# Calculating excess returns

I would like to know if I am calculating these excess returns correctly. I have here an R dataframe with weekly 3-month treasury bill rates, and the arithmetic returns (multiplied by $$100$$) of a few exchange traded funds. The t-bill rates have been de-annualized, and the date stamp refers to the Friday of each week.

                tbill       XLB        XLE        XLF
2005-12-23 0.01514498        NA         NA         NA
2005-12-30 0.01525957 -0.740537 -1.8008919 -1.5890348
2006-01-06 0.01594643  3.425060  5.8000348  2.4446630
2006-01-13 0.01632750 -2.475572  1.8521470  0.1528332
2006-01-20 0.01655597 -1.450267  3.6638244 -3.6669816
2006-01-27 0.01678431  4.597595  0.4401128  2.5713384


I am supposed to lag the tbill rates before I subtract them from the ETF returns, correct?

My reasoning is that on Friday, after the treasury bills have been auctioned, the interest rate is known, and can be earned risk free starting at the end of that Friday.

So I should have something like this:

                tbill       XLB        XLE        XLF
2005-12-23 0.01514498        NA         NA         NA
2005-12-30 0.01525957 -0.755682 -1.8160369 -1.6041798
2006-01-06 0.01594643  3.409800  5.7847752  2.4294034
2006-01-13 0.01632750 -2.491518  1.8362006  0.1368868
2006-01-20 0.01655597 -1.466595  3.6474969 -3.6833091
2006-01-27 0.01678431  4.581039  0.4235568  2.5547824


I'm not familiar with the specifics of the TBill market, but it seems that if they want to earn the risk free rate, they're locked in for three months, not a week. Otherwise, as long as one has the right type of account/broker, he/she can sell back the bonds at whatever the market price is. I guess we're ignoring that, though.

With regards to your lag, you are right that if the 1W FFOIS rate on 1st-Jan-2018 is 1% then you can acquire a 1% (yearly) return between 1st-Jan and 8-Jan, which is equivalent to a $$\frac{7}{365}*1\%$$ weekly return. If your fund rises in value between 1-st Jan and 8-th Jan from 100 to 100.05 then the excess weekly return is $$0.0005 - 0.00019 = 0.00031$$ or 0.031% (1.61% per annum excess)