its arguments (in the present case, the portfolio weights) multiplied by their first order partial derivative.
- 4.
With long-only portfolios, a negative total risk contribution is possible only if marginal risk is negative. Taking the formula in (5.3), it is easy to deduce that this presupposes that the sum in the numerator is negative, meaning, in turn, that at least some of the values of σ ij are negative.
- 5.
- 6.
For greater clarity, note that although the VaR is described as a potential loss, it can also be positive. Indeed, it is possible that the cut-off-point, which isolates to its left a probability of 1 − α of a certain distribution, indicates a profit. Consequently, the value below which the portfolio should most probably (probability α) not fall at the end of the time-frame incorporates in any case a positive result.
- 7.
Note that “quantile” is understood as a value of a random variable, such that a given percentage of its cumulative distribution function (cdf) lies below this value.
- 8.
This can be easily explained: all other conditions being equal, a higher confidence level means isolation of a smaller tail of the probability distribution. It is therefore easy to understand that in cases where the threshold that isolates a probability of 1 − α to its left is positive, then as α increases, the estimate of the VaR becomes progressively less positive.
- 9.
- 10.
Indeed, of the worst cases, the VaR represents the least severe loss.
- 11.
Though of no particular interest here, it should be noted that from a theoretical point of view the VaR has also been criticized for reasons other than those mentioned. There are claims in the literature that the VaR cannot be considered a coherent measure of risk as suggested by Artzner et al. (1999), where further details can be found. Here, it is sufficient to note that the reason is the possible violation of the subadditivity requirement.
- 12.
Artzner et al. (1999).
- 13.
Grossman and Zhou (1993).
- 14.
For example, starting with a series of market values for a portfolio, it is possible to construct a base 100 index series. Obviously, this involves eliminating those phenomena that represent external cash flows; see Chap. 7.
- 15.
Here, we use the most common definition of drawdown. There is an alternative definition according to which calculation is based on the series of accumulated returns, rather than the values/prices series. In other words, the alternative definition requires that at each time-point t, the accumulated performance of the portfolio from the beginning up to that time-point is compared with the maximum potential performance of the portfolio accumulated at any time-point prior to t.
- 16.
Leibowitz et al. (1996).
- 17.
1.28155 is the volatility multiplier that isolates a probability density of 10 % in the tail of a Gaussian distribution.
- 18.
As we have seen, this involves an expected return that a portfolio should provide, together with a given standard deviation.
- 19.
Ibbotson and Sinquefield (1976).
- 20.
- 21.
Note that a random variable with a log-normal distribution can take values between 0 and + ∞.
- 22.
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Braga, M.D. (2016). Methods and Tools for Portfolio Selection. In: Basile, I., Ferrari, P. (eds) Asset Management and Institutional Investors. Springer, Cham. https://doi.org/10.1007/978-3-319-32796-9_5
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