11

For the purposes of this question, let's assume the following:

  1. Inflation is and will be a steady 2%
  2. I am in the 30% tax bracket and there is a home mortgage interest deduction
  3. I have a 30-year fixed mortgage at an interest rate of 4%
  4. Housing values are constant except for rising at the rate of inflation

(If any of these assumptions are untenable in a way that significantly changes the answer, please point that out.)

Assuming I take advantage of the tax deduction, my effective interest rate is 4% - (4%*30%) = 2.8%. For 30 years on a $500,000 mortgage, this gives (according to the mortgage calculator that Google shows) a total mortgage cost of $739,610. But much of that is paid with inflated dollars, so the true cost is not actually 1.48 times the principal, but something less.

Subtracting the rate of inflation gives a real interest rate of 0.8%, such that the real cost of the mortgage in 2015 dollars is $562,564, or just 1.125 times the principal.

Is it correct to simply subtract the rate of inflation like this?

  • On further reflection, I think that fourth assumption is unnecessary for what follows, but I'll leave it just in case. (It was necessary for a previous approach to the problem I took, which I found the flaw in.) – reo katoa Mar 19 '15 at 23:19
10

The rates can't just be subtracted. You have to discount each future payment for inflation to find the total inflation-adjusted cost.

First though, the calculator you are using is assuming the input rate is a nominal rate, compounded monthly, not an effective rate. I'll proceed with nominal annual rates.

So, using the loan formula to arrive at the figure you calculated.

pv = 500000
n = 30*12
r = 0.028/12 = 0.00233333 (0.233333% per month)

p = r*pv/(1 - (1 + r)^-n) = 2054.47

p*n = 739610.00

Checking the discounting is working, because we'll use this for discounting inflation.

(p - (1 + r)^-n*p)/r = 500000. = pv

Yes, discounting at the interest rate gets us back to the mortgage present value.

It is basically the summation of the payments, each discounted at the interest rate.

pv = Σ p (1 + r)^-k for k = 1 to n
∴ pv = 500000.

enter image description here

This is the same method as can be used to discount for inflation.

Taking inflation at 2% nominal compounded monthly to get the inflation adjusted total.

inf = 0.02/12 = 0.00166667

adjusted = (p - (1 + inf)^-n*p)/inf = 555834.41

The total inflation-adjusted cost of the mortgage is $555,834.41

Now to try subtracting the inflation rate from the interest rate to see what the total amount paid with an adjusted rate (r2) would be.

r2 = r - inf = 0.00233333 - 0.00166667 = 0.00066666

p2 = r2*pv/(1 - (1 + r2)^-n) = 1562.67

p2*n = 562562.90

It comes out different from the inflation adjusted total, so subtracting the rates does not work.

To see what is happening in more detail, here is the same procedure simplified, with just three compounding periods.

pv = 500000
n = 3
r = 0.1 (10% per month)

p = r*pv/(1 - (1 + r)^-n) = 201057.40

pv = Σ p (1 + r)^-k for k = 1 to n
∴ pv = (p (1 + r)^-1) + (p (1 + r)^-2) + (p (1 + r)^-3) = 500000.

If inflation is 4% per month can the rates be subtracted?

inf = 0.04 (4% per month)
adjusted = (p (1 + inf)^-1) + (p (1 + inf)^-2) + (p (1 + inf)^-3) = 557952.59

Seeing if the total comes out the same starting with the rate minus inflation.

pv = 500000 
n = 3 
r2 = 0.1 - 0.04 = 0.06 (6% per month)
p2 = r2*pv/(1 - (1 + r2)^-n) = 187054.90 *
p2*n = 561164.72

So again, the total with the rate minus inflation ($561,164.72) is not the same as the full rate total with the payments properly discounted for inflation ($557,952.59).

* Check
  pv = Σ p2 (1 + r2)^-k for k = 1 to n
  ∴ pv = (p2 (1 + r2)^-1) + (p2 (1 + r2)^-2) + (p2 (1 + r2)^-3) = 500000.
  • Thanks for the walkthrough. I see that the result is different, but the message is roughly the same -- a mortgage under the conditions given is only about 11% more expensive than cash, not the 40% one might suppose. – reo katoa Mar 20 '15 at 17:05
  • Yes, it's quite an observation. In your example the inflation-adjusted total cost is quite low. Furthermore, if the savings deposit rate was, say 1.5%, and inflation was 2% it would be more cost effective to be in debt with a mortgage than it would be to have savings deposits. – Chris Degnen Mar 20 '15 at 23:06
  • And savings deposit rates on a savings account are hardly over 0.9% today. – A.R. Nov 7 '17 at 17:44

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.