Joel R. Anderson

18 Jan 1997 

Richard H. Koch, Director
Margaret H. Carozza
Division of Customer Service
Bureau of the Public Debt
Parkersburg WV 26106-0426

Dear Director Koch and Ms Carozza:

Re the upcoming revision of "Buying Treasury Securities;" publication BD P 009 rev 8/95 by the Bureau of the Public Dept (BPD); page 10, and 11, "Prices, Rates and Yields." A follow up to your letter of 23Dec96.

With your letter came Treasury's formulas in:
Pt. 356, App. B 31 CFR Ch. II (7-1-94 Edition).

With the formulas I was able to reproduce the rate calculations in: "Public Debt News," ("Buying Treasury Securities" (PD P 009, rev 8/95), page 11).

COMMENTs on "Buying Treasury Securities" and the Treasury's rate formulas; something needs to be done:

1. The use of the term *Yield* in "Prices, Rates and Yields" adheres to no convention for yield outside of Treasury, and Treasury's two formulas define yield two ways.

Defining yield in a nonstandard way is deceptive. Defining yield two ways in a nonstandard way is really deceptive.

Using the same nomenclature "investment rate" for two formulas corrupts the discourse, like calling horses *and* cows, ''cows.''

2. "Yields" on 3 and 6 month bills are really APRs, periodic yields expanded to annual by proportional time. APRs are not compound interest, not yields.

3. "Yields" on bills of more than 1/2 year use a formula that, apparently, was invented BC (before computers) to restrict the need for exponentiation required by (1+r)^n, the standard compound interest formula. Not compound interest, this formula is inaccurate, and wildly inaccurate when applied to periods longer than a year.

4. "Yields" are optionally stated on a 365 or 366 day basis, thus the same interest paid over the same number of days can be stated two ways. A "rubber" timebase is bad practice. Users must *know* (figure out) which timebase was used, then recalculate the rate to their own standard (usually a 365 day timebase). (Which they can't anyway, given the other flaws in the rate formulas.)

''y = number of days in year following the issue date: normally 365 but, if the year following the issue date includes February 29, then y is 366.''
          [31 CFR II, page 346/347]

This language seems to imply that bills issued in 1995 a non-leapyear, would have their "yield" computed to a 366-day timebase because 1996 was a leapyear. Treasury's language is ambiguous. In any case, rubber timebases are bad practice.

5. "Yields" are stated to two decimal places. This is not enough accuracy to permit reverse engineering to the original price.

Using APRs, or the more complex formula yields, require knowing the period and the timebase. Rates are stated imprecisely.

The "yield," "investment rates," published by Treasury cannot be compared to APY, the Federal Reserve standard for Truth in Savings. APR agrees with Truth in Lending, but APR is not a compound rate - Truth in Lending isn't truthful. Your long-bill rate agrees with nothing.

[It would be better if Treasury revised its "investment rate" formulas to one formula, a formula reflecting the miracle of compound interest, (1+r)^n on a single timebase, published to several decimal places.]

Treasury "investment rates" aren't yields as the term is commonly used. All the more reason for you to disclose Treasury's actual rate formulas, immediately and up-front, in "Buying Treasury Securities."

Very truly yours, 

Joel R. Anderson

Encl:

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Encl:

Pt. 356, App. B 31 CFR Ch. II (7-1-94 Edition, page 347, for bills of more than 1/2 year.

P*(1+(r-y/2)*(i/y))*(1+i/2)=100

In English:

InitBal*(1+(DaysMat-DaysTimebase/2)
          *(rate/DaysTimebase))*(1+rate/2)=EndBal

P=100/((1+(r-y/2)*(i/y))*(1+i/2))

For: ''investment rate'' 7%, all years 365 days, 365-day timebase:

Year	DaysMat	Price	APY_Fed_365
1	365	93.35107	7.1225
2	730	87.43742	6.942741
3	1095	82.22838	6.739723
4	1460	77.60510	6.543614
5	1825	73.47403	6.358749

Reconverting Price to Treasury's ''Investment rate'' [1]
Year	1	2	3	4	5
Price=	93.351070	87.43742	82.22838	77.60510	73.47403
a=	0.25	0.75	1.25	1.75	2.25
b=	1	2	3	4	5
c=	-0.071225	-0.14367	-0.21612	-0.28857	-0.36102
i=	0.07	0.07	0.07	0.07	0.07
Extract compound rate from Price:
APY=	7.1225	6.942741	6.739723	6.543614	6.358749

1. PROTOTYPE: Treasury, over 1/2 year
a=	DaysMat/(2*DaysTimebase)-0.25
b=	DaysMat/DaysTimebase
c=	(Price-100)/Price
i=	(-b+@SQRT((b^2)-4*a*c))/(2*a)