# Calculating and Comparing Treasury Bill Returns

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Just like when teaching class, if one person asks question, usually multiple people have the same question. Unless it’s that annoying person who always ask questions like “Do I have to put my middle name on the exam? Will you mark down for that?”

Anyways, someone asked how to calculate and compare T-Bill returns to online banks, even without adjusting for tax. I’ll base this off of the recent Treasury Bill auction results. Let’s take the most recent one:

Term – How long until the T-bill matures
Issue Date – When the T-bill is actually issued, and when the ‘Price per \$100’ is taken out of your bank account for online payment.
Maturity Date – When the T-bill matures, and you get paid full par value (multiple of \$1000) of your T-bill into your bank account.

Discount rate – This is the rate actually set at auction, which is why it’s always a nice round number like 4.540 in this case. You really don’t need to pay attention to this rate, but here is how it’s figured out anyways. The discount rate is the annualized rate of return based on the par value of the bills and is calculated on a 360-day year.

Price per \$1 of T-bill = 1 – (discountRate * term / 360)

or in this case = 1 – (.04540 * 28 days / 360 days) = .996468888…

which matches the listed Price Per \$100 of \$99.646889.

This means that for a T-Bill with a maturity value of \$1,000, you would pay \$1000 x .996468888 = \$996.47.

Investment rate – This is the rate you should really look at. It is APR, not APY. Therefore it is based on our regular 365-day year, and reflect the actual annualized rate to maturity.

Return per \$1 invested = investmentRate * Term/365

or in this case = .04619 x 28 days/365 days = \$0.00353.

So if you bought a \$1,000 T-Bill, you would get \$3.53 of interest after 28 days, and pay (\$1000-\$3.53) = \$996.47 upfront. Notice how that matches the number above?

The CUSIP is just an identification number for securities.

But wait – we always compare APYs of banks! So how do we find the APY for T-Bills?

APR = PeriodicRate x Periods in a Year

.04619 = 0.00353 x (365/28)

APY = (1 + PeriodicRate)^(Periods in a Year) – 1

= 1.00353^(365/28) – 1 = .0472 = ~4.70% APY

Of course, to actually get this rate for a whole year, this assumes that you have a year of consecutive 28-day T-Bills at the same rate, which won’t happen. [Added: It also assumes you can reinvest the interest at the same rate seamlessly, which you can’t. But you can reinvest them somewhere else. See comments.] But this is what online banks are like as well, since they change their rates all the time. Again, we are ignoring any tax benefits as T-Bills returns are exempt from state and local taxes.

So there you go. I hope this helps people better understand the auction results page, and realize that you can do a bit better than the APR, but probably not as good as the APY, if you reinvest the interest from your T-Bills.

I’ve also used this information to make a 28-Day Treasury Bill APY Calculator.

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1. CM says

The APY calculation you used for the t-bills assumes that you reinvest the interest each time a bill matures, which may not be a realistic assumption. For a lot of people, the APY probably equals the APR. Thanks for all the great info on treasury securities!

2. Jonathan says

Yes, that’s a good point. Hopefully it’s earning interest somewhere if it’s a significant sum of money.

3. sid says

thank you!

4. bigmouth says

So, how about we run a historical chart comparing ING direct (or any other high yield money market banking) with US Treasury? If the one year track showing T-bill is higher, I will certainly go for that. Also, what kind of factors will drive T-Bill up or down? I assume it is affected by totally different factors as that of bank rates.

5. Kevin says

Great work. Yes, APR is what we should look at, because all the earnings are usually not reinvested – you can not purchase any tbills with \$3.54 every 28 days unless you have sufficient funds.

6. Dan says

Well, the T-bill rates are determined at auction while each bank can decide it’s own rate.

But both are certainly affected by Fed Fund rate decisions, just like most other shorter term rates.

As many have probably noticed over the long chain of Fed rate hikes since 2004, banks have obviously (and understandably) been slower in increasing their rates than the market has been with T-bills. Although I don’t have numbers in front of me, my guess would be banks would be a little quicker when rates are headed the other direction, as we’re likely to see in the near future.

