The Politics of the Annual Percentage Rate (APR)

THE POLITICS OF THE ANNUAL PERCENTAGE RATE (APR)

The APR supposed to reflect the "true" rate of interest, to enable consumers to compare interest rates. [See www.washingtonpost.com/wp-srv/business/longterm/glossary/a_m/annual_percentage_rate.htm for an example of this claim. Or www.mindspring.com/~loan/art04.htm.]

It may enable consumers to compare rates when terms are equal, but it does not do so across differing terms, nor does it give a "true" rate of interest.

Definition: Annual Percentage Rate. According to [whatever the authority is], for a loan repaid N times per year, with the interest at each period assessed at i%, the A.P.R. is defined as follows:

APR = N x i

Thus if a consumer borrowed $10,000 to be repaid in monthly installments over five years, with 1% interest being assessed each month on the remaining balance, the APR would be 12%: 12 months/year times an interest rate of 1%. If a homeowner borrowed $100,000 to be repaid in monthly installments over 30 years, with 1% being assessed each month on the remaining balance, the APR would again be 12%. The length of the loan (30 years vs. five years) makes no difference.

Definition: Consistency. An interest rate and repayment amount are said to be consistent iff (mathematical shorthand for "if and only if") one could lend money to oneself, assess oneself interest at the given rate each period, repay oneself for the loan at the specified amount (for the specified number of payments at the specified frequency), have one's total debt be zero at the end, and have one's total principal and interest be equal to the amount one would receive if one simply left the principal intact at the given interest rate for the given period.

For example, let's assume we are going to repay a $10,000 loan with 60 monthly payments (i.e., over a five-year period). Ignoring the $.37 cents owed at the end, repaying the loan at $222.44/month will discharge the debt, as shown below in the yellow columns. Notice that each month, the borrower is charged 1% interest on the outstanding balance, meaning that this loan would have an APR of 12%.⁽¹⁾

Time BORROWER LENDER
SAVER
$$ KEPT IN THE BANK ACCUMULATING 1% INTEREST / MONTH

1% of outstanding principal payments made to lender outstanding principal interest paid payments received current balance of lender's account initial amount interest paid final balance

