So I was trying to figure out the amount paid for a loan in today's dollars using two different methods but they give me different results. I was hoping that someone could explain which method is right (or if neither are) and what mistakes I was making in the other method.
The problem assumes a constant rate of inflation.
Method 1
- Calculate the real interest rate using the Fisher equation: $i_{real} = \frac{i_{nominal} - inflation}{1 + inflation}$
- Use the real interest rate instead of the nominal rate in the loan payment formula: $payment = \frac{i * A}{1 - (1 + i)^{-n}}$ where i is the interest rate, A is the amount borrowed, and n is the number of payments
- Find the total amount paid by multiplying the payment by the number of payments $total = payment * n$
Method 2
- Find the payment using the nominal interest rate. Again the payment formula I used is $payment = \frac{i * A}{1 - (1 + i)^{-n}}$
- Convert each year's payment to today's dollars. To convert to today's I did $dollars_{today} = dollars_{future} * (\frac{1}{1 + inflation})^n$. n is the number of inflation periods.
- Sum up the results from step 2 to get the total paid.
An example where the methods give different answers
- Loan Amount: = $1000
- Nominal Interest: 10%
- Number of payments: 20
- Inflation: 5%
Using Method 1 we have
- Real Interest Rate = $\frac{0.10 - 0.05}{1 + 0.05} = 0.048$
- Each Payment = $\frac{0.048 * \$1000}{1 - (1 + 0.048)^{-20}} = \$78.63$
- Total Paid = $\$78.63 * 20 = \$1572.61$
Using Method 2 we have
- Each payment = $\frac{0.10 * \$1000}{1 - (1 + 0.10)^{-20}} = \$117.46$
- Here is the spreadsheet with the work and a picture of it
- The total paid is as you can see $1463.81
So you can see that the methods differ in amount by $108.80.
So can anyone explain which is right (if either of them is) and why the wrong one is wrong? My only guess so far is that
the loan payment formula always gives results in nominal dollars and all I did is change the interest rate by using the real interest rate. Not sure if this is really the case because I studied CS when I was in school, not Econ.
Thank you for your help in advance.