# How does Sallie Mae calculate its payment?

## 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

• 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:

## 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?