Why Covariance Matrices Use Returns Instead of Asset Prices
Summary
The document explains why covariance and correlation between assets are generally estimated from returns or price changes rather than price levels. Price levels can be nonstationary, with changing means and trends, so their measured co-movement may reflect shared drift instead of meaningful dependence. A covariance matrix built from such levels may therefore misrepresent the relationship relevant to portfolio analysis.
A second explanation assumes prices evolve as cumulative sums of independent changes, as in a Wiener-process model. Covariance between the resulting price levels combines many past increments, making early changes disproportionately influential rather than measuring contemporaneous co-movement. The discussion also notes that price variance can be misleading across assets with different price scales. These arguments motivate using stationary changes or returns, with log returns appropriate under a multiplicative price model. The document gives conceptual and algebraic reasoning, but does not report a direct empirical comparison of portfolio performance using price-level and return-based covariance estimates.
Key ideas
- Price levels often have changing means, making level-based correlation potentially spurious.
- Covariance describes co-movement around means, which is hard to interpret when the means drift.
- Under a cumulative-change price model, level covariance aggregates past changes and can overweight early increments.
- Returns or price changes are usually more suitable inputs for asset covariance estimates.
- The discussion motivates this choice conceptually but provides no practical performance comparison.
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Full text
# Estimate covariance matrix using prices
# Estimate covariance matrix using prices
We generally estimate the covariance matrix of assets using their returns instead of prices. Why is that the case?
I can think of two possible reasons and would appreciate comments/feedback regarding them:
- Correlation of two non-stationary time series' is spurious since they have trends embedded in them.
- Variance of prices doesn't make sense. Consider two assets with price sequences of {100, 105, 101, 104, 102, 103} and {100, 101, 102, 103, 104, 105}. Clearly the first asset is more "variable". But the variance of prices for both is precisely the same.
In practical applications, does a covariance matrix estimated using prices actually perform worse?
## Answer by Chris (score 3, accepted)
https://quant.stackexchange.com/a/50958
For the same reason you can't meaningfully measure covariance/correlation using price of individual assets...correlation (covariance by extension) represents the comovement in deviations from individual means. You can't represent that if the mean continues to change (ie, series considered aren't stationary). Same goes for multiple assets as is represented in a covariance matrix.
## Answer by Attack68 (score 8)
https://quant.stackexchange.com/a/50931
If you assume that a financial asset price has a change that is a wiener process then you can view the future value of that asset as the initial value plus the sum of the independent daily changes (for equity or returns based then you would need log version of this):
$$ S_t = S_0 + \sum \Delta S_i $$
where $\Delta S_i = S_i - S_{i-1} $ is a wiener process.
You could also state the same for a second asset $T_t$.
If you were to evaluate the covariance of the absolute prices (rather than the changes) then you have:
$$ Cov(S,T) = \frac{1}{n}\sum_{t=1}^n (S_t - E[S])(T_t - E[T]) $$
If you recognise (under the wiener process) that $E[S] = S_0$ and same for T then you expand this out to get:
$$ \frac{1}{n} \sum_{t=1}^n (\sum_{i=1}^t \Delta S_i)(\sum_{i=1}^t \Delta T_i) $$
so the covariance result is dominated by the initial changes rather than any of the latter, which is not valid under the assumed pricing process.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.