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Generating a variance covariance matrix for a portfolio of assets.

Any time we are dealing with portfolios that contain more than 2 assets  that we want to perform calculations on such as evaluating risk, returns and so on it is better to construct a Variance-Covariance matrix. Let’s assume our portfolio P consists of N assets and we have k periods worth of return data for each of the assets. To generate a variance-covariance matrix we do the following steps:

For each of our N assets we obtain a time-series of historical price data, starting with the earliest first. We aim to get as long a series as possible.

1) Portfolio P

2)  For each of the assets calculate the continuously compounded returns for each of          the kth periods as

LN(Asset1 Value (k+1)/Asset Value1 (k) etc …..

3) For each of the assets calculate the average of the values obtained from step (2).

This will result in an N x1 Matrix , let's call it R

4) For each of the assets for each of the kth periods calculate the excess return over the average value calculated in step (3). This will result in an N x k matrix, let’s call it X

The variance covariance matrix  lets say is given by

∑ = 1/k * X’ X  

i.e the transpose of the matrix of excess returns X’ multiplied by matrix of excess returns X divided by the number of periods. This will result in an N x N variance covariance matrix .

So now we have we can use it to calculate some useful portfolio values.

e.g The portfolio volatility σ =  √X’∑X

And now that we have

a)   the portfolio volatility σ,

b)   the average of each of the asset returns R

      ( the1xN matrix calculated in step 3 earlier )

c)   and if we assume we have an Nx1 matrix of the asset weights, say W, in the portfolio we can calculate the expected return E(r) of the portfolio using:-

E(r) = W’R = {W1, W2,… WN}         {R(1)}

If we happen to know or make an assumption about the risk-free rate rf we can also compute the Sharpe ratio  as

Sr   =  (E(r) – rf)/σ

Finally we can also compute the Value At Risk VaR as for example 99% VaR

VaR = NORMINV(99%,E(r),σ)