The constant amortization scheme with multiple contracts

Faro, Clovis de

doi:10.5935/0034-7140.20220007

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Revista Brasileira de Economia

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Artigos • Rev. Bras. Econ. 76 (2) • Apr-Jun 2022 • https://doi.org/10.5935/0034-7140.20220007 copy

The constant amortization scheme with multiple contracts^* * Section 7 was contributed by an anonymous referee.

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract

For the case of a loan with constant payments, De-Losso et al. (2013) has shown that substituting a single contract by subcontracts, one for each of the n payments of the single contract, may result, depending on the financial institution opportunity cost, in substantial fiscal gain. The present paper extends its analysis to the case of the constant amortization scheme of debt financing. It is shown that the fiscal gain can be even greater.

constant amortizations; multiple contracts

k	S_k	p_k	${\hat{A}}_{k}$	J_k	${\hat{J}}_{k}$	d_k
0	1,200,000.00	–	–	–	–	–
1	1,100,000.00	124,000.00	121,568.63	24,000.00	2,431.37	21,568.63
2	1,000,000.00	122,000.00	117,262.59	22,000.00	4,737.41	17,262.59
3	900,000.00	120,000.00	113,078.68	20,000.00	6,921.32	13,078.68
4	800,000.00	118,000.00	109,013.76	18,000.00	8,986.24	9,013.76
5	700,000.00	116,000.00	105,064.77	16,000.00	10,935.23	5,064.77
6	600,000.00	114,000.00	101,228.74	14,000.00	12,771.26	1,228.74
7	500,000.00	112,000.00	97,502.74	12,000.00	14,497.26	-2,497.26
8	400,000.00	110,000.00	93,883.94	10,000.00	16,116.06	-6,116.06
9	300,000.00	108,000.00	90,369.57	8,000.00	17,630.43	-9,630.43
10	200,000.00	106,000.00	86,956.92	6,000.00	19,043.08	-13,043.08
11	100,000.00	104,000.00	83,643.36	4,000.00	20,356.64	-16,356.64
12	0.00	102,000.00	80,426.30	2,000.00	21,573.70	-19,573.70
∑	–	1,356,000.00	1,200,000.00	156,000.00	156,000.00	0.00

n_a
n_a	5	10	15	20	25	30
1	1.5	2.9	4.2	5.5	6.8	8.0
5	7.1	14.3	21.6	28.9	36.3	43.4
10	12.2	26.4	40.3	54.6	69.0	83.4
15	17.4	35.9	55.1	74.7	94.2	113.1
20	29.9	43.4	66.4	89.6	112.3	134.5
25	23.8	49.1	74.9	100.4	125.2	149.1
30	26.4	53.6	81.3	108.3	139.4	159.4

n_a
n_a	5	10	15	20	25	30
1	1.4	2.8	4.1	5.3	6.6	7.8
5	6.2	12.4	18.6	24.8	30.9	37.0
10	10.0	20.3	30.6	40.9	51.2	61.3
15	12.5	25.2	38.1	50.7	63.2	75.3
20	14.1	28.5	42.8	56.8	70.4	83.5
25	15.3	30.7	45.9	60.7	74.9	88.6
30	16.1	32.3	48.1	63.3	78.0	92.0

k	${J^{'}}_{k}$	${\hat{J^{'}}}_{k}$	${d^{'}}_{k}$	${\hat{J}}_{k}$	${d^{″}}_{k}$
1	24,000.00	2,224.93	21,775.07	2,431.37	206.44
2	22,210.57	4,406.24	17,804.33	4,737.41	331.17
3	20,385.35	6,544.77	13,840.58	6,921.32	376.55
4	18,523.63	8,641.37	9,882.26	8,986.24	344.87
5	16,624.67	10,696.87	5,927.80	10,935.23	238.36
6	14,687.73	12,712.06	1,975.67	12,771.26	59.20
7	12,712.06	14,687.73	−1,975.67	14,497.26	−190.47
8	10,696.87	16,624.67	−5,927.80	16,116.06	−508.61
9	8,641.37	18,523.63	−9,882.26	17,630.43	−893.20
10	6,544.77	20,385.35	−13,840.58	19,043.08	−1,342.27
11	4,406.24	22,210.57	−17,804.33	20,356.64	−1,853.93
12	2,224.93	24,000.00	−21,775.07	21,573.70	−2,426.30
∑	161,658.19	161,658.19	0.00	156,000.00	−5,689.19

