True cost of a mortgage when considering inflation

Question

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?

Answer 1

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.

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.

Answer 2

The current accepted answer is essentially correct—the inflation rate cannot simply be deducted from the interest rate. However, it underestimates the real cost of the loan by about the same amount as the naïve method overestimates when the mortgage interest deduction is taken into account. The mistake is that it applies inflation discounting to a fixed nominal payment based on a tax adjusted interest rate. This is incorrect because the true tax adjusted nominal payment is not fixed; with a fixed pre-tax nominal payment, the tax adjusted nominal payment increases over the life of the loan as the interest paid in each period (and, correspondingly, the value of the tax deduction) decreases.

The correct way to account for both inflation and the tax deduction is to separately compute the real (inflation adjusted) payments on the loan and the real benefit of deduction.

The fixed nominal payment is computed with the loan formula:

pv = 500000
n = 30*12
r = 0.04/12 = 0.0033333 (0.333333% per month)

p = r*pv/(1 - (1 + r)^-n) = 2387.08

As in the other answer, we assume that the nominal inflation rate is 2% compounded monthly and discount the nominal monthly payments to get the inflation adjusted total payment.

inf = 0.02/12 = 0.00166667

adjusted = (p - p*(1 + inf)^-n)/inf = 645821

The value of the mortgage interest deduction is based on the interest paid in each period. Assuming a 30% tax rate*, the deduction's nominal value is 30% of the nominal interest paid. Specifically, in the kth period, the nominal value of the deduction is ded_nom[k] = fv[k] * r * r_tax where fv[k] is the principal balance to which the interest is applied.

To compute the real value of the deduction, we must discount each of these nominal values for inflation:

ded_real[k] = fv[k] * r * r_tax * (1 + inf)^-k 

The total real value of the deduction is then

ded_real = Σ ded_real[k] for k = 1 to n
         = Σ fv[k] * r * r_tax * (1 + inf)^-k
         = 87492

Subtracting this from the real total of the payments, we have the real total cost of the loan: $558,329, or about 1.117 times the principal.

You'll notice that the naïve method does provide a decent approximation ($562,564). That's because it applies the Fisher approximation, which is valid as long as the nominal interest rate, expected inflation rate, and number of periods are reasonably small. Thus, for a back-of-the-envelope estimate under such conditions, it's perfectly appropriate to simply deduct the inflation rate from the nominal interest rate.

* This calculation ignores the distinction between marginal and effective tax rates in the U.S. individual income tax system, which will lead to overestimating the benefits of the deduction in some cases.