7. Lazy Indexer says

Great post! This info should be on the Treasury website.

One thing…the investment rate should be applied to the amount invested, not the full face amount of the T-Bill. So for the \$1000 T-Bill, the return is:

\$1000 * [Price per \$1] * 4.619% * 28 days / 365 days

= \$3.53 (not \$3.54)

8. Jonathan says

Thanks, fixed to make more clear. I think there are some other rounding errors too, but I’ll fix those later.

9. LZ says

Hey All,
I am a new comer. I have two quick questions.
1. How to associate an Emigrant saving account with TD without mailing a check to Emigrant?
2. Are there any charges if I use the emigrant saving account to buy T-bills?

Thanks.

LZ

10. Gonzalo says

Great post!!
I’ve been buying T-bills for more than a year, and always had a few cents difference between my numbers and actual values…now I know how to check it better.

Thanks!!

11. Dan says

The TD website lets you add additional banks. You just need the routing number for Emigrant Direct and your account number. Click on “Manage Direct” and then “Update my bank information” to get to these settings.

Emigrant Direct won’t charge you TD purchases, but you do need to keep track of your total number of withdrawls per month. If you exceed 6 withdrawls (of any type) in a single month, they can penalize you or even close your account.

For example, if you were to buy a T-bill every week from Emigrant, you would eat up four of that month’s allowed withdrawls right away. (Of course, you could just use the TD’s C of I for reinvesting instead of using your limited withdrawls at Emigrant.)

12. LZ says

Dan, Thanks a lot for your sincere help. Your post really solve the puzzle I had. I will give T bill a try next week.

LZ

13. CM says

In regard to a comment from one of the previous posts, the bank rate required to equal the tax-advantaged t-bill rate, assuming you itemize deductions, is simply:

t-bill rate * 1/(1-ST)

where ST is your marginal state tax rate.

14. SteveK says

Re. the upcoming calculator
Yes, a calculator would be handy comparing the APR of T-Bills and APY of savings/MM accounts at banks. I don’t know what would be more useful, convert APR to APY (with the assumption that one can re-invest the interest from a T-Bill like one can with a savings account) or convert APY to APR.

15. Dean says

The APY calculation you used for the t-bills assumes that you reinvest the interest each time a bill matures, which may not be a realistic assumption. For a lot of people, the APY probably equals the APR.

As long as you sweep out the “leftover” amounts that are not re-invested into a high yield savings account, you will be getting an actual yield much close to APY rather than APR.

16. SteveK says

Are you sure that the Investment Rate is APR not APY? Reading today’s 4-week T-Bill auction results, the asterisk next to the Investment Rate says “Equivalent coupon-issue yield.”

It is a bit confusing when they use “rate” and “yield” to describe the same number. Then again, maybe in this case since the interest is not re-investible, they are the same?

17. Dean says

Are you sure that the Investment Rate is APR not APY? Reading today’s 4-week T-Bill auction results, the asterisk next to the Investment Rate says “Equivalent coupon-issue yield.”

Correct, the Investment Rate APR is not equivalent to the APY that banks quote for checking, savings, and CDs. Essentially, the Investment Rate APR calculation ignores compounding. APY calculation includes compounding.

The only way your actual return will be equal to APR is if you invest the interest in a 0% account for the rest of the year, which is not realistic (hopefully).

On the other hand, you may not be able to achieve APY either, since you can’t re-invest the interest in T-Bills since they’re in \$1000 increments.

But if you sweep the interest paid into a high yield savings that’s at 4.5%, your actual return is going to be much close to APY than APR.

18. Jonathan says

The term “yield” is used differently when talking about bonds than when talking about APY. The investment rate listed is definitely the equivalent to an APR rate. No compounding is taken into account.

19. Michael says

I agree with the poster who said this stuff should be on the Treasury Direct website, it isn’t user friendly with respect to “help” and there is no way to call anyone. Just email, which they don’t seem to reply to.

