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The standard representation of the diversification effect applies to portfolio and asset returns. The principles date from Markowitz principles[1]. The portfolio return varies with the equity index as risk factor. The specific risk of each asset is independent of this common factor.

Extending the number of assets creates diversification because the specific risks of individual asset returns offset each other. The undiversifiable risk is the general risk common to all assets. This well-known result applies notably to stock returns and to asset returns in the structural model of default, and is the basis for modeling diversification effect for credit risk.

When looking at values rather than returns, the same basic mechanisms of diversification apply. Nevertheless, unlike return volatility that tends to decline down to a floor corresponding from general risk, the volatility of the value increases with each new asset. The portfolio value volatility increases with the number of assets and portfolio average return correlation.

A simple way of illustrating the mechanism of diversification effect is to refer to a simple uniform portfolio, and see what happens when the number of assets increases. A uniform portfolio is such that all assets have the same weights, with same return volatility and same pair correlation between returns. All values V and all weights w are now equal: Vj0 — VQ. The portfolio value is Vn =NVQ. The value of each asset is €1. The common weight is w — lln. All variances, covariances and correlations of returns are equal as well. The volatility of each asset return is a(R) — 30%. The correlation between pairs of assets is p 30%.

The notations are now simpler:

The portfolio return volatility sums up n variance terms weighted by the constant weights lln2 and n(n - 1) covariance terms equally weighted by lln2. The variance of the portfolio return is:

The portfolio return variance and volatility now simplify to:

Sample calculations are shown below for the portfolio return volatility and its changes when n increases (Table 30.2).

With the above inputs, we observe the decrease of portfolio return risk due to diversification when the number of assets increases. The above relations also show that with small values of n, the portfolio return volatility increases with n and with the uniform volatility. It is also proportional to the uniform correlation. When the uniform correlation increases, the portfolio

TABLE 30.2 Variance and volatility of the uniform portfolio

Variance and volatility of the uniform portfolio

return volatility also increases, as well as the minimum value of portfolio return volatility when n increases (Figure 30.3).

Note that this example, using as target variable the portfolio return, is very classical. But the same basic formulas apply in other contexts, when the variable of interest changes. The volatility of the P & L of a trading portfolio P & L is a critical parameter for market VaR determination. For credit VaR, the target variable is the portfolio loss volatility due to credit risk and its variation when adding a facility or expanding the exposure to any one obligor.

The portfolio return volatility decreases quickly from the initial 30% to a minimum value equal to systematic risk, which, in this example, is around 10%. The threshold is approximately reached around 30 assets, a well-known result in portfolio theory. This implies that maximum diversification is almost reached with around 30 assets in the portfolio.

Variation of the portfolio return volatility with the number of assets

FIGURE 30.3 Variation of the portfolio return volatility with the number of assets


Transpose Matrix

Consider any square matrix M.

The transposed matrix is M1:

Consider the column vector:

The transpose V1 is a row vector of which terms are that in rows in the same order:

Variance-Covariance Matrix and Correlation Matrix

This form is fully expanded, but it can be written in a more compact form using matrix notations and using the correlation matrix, p, a square matrix with coefficients 1 along the diagonal and correlations between asset returns in off-diagonal terms. We need to introduce O, a diagonal matrix of standard deviations, with standard deviations along the diagonal and all off-diagonal terms equal to zero. The following matrix product equals the variance-covariance X matrix.

The variance-covariance matrix can also be expressed, noting that the transpose Ot of the diagonal matrix o is identical to o:

Variance Formula and Correlation Matrix

Note that we can replace the (m x m) variance-covariance matrix X by the (m x m) correlation matrix p using:

Since the transpose of the product wo, or (wo)T is the product of the transpose in reverse order, (w o)T = oTwT, the variance of a weighted asset portfolio is:

  • [1] See Markowitz, H. M. (1952), Portfolio selection, Journal of Finance, 7, 77-91. Also the textbook of Bodie, Kane and Marcus is an excellent reference on "Investments" and portfolio management [14].
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