MATH2070 Optimisation and Financial Mathematics

Correlations and the covariance matrix

1. Export the data into Matlab using csvread and/or readtable. This question investigates the correlations of the return rates of the 29 stocks. When analysing return rate data one has several choices. A commonly used variable is the logarithmic change of price or the so called log return rate: Let Skt be the price of k-th stock at time t, then consider Ykt = log Skt ?log Sk(t?1) (wrt the natural base).

(i) Calculate the maximal correlation between the Yk, name and plot the two stock prices associated with the highest correlation as a function of time. On the graph present normalised time series so that they start from the same value 100 on 2/01/2013.

(ii) Calculate the minimal correlation between the Yk, name and plot the two stock prices associated with the smallest correlation as a function of time. On the graph present normalised time series so that they start from the same value 100 on 2/01/2013.

(iii) Visualise the correlation matrices for two subperiods: 1/12/2014–1/09/2016 and 1/09/2016 1/02/2018 (you may use Matlab’s command imagesc). Can you spot differences? Plot the price of Dow Jones Industrial Average in the whole period. Can you relate it to your observations about the correlation matrices?

(iv) Plot the histogram of the correlation coefficients ij for the two periods from the previous point. Comment on your result.

(v) Find the maximal risk (expressed via standard deviation) of the 29 stocks and the most risky stock corresponding to this value. Calculate the maximal correlation between the most risky stock and one of the remaining stocks, name and plot these two stock prices as a function of time. On the graph present normalised time series so that they start from the same value 100 on 2/01/2013. (Here work with the entire time series.)

(vi) Find the minimal risk (expressed via standard deviation) of the 29 stocks and the least risky stock corresponding to this value. Calculate the maximal correlation between the least risky stock and one of the remaining stocks, name and plot these two stock prices as a function of time. On the graph present normalised time series so that they start from the same value 100 on 2/01/2013. (Here work with the entire time series.)