# Leverage and Tracking difference

I am a little confused. I have calculated the tracking difference of an Index and a n ETF using the return getting 0,4% tracking difference per year. I have then leveraged both, the Index and the ETF to a lever of 2 getting 0.63% tracking difference per year. I have then done some testing with hypothecical value and got 10% unleveraged and 20% levaraged with an lever of 2. So, $$leveraged\ tracking\ difference = lever*(unleveraged\ tracking\ difference)$$ in the hypothetical case. In the real case, I am not getting an equality.

However the real case used a 5-year period and I calculated the annual return using the geometric mean. In the hypothetical case, I only simulated one year of return.

My question is. Should the tracking difference always be equal to my formula when leverage? if that is the case, I might have done some mistake in the real case. Else, it differs when a 5-period is used along with the geomtric mean.

• Hmm... the AM-GM inequality WILL cause some divergence, but the difference between 2x and root-2x would require a VERY volatile underlying. Are your daily/monthly tracking errors correlated with the index returns? Then the leverage could create time-diversification effects to amplify AM-vs-GM effects? – demully Mar 21 at 22:43

## 1 Answer

It is probably coming from AM-GM inequality, i.e. from the fact that geometric mean is a concave function