Portfolio Wealth Variance from Asset Returns and Covariances
Summary
The document explains how the variance of a trader’s wealth can be calculated from the risks and co-movements of the assets held. If wealth is a fixed dollar-weighted sum of asset prices, its variance is the sum of each holding’s price variance scaled by squared exposure, plus covariance terms for pairs of holdings. The answer then gives the corresponding return-based expression for a portfolio with fixed weights.
For initial wealth multiplied by the portfolio’s gross return, wealth variance equals initial wealth squared times portfolio return variance. Portfolio return variance depends on the weighted asset return variances and their pairwise covariances. This describes the mathematical equivalence between measuring the wealth series and combining underlying asset risk under the stated fixed-exposure setup. The document does not discuss changing weights, cash flows, leverage adjustments, or the distinction between price levels and returns in a time-series volatility estimate.
Key ideas
- Wealth formed as a fixed dollar-weighted sum of asset prices has variance determined by asset variances and covariances.
- Portfolio covariance terms capture the joint movement of different holdings.
- With fixed portfolio weights, return variance is calculated from weighted asset return variances and covariances.
- Scaling portfolio returns by initial wealth scales wealth variance by initial wealth squared.
- The formulas presume the stated portfolio setup and do not address changing exposures or cash flows.
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Full text
# Is the volatility of a trader's wealth equal to the volatility of the underlying assets traded?
# Is the volatility of a trader's wealth equal to the volatility of the underlying assets traded?
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!
## Answer by pbr142 (score 1)
https://quant.stackexchange.com/a/10700
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 $W$: $$ Var(W) = W_0^2 \left( \sum_i w_i Var(r_i) + \sum_i \sum_{j,j\neq i} w_i w_j Cov(r_i, r_j) \right).$$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.