Someone help me understand why for portfolio variance or Parametric Value at Risk we have to compute the covariance matrix?

Question

I understand that portfolio variance is computed through $w'Cw$, where w is the vector of weights, $C$ being the covariance matrix. However, what I don't get is this: why can't this portfolio variance simply be calculated by observing the daily portfolio level returns and simply taking the variance of it? That would save the problem of having to compute a unwieldy large portfolio variance no?

I am asking in the context of Value-At-Risk (Parametric Method) - it is so named the variance-covariance method because it uses the covariance matrix explicitly - what I don't understand is why can't we do it in similar fashion to historical simulation (where correlation is factored implictly by just taking daily portfolio value data) and assuming a distribution over it. This is clearly simpler, but no finance book has proposed doing this for parametric VaR?

Please help me understand the difference thanks!

Answer 1

Consider you have positions in $N$ assets, with market values $S$, and that the daily PnL is acquired via multiplying the daily returns vector, which is a random vector with some unknown joint probability distribution.

$$ p = S^T R $$

You are interested in variance of $p$ for constant $S$:

$$ Var(p) = E[(p-E[p])^2] = E[(S^TR - E[S^TR])^2] $$ $$ Var(p) = E[(p-E[p])^2] = S^T C S $$

where $C$ is the covariance matrix of $R$.

The advantage of now applying a parametric model to $R$ is that you can achieve and derive many useful results in terms of the underlying assets, e.g. VaR allocation and optimization. The disadvantage is that you have to assume a (possibly incorrect) statistical joint distribution of returns and parameterize it.

On the other hand you can obtain a non-parametric measure of VaR by evaluating historical values of $p$. This has the advantage of not requiring any statistical distribution being assumed for any parametric model of p (since we use empirical data), so maybe closer to truth. However its use to derive any meaningful results is limited - you only get a VaR metric for some confidence interval.

The in between approach of applying a parametric model to $p$ seems to offer no advantage over either. You need to assert a statistical distribution which may be false, and you don't get any ability to evaluate useful results in terms of individual assets. I have only ever seen this applied where the VaR is extrapolated out to predict very small confidence values, e.g. the 99.97% confidence interval. The problem there is that even the slightest error in the assumed distribution (and most choose a normal!) will vastly affect the results in this tail, to the point where it is probably entirely unreliable.