> Joel R. Anderson
18 Jan 1997
Richard H. Koch, Director
Margaret H. Carozza
Division of Customer Service
Bureau of the Public Debt
Parkersburg WV 261060426
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 (7194 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 nonleapyear, would have their "yield"
computed to a 366day 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 longbill 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 upfront, in "Buying Treasury Securities."
Very truly yours,
Joel R. Anderson
Encl:
Encl: =======================================
Pt. 356, App. B 31 CFR Ch. II (7194 Edition,
page 347, for bills of more than 1/2 year.
P*(1+(ry/2)*(i/y))*(1+i/2)=100
In English:
InitBal*(1+(DaysMatDaysTimebase/2)*
(rate/DaysTimebase))*(1+rate/2)=EndBal
P=100/((1+(ry/2)*(i/y))*(1+i/2))
For: "investment rate" 7%, all years 365 days, 365day 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= (Price100)/Price
i= (b+@SQRT((b^2)4*a*c))/(2*a)

