Question

1. In the fictional land of Westeros, House Lannister vies with rival Houses for control of the throne. War breaks out. To finance its war operations, House Lannister borrows 900,000 gold pieces from the “Iron Bank.” The loan is structured as a balloon mortgage with the following details: a term of 15 years, with amortization based on a 30-year loan with monthly payments. The Iron Bank offers the Lannisters a relatively low APR of 6%, because “a Lannister always repays his debt.”

a) Determine the EAR associated with the loan and the remaining principal that will be due in 15 years. EAR: ___________________ Principal due in 15 years: ________________

b) Complete the following portion of the loan amortization schedule: Month Beginning balance Interest paid Principal paid Ending balance _______________________

c) Compute the total interest paid on the loan (over 15 years), and the interest paid in the first month of the 10th year of the loan. Total interest paid on loan: ____________ Interest paid in first month of 10th year: ____________ Reasoning:

d) (3 points) Now suppose that the Iron Bank charges an origination fee of 2%, which is rolled into the loan value. Determine the EAR associated with the loan when the fee is incorporated. New EAR: ___________

Answer 1

a) Calculation of effective annual rate (EAR)

Lets assume i = EAR, r = APR (interest rate), m = Month

i = (1 + r/m)^m - 1

i = (1 + 0.06/12)¹² - 1

Hence i = 0.061678 i.e 6.1678% lets say 6.17%

Calculation of Principle due in 15 Years

Lets calculate EMI first :

P = Principal, n = Number of monthly installments (30 years i.e 360 months)

EMI = [P x r x (1+r)ⁿ] / [(1+r)ⁿ - 1]

EMI = [900000 x (0.06/12)] x [( 1 + 0.06/12 )³⁶⁰ - 1 ]

Therefore EMI = 5395.95

Please refer below table for answer of a, b & c

1. Principle due in 15 Years = 2,60,502

2. Interest paid on the loan over 15 years = 7,38,164

3.Interest paid in the first month of the 10th year = 3860.56

4. Total interest paid on loan = 10,42,543.70

Month	Installment paid	Interest Paid (o/s loan x rate of interest)	Principle Paid (EMI - Interest)	Balance due
108	5,396	3,868	1,528	7,72,112
109	5,396	3,861	1,535	7,70,576
110	5,396	3,853	1,543	7,69,033
111	5,396	3,845	1,551	7,67,482
112	5,396	3,837	1,559	7,65,924
113	5,396	3,830	1,566	7,64,357
114	5,396	3,822	1,574	7,62,783
115	5,396	3,814	1,582	7,61,201
116	5,396	3,806	1,590	7,59,611
117	5,396	3,798	1,598	7,58,013
118	5,396	3,790	1,606	7,56,407
119	5,396	3,782	1,614	7,54,794
120	5,396	3,774	1,622	7,53,172

Yearly calculation with rate of 6%

Year	EMI	Interest	Principle	Balance due
1.00	66,577.77	55,530.00	11,047.77	8,88,952.23
2.00	66,577.77	54,848.35	11,729.42	8,77,222.81
3.00	66,577.77	54,124.65	12,453.13	8,64,769.68
4.00	66,577.77	53,356.29	13,221.48	8,51,548.20
5.00	66,577.77	52,540.52	14,037.25	8,37,510.95
6.00	66,577.77	51,674.43	14,903.35	8,22,607.60
7.00	66,577.77	50,754.89	15,822.88	8,06,784.72
8.00	66,577.77	49,778.62	16,799.16	7,89,985.56
9.00	66,577.77	48,742.11	17,835.66	7,72,149.90
10.00	66,577.77	47,641.65	18,936.12	7,53,213.77
11.00	66,577.77	46,473.29	20,104.48	7,33,109.29
12.00	66,577.77	45,232.84	21,344.93	7,11,764.36
13.00	66,577.77	43,915.86	22,661.91	6,89,102.45
14.00	66,577.77	42,517.62	24,060.15	6,65,042.29
15.00	66,577.77	41,033.11	25,544.66	6,39,497.63
16.00	66,577.77	39,457.00	27,120.77	6,12,376.86
17.00	66,577.77	37,783.65	28,794.12	5,83,582.74
18.00	66,577.77	36,007.06	30,570.72	5,53,012.02
19.00	66,577.77	34,120.84	32,456.93	5,20,555.09
20.00	66,577.77	32,118.25	34,459.52	4,86,095.57
21.00	66,577.77	29,992.10	36,585.68	4,49,509.89
22.00	66,577.77	27,734.76	38,843.01	4,10,666.88
23.00	66,577.77	25,338.15	41,239.63	3,69,427.25
24.00	66,577.77	22,793.66	43,784.11	3,25,643.14
25.00	66,577.77	20,092.18	46,485.59	2,79,157.55
26.00	66,577.77	17,224.02	49,353.75	2,29,803.80
27.00	66,577.77	14,178.89	52,398.88	1,77,404.92
28.00	66,577.77	10,945.88	55,631.89	1,21,773.03
29.00	66,577.77	7,513.40	59,064.38	62,708.65
30.00	66,577.77	3,869.12	62,708.65	0.00

d) Calculation of effective annual rate (EAR) if 2% are bank charges (It is assumed that EMI & tenure will be same)

P = Principal, n = Number of monthly installments (30 years i.e 360 months)

EMI = [P x r x (1+r)ⁿ] / [(1+r)ⁿ - 1]

5395.95 = [918000 x r] x [( 1 + r)³⁶⁰ - 1 ]

Therefore r = 5.82%