Converting Treasury Bill Discount Quotes into Daily Risk-Free Returns
Article Quant Q&A · Author: user233051
Summary
The document explains how to turn a 13-week Treasury bill’s quoted bank discount rate into a daily risk-free return for calculating excess returns. The quote is divided by 100, then translated using the bill’s price and days remaining to maturity. Since the recently auctioned bill’s maturity declines between auctions, an exact conversion requires estimating the remaining days; the text suggests a representative maturity as a practical approximation.
It also compares simpler conventions. For daily returns that accrue across weekends, the author finds that using the discount rate divided by 360 is close to the more exact conversion across the examples shown. Dividing by 252 is presented as less accurate for that purpose. A more precise alternative is to model a specific bill’s daily marked-to-market returns and roll into a new bill monthly, which requires accurate prices. The monthly-rate shortcut of dividing by 252 is described as adequate for many applications, though less precise.
Key ideas
- A Treasury bill bank discount quote is based on par value and a 360-day year, so it is not itself a daily return.
- An exact daily conversion depends on the bill’s remaining days to maturity.
- Using the discount rate divided by 360 is a practical approximation when returns accrue over weekends.
- A marked-to-market bill strategy that rolls into a new bill each month requires accurate price data.
- Using a prior month-end rate divided by 252 is a simpler, less precise convention.
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Full text
# how to calculate daily risk free rate using 13 week treasury bill
# how to calculate daily risk free rate using 13 week treasury bill
I want to calculate excess return for AAPL plus the S&P 500. I have computed monthly and daily logarithmic returns for every stock and for the market, I now need to calculate the risk free interest rate in order to be able to compute the excess return for every stock and the market.
Interest rates on three months T-Bills are a good proxy for the risk-free rate of return, but I have a lot of doubts on how to use data provided by Yahoo! Finance in order to compute the daily risk-free. I use the 13 week treasury bill (ticker: ^IRX) historical quotes provided by Yahoo! Finance. I divided t-bill rate by 100 and calculate the daily excess returns.
```
t-bill AAPL Excess
2017-01-31 0.500 -0.00230205 -0.00730020
2017-01-30 0.495 -0.00262402 -0.00762402
```
However, I realized that monthly t-bill rate units are the same as daily rates.
```
t-bill AAPL
Jan 2017 0.500 0.0477
Dec 2016 0.480 0.0479
```
I am making a mistake while computing daily and monthly t-bill rate units. The Federal Reserve (https://fred.stlouisfed.org/series/DTB3) also provide the same t-bill rates for daily and monthly frequency as in Yahoo Finance. How can I correctly calculate the excess returns? Thanks for any answer.
## Answer by Mike O'Connor (score 4)
https://quant.stackexchange.com/a/41168
user233051 notes that ^IRX is indeed the official discount rate of the US Treasury. So to answer his question we need to exactly understand how the Treasury computes the discount rate. My answer is based on www.treasury.gov pages here, here and here.
The official way of calculating the discount rate $d$ is $d = \frac{100-P}{100}\frac{360}{n}$ where $P$ is the price per \$100 of par (face) value and $n$ is the number of days until expiration. In order to get the $d$ of this formula we would divide the ^IRX by 100 because it is stated as a percent. Yes, it's a somewhat arbitrary way of compounding the return, but that's what they do. The Treasury says "The Bank Discount rate is the rate at which a Bill is quoted in the secondary market and is based on the par value, amount of the discount and a 360-day year." The missing 5 or 6 days are for bankers' holidays.
But we want instead $r = \frac{100}{P}^\frac{1}{n}$ where $r$ is the daily risk-free return ratio. So we have some simple algebra to do. We solve the first equation for $P$ and substitute that value in the equation for $r$ and get $r = \frac{ 1 }{ 1 - \frac{dn}{360}}^{1/n}$ where $dn$ is the product of $d$ and $n$.
So we have our answer but we really need to know what $n$ is, for each day. The Treasury says that their discount rates are determined by prices in the secondary market--- for the most recently auctioned bill. New 13-wk bills are auctioned-off every Monday. So this means that at first $n$ is approximately 91 days, but it will decline to about 85 days before a new bill is auctioned. So unless our code is responsive to the days of the week (and we do some further research about the details) we may instead want to consider making an approximation.