12/31/00 -- 0 0 -- -- 10,000 10,000 -- 10,000

1/1/01 -- 0 10,000.00 -- 0 0 10,000 -- 10,000

2/1/01 100.00 222.44 9877.56 -- 222.44 222.44 10,000 100 10,100

3/1/01 98.78 222.44 9753.90 2.22 222.44 447.10 10,100 101 10,201

4/1/01 97.54 222.44 9629.00 4.47 222.44 674.01 10,201 102.01 10303.01

5/1/01 96.29 222.44 9502.85 6.74 222.44 903.19 10303.01 103.03 10406.04

6/1/01 95.03 222.44 9375.44 9.03 222.44 1,134.66 10406.04 104.06 10510.10

7/1/01 93.75 222.44 9246.75 11.35 222.44 1,368.45 10510.10 105.10 10615.20

8/1/01 92.47 222.44 9116.78 13.68 222.44 1,604.57 10615.20 106.15 10721.35

9/1/01 91.17 222.44 8985.51 16.05 222.44 1,843.06 10721.35 107.22 10828.57

10/1/01 89.86 222.44 8852.93 18.43 222.44 2,083.93 10828.57 108.28 10936.85

11/1/01 88.53 222.44 8719.02 20.84 222.44 2,327.21 10936.85 109.37 11046.22

12/1/01 87.19 222.44 8583.77 23.27 222.44 2,572.92 11046.22 110.46 11156.68

1/1/02 85.84 222.44 8447.17 25.73 222.44 2,821.09 11156.68 111.57 11268.25

2/1/02 84.47 222.44 8309.20 28.21 222.44 3,071.74 11268.25 112.68 11380.93

3/1/02 83.09 222.44 8169.85 30.72 222.44 3,324.90 11380.93 113.81 11494.74

4/1/02 81.70 222.44 8028.11 33.25 222.44 3,580.59 11494.74 114.95 11609.69

5/1/02 80.28 222.44 7886.96 35.81 222.44 3,838.84 11609.69 116.10 11725.79

6/1/02 78.87 222.44 7598.35 38.39 222.44 4,099.67 11725.79 117.25 11843.04

7/1/02 75.98 222.44 7451.92 41.00 222.44 4,363.11 11843.04 118.43 11961.47

8/1/02 74.52 222.44 7304.00 43.63 222.44 4,629.18 11961.47 119.62 12081.09

9/1/02 73.04 222.44 7154.60 46.29 222.44 4,897.91 12081.09 120.81 12201.90

10/1/02 71.55 222.44 7003.71 48.98 222.44 5,169.33 12201.90 122.02 12323.92

11/1/02 70.04 222.44 6851.31 51.69 222.44 5,443.46 12323.92 123.24 12447.16

12/1/02 68.51 222.44 6697.38 54.43 222.44 5,720.33 12447.16 124.47 12571.63

1/1/03 66.97 222.44 6541.91 57.20 222.44 5,999.97 12571.63 125.72 12697.35

2/1/03 65.42 222.44 6384.89 60.00 222.44 6,282.41 12697.35 126.97 12824.32

3/1/03 63.85 222.44 6226.30 62.82 222.44 6,567.67 12824.32 128.24 12952.56

4/1/03 62.26 222.44 6066.12 65.68 222.44 6,855.79 12952.56 129.53 13082.09

5/1/03 60.66 222.44 5904.34 68.56 222.44 7,146.79 13082.09 120.82 13212.91

6/1/03 59.04 222.44 5740.94 71.47 222.44 7,440.70 13212.91 132.13 13335.04

7/1/03 57.41 222.44 5575.91 74.41 222.44 7,737.55 13335.04 133.35 13478.49

8/1/03 55.76 222.44 5409.23 77.38 222.44 8,037.37 13478.49 124.78 13613.27

9/1/03 54.09 222.44 5240.88 80.37 222.44 8,340.18 13613.27 136.13 13749.41

10/1/03 52.41 222.44 5070.85 83.40 222.44 8,646.02 13749.41 137.49 13886.90

11/1/03 50.71 222.44 4899.12 86.46 222.44 8,954.92 13886.90 138.87 14025.77

12/1/03 48.99 222.44 4725.67 89.55 222.44 9,266.91 14025.77 140.26 14166.03

1/1/04 47.26 222.44 4550.49 92.67 222.44 9,582.02 14166.03 141.66 14307.69

