T-Bill Coupon-Equivalent Yield

yogibearbull

T-Bill Coupon-Equivalent Yield

According to several web sources, Treasury simply uses this formula for Coupon-Equivalent Yield of T-Bills,

Coupon-Equivalent Yield = 100*[(Par Value - Purchase Price)/Purchase Price]* 360/d, where d = days to maturity.

So, Coupon-Equivalent Yield = Total Return * 360/d.

d must be counted properly taking into account specific T-Bill issue and maturity dates. Just because their name says 13-wk, 26-wk, 52-wk, that doesn't mean 13*7, 26*7, 52*7 days.

Don't ask my why the Treasury wants to put all T-Bills on 360 day standard.

The 52-wk T-Bill today (8/8/23) will be issued on 8/10/23 but will mature on 8/8/24, so that is 2-3 days less than a full year (or, 362-363 days). It seems that Treasury used d = 362.76.

Price was 94.883778, so TR = (100 - 94.883778)/94.883778 = 5.392%.

So, Coupon-Equivalent Yield = 5.392*360/362.76 = 5.351%. That is what Treasury provides, but I don't like this at all. It doesn't related to any realistic TR or YTM that one may calculate.

To me, if the TR is 5.392% for 362.76 days (implied by Treasury), I would annualize it as 5.392*365/362.76 = 5.425% annualized.


52-Wk T-Bill Auction on 8/8/23 https://www.treasurydirect.gov/instit/annceresult/press/preanre/2023/R_20230808_2.pdf
https://www.investopedia.com/terms/c/couponequivalentrate.asp
https://www.bogleheads.org/forum/viewtopic.php?t=248337

Sven

Thanks for the info. We bought some 12 months T bill this week as other T bills matured. Yields are rising after Fitch’s downgrade of US Treasuries. For taxable account we prefer T bills due to the state tax exemption. Otherwise, brokered CDs are very competitive.

By the way, Warren Buffet buys $6 billions worth of 3 or 6 months T bill every week. He does not like longer term T bills.

yogibearbull

I bought 52-wk this week after a LONG time. Next week, I am back to rolling into 26-wk.

Sven

Same for us too.

rforno

Threw a very large chunk of cash from recent stock sales into another 2-week bill today.

Tarwheel

I bought 26 and 52-week T-bills this week. I had a lot of cash from liquidating my muni funds. I’ve also got a number of CDs and T-bills that will be maturing between now and December, as part of ladders. Hopefully, there will still be some good options at that time. If interest rates finally stabilize, then I’ll start putting some of this cash back in bond funds.

rforno

I was at my CU today to open up a MM account to locally store some $$$ at my bank outside of my checking account -- mostly to reduce the chances of potential significant direct-debit/ACH losses should any of my vendors get compromised.

Amazing that even at the CU their MM is only paying 1.X% interest when I would make 4.X or more parking money in USTs ... that's a pretty *big* spread, imho.

Derf

@rforno : Not a shocker here. Thursday purchased cd at BMO for apy 5.15% 13 month cd. 3-4-6 month cds paying less than 1% .01 &.05 apy !!!! I really got a GREAT Deal.

msf

According to several web sources, Treasury simply uses this formula for Coupon-Equivalent Yield of T-Bills,

Coupon-Equivalent Yield = 100*[(Par Value - Purchase Price)/Purchase Price]* 360/d, where d = days to maturity.

Which just goes to show that you can't believe everything you read on the web. (In all fairness, the second post in the Bogleheads thread correctly says that 365 days are used and references the same Treasury sources I'm relying on below. Except it misses an added complication for T-bills maturing in more than six months.)

According to the Treasury (the authoritative source), the above formula is not what is used for Coupon Equivalent Yield of T-bills. It is almost correct for T-bills maturing in six months or less, except that the correct formula uses 365 or 366 day years. For T-bills with longer maturities (still under a year), a quadratic equation must be solved.

Some people may not be clear about what Coupon Equivalent Yield represents.

The Coupon Equivalent, also called the Bond Equivalent, or the Investment Yield, is the bill's yield based on the purchase price, discount, and a 365- or 366-day year. The Coupon Equivalent can be used to compare the yield on a discount bill to the yield on a nominal coupon security that pays semiannual interest with the same maturity date.

https://home.treasury.gov/resource-center/data-chart-center/interest-rates/TextView?type=daily_treasury_bill_rates&field_tdr_date_value_month=202209

In plain English, if you have a bond with a 4% coupon purchased at par ($100), then every six months it pays $2. To compute the annualized rate of return one assumes that the coupons are reinvested at the original rate. So after one year, one would have:
$100 x (1 + 2%) x (1 + 2%) = $100 x 1.0404 = $104.04. That's an annual yield of 4.04%.

T-bills don't have coupons, but if they did, this particular T-bill would pay coupons at an annual rate of 5.351%. That is, a six month coupon would pay half of that, or 2.6755%.

When one applies the same compounding as above (though reducing the second coupon by 2 days simple interest), one finds that the government figures are correct:

Rate (i)		5.351%
Price per $100		$94.883778
Days in 2024 (y)	366
1/2 year coupon		2.6755%		(1/2 X 5.351%)
Total days to maturity	364
2nd half frac of year	0.494535519	(364 - 366/2) / 366
2nd half coupon		2.6463%		(0.494... x 5.351%)

Compounding the coupons as before (except the second coupon isn't for a complete half year):
(1 + 2.6755%) x (1 + 2.6463%) = 105.3926%.

As in the OP, 5.392% is the actual total return. It is higher than the six month (coupon equivalent) yield, because coupons compound.

This is important. New issue T-bills have APYs greater than their coupon equivalent yields.

If a new six month T-bill has a 5.0% "coupon equivalent yield" it will pay 2.5% (half of 5%) after six months by definition. That would compound to 5.06¼% APY if reinvested at the same rate for another six months. That beats a six month CD with a 5.0% APY, paying just 2.47% at its six month maturity.

Treasury page with coupon equivalent yield formulae, examples

Treasury Regulation (Code of Federal Regulations) rules on calculating T-bill discount rates.

Note that the 360 day calendar is used when calculating the bank discount rate, but a 365 or 366 day calendar is used when calculating the "true discount" rate.

yogibearbull

@msf, thanks for Treasury and CFR links. I will go over them.

Interesting that calculations are different for < 6 months (an approximation) and 6-12 months (exact calculations).

There are 4 examples presented in the Treasury link. The first two (#1, #2) use 360-day year for price and discount rate calculations, the other two (#3, #4) use 365/366 days for coupon-equivalent yield.

In examples #3 and #4, the first full 6-month period is defined as 365/2 =182.5 or 366/2 = 183. So, it is possible that Treasury just defaults to 360-day year convention in the approximation used for less than 6 months rather than using "odd" 182.5 or 183 days.

I was also solving quadratic equation for interest rate in my notes (and shown in this BB LINK), but I didn't present those details elsewhere as the results weren't matching anyway with the Treasury calculations. But I see that Treasury is using 2 unequal compounding periods while I was using 2 identical periods for compounding (and the same as you did in your plain English bond example.

Lot of food for thought.

msf

deleted (for now)

yogibearbull

Many may not be following these details. But using the formula in Treasury LINK provided by @msf, I was able to verify the data for the recent 52-wk Auction.

In the recent 52-week Auction example in the OP, TR = 5.392% (calculated) and i = 5.351%, d = 363 (Treasury data).

So, 1 + 0.01*TR = (1 + 0.05351/2)*[1 + 0.05351*(363 - 365/2)/365],

or, TR = 5.3925%, and that is close enough to 5.392%.

LINK2