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Given portfolio risk is $\mathbf{w}\boldsymbol{\Sigma}\mathbf{w}$ where $\boldsymbol{\Sigma}$ is the covariance matrix whose diagonal elements $\sigma^2_{n}$ are individual asset return variances and whose off-diagonal elements are pair-wise covariances of assets, $\sigma_{n,\neg n}$

what is the interpretation of element $\sigma_{1,2}$ in $\boldsymbol{\Sigma}$, and how would you describe its units?

If $\sigma_{1,2}=0.1$ would it be correct to say the following?

"movements in asset 1 returns on average co-vary with asset 2 return movements by 10% standard deviations and vice versa"

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    $\begingroup$ You've been asking insightful questions. Out of curiosity, are you working on a personal project exploring portfolio risk or what is the motivation? $\endgroup$ – Quantoisseur Aug 17 at 13:57
  • $\begingroup$ I'm sorry Develarist: I've been confusing myself but I think that I finally got it now. First, it's easier if you assume that $\sigma_{1,2}$ is a correlation ( for understanding purposes ). So, assume that $\sigma_{1,2}$ has already been divided by the square root of $(\sigma_{1,1} \times \sigma_{2,2})$. Next, suppose that we did a scatterplot of the returns of stock 1 and the returns of stock 2. Finally, suppose we fit a regression line to the scatterplot. Then, the resulting $R^2$ from the regression would be $\sigma_{1,2}^2$ so $0.1^2$. $\endgroup$ – mark leeds Aug 17 at 14:28
  • $\begingroup$ Can someone who knows the answer, comment or correct my answer above ? I don't think it can be correct because, by definition, $\sigma_{1,2} = \sigma_{2,1}$ but a regression of $Y$ versus $X$ won't give the same $R^2$ as the regression of $X$ versus $Y$. ( unless you used partial least squares ). I think something is wrong in my thinking. Thanks. $\endgroup$ – mark leeds Aug 18 at 0:27
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The interpretation and units problem, ie the lack of an easily intuitive answer, is precisely why quants/econometricians etc. tend to shy away from talking too much about covariances [even if they are absolutely necessary; and frequently used]. Thus if anything involving covariances has to interpreted, let alone explained, the default is usually to express it in terms of correlation, which does have intuitive units: bounded [-1,1] with 0 = independence, etc.

Cor(1,2) = Cov(1,2) / ( sd(1) * sd(2) )

Cov(1,2) = Cor(1,2) * sd(1) * sd(2)

So the "units" here is a product blend of three measures, each with their own units: two volatilities and a bounded measure of association. As such, they exist but lack an intuitive explanation.

The closest one can do is to express the covariance as a marginal change in portfolio variance per unit change in the product of Weights 1 & 2. Which remains inelegant in the extreme, to be polite ;-)

Recall also that the traditional OLS beta can be expressed as:

Beta(1|2) = Cov(1,2) / Var(2) = E(d1) / d2

E(d1) = Cov(1,2) * d2/Var(2)

So a change of +1 in Asset2 has a +0.1 divided by its variance effect on Asset1. Which is the same as saying that a +1 sigma move in Asset2 has a 0.1 divided by its standard deviation on Asset1. Which is the same as saying (where Z=1 is a 1 sigma shock):

d1/d2 = Cov(1,2) / Var(2)

d1/z2 = Cov(1,2) / SD(2)

z1/z2 = Cov(1,2) / (SD(1) * SD(2)) = Cor(1,2)!

So the way to make the kind of statement you try to make above intuitive remains to translate your covariances into (intuitive) unitless correlations. A one sigma move in either 1 or 2 will have a marginal Cor(1,2) sigma effect on the other.

However you approach this, you always need to process the covariance via an additional metric (with its own units, whether absolute returns, vol-adjusted returns, or weights) to generate any intuitive explanatory outcome here. The traditional w.Cov.w formulation is efficient for predicting portfolio risk; but when it comes to interpretation and explanation, it fails big time. Which is why publications inevitably show the associated correlation matrices in preference. The two will always give you the same outputs/forecasts; with the choice between the two ultimately a question of prediction vs interpretation (ie presentational in nature).

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So let us assume that the portfolio is entirely made up of consols or single period discount bonds. This would be dubious for equities because $$_iR_t=\frac{_ip_{t+1}}{_ip_t}\times\frac{_iq_{t+1}}{_iq_t}-1$$ and $$_jR_t=\frac{_jp_{t+1}}{_jp_t}\times\frac{_jq_{t+1}}{_jq_t}-1$$ if you ignore the effect of dividends. That makes returns the product distribution of two ratio distributions. Models like the CAPM escape this issue by assuming that all parameters are known and that nobody is doing any estimation. Under mild assumptions, these returns would not have a defined covariance matrix even in log space.

