How does Sallie Mae calculate its payment?

Question

Short Version

Can anyone figure out how Sallie Mae comes up with its personal loan payment amount?

Long Version

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 statement, and while it is very clear on everything you'll pay:

• they don't explain how they come up with their payment amount
• their payment amount doesn't match any other loan calculator
• or the math of any introduction to economics textbook

Details

I'll start with the facts:

• Loan Amount: $15,000
• APR: 24.99%
• Term: 36 months
• Monthly Payment: $602.03 ($507.80 for final month)
• Theoretical Monthly Payment: $596.32

Nowhere in the full set of 6 PDF documents I reviewed did it mention how they come up with their payment schedule (e.g. 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.

But assuming the simple answer of compounded monthly:

• Monthly interest rate: 24.99% / 12 = 2.0825% per month
• Effective annual rate: (1 + 2.0825%)^12 = 28.606% EAR

Payment Calculation using Excel

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:

| Period | Starting Balance | Interest | Payment | New Balance |
|--------|------------------|----------|---------|-------------|
| 1      |       $15,000.00 |  $312.38 | $596.32 |  $14,716.06 |
| 2      |       $14,716.06 |  $306.46 | $596.32 |  $14,426.20 |
| 3      |       $14,426.20 |  $300.43 | $596.32 |  $14,130.31 |
| 4      |       $14,130.31 |  $294.26 | $596.32 |  $13,828.25 |
| 5      |       $13,828.25 |  $287.97 | $596.32 |  $13,519.91 |
| 6      |       $13,519.91 |  $281.55 | $596.32 |  $13,205.14 |
...
| 30     |        $3,847.16 |   $80.12 | $596.32 |   $3,330.95 |
| 31     |        $3,330.95 |   $69.37 | $596.32 |   $2,804.00 |
| 32     |        $2,804.00 |   $58.39 | $596.32 |   $2,266.08 |
| 33     |        $2,266.08 |   $47.19 | $596.32 |   $1,716.95 |
| 34     |        $1,716.95 |   $35.76 | $596.32 |   $1,156.39 |
| 35     |        $1,156.39 |   $24.08 | $596.32 |     $584.15 |
| 36     |          $584.15 |   $12.16 | $596.32 |       $0.00 |

• Conclusion: monthly payment of $596.32
• Total repayment:: $596.32 * 36 = $21,467.52

Solve it algebraically

The above 36 term equation has been solved by mathematicians:

• P: $15,000 (present value)
• i: 2.0825% (rate per period)
• N: 36 (number of periods)
• A: ? (amount)

The formula is given as:

A =     P * [ i(1+i)^N / ((1+i)^N - 1 ]
  = 10000 * [ 0.020825(1.020825)^36 / (1.020825^36-1) ]
  = 10000 * [ 0.04373526 / 1.100132547 ]
  = 10000 * [ 0.039754537 ]
  = $596.32

• Conclusion: monthly payment of $596.32
• Total repayment:: $596.32 * 36 = $21,467.52

Solve using PMT function

We can try solving it using the PMT function of every spreadsheet ever.

=PMT(2.0825%, 36, 15000, 0, 0)

• Conclusion: monthly payment of $596.32
• Total repayment:: $596.32 * 36 = $21,467.52

Solve using online calculator

We can try solving it using online calculators:

• The Calculator Site: $596.32
• Calculator.net: $596.32

• TD Canada Trust: $596.73

• Conclusion: monthly payment of $596.32 (ish)

• Total repayment: $596.32 * 36 = $21,467.52 (ish)

Sallie Mae come up with a loan amount much higher

• Numerically: $596.32
• Algebraically: $596.32
• PMT function: $596.32
• Online calculators: $596.32 (ish)
• Sallie Mae: $692.03 (for 35 months, $507.80 for final month)

Sallie Mae seems to have a higher amount than they should:

| Item                |My calculations | Theirs     |
|---------------------|----------------|------------|
| Payment             |        $596.32 |    $602.03 |
| Total amount repaid |     $21,467.52 | $21,578.85 |
| Cost to borrow      |      $6,467.52 |  $6,578.85 |
|                     |                |   +$111.33 |

From their Truth in Lending Statement:

• Interest Rate: 24.990%
• Disbursement Amount: $15,000
• Annual Percentage Rate: 25.02% (The cost of your credit as a yearly rate.)
• Finance Charge: $6,578.85 (The dollar amount the credit will cost you.)
• Total Payments: $21,578.85 (The amount you will have paid when you have made all payments as scheduled)

Can anyone explain the difference?

Their examples match perfectly

I've repeated this exercise on two personal loan examples they give on their web-site (archive):

For a typical 60-month term loan of $20,000 at a 15.99% fixed APR,
you will make 59 monthly payments of $487.32
and one monthly payment of $387.45.

For a typical 36-month term loan of $10,000 at 11.99% fixed APR,
you will make 35 monthly payments of $332.64
and one monthly payment of $308.59.

I'll omit the entire exercise, but sufficient to say that it does match the theoretical values.

I realize I'm only talking about $111.33 extra at the end of 36 months; but can anyone explain the difference?

Answer 1

I'd wager that the discrepancy you are seeing is due to a difference between the loan disbursement date and the payment due date, such that the first loan payment is/was due more than a month from the loan disbursement date. Interest accrual begins immediately, but repayment doesn't always begin exactly a month after disbursement. If you check the origination documents you'll likely be able to confirm this.

Lenders do this very frequently, they may let the borrower choose a payment due date, or automatically make the due date the first of each month, or adjust by just a few days to avoid having payments due after the 28th of each month since not all months have more than 28 days.

A common example, if I closed on my house the 15th of May, that's when my loan was disbursed, but my first payment is due July 1st. Since interest starts accruing from disbursement that extra half month of interest gets factored in to my repayment. Rather than make you speculate about disbursement dates and loan payment due dates, they just offer a simplified version in their calculator.