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Assume that a trader trades in several stocks with different volatilities. The return of the trader's portfolio would be the weighted average of returns and the risk would be a function of the the underlying assets' volatilities and correlation as stated by the Modern Portfolio Theory. Assume also that the total wealth of the trader is also recorded daily and can thus be regarded as an "index". Would the volatility of this "index" given by the standard deviation of the changes in the trader's wealth $$ Var(W)= \frac{\sum_i^n(W_i - \bar{W})^2}{n} $$ be the same as the volatility of the underlying assets given by

$$ Var(W)=\sum_iπ_iVar(P_i)+\sum_i \sum_j,j≠iπ_iπ_jCov(P_i,P_j) $$

I know that it would be different if the index is traded such as closed-ended mutual funds. But what if the index is not traded and is simply a reflection of the underlying profits/losses?

Thanks!

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I don't understand the distinctions between the trader's portfolio, wealth, and "index". They are all portfolios of assets. And it's not a necessary condition that the volatility be different if the "index" is traded, but it could be different. –  Joshua Ulrich Mar 25 at 12:27

1 Answer 1

Sure, the variance of the total wealth can be expressed in terms of the variances and covariances of the prices of the assets. If $$ W = \sum_{i} \pi_i P_i $$ where $\pi_i$ is the total dollar amount invested in asset $i$ with price $P_i$. The variance of total wealth is then $$ Var(W) = \sum_i \pi_i Var(P_i) + \sum_i \sum_{j, j\neq i} \pi_i \pi_j Cov(P_i, P_j) $$.

Edit:

You can also express the variance of final wealth in terms of the variance of the returns of the assets in the portfolio. If $W$ is the final wealth and $W_0$ is the initial wealth, then $$ W = R_P W_0$$ where $R_P = 1 + r_P$ is the gross return of the portfolio and $r_P$ the rate of return. If $w_i$ is the portfolio weight of asset $i$, then $$ r_P = \sum_i w_i r_i $$ and $$ Var(r_P) = \sum_i w_i Var(r_i) + \sum_i \sum_{j,j\neq i} w_i w_j Cov(r_i, r_j). $$ Since $Var(R_P) = Var(1+r_p) = Var(r_p)$, you can use this to calculate the variance of $W$: $$ Var(W) = W_0^2 \left( \sum_i w_i Var(r_i) + \sum_i \sum_{j,j\neq i} w_i w_j Cov(r_i, r_j) \right).$$

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The notation isn't clear. What is $Var(P)$? $P$ is a price. Do you mean to say var of returns? –  user2763361 Mar 26 at 8:50
    
No, Var(P) is the variance of the price of the asset, not of the returns. –  pbr142 Mar 26 at 11:25
    
Would a variance of a non-stationary process like Price make sense ? –  silencer Mar 26 at 12:49
    
Stationarity is important to estimate the variance based on historical observations. But a price is a random variable and its distribution has a variance, stationary or not. The question did not state whether an estimate is required or not so I did not really think about any estimation issues. –  pbr142 Mar 26 at 15:25

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