2/1/04 45.50 222.44 4373.55 95.82 222.44 9,900.28 14307.69 143.08 14450.76

3/1/04 43.74 222.44 4194.85 99.00 222.44 10,221.72 14450.76 144.51 14595.27

4/1/04 41.95 222.44 4014.36 102.22 222.44 10,546.38 14595.27 145.95 14741.23

5/1/04 40.14 222.44 3832.06 105.46 222.44 10,874.28 14741.23 147.41 14888.64

6/1/04 38.32 222.44 3647.94 108.74 222.44 11,205.46 14888.64 148.88 15037.52

7/1/04 36.48 222.44 3461.98 112.05 222.44 11,539.95 15037.52 150.38 15187.90

8/1/04 34.62 222.44 3274.16 115.39 222.44 11,877.79 15187.90 151.88 15339.78

9/1/04 32.74 222.44 3084.46 118.78 222.44 12,219.01 15339.78 153.40 15493.18

10/1/04 30.84 222.44 2892.86 122.19 222.44 12,563.64 15493.18 154.93 15648.11

11/1/04 28.93 222.44 2699.35 125.64 222.44 12,911.72 15648.11 156.48 15804.59

12/1/04 26.99 222.44 2503.90 129.12 222.44 13,263.28 15804.59 158.04 15962.63

1/1/05 25.04 222.44 2306.50 132.63 222.44 13,618.35 15962.63 159.63 16122.26

2/1/05 23.07 222.44 2107.13 136.18 222.44 13,976.97 16122.26 161.22 16283.48

3/1/05 21.07 222.44 1905.76 139.77 222.44 14,339.18 16283.48 162.84 16446.32

4/1/05 19.06 222.44 1702.38 143.39 222.44 14,705.01 16446.32 164.46 16610.78

5/1/05 17.02 222.44 1496.96 147.05 222.44 15,074.50 16610.78 166.11 16776.89

6/1/05 14.97 222.44 1289.49 150.75 222.44 15,447.69 16776.89 167.77 16944.66

7/1/05 12.89 222.44 1079.74 154.48 222.44 15,824.61 16944.66 169.44 17114.10

8/1/05 10.80 222.44 868.30 1158.25 222.44 16,205.30 17114.10 171.15 17285.25

9/1/05 8.68 222.44 654.54 162.05 222.44 16,589.79 17285.25 172.85 17458.10

10/1/05 6.55 222.44 438.65 169.78 222.44 16,978.13 17458.10 174.58 17632.68

11/1/05 4.39 222.44 220.60 173.70 222.44 17,370.35 17632.68 176.33 17809.01

12/1/05 2.21 222.44 .37 173.70 222.44 17,766.49 17809.01 178.09 17,987.10

The important figures are the ones highlighted/enlarged in the last row of the above table. The $.37 (in the third yellow column) shows that at the rate of $222.44/month, the borrower would still owe $.37. (A rate of $222.45 would wind up with the borrower being owed $.41, so it really makes little difference.) The $17,766.49 figure (in the third pink column) says that if the monthly payments were simply saved in an account paying interest at the rate of 1%/month, the account balance at the end would be $17,766.49. The $17,987.10 figure (in the third green column) says that if the lender chose not to give the loan and instead just saved the $10,000 in an account paying 1%/month, the account balance at the end would be $17,987.10.

I'm not sure of my figures here. They show that the two methods aren't equivalent, but I would expect to show the lender having more at the end than the saver. I think I have my logic backwards. [... To be continued.]

