Why Annual Treasury Bill Yields Cannot Recover Daily Returns
Summary
The document asks how to convert annualized three-month Treasury bill discount rates into daily risk-free returns for portfolio simulation. The proposed approach converts an annual rate into a daily compounded rate, then takes its logarithm. The response warns that annual or yearly return data cannot reveal the actual path or volatility of daily returns; daily observations are needed to measure those properties.
A stock example illustrates the distinction: an annual gain may imply an average daily gain, but that average does not describe the stock’s actual daily movements. The same limitation applies when using annualized bill yields as though they were observed daily returns. The discussion offers no bill-specific convention for interpreting discount-basis rates or a formula for converting them. Its practical lesson is about missing information: spreading an annual figure across days can create a synthetic series, but cannot recover the historical daily yields or their variability.
Key ideas
- Annualized rates do not contain the information needed to reconstruct observed daily returns.
- Dividing an annual return across days gives an average, not the actual daily return path.
- Synthetic daily values can understate variability when treated as historical observations.
- Portfolio analysis requiring daily risk-free returns needs data sampled at the daily frequency.
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# Conversion of yearly values of 3M Tbills to daily rates # Conversion of yearly values of 3M Tbills to daily rates I am trying to convert a time series of yearly values of 3M treasury bills into daily returns (yields) since I want to simulate a portfolio based on them. The data is downloaded from this link: https://fred.stlouisfed.org/series/RIFSGFSM03NA and it has a description : "3-Month Treasury Bill Secondary Market Rate, Discount Basis" I am not sure how this discount basis is computed but I assume would be the deviation between the face value and the purchase price in percentage hence the nominal rate and not continuously compounded effective rate. Based on this assumption I am using the following formula to convert to daily: Daily_yield = exp(TB_yearly_value / 360) - 1 which would be equal to the continuous effective daily rate of the treasury bills. Then I am using these daily yields inside a portfolio as the risk free asset rate and converting them again to Log_returns: Daily_log_return = ln(Daily_yield + 1) Is my assumption and the whole approach correct? ## Answer by phdstudent (score 3) https://quant.stackexchange.com/a/81668 No. Not at all. There is no way of converting yearly yields to daily yields. You would need the actual daily data to do that. You would actually be making a huge mistake, since yields at the daily level would look way less volatile than they actually are. With a stock market return is easier to understand the mistake you are making. Let us say that stock XYZ was trading at \$100 and after a year it was trading at \$110. There were no dividends. This stock had an annual return of 10%. It of course had an average daily return of $\approx 10\%/252 = 0.04\%$. But I cannot use a daily series of 0.04% for anything, specially not to compute daily betas and so on. In fact, this stock may be super volatile and you would not be able to capture that.
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