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# Untitled

a guest Jun 30th, 2018 3 Never
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1. Short Version
2. ---------------
3.
4. Can anyone figure out how Sallie Mae comes up with its loan payment amount?
5.
6. Long Version
7. -------------
8.
9. I have a friend who took out a personal loan with Sallie Mae, and their payment amount is higher than any math i can figure out. I looked over their [Truth in Lending](https://en.wikipedia.org/wiki/Truth_in_Lending_Act) statement, and while it is very clear on everything you'll pay:
10.
11. - they don't explain how they come up with their payment amount
12. - their payment amount doesn't match any other loan calculator
13. - or the math of any [introduction to economics textbook][1]
14.
15. Sallie Mae gives an example
16. --------------------
17.
18. Rather than giving out details of my friend's own personal loan, we can use two [examples from their own web-site][2] ([archive][3])
19.
20. > For a typical 60-month term loan of \$20,000 at a 15.99% fixed APR,
21. > you will make 59 monthly payments of \$487.32
22. > and one monthly payment of \$387.45.
23. >
24. > For a typical 36-month term loan of \$10,000 at 11.99% fixed APR,
25. > you will make 35 monthly payments of \$332.64
26. > and one monthly payment of \$308.59.
27.
29.
30. - **Loan Amount:** \$10,000
31. - **APR**: 11.99%
32. - **Term**: 36 months
33.
34. Nowhere in the full set of 12 PDF documents i reviewed did it mention how they come up with their payment schedule (e.g. the if the effective annual rate assumes 6-month compounding) - so a consumer who has signed a loan has as much information as you do right now.
35.
36. But assuming the simple answer of compounded monthly:
37.
38. - **Monthly interest rate:** 11.99% / 12 = 0.9992% per month
39. - **Effective annual rate:** `(1 + 0.9992%)^12 = 12.67% EAR
40.
41. Payment Calculation using Excel
42. ----------------
43.
44. The easiest way to solve it is to create a payment schedule in Excel, and solve for the payment that causes the loan outstanding amount to hit zero at the end of month 36:
45.
46.     | Period | Starting Balance | Interest | Payment | New Balance |
47.     |--------|------------------|----------|---------|-------------|
48.     | 1      |       \$10,000.00 |   \$99.92 | \$332.10 |   \$9,767.82 |
49.     | 2      |       \$9,767.82  |   \$97.60 | \$332.10 |   \$9,533.32 |
50.     | 3      |       \$9,533.32  |   \$95.25 | \$332.10 |   \$9,296.48 |
51.     | 4      |       \$9,296.48  |   \$92.89 | \$332.10 |   \$9,057.27 |
52.     | 5      |       \$9,057.27  |   \$90.50 | \$332.10 |   \$8,815.68 |
53.     | 6      |       \$8,815.68  |   \$88.08 | \$332.10 |   \$8,571.66 |
54.     | 7      |       \$8,571.66  |   \$85.65 | \$332.10 |   \$8,325.21 |
55.     | 8      |       \$8,325.21  |   \$83.18 | \$332.10 |   \$8,076.30 |
56.     | 9      |        \$8,076.30 |   \$80.70 | \$332.10 |   \$7,824.90 |
57.     | 10     |        \$7,824.90 |   \$78.18 | \$332.10 |   \$7,570.99 |
58.     | 11     |        \$7,570.99 |   \$75.65 | \$332.10 |   \$7,314.54 |
59.     | 12     |        \$7,314.54 |   \$73.08 | \$332.10 |   \$7,055.53 |
60.     | 13     |        \$7,055.53 |   \$70.50 | \$332.10 |   \$6,793.93 |
61.     | 14     |        \$6,793.93 |   \$67.88 | \$332.10 |   \$6,529.72 |
62.     | 15     |        \$6,529.72 |   \$65.24 | \$332.10 |   \$6,262.87 |
63.     | 16     |        \$6,262.87 |   \$62.58 | \$332.10 |   \$5,993.35 |
64.     | 17     |        \$5,993.35 |   \$59.88 | \$332.10 |   \$5,721.14 |
65.     | 18     |        \$5,721.14 |   \$57.16 | \$332.10 |   \$5,446.20 |