To that end, we can do some simple algebra that shows that the quantity that we want to compute is rather insensitive to small differences in the assumed value of $n$, to such an extent that we almost don't really need to have any idea what the value of $n$ is (because it will always be fairly large, at least 85). Since $d$ is quite small and is also divided by $360$, $\frac{1}{1 - \frac{dn}{360}}\approx 1 + \frac{dn}{360}$. And we recognize $1 + \frac{dn}{360}$ to be the leading term in $d$ in $(1 + \frac{d}{360})^n$. Therefore $r\approx 1 + \frac{d}{360}$!
The following Python code explores which approximation is best. In the printout `rf` is the annualized risk-free return ratio, annualized so as to magnify the differences over a respectable period of time.
```
for i in range(1,5):
d = ( float(i) * 0.5 ) / 100.0 # discount rate
nhi = 91.0 # days until maturity
nlow = 91.0 - 6.0
rhi = ( 1.0 / (1 - d * nhi/360.0) )**(1/nhi) # corresponding daily risk-free return ratio (exact)
rlow = ( 1.0 / (1 - d * nlow/360.0) )**(1/nlow)
print( 'discount rate, d = {}'.format(d, '') )
print( 'rf = {} {}'.format(rhi**365.0, 'exact: n = 91 days') ) # rf is the annualized risk-free return ratio
print( 'rf = {} {}'.format( (1+d/252.0)**252.0, '(1+d/252.0)**252.0' ) )
print( 'rf = {} {}'.format( (1+d/360.0)**365.0, '(1+d/360.0)**365.0' ) )
print( 'rf = {} {}'.format(rlow**365.0, 'exact: n = 85 days') )
print('')
discount rate, d = 0.005
rf = 1.00508553843 exact: n = 91 days
rf = 1.00501247101 (1+d/252.0)**252.0
rf = 1.00508228044 (1+d/360.0)**365.0
rf = 1.00508532578 exact: n = 85 days
discount rate, d = 0.01
rf = 1.01020342853 exact: n = 91 days
rf = 1.01004996668 (1+d/252.0)**252.0
rf = 1.01019031932 (1+d/360.0)**365.0
rf = 1.01020257221 exact: n = 85 days
discount rate, d = 0.015
rf = 1.01535391748 exact: n = 91 days
rf = 1.01511261146 (1+d/252.0)**252.0
rf = 1.01532424685 (1+d/360.0)**365.0
rf = 1.01535197778 exact: n = 85 days
discount rate, d = 0.02
rf = 1.02053725469 exact: n = 91 days
rf = 1.02020053039 (1+d/252.0)**252.0
rf = 1.02048419386 (1+d/360.0)**365.0
rf = 1.02053378305 exact: n = 85 days
```
Note that the calculated values that are exact for 91 days and 85 days are almost equal. Next to those the $1 + \frac{d}{360}$ estimate isn't too bad. Those possibilities should be used with code that pays interest over the weekend. If you're just doing bars without regard for the day of the week, you'd like to use $1 + \frac{d}{252}$ but it doesn't fare as well. All things considered, for those of us who do go so far as to give proper consideration to the weekend the most feasible thing to do would seem to be to approximate $n$ by, say, $88$.
## Answer by Helin (score 2)
https://quant.stackexchange.com/a/33080
If you want to do it super precisely, the convention for building fixed-income total return index is as follows:
- You assume at the end of the month, you buy the instrument (in this case a 3-month T-bill).
- You hold this exact T-bill over the course of the next month and mark it to market daily and calculate the daily returns ($P_t / P_{t-1} - 1$, which is standard performance calculation).
- At the end of the month, you sell the T-bill and buy the next 3-month bill.
This of course requires that you have accurate bill pricing data. You can also look at the Bloomberg Barclays 3-Month Treasury Bellwether Index, or the BofA Merrill Lynch 3-Month Bill Index. Both are calculated this way.
Alternatively, a lot of people just use the previous month-end bill rate, divide it by 252, and use that as the daily bill return for the current month. Not as precise, but more than sufficient for most purposes.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.