Calculating Weekly Log Excess Returns and Choosing a Risk-Free Rate
Summary
The document discusses converting a quoted three-month Treasury bill rate into a weekly risk-free return and subtracting it from an asset’s weekly log return. It endorses scaling the log of one plus the quoted annual rate by the weekly fraction of a 360-day year. At low rates, this is nearly the same as scaling the quoted rate directly, though the logarithmic conversion is mathematically consistent with log-return calculations.
The central caveat is that the risk-free rate’s tenor should match the period over which the investor can leave cash invested before reconsidering the portfolio. A three-month bill can fit a quarterly decision horizon, while a shorter or longer horizon may call for a different rate. The answer also notes that the appropriate funding benchmark can depend on the application, and that a longer-maturity yield brings price risk if the investment is sold early.
Key ideas
- Weekly log excess returns subtract a weekly risk-free log return from the asset’s weekly log return.
- A quoted annual bill rate can be converted using its day-count convention and the relevant weekly fraction.
- For low rates, the logarithmic conversion is close to simple proportional scaling.
- Choose a risk-free tenor that reflects how long cash can remain invested before the next decision.
- Longer-maturity yields carry price risk when the investment horizon is shorter than the maturity.
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Full text
# Calculating excess returns with 3M T-Bill
# Calculating excess returns with 3M T-Bill
I have to calculate weekly log excess returns using the 3-month T-bill. However I am not really sure if I am doing this correctly. This is what I did:
- first I calculated the returns with ln(price/price on previous week)
- then I did this with the 3M T-bill rate: ln((rate/100) +1) / 52. I divided the rate by 100 because it was presented in percentages (so 2% was just written as 2). Divided by 52 because of the amount of weeks in a year.
- after this I subtracted the number from the second step from the first step
Is this the correct way to calculate the excess returns? I'm not really sure about the second step. Is this how to calculate the weekly risk free rate?
Thanks for your time
Edit: because the T-bill is based on a 360 day year, I adjusted my calculations to ln((rate/100) +1) * 7/360
## Answer by kurtosis (score 1)
https://quant.stackexchange.com/a/57480
Is log(1+r/100) Reasonable?
What you have done is reasonable -- with a possible caveat.
For low interest rates, using $\log(1+\frac{\text{quoted rate}}{100})\cdot\frac{7}{360}$ will be almost identical to $\text{quoted rate}\cdot\frac{7}{360}$. However, good on you for doing the correct math.
Which Tenor to Use?
The bigger issue is that the tenor of your risk-free rate should match your investment horizon.
If I were a market maker, my alternative to trading would be sitting out a day and earning a day's interest at an overnight rate. If I make the decision to trade or not quarterly, I should use a 3-month rate. If I rebalance my portfolio annually or intend to hold my investments for 5 years (so not revisiting that decision in 3 months), then a 1-year or 5-year rate would be sensible.
The key governing factor is the opportunity cost: how long would you leave money in cash before possibly changing what you hold? Some people might say "I rebalance quarterly but I put cash into a 10-year note." Great, but the implication is that you can earn 10-year yields for any given 3-month investment. That completely ignores the (very real) risk that interest rate fluctuations could leave you selling at a loss after 3 months. If that were not the case, there would be a huge arbitrage by trading the 3M vs 10Y.
I assume you are proposing to rebalance your portfolio quarterly -- in which case the 3M T-bill rate is the correct rate to use.
## Answer by demully (score 0)
https://quant.stackexchange.com/a/58087
Your approach is how I would it.
I share @kurtosis's questions about whether 3m Govt is the correct funding rate for a risk asset (for slightly different reasons); but your method is 100% kosher if you are supposed to use T-Bills here. I would prefer 3m (or 6w) Eurodollar/LIBOR/swap myself; but the difference between this and Bills will likely be a rounding error compared to the volatility of the risky asset you want to convert from total to excess.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.