Calculations Relating To Loans

Which Interest Rate?

How to Calculate Payments on an Amortizing Loan

\begin{align} A = P*\frac{r(1+r)^n}{(1+r)^n-1} \end{align}
  • A = Payment
  • n = total number of payments or periods
  • P = initial Principal (loan amount)
  • r = effective interest rate per payment period
\begin{align} r = (1+\frac{i}{n})^{p} - 1 \end{align}
  • i = nominal interest rate
  • n = compounding periods in nominal period
  • p = compounding periods in payment period
  • r = effective interest rate per payment period

How to Calculate Payments on an Interest Only Loan

An interest only loan is the kind of loan where the payments only cover the 'rent' on the money you borrow; you still owe the full amount that you borrowed after you make the last payment.

Payments on interest-only loan are calculated using the following formula:

A =(P * I)

A - monthly payment
P - amount borrowed
I – effective interest rate per payment


$200,000 mortgage at 6.0% interest. Please note that with US mortgages the interest rate is assumed to be an annual statement that compounds monthly.

A= $200,000 (0.06/12)
=$1000 a month

How to Calculate the Balance Remaining on a Loan

There are two ways of calculating the balance remaining on a loan, the long tabular way, and the short way that uses a time value of money calculation.

In the long tabular way, each payment, can be broken up into two parts, the interest payment and the principle payment. The interest payment is the 'rent' on the borrowed money. This is just equal to the current balance remaining, B, times the effective interest rate per payment period, r.

\begin{equation} I = B r \end{equation}

Please note that for the first payment the balance remaining is the amount originally borrowed.

The principal payment is just the difference between the full payment, A, and the interest expense, I. This is the amount that the payment reduces the balance remaining. Over time the balance remaining, interest expense and principal payment evolve, with the interest expense falling with each payment and the principal payment increasing with each payment.

\begin{equation} B_{t+1}=B_{t}- (A- I_{t}) \end{equation}
\begin{equation} I_{t}=B_{t} r \end{equation}

That works, and is great if you want a full amortization table, but sometimes you only want one value. In that case what you want to do is find the present value of the remaining payments.

\begin{align} \textstyle{Balance~Remaining}= Payment(P|A,i=\textstyle{effective~per~period}, N= \textstyle{Payments~remaining}) \end{align}

How to Recalculate Payments When Interest Rates Change

*Variable Loan Calculations

What is an Amortization Table?

  • An amortization table lays out how much of each payment goes towards paying off the interest and how much goes towards paying off the actual principal (the loan).
  • Ex:If we took out a $10,000 loan at a rate of 11% that had to be paid back in 5 years, one payment per year.

To calculate the payments. 10k(A|P, i=11%, n=5) => 10k(.2706) = $2,706

To calculate the interest payment just multiply the remaining loan balance by the interest rate

To calculate the principal payment it is size of the payment less the interest payment

The loan balance is the previous balance less the principal payment

Year Payment Size Interest Payment Principal Payment Loan Balance
1 0 0 0 10,000
2 2,706 10,000 * .11 = 1,100 2,706 - 1,100 = 1,606 10,000 - 1,606 = 8,394
3 2,706 8,394 * .11 = 923.34 2,706 - 923.34 = 1,782.66 8,394 - 1,782.66 = 6,611.34
4 2,706 6,611.34 * .11 = 727.25 2,706 - 727.25 = 1978.75 6,611.34 - 1978.75 = 4,632.59
5 2,706 4,632.59 * .11 = 509.59 2,706 - 509.59 = 2,196.41 4,632.59 - 2,196.41 = 2,436.18
6 2,706 2,436.18 * .11 = 267.98 2,706 - 267.98 = 2,438.02 2,436.18 - 2438.02 = -1.84

*Note this should have come out to exactly 0, but due to rounding errors we got an overpayment of $1.84

You can also make amortization tables for interest only loans. The big difference is that there are no principal payments during the course of the loan. There seems to be no convention with how you represent the repayment of the principle. Sometimes you see a payment after the last payment which is the full principle. Other times you will see the principle added into the last payment so it looks larger than the others.


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