Concept

• Two ways to repay a loan:
  • Amortization
    • Taking the payment and splitting it up into parts—some goes towards the principal and part goes towards the interest
    • Always pay off the interest first and whatever is left over pays the principal
  • Sinking fund
    • Only the interest is paid off until the balance can be paid as a lump-sum
    • You would have to assume there’s some balance in another account that’s growing in order to pay off the balance on the loan

Amortization

Prospective Method

• The outstanding balance at $t=0$ is the present values of the remaining payments
• The loan amount is just the PV of all the payments

Retrospective Method

• The outstanding balance at some non-zero $t$ is the accumulated loan amount (including interest) minus the accumulated value of all previous payments

Amortization Schedules

• Splits up recurring payments into interest and principal
  • You will always have to pay off the interest, and then the rest goes towards the principal
• Create a table with the time value, payment, interest paid, principal paid, and outstanding balance
• Outstanding Balance for each payment:

• Interest Paid for each payment:

• Principal Paid for each payment:
  • Basically whatever is left from the payment when you subtract the interest

• TOTALS at $t=n$:
  • $OB = 0$
  • $I_{P, \text{total}} =Pn - A_{P, \text{total}} = Pn - Pa_{\left< n \right>}$
  • $A_{P, \text{total}} =Pa_{\left< n \right>}$

• Per the examples above, my answers are off by a couple of decimal points, so I don’t know 100% if I did them correctly. I will have to double-check in Academic GOAL whether this is the case via pure problem-solving

Sinking Fund

• All payments go towards your interest and then something else that builds towards the loan and accrues interest
• Since the loan is going to be paid off all at once at the end, there aren’t gonna be any in-between payment contributions
  • Therefore, the interest payments will also remain the same
• You should be careful as the sinking fund interest rate may not be the same as the loan interest rate