A-70503, MARCH 14, 1936, 15 COMP. GEN. 793

A-70503: Mar 14, 1936

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COMPOUND - COMPUTATION - INTERMEDIATE INTEREST DATES IN THE VALUATION OF SECURITIES OF A FEDERAL AGENCY SOLD TO A FEDERAL CORPORATION THERE IS FOR APPLICATION. INSOFAR AS VALUATION BETWEEN ANY TWO ESTABLISHED SEMIANNUAL COMPOUND INTEREST PERIODS IS CONCERNED. 360 THE FINAL RESULT IS OBTAINED BY USE OF THE FOLLOWING FORMULA: (1 TIMES PA) PA P EQUALS . THE QUESTION IS ARISING CONTINUALLY IN OUR TRANSACTIONS WITH THE RECONSTRUCTION FINANCE CORPORATION. UPON WHICH A DISCOUNT IS ALLOWED FOR THE INTEREST FREE PERIODS. AN ARITHMETICAL CHECK ON THIS FORMULA PROVES THE METHOD WILL PRODUCE THE PROPER RETURN. A RULING ON THIS MATTER WILL BE APPRECIATED. WHICH WILL BE DISCUSSED HEREAFTER. THE DATE OF THIS TRANSACTION IS BETWEEN THE EIGHTH AND NINTH 6-MONTH PERIODS PRIOR TO MATURITY.

A-70503, MARCH 14, 1936, 15 COMP. GEN. 793

INTEREST, COMPOUND - COMPUTATION - INTERMEDIATE INTEREST DATES IN THE VALUATION OF SECURITIES OF A FEDERAL AGENCY SOLD TO A FEDERAL CORPORATION THERE IS FOR APPLICATION, INSOFAR AS VALUATION BETWEEN ANY TWO ESTABLISHED SEMIANNUAL COMPOUND INTEREST PERIODS IS CONCERNED, THE SO- CALLED TRUE-VALUE BASIS, THE COMPOUNDING OF INTEREST BEING A PROCESS OF CONTINUING DEVELOPMENT DURING THE INTEREST PERIOD AND NOT A SPASMODIC SEMIANNUAL OCCURRENCE.

ACTING COMPTROLLER GENERAL ELLIOTT TO THE ADMINISTRATOR, FEDERAL EMERGENCY ADMINISTRATION OF PUBLIC WORKS, MARCH 14, 1936:

CONSIDERATION HAS BEEN GIVEN YOUR LETTER OF JANUARY 23, 1936, AS FOLLOWS: RE: STATE OF GEORGIA HIGHWAY REFUNDING CERTIFICATES, DOCKETS 3530, 4699,

4143, 1602, 2055, 2055-A, 2944, 4688, 5988, 6748, 4687, AND 3523

ON DECEMBER 28, 1935, THE PUBLIC WORKS ADMINISTRATION SOLD TO THE RECONSTRUCTION FINANCE CORPORATION $573,871.92, PAR AMOUNT OF THE ABOVE CERTIFICATES ON A 4 PERCENT TRUE DISCOUNT BASIS TO MATURITY.

THE MATURITIES AND AMOUNTS INVOLVED IN THIS TRANSACTION FOLLOW:

TABLE

DATE OF MATURITY TIME PAR AMOUNT

3-25-36 ------------------- 87 DAYS --------------------- $31,500.00

3-25-37 ------------------- 1 YEAR 87 DAYS -------------- 54,200.00

3-25-38 ------------------- 2 YEARS 87 DAYS ------------- 67,600.00

3-25-39 ------------------- 3 YEARS 87 DAYS ------------- 64,200.00

3-25-40 ------------------- 4 YEARS 87 DAYS -------------93,000.00

3-25-41 ------------------- 5 YEARS 87 DAYS ------------- 88,000.00

3-25-42 ------------------- 6 YEARS 87 DAYS ------------- 57,000.00

3-25-43 ------------------- 7 YEARS 87 DAYS ------------- 55,000.00

3-25-44 ------------------- 8 YEARS 87 DAYS ------------- 43,185.95

3-25-45 ------------------- 9 YEARS 87 DAYS ------------- 20,185.97

PUBLIC WORKS ADMINISTRATION METHOD OF COMPUTING THIS DISCOUNT USES THE FOLLOWING FORMULA:

1

------- TIMES PAR AMOUNT EQUALS PROCEEDS

(102/N

TD EQUALS PA MINUS P

THE RECONSTRUCTION FINANCE CORPORATION FORMULA ACCEPTS OUR VALUES UP TO 6 MONTHS BEFORE MATURITY; FOR THE SPLIT PERIOD OF 87 DAYS THEY USE THE FOLLOWING FORMULA:

PA PA

------------ EQUALS P EQUALS --------------

87 TIMES 04 1.0096666 1 PLUS ------------ 360

THE FINAL RESULT IS OBTAINED BY USE OF THE FOLLOWING FORMULA:

(1 TIMES PA) PA

P EQUALS ------------ TIMES -----------------

(102/N 1.0096666

THE COMPUTATIONS BY THE TWO METHODS DEVELOPED A DIFFERENCE OF $23.60 FOR ALL MATURITIES.

