Ok, it appears student loan repayment is different from mortgage loan repayment (i.e., different from what Excel PMT assumes).

See

https://studentaid.ed.gov/types/loans/interest-rates:

"The amount of interest that accrues (accumulates) on your loan from month to month is determined by a simple daily interest formula. This formula consists of multiplying your loan balance by the number of days since the last payment times the interest rate factor.

Simple daily interest formula:

Outstanding principal balance

x number of days since last payment

x interest rate factor

= interest amount"

Note the loan service e-mail quote (emphasis added): "As indicated on your promissory note(s), interest accrues

**daily** based on the current principal balance."

The reason the interest payments do not monotonically decrease is because you are being charged interest

*daily*, not monthly. Payments in March show the lowest interest amounts, because February has the least number of days: the time between your Feb and March payments is the shortest of the year.

One should be able to reproduce the payment history in Excel, based on

http://www.myfedloan.org/billing-payment/about-interest/index.shtml:

"When you make a payment, our computer system counts the number of days since we processed your last payment.

Then we apply your payment as follows:

Accrued interest—The amount of interest that accrued every day between the date of the last payment and the new payment must be satisfied first.

Current principal balance—The remainder then applies toward your current principal balance."

And so we can: if you copy and paste the following table into Excel, with "Annual interest rate" going in cell A2, it reproduces your payment history "almost" exactly (you'll need to reformat some, e.g. column A as "dates", etc.). The "almost" is likely because the loan service rounds calculations in a slightly different manner.

This doesn't explain how $34,888 became $40,791 but at least it gives you something....

