# 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.

## 2 Answers

The reason for subtracting the tbill rate is to incorporate the general level of short term rates in the analysis (rates were considerably higher in 2005 than now, for ex). Your idea of lagging the rate by one week is a good one IMO. Further refinements (eg trying to simulate buying and selling of tbills on a weekly basis ) are completely unnecessary and will add nothing to your study except complications. All you need is a rough proxy for short term rates. 3 month tbill rate lagged by one week is fine. (3 month tbill rate held constant for 3 months is probably also fine). Notice how in your figures the tbill return is one or 2 orders of magnitude smaller than the stock returns, it is almost negligible. Ant it is probably uncorrelated to the stock returns. So the exact method used to incorporate short term rates will make almost no difference to your results.

Ok so first I would ask you to take a look at this: Why is the 1 month OIS rate so stable?

It seems to me that in calculating excess returns you are effectively trying to derive information, or I suppose strategies, on the underlying funds. Since all funds will be using the same risk-free rate as their weekly benchmark I cannot really envisage a scenario where your analysis is enhanced by having a volatile underlying series such as the 1W treasury bill rate, or in your case proxying the 1W treasury bill rate by a 3M treasury bill rate.

I strongly suspect that there will be no correlation between these values and your underlying funds so what it amounts to is you just adding noise for no benefit. If you are interested in excess returns I would advise considering the 1W FFOIS rate instead. For one this is a risk free rate in the sense that banks can deposit cash with the federal reserve and attain this rate, for two it matches the maturity you want and for three it is a much more stable series avoiding the noise issue.

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)