THE QUESTION IS ARISING CONTINUALLY IN OUR TRANSACTIONS WITH THE RECONSTRUCTION FINANCE CORPORATION, PARTICULARLY IN THE SALE OF RAILROAD BONDS, UPON WHICH A DISCOUNT IS ALLOWED FOR THE INTEREST FREE PERIODS.

THE RECONSTRUCTION FINANCE CORPORATION METHOD APPEARS TO US TO BE CORRECT AS IT REPRESENTS DISCOUNT COMPOUNDED SEMI-ANNUALLY UP TO 6 MONTHS OF MATURITY, PLUS SIMPLE DISCOUNT FOR THE SPLIT PERIOD. AN ARITHMETICAL CHECK ON THIS FORMULA PROVES THE METHOD WILL PRODUCE THE PROPER RETURN, WHEREAS THE FORMULA

1

----------- TIMES PA

(102/N

DOES NOT QUITE EQUAL THE 4 PERCENT RETURN REQUIRED BY THE RECONSTRUCTION FINANCE CORPORATION.

A RULING ON THIS MATTER WILL BE APPRECIATED.

THERE APPEARS NO QUESTION BETWEEN THE PUBLIC WORKS ADMINISTRATION AND THE RECONSTRUCTION FINANCE CORPORATION AS TO THE VALUATION OF THE SECURITIES AS AT ANY 6-MONTH PERIOD PRIOR TO MATURITY. THE QUESTION PRESENTED INVOLVES THE COMPUTATION OF VALUE FOR THE SPLIT PERIOD WHEN THE TRANSACTION TAKES PLACE BETWEEN ANY TWO 6-MONTH PERIODS PRIOR TO MATURITY.

FOR THE PURPOSE OF ILLUSTRATING THE VARIOUS BASES OF VALUATION, WHICH WILL BE DISCUSSED HEREAFTER, THERE HAS BEEN SELECTED ONE OF THE TRANSACTIONS PRESENTED IN THE SUBMISSION, VIZ: $93,000 NONINTEREST BEARING BONDS DUE MARCH 25, 1940, OR IN 4 YEARS 87 DAYS. THE DATE OF THIS TRANSACTION IS BETWEEN THE EIGHTH AND NINTH 6-MONTH PERIODS PRIOR TO MATURITY; THAT IS TO SAY, 93 DAYS SUBSEQUENT TO THE NINTH 6-MONTH PERIOD OR 87 DAYS PRIOR TO THE EIGHTH 6-MONTH PERIOD.

THE VALUE AT ANY 6-MONTH PERIOD PRIOR TO MATURITY IS OBTAINED BY USE OF THE FOLLOWING FORMULA:

1

PAR AMOUNT TIMES ---------

(1.02) TO THE NINTH POWER

THEREFORE, THE VALUE OF THE SECURITIES ABOVE-MENTIONED AT THE EIGHTH AND NINTH 6-MONTH PERIODS PRIOR TO MATURITY, RESPECTIVELY, IS AS FOLLOWS: EIGHTH PERIOD EQUALS

1 PA TIMES -------- EQUALS $93,000 TIMES .85349037 EQUALS $79,374.60441

(1.02) TO THE EIGHTH POWER NINTH PERIOD EQUALS

1 PA TIMES -------- EQUALS $93,000 TIMES .83675527 EQUALS $77,818.24011

(1.02) TO THE NINTH POWER

A DISCUSSION OF THE VARIOUS BASES OF VALUATION FOR THE FRACTIONAL PERIOD FOLLOWS:

1. BASIS PROPOSED BY THE R.F.C.--- UNDER THIS BASIS THE VALUE OF THE SECURITIES AS AT THE NEXT 6-MONTH-INTEREST PERIOD (EIGHTH PERIOD) IS DISCOUNTED AT SIMPLE DISCOUNT FOR 87 DAYS BY THE USE OF THE FOLLOWING FORMULA:

1 1

PA TIMES ----------- TIMES ------------------- EQUALS

PROCEEDS(1.02) TO THE 1 PLUS .02 TIMES 87

EIGHTH POWER ------------

180 1

OR 79374.60441 TIMES ---------------- EQUALS 78,614.67

1.0096666

THIS BASIS IS INACCURATE, AND IN THIS INSTANCE OPERATES TO THE BENEFIT OF THE PURCHASER (THE R.F.C.) AND TO THE DETRIMENT OF THE SELLER (THE P.W.A.).

