Portfolio¶

Pool of securities combined such that

• Maximizes expected returns
• Minimizes unsystematic risk

Concept of hedging

Try to diversify on all 4 pillars of GEIC

Aspects¶

• What set of securities to be selected
• What proportions
• Selection of optimum portfolio

Characteristics¶

Let \(w_i\) be the fraction of investment allocation to security \(i\)

• \(w_i \in [0, 1]\)
• \(\sum_i w_i = 1\)
• \(w_i<0 \implies\) Taking loan?

Note: We assume Gaussian distribution of returns for all securities. If violated, then analyze accordingly $$ \begin{aligned} E[R_p] &= \sum_i^n w_i R_i \

\sigma^2_{R_p} &= \sum_i^n (w_i \sigma_i)^2 + 2 \sum_{i=1}^{\lceil n/2 \rceil} \sum_{j>i}^n w_i w_j \sigma_i \sigma_j \rho_{ij} \ \beta_p &= \sum_i^n w_i \beta_i \end{aligned} $$

where

• \(\rho_{ij} =\) correlation between 2 securities \(i\) and \(j\)
• \(\beta_i =\) \(\beta\) of security \(i\)
• Given +ve portfolio weights on 2 shares, the lower the correlation between them, the lower the variance of the portfolio

Minimum Variance Portfolio¶

A portfolio of group of shares that minimizes the return variance is the portfolio that has equal variance with every share return $$ w^* = \arg \min \sigma^2_{R_p} \ \implies w^* = w @ \dfrac{d \sigma^2_{R_p}}{dw} = 0 $$

2 Securities¶

Types of Portfolios¶

Benchmark¶

• 60% Equity, 40% Bonds

India¶

Nifty50 makes a 12% average return, but actually, entire pool of Indian stock market makes a negative return

Retail investors lose money due to single-stock investment

Exane, Expose

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