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Author: imdajunkman Three stars, 500 posts Add to my Favorite Fools Ignore this person (you won't see their posts anymore) Number: of 35351  
Subject: Yield-Math Madness Date: 6/19/2006 5:14 PM
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It's amazing what can be discovered once one starts questioning conventional assumptions and practices.

Wendy's thanking to me (for having laid out some of the steps by which I discovered that E*Trade's yield calculations shouldn't be trusted) was appreciated. But her appreciation was a bit premature. The problem is worse than I suspected, as can be discovered if you grind through an estimate of the YTM and YTC that E*Trade is reporting for the Freddie MAC zeros of '36, which they say are 6.724% and 7.3%, respectively, but which appear to be higher.

But let's consider another example instead. For the most recently auctioned 6-month T-bill, to be issued 6/22 and maturing 12/21, TreasDirect is reporting a discount rate of 5.055% and an investment rate of 5.260%. That's a fat return for short-term, high-quality paper. But are those returns actually better than they are reporting? In other words, what is their APR/APY so that comparisons can be made with competing instruments (setting aside, for now, the fact of their tax advantagement, which will kick the effective yield higher for most purchasers).

Excel happens to have a formula, called YIELDDISC, in its Analysis Toolpak which seems ideally suited to calculating the yields on T-Bills. The formula asks for various inputs: settlement date; maturity date; price; par; etc., all of which can have only a single answer. The only problematic portion of the formula is selecting the calendar “basis”, which can be any of the following: 30/360, Actual/actual, Actual/360, Actual/365. (For full details, see the “Help Files” in Excel.)

Below is an example of what the formula would look like (which, if copied from this post and pasted into your own Excel program, should work just fine. But it won't, due to hassles with cell formatting which I haven't yet resolved.)

=YIELDDISC(“06/22/2006”, “12/21/2006”, 97.444417, 100, 1)

Also note that I've selected as the basis choice number 1, “Actual/actual”, as being the most robust. In other words, leap year or not, the formula “should” reflect the actual holding period. (It won't, but I'll get to that point in a minute.) For now, trust me that if you type the formula into Excel (instead of copy & pasting it), it will return an answer which can then be converted to a percentage with two decimal places. And if you select each of the five calendar basis choices in turn and do the percentage trick, you will end up will the following:

0.052745 5.27% 0 US (NASD) 30/360
0.052596 5.26% 1 Actual/actual
0.051876 5.19% 2 Actual/360
0.052596 5.26% 3 Actual/365
0.052745 5.27% 4 European 30/360

5.26% is what TreasDirect is reporting as an investment yield. 5.26% is what Excel is reporting when an “actual” calendar basis is selected. So it seems like there is nothing further to discuss.

But I'm a simple minded fellow, and when I look at a calendar and start counting days, I don't end up with the same results. There are 7 days in a week, and in 26 weeks --the customary holding period for a 6-month bill-- there are 183 days, not the customarily reported 182. (Yes, this number can vary and is reported by the Treasury itself as being anywhere from 181 to 183 depending on how holidays impact auction, issue, and maturity dates.) But “the 182-bill” is how the bill is talked about most generally, and it is the bill that was most recently issued. However, when I buy a 182-bill with funds that were returned to me by a maturing 182-bill, the money comes into my bank account from the maturing bill and then, the same day goes out again to cover the new purchase. So, call the bill whatever you want to, I do not have use of that money for the next 183 days. That 183 days is my effective holding period and that's the numerator that has to be used when calculating year fractions. So, let's do it.

183/365 = 0.501369863.

Store that number in your calculator's memory and then do this calculation:

Gross Return = Par / Purchase Price, or 100/97.444417 = 1.026226059

That number is the gross return obtained over a holding period of 183 days. Now let's annualize it to obtain an APR/APY, which then becomes 5.2990952%, which can be rounded to 5.30%, or a difference of 4 basis points, which matches none of Excel's answers, nor the Treasury's.

So, who is right?

Who knows? Who cares? Run your own numbers. Those are the ones you have to live with. I prefer my method, because I can obtain consistency across any debt instrument of any maturity and make “apples-to-apples” comparisons.

Charlie

PS Even at 5.25%, a 9% state-tax rate kicks the effective yield to 5.78%. That is serious money for short-term, high quality paper. If the market is demanding that kind of return, they're worried, and so should you be.
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