BIBLIOGRAPHY

FOOTNOTES

1. The loan amount of $10,000 is used for convenience; consistency does not depend on the loan amount.

Page URL: http://www.d.umn.edu/~schilton/Articles/APR.html
Page Author: Stephen Chilton
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Last Modified: November 27, 1999
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Time	BORROWER			LENDER			SAVER $$ KEPT IN THE BANK ACCUMULATING 1% INTEREST / MONTH
Time	1% of outstanding principal	payments made to lender	outstanding principal	interest paid	payments received	current balance of lender's account	initial amount	interest paid	final balance
12/31/00	--	0	0	--	--	10,000	10,000	--	10,000
1/1/01	--	0	10,000.00	--	0	0	10,000	--	10,000
2/1/01	100.00	222.44	9877.56	--	222.44	222.44	10,000	100	10,100
3/1/01	98.78	222.44	9753.90	2.22	222.44	447.10	10,100	101	10,201
4/1/01	97.54	222.44	9629.00	4.47	222.44	674.01	10,201	102.01	10303.01
5/1/01	96.29	222.44	9502.85	6.74	222.44	903.19	10303.01	103.03	10406.04
6/1/01	95.03	222.44	9375.44	9.03	222.44	1,134.66	10406.04	104.06	10510.10
7/1/01	93.75	222.44	9246.75	11.35	222.44	1,368.45	10510.10	105.10	10615.20
8/1/01	92.47	222.44	9116.78	13.68	222.44	1,604.57	10615.20	106.15	10721.35
9/1/01	91.17	222.44	8985.51	16.05	222.44	1,843.06	10721.35	107.22	10828.57
10/1/01	89.86	222.44	8852.93	18.43	222.44	2,083.93	10828.57	108.28	10936.85
11/1/01	88.53	222.44	8719.02	20.84	222.44	2,327.21	10936.85	109.37	11046.22
12/1/01	87.19	222.44	8583.77	23.27	222.44	2,572.92	11046.22	110.46	11156.68
1/1/02	85.84	222.44	8447.17	25.73	222.44	2,821.09	11156.68	111.57	11268.25
2/1/02	84.47	222.44	8309.20	28.21	222.44	3,071.74	11268.25	112.68	11380.93
3/1/02	83.09	222.44	8169.85	30.72	222.44	3,324.90	11380.93	113.81	11494.74
4/1/02	81.70	222.44	8028.11	33.25	222.44	3,580.59	11494.74	114.95	11609.69
5/1/02	80.28	222.44	7886.96	35.81	222.44	3,838.84	11609.69	116.10	11725.79
6/1/02	78.87	222.44	7598.35	38.39	222.44	4,099.67	11725.79	117.25	11843.04
7/1/02	75.98	222.44	7451.92	41.00	222.44	4,363.11	11843.04	118.43	11961.47
8/1/02	74.52	222.44	7304.00	43.63	222.44	4,629.18	11961.47	119.62	12081.09
9/1/02	73.04	222.44	7154.60	46.29	222.44	4,897.91	12081.09	120.81	12201.90
10/1/02	71.55	222.44	7003.71	48.98	222.44	5,169.33	12201.90	122.02	12323.92
11/1/02	70.04	222.44	6851.31	51.69	222.44	5,443.46	12323.92	123.24	12447.16
12/1/02	68.51	222.44	6697.38	54.43	222.44	5,720.33	12447.16	124.47	12571.63
1/1/03	66.97	222.44	6541.91	57.20	222.44	5,999.97	12571.63	125.72	12697.35
2/1/03	65.42	222.44	6384.89	60.00	222.44	6,282.41	12697.35	126.97	12824.32
3/1/03	63.85	222.44	6226.30	62.82	222.44	6,567.67	12824.32	128.24	12952.56
4/1/03	62.26	222.44	6066.12	65.68	222.44	6,855.79	12952.56	129.53	13082.09
5/1/03	60.66	222.44	5904.34	68.56	222.44	7,146.79	13082.09	120.82	13212.91
6/1/03	59.04	222.44	5740.94	71.47	222.44	7,440.70	13212.91	132.13	13335.04
7/1/03	57.41	222.44	5575.91	74.41	222.44	7,737.55	13335.04	133.35	13478.49
8/1/03	55.76	222.44	5409.23	77.38	222.44	8,037.37	13478.49	124.78	13613.27
9/1/03	54.09	222.44	5240.88	80.37	222.44	8,340.18	13613.27	136.13	13749.41
10/1/03	52.41	222.44	5070.85	83.40	222.44	8,646.02	13749.41	137.49	13886.90
11/1/03	50.71	222.44	4899.12	86.46	222.44	8,954.92	13886.90	138.87	14025.77
12/1/03	48.99	222.44	4725.67	89.55	222.44	9,266.91	14025.77	140.26	14166.03
1/1/04	47.26	222.44	4550.49	92.67	222.44	9,582.02	14166.03	141.66	14307.69
2/1/04	45.50	222.44	4373.55	95.82	222.44	9,900.28	14307.69	143.08	14450.76
3/1/04	43.74	222.44	4194.85	99.00	222.44	10,221.72	14450.76	144.51	14595.27
4/1/04	41.95	222.44	4014.36	102.22	222.44	10,546.38	14595.27	145.95	14741.23
5/1/04	40.14	222.44	3832.06	105.46	222.44	10,874.28	14741.23	147.41	14888.64
6/1/04	38.32	222.44	3647.94	108.74	222.44	11,205.46	14888.64	148.88	15037.52
7/1/04	36.48	222.44	3461.98	112.05	222.44	11,539.95	15037.52	150.38	15187.90
8/1/04	34.62	222.44	3274.16	115.39	222.44	11,877.79	15187.90	151.88	15339.78
9/1/04	32.74	222.44	3084.46	118.78	222.44	12,219.01	15339.78	153.40	15493.18
10/1/04	30.84	222.44	2892.86	122.19	222.44	12,563.64	15493.18	154.93	15648.11
11/1/04	28.93	222.44	2699.35	125.64	222.44	12,911.72	15648.11	156.48	15804.59
12/1/04	26.99	222.44	2503.90	129.12	222.44	13,263.28	15804.59	158.04	15962.63
1/1/05	25.04	222.44	2306.50	132.63	222.44	13,618.35	15962.63	159.63	16122.26
2/1/05	23.07	222.44	2107.13	136.18	222.44	13,976.97	16122.26	161.22	16283.48
3/1/05	21.07	222.44	1905.76	139.77	222.44	14,339.18	16283.48	162.84	16446.32
4/1/05	19.06	222.44	1702.38	143.39	222.44	14,705.01	16446.32	164.46	16610.78
5/1/05	17.02	222.44	1496.96	147.05	222.44	15,074.50	16610.78	166.11	16776.89
6/1/05	14.97	222.44	1289.49	150.75	222.44	15,447.69	16776.89	167.77	16944.66
7/1/05	12.89	222.44	1079.74	154.48	222.44	15,824.61	16944.66	169.44	17114.10
8/1/05	10.80	222.44	868.30	1158.25	222.44	16,205.30	17114.10	171.15	17285.25
9/1/05	8.68	222.44	654.54	162.05	222.44	16,589.79	17285.25	172.85	17458.10
10/1/05	6.55	222.44	438.65	169.78	222.44	16,978.13	17458.10	174.58	17632.68
11/1/05	4.39	222.44	220.60	173.70	222.44	17,370.35	17632.68	176.33	17809.01
12/1/05	2.21	222.44	.37	173.70	222.44	17,766.49	17809.01	178.09	17,987.10