Calculating the amount paid for a loan in today's dollars 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.
The monthly payment formula tries to ensure that you pay a constant amount each month. Hence, in real terms, you are actually paying more up front as $117.46 today is worth more than in 20 years assuming positive inflation. The more you pay up front, the less you have to pay in total. Even if you managed to correctly calculate payments such that they were constant in real terms, you would end up paying more in nominal terms than if payments were constant in nominal terms.