Annual interest rate | 0.03125 | | | | |

Daily interest rate | =B2/365 | | | | |

- | - | - | - | - | - |

Date | Days since last pmt | Interest | Payment | Principal | Loan amount |

40557 | | | | | 39439.36 |

=EDATE(A6,1) | =A7-A6 | =ROUND(B7*$B$3*F6,2) | 235.9 | =D7-C7 | =F6-E7 |

=EDATE(A7,1) | =A8-A7 | =ROUND(B8*$B$3*F7,2) | =D7 | =D8-C8 | =F7-E8 |

=EDATE(A8,1) | =A9-A8 | =ROUND(B9*$B$3*F8,2) | =D8 | =D9-C9 | =F8-E9 |

=EDATE(A9,1) | =A10-A9 | =ROUND(B10*$B$3*F9,2) | =D9 | =D10-C10 | =F9-E10 |

=EDATE(A10,1) | =A11-A10 | =ROUND(B11*$B$3*F10,2) | =D10 | =D11-C11 | =F10-E11 |

=EDATE(A11,1) | =A12-A11 | =ROUND(B12*$B$3*F11,2) | =D11 | =D12-C12 | =F11-E12 |

=EDATE(A12,1) | =A13-A12 | =ROUND(B13*$B$3*F12,2) | =D12 | =D13-C13 | =F12-E13 |

=EDATE(A13,1) | =A14-A13 | =ROUND(B14*$B$3*F13,2) | =D13 | =D14-C14 | =F13-E14 |

=EDATE(A14,1) | =A15-A14 | =ROUND(B15*$B$3*F14,2) | =D14 | =D15-C15 | =F14-E15 |

=EDATE(A15,1) | =A16-A15 | =ROUND(B16*$B$3*F15,2) | =D15 | =D16-C16 | =F15-E16 |

=EDATE(A16,1) | =A17-A16 | =ROUND(B17*$B$3*F16,2) | =D16 | =D17-C17 | =F16-E17 |

=EDATE(A17,1) | =A18-A17 | =ROUND(B18*$B$3*F17,2) | =D17 | =D18-C18 | =F17-E18 |

=EDATE(A18,1) | =A19-A18 | =ROUND(B19*$B$3*F18,2) | =D18 | =D19-C19 | =F18-E19 |

=EDATE(A19,1) | =A20-A19 | =ROUND(B20*$B$3*F19,2) | =D19 | =D20-C20 | =F19-E20 |

=EDATE(A20,1) | =A21-A20 | =ROUND(B21*$B$3*F20,2) | =D20 | =D21-C21 | =F20-E21 |

=EDATE(A21,1) | =A22-A21 | =ROUND(B22*$B$3*F21,2) | =D21 | =D22-C22 | =F21-E22 |

=EDATE(A22,1) | =A23-A22 | =ROUND(B23*$B$3*F22,2) | =D22 | =D23-C23 | =F22-E23 |

=EDATE(A23,1) | =A24-A23 | =ROUND(B24*$B$3*F23,2) | =D23 | =D24-C24 | =F23-E24 |

=EDATE(A24,1) | =A25-A24 | =ROUND(B25*$B$3*F24,2) | =D24 | =D25-C25 | =F24-E25 |

=EDATE(A25,1) | =A26-A25 | =ROUND(B26*$B$3*F25,2) | =D25 | =D26-C26 | =F25-E26 |

=EDATE(A26,1) | =A27-A26 | =ROUND(B27*$B$3*F26,2) | =D26 | =D27-C27 | =F26-E27 |

=EDATE(A27,1) | =A28-A27 | =ROUND(B28*$B$3*F27,2) | =D27 | =D28-C28 | =F27-E28 |

=EDATE(A28,1) | =A29-A28 | =ROUND(B29*$B$3*F28,2) | =D28 | =D29-C29 | =F28-E29 |

=EDATE(A29,1) | =A30-A29 | =ROUND(B30*$B$3*F29,2) | =D29 | =D30-C30 | =F29-E30 |

=EDATE(A30,1) | =A31-A30 | =ROUND(B31*$B$3*F30,2) | =D30 | =D31-C31 | =F30-E31 |

=EDATE(A31,1) | =A32-A31 | =ROUND(B32*$B$3*F31,2) | =D31 | =D32-C32 | =F31-E32 |

=EDATE(A32,1) | =A33-A32 | =ROUND(B33*$B$3*F32,2) | =D32 | =D33-C33 | =F32-E33 |

=EDATE(A33,1) | =A34-A33 | =ROUND(B34*$B$3*F33,2) | =D33 | =D34-C34 | =F33-E34 |

=EDATE(A34,1) | =A35-A34 | =ROUND(B35*$B$3*F34,2) | =D34 | =D35-C35 | =F34-E35 |

=EDATE(A35,1) | =A36-A35 | =ROUND(B36*$B$3*F35,2) | =D35 | =D36-C36 | =F35-E36 |

=EDATE(A36,1) | =A37-A36 | =ROUND(B37*$B$3*F36,2) | =D36 | =D37-C37 | =F36-E37 |

=EDATE(A37,1) | =A38-A37 | =ROUND(B38*$B$3*F37,2) | =D37 | =D38-C38 | =F37-E38 |

=EDATE(A38,1) | =A39-A38 | =ROUND(B39*$B$3*F38,2) | =D38 | =D39-C39 | =F38-E39 |

=EDATE(A39,1) | =A40-A39 | =ROUND(B40*$B$3*F39,2) | =D39 | =D40-C40 | =F39-E40 |

=EDATE(A40,1) | =A41-A40 | =ROUND(B41*$B$3*F40,2) | =D40 | =D41-C41 | =F40-E41 |

=EDATE(A41,1) | =A42-A41 | =ROUND(B42*$B$3*F41,2) | =D41 | =D42-C42 | =F41-E42 |

=EDATE(A42,1) | =A43-A42 | =ROUND(B43*$B$3*F42,2) | =D42 | =D43-C43 | =F42-E43 |

=EDATE(A43,1) | =A44-A43 | =ROUND(B44*$B$3*F43,2) | =D43 | =D44-C44 | =F43-E44 |

=EDATE(A44,1) | =A45-A44 | =ROUND(B45*$B$3*F44,2) | =D44 | =D45-C45 | =F44-E45 |