n (years)	ρ_a (%)
n (years)	0	5	10	15	20	25	30
1	1.82	1.79	1.76	1.73	1.69	1.67	1.64
5	9.73	8.95	8.20	7.48	6.78	6.11	5.48
10	19.28	16.39	13.66	11.12	8.78	6.64	4.69
15	28.21	22.19	16.79	12.05	7.99	4.54	1.63
20	36.32	26.48	18.21	11.52	6.24	2.13	−1.08
25	43.50	29.43	18.49	10.39	4.54	0.31	−2.79
30	49.75	31.26	18.05	9.14	3.21	−0.81	−3.62

n (years)	ρ_a (%)
n (years)	0	5	10	15	20	25	30
1	2.33	3.56	3.50	3.43	3.27	3.32	3.26
5	19.03	17.60	16.20	14.89	13.65	12.47	11.35
10	36.04	30.98	26.41	22.30	18.62	15.36	12.46
15	49.44	39.65	31.37	24.48	18.82	14.18	10.40
20	59.41	44.39	32.68	23.79	17.10	12.07	8.25
25	66.64	46.32	31.85	21.84	14.94	10.11	6.34
30	71.91	46.45	29.95	19.55	12.92	8.23	5.49

k	${\hat{J}}_{k}$	${J^{'}}_{k}$	${\hat{J^{'}}}_{k}$	${J^{'}}_{k} - {\hat{J^{'}}}_{k}$	${\hat{J}}_{k} - {\hat{J^{'}}}_{k}$
1	2,431.37	23,131.32	2,136.39	20,994.93	294.98
2	4,737.41	21,399.63	4,232.39	17,167.24	505.02
3	6,921.32	19,634.55	6,288.74	13,345.81	632.58
4	8,986.24	17,835.45	8,306.20	9,529.25	680.04
5	10,935.23	16,001.68	10,285.52	5,716.16	649.71
6	12,771.26	14,132.55	12,227.40	1,905.16	543.86
7	14,497.26	12,227.40	14,132.55	−1,905.16	364.71
8	16,116.06	10,285.52	16,001.68	−5,716.16	114.38
9	17,630.43	8,306.20	17,835.45	−9,529.25	−205.02
10	19,043.08	6,288.74	19,634.55	−13,345.81	−591.47
11	20,356.64	4,232.39	21,399.63	−17,167.24	−1,042.99
12	21,573.70	2,136.39	23,131.32	−20,994.93	−1,557.02

n	ρ_a
n	5	10	15	20	25	30	60
1	−0.04	−0.04	−0.04	−0.04	−0.04	−0.04	−0.04
5	−0.03	−0.03	−0.03	−0.03	−0.03	−0.03	−0.04
10	−0.01	−0.02	−0.03	−0.03	−0.04	−0.04	−0.06
15	−0.01	−0.02	−0.03	−0.04	−0.05	−0.06	−0.09
20	0.00	−0.02	−0.04	−0.06	−0.07	−0.08	−0.12
25	0.00	−0.03	−0.05	−0.08	−0.09	−0.11	−0.15
30	0.01	−0.04	−0.07	−0.10	−0.12	−0.24	−0.17
90	−0.05	−0.16	−0.20	−0.22	−0.23	−0.24	−0.25

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The constant amortization scheme with multiple contracts* * Section 7 was contributed by an anonymous referee.

Abstract

The constant amortization scheme with multiple contracts^* * Section 7 was contributed by an anonymous referee.