2. ORDINARY INTERPOLATION BASIS.--- A BASIS MORE COMMONLY USED FOR THE VALUATION OF SECURITIES SOLD OR PURCHASED AT INTERMEDIATE DATES IS THE ORDINARY INTERPOLATION METHOD, WHEREBY THE APPROPRIATE PROPORTION OF THE DIFFERENCE BETWEEN THE VALUES AS AT THE TWO 6-MONTH PERIODS INVOLVED IS SUBTRACTED FROM THE VALUE AT THE NEXT PERIOD OR ADDED TO THE VALUE AT THE LAST PERIOD, AS FOLLOWS:

TABLE

VALUE AT EIGHTH PERIOD EQUALS ------------- 79,374.60441

VALUE AT NINTH PERIOD EQUALS -------------- 77,818.24011

DIFFERENCE -------------------------- 1,556.36430

( 87 (

79,374.60 MINUS (--- OF DIFFERENCE) OR 752.24 EQUALS 78,622.36

(180 ( OR

( 93 (

77,818.24 PLUS (--- OF DIFFERENCE) OR 804.12 EQUALS 78,622.36

(180 (

THE METHOD OF ORDINARY INTERPOLATION HAS ALSO BEEN RECOGNIZED AS BEING INACCURATE, AND IN THIS INSTANCE OPERATES TO THE BENEFIT OF THE SELLER. THE SAME RESULT IS OBTAINED BY ALLOWING THE SELLER SIMPLE INTEREST FOR THE FRACTIONAL PERIOD ON THE VALUE AT THE LAST INTEREST PERIOD, AS FOLLOWS:

( .02 TIMES 93)

77,818.24011 TIMES (1 PLUS ------------) OR 1.0103333 EQUALS 78,622.36

( 180 (

3. TRUE VALUE BASIS.--- THE METHOD USED BY THE PUBLIC WORKS ADMINISTRATION IN THE VALUATION OF THE SECURITIES IN QUESTION, FOR THE FRACTIONAL PERIOD, IS KNOWN AS THE TRUE VALUE BASIS. UNDER THIS METHOD, INSTEAD OF SIMPLE INTEREST OR DISCOUNT FOR THE FRACTIONAL PERIOD, THERE IS USED THE FRACTIONAL POWER OF THE RATIO OF INTEREST ACCUMULATION FOR ONE PERIOD, WHICH PRODUCES THE SAME EFFECTIVE RATE OF RETURN TO BOTH BUYER AND SELLER. TABLES OF TRUE VALUE FACTORS FOR MONTHS AND DAYS ARE PUBLISHED BY THE FINANCIAL PUBLISHING CO.

THE VALUE OF SECURITIES FOR THE INTERMEDIATE PERIOD, ON A TRUE VALUE BASIS, IS OBTAINED BY THE USE OF THE FOLLOWING FORMULAS:

1 1

(A) PA TIMES ------- TIMES ------------- EQUALS PROCEEDS

(1.02) TO (1.02) TO THE EIGHTY-SEVEN ONE

THE EIGHTH HUNDRED EIGHTIETHS POWER

POWER 1

$79,374.60441 TIMES ---------- EQUALS $78,618.51

1.00961722 OR 1

(B) PA TIMES ------ TIMES (1.02) TO EQUALS PROCEEDS.

THE NINETY-THREE ONE HUNDRED

EIGHTIETHS POWER (1.02)

TO THE NINTH POWER

$77,818.24011 TIMES 1.010283878 EQUALS $78,618.51

METHODS 1 AND 2, ABOVE MENTIONED, FOR VALUING SECURITIES AT INTERMEDIATE PERIODS, ARE PREDICATED ON THE ERRONEOUS ASSUMPTION THAT THE COMPOUNDING OF INTEREST OR DISCOUNT OPERATES SPASMODICALLY TWICE A YEAR--- AT THE EXACT 6-MONTH PERIODS--- AND THAT THE GROWTH OR DIMINUTION OF VALUE BETWEEN THE 6-MONTH PERIODS OPERATES AT SIMPLE INTEREST OR DISCOUNT. WILL BE NOTED THAT DIFFERENT RESULTS ARE OBTAINED BY THE USE OF THESE METHODS--- $78,614.67 BY SIMPLE DISCOUNT, AND $78,622.36 BY SIMPLE INTEREST OR ORDINARY INTERPOLATION--- OPERATING TO THE DETRIMENT OF THE SELLER OR THE PURCHASER AS THE CASE MAY BE. THE COMPOUNDING OF INTEREST IS A PROCESS OF CONTINUING DEVELOPMENT, AND THE TRUE VALUE BASIS (METHOD 3, ABOVE) IS THE CORRECT METHOD FOR ADJUSTING THE PRICE OF SECURITIES AT INTERMEDIATE DATES. THIS METHOD, AS HERETOFORE STATED, PRODUCES THE SAME EFFECTIVE YIELD TO BOTH BUYER AND SELLER, AND APPEARS TO BE MORE AND MORE COMMONLY USED BY INVESTORS IN LARGE TRANSACTIONS.

THE TRUE VALUE BASIS, EXAMPLE NO. 3 ABOVE, HAS BEEN ADOPTED BY THIS OFFICE IN THE AUDIT OF SUCH TRANSACTIONS AND APPEARS TO BE FOR APPLICATION TO THE TRANSACTIONS BETWEEN YOUR ADMINISTRATION AND THE RECONSTRUCTION FINANCE CORPORATION. SEE DECISION TO YOU OF FEBRUARY 27, 1936, A-44023, A -59870.