Recall that the variance improvement is given by:

$$1 - \left(\frac{\text{Tracking Error VaR}}{\text{Absolute risk of the index}}\right)^2 = 1 - \left(\frac{1.09}{2.08}\right)^2 = 0.725 = 72.5\%$$

Note:

To benchmark a portfolio, we measure the VaR of the portfolio relative to the VaR of a benchmark. The VaR of the deviation between the two portfolios is referred to as a tracking error VaR. The difference comes in because it is possible to construct portfolios that match the risk factors of a benchmark portfolio but have either a higher or a lower VaR.

If $x$ is the vector position of the portfolio and $x_0$ the vector position of the index, then the tracking error VaR is given by:

$$\text{Tracking Error VaR} = \alpha \sqrt{(x - x_0)' \Sigma (x - x_0)}$$

Now, once we have the tracking error VaR, and if the absolute risk of the index is known, we can now go ahead to calculate the variance improvement as we have done above.