However, with regard to your question, it is important to remember that parameters such as $\{\mu_i,\mu_j,\sigma_{i,j},\sigma_{i,i},\sigma_{j,j}\}$ are thought of as fixed points in Frequentist theory. Models like the CAPM don't work in a Bayesian space because the parameters are random variables.

So, in answer to your question, the units of $\sigma_{i,j}$ are in directionally signed square excess/deficit returns from the joint expectation. It could be thought of as an area with direction.

The usual interpretation is always scaled by the variance by noting that $\beta_{i,j}=\frac{\sigma_{i,j}}{\sigma_{i,i}}.$

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  • $\begingroup$ "movements in asset 1 returns, scaled by their variance, co-vary with asset 2 return movements by 10% square excess/deficit returns from the joint expectation of the distributions of assets 1 and 2, and vice versa" ? $\endgroup$ – develarist Aug 18 at 5:50
  • $\begingroup$ No, asset 1 moves $\beta$ times more than asset 2 (or vice versa depending on where you are putting the coefficient), In the case of the CAPM, the excess returns above the risk free rate from one asset move $\beta$ times more than the excess returns of another asset. above the risk free rate. $\endgroup$ – Dave Harris Aug 18 at 6:10
  • $\begingroup$ However, $\sigma_{i,j}$ in the CAPM is the expected squared deviation from the joint expectation. It is better to think in terms of being a signed area so that you can have negative areas. It is a squared term. It does not depend on the risk free rate in the CAPM. The covariance of an risky asset and the risk free asset must be zero. $\endgroup$ – Dave Harris Aug 18 at 6:12
  • $\begingroup$ the question though was in terms of mean-variance portfolio risk, which was originally introduced outside of the auspices of CAPM. It was not until later that the maximal Sharpe ratio portfolio was introduced that CAPM was assimilated into the model. I'm not saying that the only answer must therefore be in a CAPM framework, but was looking for something directly inferrable from the minimum variance portfolio alone which didn't depend on a CAPM (beta) interpretation. So, without $\beta$, what can we say about $\sigma_{1,2}$? $\endgroup$ – develarist Aug 18 at 6:16
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    $\begingroup$ @develarist outside the framework of the CAPM, but inside mean-variance finance, as if valid, or with known parameters, then it is the expected product of the joint deviation from the joint mean. It is actually a rectangle, not actually a squared term, where the product could be negative. It does not lend itself to nice interpretations unless you possibly think about it as a contributor to total variation. $\endgroup$ – Dave Harris Aug 18 at 6:22
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@develarist: I did some more reading and it goes like this. ( not talking about this with respect to CAPM nor commenting on your current discussion with Dave ). Suppose you have $\sigma_{(1,2)}$ which denotes the covariance (of the returns) of stock 1 and stock 2. Denote $x$ as the returns ( in the sample ) of stock 1 and $y$ as the returns ( in the sample ) of stock 2.

The first step towards interpretation is to take $\sigma_{(1,2)}$ and divide it by the sample variance of the returns of stock 1. Call this $\beta_{(1,2)}$. Then, once you do this, $\beta_{(1,2)}$ can be interpreted as the coefficient ( not the intercept. the other one ) of a simple regression of the returns of stock 1 versus the stock returns of stock_2 where the returns of stock 2 are the response ($y$) and the returns of stock 1 are the predictor ($x$).

The fact that $\sigma_{(1,2)}$ is 0.1 really doesn't mean much because that has to be divided by the sample variance of the stock returns of stock 1 in order for it to have the regression interpretation described. Of course, if the sample variance of the returns of stock 1 happened to be 1.0, then one could interpret the covariance as the estimated amount that the return of stock 2 increases for each unit increase in the return of stock 1.

Note that the seeming contradiction that I referred to in my original post (which confused me ) doesn't exist because if we flipped the regression and made stock 1's returns (x) the response and stock 2's returns ( y) the predictor, then one would need to divide the covariance, $\sigma_{(1,2)}$ by the sample variance of stock 2's returns (y) rather than the sample variance of stock 1's returns (x). So, there is no inconsistency in the definition. I hope this clarifies things.

Oh, also, as far as I can tell, there also does not seem to be any relation between the covariance and the R^2 of the regression which I mistakenly thought was the case. My apologies for the confusion there.

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