By Robert R. Prechter
Pioneering reports in Socionomics is the follow-up to Prechter's ground-breaking introductory textual content on Socionomics, The Wave precept of Human Social habit. This new identify is made from a set of attractive essays representing over twenty years worthy of study into this new version of idea.
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N. If there are only two risky assets, then π π = π12 var(R˜ 1 ) + π22 var(R˜ n ) + 2π1 π2 cov(R˜ 1 , R˜ 2 ) . In general, n π n n πi2 var(R˜ i ) + 2 π= i=1 πi πj cov(R˜ i , R˜ j ) . i=1 j=i+1 The number of covariance terms is n(n − 1)/2. This grows large quite rapidly. For example, with 1,000 stocks, there are 499,500 covariance terms. 4) to reduce the number of covariances that must be estimated. Because variances φ φ must be nonnegative, any covariance matrix is positive semidefinite. If a symmetric positive semidefinite matrix is nonsingular, then it is positive definite (see, for example, Pemberton and Rau, 2011).
I) ρ > 1. The utility is proportional to −(w − ζ )−η , where η = ρ − 1 > 0. It is defined for w > ζ and is monotone increasing up to zero as w → ∞. (ii) 0 < ρ < 1. The utility is proportional to (w − ζ )η , where η = 1 − ρ ∈ (0, 1). It is zero at ζ and is monotone increasing up to infinity as w → ∞. (iii) ρ < 0. The utility is proportional to −(ζ − w)η , where η = 1 − ρ > 1. It is defined for w ≤ ζ and is monotone increasing up to zero as w ↑ ζ . Quadratic Utility A special case of category (iii) of the shifted power utility function is ρ = −1, in which case the utility is 1 1 1 − (w − ζ )2 = − ζ 2 + ζ w − w2 .
In this example, portfolio risk can be made negligible through diversification if there is a large number of assets. Setting πi = 1/n for each i, we have n π π =σ 2 i=1 1 σ2 = →0 n2 n as n → ∞. The unrealistic feature of this example is that the assets are uncorrelated. Generally, we find positive correlations between assets; for example, all stocks tend to go up when the market goes up. Thus, risk cannot generally be eliminated by diversification. However, this example shows that the risk coming from the term n πi2 var(R˜ i ) i=1 can generally be made small by diversification.
Pioneering Studies in Socionomics by Robert R. Prechter