Python Tutorial: Measuring risk of a portfolio

Key Takeaways

let's talk about portfolio risk when you invest in stocks you don't know beforehand what your return will be prices go up and down so there is a degree of uncertainty which implies that stock return is a random variable the extent to which the actual returns are spread around their mean value is called variance here is the official formula for variant it is a great indication of stocks riskiness or volatility you might have come across variants in your statistics class certain stocks have a small variance that means their returns are always close to the mean like the returns distribution here in red sometimes stocks have a high variance and are widely spread around the mean like the distribution here in blue this might be easy to understand for a single asset but how does this work for a portfolio well portfolio variance isn't simply the sum of all variances of the underlying stocks and due to the correlation between the assets it becomes more complicated since the assets in your portfolio correlate ie move together or in the opposite direction you intuitively understand that this will influence the riskiness of your investment that implies that correlation should be an ingredient in your portfolio variance also the individual risk levels of the stocks are part of the calculation as well as the portfolio weights lastly you might come across standard deviation used as an indication of risk this is simply the square root of the variance and both are used in practice suppose I calculate the variance of a portfolio with two stocks the portfolio variance is simply calculated by taking the weights times the variance is Sigma 1 and 2 for stock 1 and 2 respectively I need to add a term to account for correlation between the stocks and that's why I multiply W 1 W 2 Rho the correlation coefficient and the variance is Sigma 1 and 2 this last term is actually what we call the covariance so let's rewrite the formula and insert the covariance instead of the correlation times the variances let's take that last equation from the previous slide this one is long and difficult to work with hence we can write it shorter and smarter if we use some matrix notation it then becomes weights transposed times the covariance matrix times the weights the covariance matrix depicted here in the middle contains two variances on the diagonal and the covariances between acid one and two on the off diagonal terms this is something we can actually implement easily in Python let's start from the beginning by taking the price data again first remember to calculate the daily returns using the percentage change function as we need to calculate variance from our set of returns not from prices then we can let Python calculate our covariance matrix for us very easily so now we almost have all ingredients to calculate the portfolio variance we need to take a short additional step which is to analyze our volatility by multiplying it with 250 which is the amount of trading days in a year don't worry about it for now and you'll learn more about this in the next chapter last we need to weight of our portfolio we have five stocks here so let's create a simple equal weight of portfolio now apply the formula and multiply the transpose weights with the covariance matrix and then with the normal weights again make sure to use the dot multiplier here the numpy dot function takes only two arguments to multiply so start with the covariance matrix with normal weights and then multiply that whole thing with the weights transposed and that gives you the portfolio variance last let's take the square root of the variance to calculate the standard deviation like this let's break