Do you happen to know the answer to this question: Assume the purchase of the recent 13-week Bill that I set to mature into the Zero C of I upon its maturity date of Thursday 8/3/06.

Can I use those funds to buy T-Bills from the Monday 7/31/06 auction that will be issued on 8/3/06 (i.e. the same day the old Bill matures)?

Assuming I can, your point about not being able to reinvest the interest is valid, because I can only purchase in increments of round \$1,000s.

So then I guess I need to redeem that C of I to my bank for the difference (interest).

20. Michael says

I don’t know if this thread is dead or what, but I heard beack from Customer Support and the answer is YES, you can bid on T-Bills and use as your funding a T-Bill that is maturing the same day the new T-Bill will be issued.

21. Noman says

Michael,

The time delay between the purchase settlement and the maturity is set so you can you the proceeds from a maturing T-bill to pay for the next one. I have a ladder of 28 day bills chugging along since November. Just have to pick-up the interest every Thursday and sweep it into savings. Overall the system seems to work.

22. Dean says

Just email, which they don’t seem to reply to

Actually, they do respond to email within 2-3 days. And I’ve asked really specific, complicated questions that they fully answered.

23. tuphat says

J — I know I am being lazy and all, but … Have you created a calcluator for easily translating T-bill auction results into APY? TIA.

You can also use Excel and the formula TBILLYIELD. You can get the official explanation here: http://office.microsoft.com/en-gb/assistance/HP052093111033.aspx

This is the calculations I came up with for the 5/11 auction.
Term Issue Date Yield

28-DAY 5/11/06 4.637%
91-DAY 5/11/06 4.797%
182-DAY 5/11/06 4.951%

25. Jonathan says

SimpleStock, looking at the formula underlying that function, it actually just finds the Investment Rate based on a 360-day year (as opposed to 365 above).

26. Doug says

I believe the posted Investment Rate example could be a bit misleading. The expression 0.04619 * 28 / 365 actually evaluates to 0.03543, not 0.0353. Hence the return per dollar invested is \$0.03543, not \$0.03530. For a \$1000 bond one then confirms that an investment return of \$3.53 is earned on the amount actually invested, i.e. on the \$996.47. I.e. \$3.53 = 0.03543 * \$996.47. Otherwise you might be giving someone the impression that one earns the \$3.53 on \$1000.

To pretend one earns the \$3.53 return on the maturity value, not the amount invested, one must invoke the Discount Rate formula. I.e. write that formula as

Return per \$1 face value of T-bill = \$1 * (discountRate * term / 360)

Perhaps an historical finance expert can explain the purpose of the Discount Rate formula?

BTW in MSFT Excel 2002 there?s a function called DISC that is based on the concept of earning the return on the amount invested rather than the maturity value. Unfortunately my Excel’s Help for DISC calls it a ?discount rate? calculation rather than an ?investment rate? calculation. The description details also seem a bit buggered.

27. Harry says

i have a problem.Can somebody help me???

My proff gave me an assignement.I have to invest 1000.000 usd in 5 different investments.

So i had invested in 5 different invst on 15/10/08

1) US T-bills 30%
2) Cac 40 20%
5)Nasdaq 15%

of total money
So i have to calculate my portfolio
using 05/12/08 prices for all these.

I dont know how to do that……

PLz help me

28. Alex says

Hi,
Can anyone tell me how I can calculate annual risk free (13 week T-bill) rate using excel?
When I download the adjusted close prices into my spreadsheet (60 monthly data points), I calculated the monthly returns which was around 2%. Then multiplied it by 12 to give me an answer of 24%. This seems to high for a risk free rate.
Can anyone tell me what I’m doing wrong?
Thank you

29. JO ANN WEST says

Thank you so much for this explanation. It was just what I was looking for.

30. Danny says

Hi everyone, I have a TD account & I am interested in purchasing a 4 week bill which is also 28 days correct? So is the example saying that you will only receive \$3.53 after maturity plus the \$1000 you invested?