66.     | 19     |        \$5,446.20 |   \$54.42 | \$332.10 |   \$5,168.52 |
67.     | 20     |        \$5,168.52 |   \$51.64 | \$332.10 |   \$4,888.07 |
68.     | 21     |        \$4,888.07 |   \$48.84 | \$332.10 |   \$4,604.82 |
69.     | 22     |        \$4,604.82 |   \$46.01 | \$332.10 |   \$4,318.73 |
70.     | 23     |        \$4,318.73 |   \$43.15 | \$332.10 |   \$4,029.79 |
71.     | 24     |        \$4,029.79 |   \$40.26 | \$332.10 |   \$3,737.96 |
72.     | 25     |        \$3,737.96 |   \$37.35 | \$332.10 |   \$3,443.21 |
73.     | 26     |        \$3,443.21 |   \$34.40 | \$332.10 |   \$3,145.52 |
74.     | 27     |        \$3,145.52 |   \$31.43 | \$332.10 |   \$2,844.85 |
75.     | 28     |        \$2,844.85 |   \$28.42 | \$332.10 |   \$2,541.18 |
76.     | 29     |        \$2,541.18 |   \$25.39 | \$332.10 |   \$2,234.48 |
77.     | 30     |        \$2,234.48 |   \$22.33 | \$332.10 |   \$1,924.71 |
78.     | 31     |        \$1,924.71 |   \$19.23 | \$332.10 |   \$1,611.84 |
79.     | 32     |        \$1,611.84 |   \$16.10 | \$332.10 |   \$1,295.85 |
80.     | 33     |        \$1,295.85 |   \$12.95 | \$332.10 |     \$976.70 |
81.     | 34     |          \$976.70 |   \$9.76  | \$332.10 |     \$654.37 |
82.     | 35     |          \$654.37 |   \$6.54  | \$332.10 |     \$328.81 |
83.     | 36     |          \$328.81 |   \$3.2 9 | \$332.10 |       \$0.00 |
84.
85. **Conclusion:** monthly payment of \$332.10
86.
87. Solve it algebraically
88. ---------------
89.
90. The above 36 term equation has been solved:
91.
92. - ***P***: \$10,000 *(present value)*
93. - ***i***: 0.9992% *(rate per period)*
94. - ***N***: 36 *(number of periods)*
95. - ***A***: ? *(amount)*
96.
97. The formula is given as:
98.
99.     A = P * [ i(1+i)^N / ((1+i)^N - 1 ]
100.       = 10000 * [ 0.009992(1.009992)^36 / (1.009992^36-1) ]
101.       = 10000 * [ 0.014292166 / 0.43036086 ]
102.       = 10000 * [ 0.033209534]
103.       = \$332.10
104.
105. **Conclusion:** monthly payment of \$332.10
106.
107. Solve using PMT function
108. -------------------
109.
110. We can try solving it using the [`PMT`][4] function of every spreadsheet ever.
111.
112.     =PMT(0.009992, 36, 10000, 0, 0)
113.
114. [![enter image description here][5]][5]
115.
116. **Conclusion:** monthly payment of \$332.10
117.
118. Solve using online calculator
119. -----------------------
120.
121. We can try solving it using online calculators:
122.
123. - [**The Calculator Site**][6]: \$332.10
124. - [**TD Canada Trust**][7]: \$332.2
125. - [**Calculator.net**][8]: \$332.10
126. - [**ScotiaBank**][9]: \$331.67
127.
128. **Conclusion**: monthly payment of \$332.10 *(ish)*
129.
130. Sallie Mae come up with a loan amount much higher
131. =================
132.
133. - my calculated monthly payment amount: \$332.10
134. - Sallie Mae's example payment amount: \$332.64
135.
136. Sallie Mae seems to have a higher amount than they should:
137.
138.     | Item         |My calculations           | Theirs                |
139.     |--------------|---------------------------|-----------------------|
140.     |
141.
142. > For a typical 36-month term loan of \$10,000 at 11.99% fixed APR,
143. > you will make 35 monthly payments of \$332.64
144. > and one monthly payment of \$308.59.
145.
146.   [1]: http://a.co/8HsfOz2
147.   [2]: https://www.salliemae.com/banking/personal-loans/
148.   [3]: https://archive.fo/qjefe
149.   [4]: https://support.office.com/en-us/article/pmt-function-0214da64-9a63-4996-bc20-214433fa6441
150.   [5]: https://i.stack.imgur.com/x15Or.png
151.   [6]: https://www.thecalculatorsite.com/finance/calculators/loancalculator.php