Advanced Portfolio Management. Giuseppe A. Paleologo

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Advanced Portfolio Management - Giuseppe A. Paleologo


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contribution of the market to the portfolio PnL. The daily volatility of the portfolio deriving from the market is

StartLayout 1st Row 1st Column left-parenthesis p o r t f o l i o m a r k e t v o l a t i l i t y right-parenthesis equals 2nd Column left-parenthesis p o r t f o l i o d o l l a r b e t a right-parenthesis 2nd Row 1st Column Blank 2nd Column times left-parenthesis m a r k e t v o l a t i l i t y right-parenthesis 3rd Row 1st Column equals 2nd Column dollar-sign 25.5 normal upper M times 1.4 percent-sign equals dollar-sign 36 normal upper K EndLayout

      The other term is the idiosyncratic PnL. The volatility of the idiosyncratic PnL of the portfolio is the sum of three terms. As in the case of two variables, the variance of the sum is the sum of the variances:

StartLayout 1st Row 1st Column left-parenthesis p o r t f o l i o i d i o v a r i a n c e right-parenthesis 2nd Column equals left-parenthesis NMV Subscript SYF Baseline times vol Subscript SYF Baseline right-parenthesis squared 2nd Row 1st Column Blank 2nd Column plus left-parenthesis NMV Subscript WMT Baseline times vol Subscript WMT Baseline right-parenthesis squared 3rd Row 1st Column Blank 2nd Column plus left-parenthesis NMV Subscript SPY Baseline times vol Subscript SPY Baseline right-parenthesis squared EndLayout

      And the volatility is

left-parenthesis p o r t f o l i o i d i o v o l a t i l i t y right-parenthesis equals StartRoot left-parenthesis 10 times 1.2 right-parenthesis squared plus left-parenthesis 5 times 0.5 right-parenthesis squared EndRoot equals dollar-sign 122 normal upper K

      Finally, the variance of the portfolio is the sum of the variances, because idio and market returns are independent of each other. The volatility is the square root:

left-parenthesis p o r t f o l i o t o t a l v o l a t i l i t y right-parenthesis equals StartRoot 3 6 squared plus 12 2 squared EndRoot equals dollar-sign 127 normal upper K

      Procedure 3.1 Compute the volatility of a portfolio.

      1 Compute the dollar betas for the individual positions;

      2 Compute the dollar portfolio beta as the sum of the individual betas;

      3 Compute the market component of the volatility as (portfolio beta) (market volatility);

      4 Compute the dollar idio volatility as the square root of the sum of the squared dollar volatilities.

# Stocks Idio Vol ($) Market Vol ($) Idio Var (% tot)
1 1M 1M 50
10 316K 1M 9.09
100 100K 1M 0.99
1000 31.6K 1M 0.01

      Given this example, you can now understand better why SPY has zero percentage idio volatility in Table 3.5. The SPY is a long-only portfolio of 500 stocks. Each stock in the portfolio has a positive beta. As a percentage of the total risk, the idiosyncractic risk is very small, and is usually approximated to zero.


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