1. GARCH volatility leads to time-varying variability of the returns
GARCH effects in volatility lead to a time-varying variability of the returns. This time-variation has a direct impact on the relationship between two asset returns. When their variance is time-varying, also their covariance changes over time.
2. GARCH covariance
This becomes immediately clear from the formula of the covariance between two asset returns. It equals the product of their correlation rho and their volatility. A time-varying volatility thus leads to a time-varying covariance.
3. GARCH covariance estimation in four steps
You can estimate this GARCH covariance in four steps. First you use ugarchfit to estimate the GARCH model of each return series. Then you compute the standardized returns using the residuals method. Third, you need to compute the correlation between the standardized returns. For the daily Microsoft and Walmart standardized returns, you obtain a positive correlation of around 30%.
4. GARCH covariance estimation in four steps
In step 4, you get the GARCH covariance by multiplying the estimated correlation with the estimated volatilities. Note that the predicted GARCH covariance of Walmart and Microsoft clearly peaks in the turbulent periods when also the volatility peaks. Since the average correlation is positive, the GARCH covariance is positive as well.
5. Applications of covariance in finance
There are numerous applications of dynamic covariances in finance. A first very important one is the optimization of the portfolio variance, as it depends on the portfolio weights, the variance of all the assets and the covariance between the asset returns.
6. Application to portfolio optimization
Here you see the formula for the portfolio variance in the simple case of a portfolio with two assets with weight w1t invested in asset 1 and 1-w1t in asset 2. The variance then equals the sum of the squared weights times their variance plus 2 times the product of the weights and the covariance.
One approach to obtain optimal weights is to set them such that the variance is minimized.
From this first order condition, we then get the optimal weights.
7. Minimum variance portfolio weights
The optimal weight of asset 1 equals the variance of asset 2 minus the covariance, divided by the sum of the variances minus 2 times the covariance.
The corresponding R code requires to compute the variances and the covariance, and then implement the formula for the weight.
8. Daily minimum variance weights in MSFT-WMT portfolio
Here you see that the portfolio manager should change his portfolio on a daily basis since the optimal weights of Microsoft change every day. The portfolio strategy is clearly dynamic: depending on the value of the variances and covariance, the optimal weight of Microsoft varies between 0 and 85% over the period.
9. Dynamic beta
A second important application is the evaluation of the beta of a stock which measures its systematic risk.
The beta equals the covariance of the stock return and the market return, divided by the market return variance.
You need this number to set the risk premium of the stock. The higher it is, the more risky the stock is and thus the higher is the required rate of return.
When computing the beta of US stocks, the S&P 500 return is used as market return.
10. The daily beta of MSFT
Here you see the calculation for the daily beta of Microsoft. We compute the covariance between the Microsoft return and the S&P 500 return as the product between their correlation and GARCH volatilities. Then we compute the S&P 500 return variance.
Computing the ratio between the covariance and the market variance yields the beta.
11. Time series plot of daily beta of MSFT
Here you see the resulting time series of daily values for the Microsoft beta. Its average value is 1, but in periods of high volatility it can increase to more than double, while in calm periods it can drop to 0-point-4.
12. Let's practice!
Now, let's get hands on experience in analyzing the GARCH effects on the covariance of two return series.