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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.  
  28. I'll start with the second example:
  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
  152. [7]: https://www.tdcanadatrust.com/loanpaymentcalc.form
  153. [8]: https://www.calculator.net/loan-calculator.html?cloanamount=10000&cloanterm=3&cloantermmonth=0&cinterestrate=11.99&ccompound=monthly&cpayback=month&x=120&y=21
  154. [9]: https://www.scotiabank.com/ca/en/0,,8166,00.html
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