Calculating the amount paid for a loan in today's dollars

Question

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

1. Calculate the real interest rate using the Fisher equation: $i_{real} = \frac{i_{nominal} - inflation}{1 + inflation}$
2. 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
3. Find the total amount paid by multiplying the payment by the number of payments $total = payment * n$

Method 2

1. Find the payment using the nominal interest rate. Again the payment formula I used is $payment = \frac{i * A}{1 - (1 + i)^{-n}}$
2. 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.
3. 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

1. Real Interest Rate = $\frac{0.10 - 0.05}{1 + 0.05} = 0.048$
2. Each Payment = $\frac{0.048 * \$1000}{1 - (1 + 0.048)^{-20}} = \$78.63$
3. Total Paid = $\$78.63 * 20 = \$1572.61$

Using Method 2 we have

1. Each payment = $\frac{0.10 * \$1000}{1 - (1 + 0.10)^{-20}} = \$117.46$
2. Here is the spreadsheet with the work and a picture of it
3. 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.

Answer 1

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.

Answer 2

We can reduce the problem down to just two payments. Suppose you take out a loan for 1000 in 2000, you pay back 576.16 in 2001 and 2002, with a nominal interest rate of 10% and inflation of 5%. If "today" is 2000, then the value of the 576.16 that you pay in 2001 is 548.75, and 522.62 for 2002, for a total of 1071.34.

The Fisher equation gives that the real interest rate is 4.76. What you're saying "You know how you calculated how large the payments should be for an interest rate of 10%? How about you do the calculation again for an interest rate of 4.76%." That gives 535.97 each payment, for a total payment of 1071.95, which is 61 cents larger than the 1071.34 calculated in the first paragraph.

In other words, in an alternative world in which there were no inflation, the equivalent interest rate would be 4.76%, so the bank would ask for payments of 535.97. But we're not in that universe. Rather than paying 535.97 real dollars each time, you paid 548.75 the first time (which is more than 535.97) and 522.62 the second time (which is less than 535.97). You paid an extra 12.77 real dollars in your first payment. Since the bank got those 12.77 dollars a year early, it didn't charge you interest on them during the final year. 12.77*4.76% gets you the 61 cent difference.

The effective interest rate is how much extra real dollars you owe. Whenever you have a formula for how much real money you owe, you can plug in the effective interest rate. But you can't plug it into formulas in general.

The loan payment formula tells you:

(1) given a nominal interest rate, if you have payments that all have the same nominal value, how much that nominal value

(2) given an effective interest rate, if the payments have the same real value, how much that real value is

You can't use (1) to tell you what the real value of the amount in (2) is. It doesn't mix and match like that.