Quantitative Analysis, Risk Management, Modelling, Algo Trading, and Big Data Analysis

Riskless Diversification

Diversification can be viewed by many as an attractive tool for risk limitation for a given portfolio currently held by investor. Dealing with numbers one can show that if one holds $N$ securities that are uncorrelated completely, the volatility of such portfolio $Vol(P)$ will decrease with an increasing number of securities $N$ as follows:
$$
Vol(P) = \left[\sum_{i=1}^{N} \left(\frac{1}{N}\right)^2 Var(y_i)^2 \right] = \frac{Var(y)}{\sqrt{N}} .
$$
What remains unspoken but important to keep in mind is that all securities $i$ where $i=1,…,N$ have the same volatility and the same expected return. Therefore, by holding uncorrelated securities, one can eliminate portfolio volatility if one holds sufficiently many of these securities.

That is a theory. The investor’s reality is usually far from this ideal situation as portfolios usually contain a number of investments mutually connected or dependable upon themselves as the feedback to the market movements or sentiments. Thus, the reminder of diversifiable risk which cannot be eliminated by the diversification process becomes a systematic risk.

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