Question 15.15: Mortgage Interest and Inflation What is the constant-dollar,...
Mortgage Interest and Inflation
What is the constant-dollar, after-tax cost (the IRR) of a mortgage loan? The loan is for $100K at 9%, with 30 equal annual payments. Assume inflation is 4% and that the taxpayer itemizes and is in the 27.5% tax bracket.
This is a simplification of a typical mortgage, which has 360 monthly rather than 30 annual payments. The most intuitive way to solve this problem is to calculate interest payments and tax savings in nominal dollars. Then, nominal ATCFs are converted to year-0 dollars. Equation 15.3 is used to compute floan(= −3.85%).
1 + f_{Δitem} = \frac{1 + f_{item}}{1 + f} (15.3)
In Exhibit 15.13, columns B and C summarize the amortization schedule over the 30 years. In column B, each year’s interest payment is computed using IPMT, or each equals 9% of the previous year’s loan balance. The annual payment of $9734, shown in the data block for column B, is computed using PMT(loan rate,N,−amount borrowed) or PMT(A3,A2,−A1). Each year’s principal payment is $9734 minus the year’s interest payment. This is the yearly change in the loan balances shown in column C.
In column D, each year’s tax savings is computed as 27.5% of the year’s interest payment. This is subtracted from the annual payment of $9734 to calculate column E, the after-tax cash flows in nominal dollars.
In column F, each of these values is converted to year-0 dollars using the −3.85% value of f_{\Delta loan} . This is the first point in this example where inflation is considered. An answer in year-0 dollars is desired (this is when the $100,000 is received), and the first payment is made a full year later, at the end of the year. Thus, the conversion formula in column F is:
ATCF_{year-0\$} = Nominal ATCF/(1 + f )^{t}
Finally, the IRR is computed using IRR(F10:F39). Note that this interest rate is only 2.43%! The dramatic drop from 9% before tax deductions and inflation is because the tax deduction is based on the full interest payment, so the home owner really only pays about (1 − 27.5%) times 9%, or 6.52%, interest after taxes. Then, the 4% inflation rate reduces the true cost of this to about 2.52%.
To calculate this rate more exactly and to match the results of Exhibit 15.13, Equation 15.2 is used to divide 1.06525 by 1.04 to get 1.0243 or 2.43%.
(1 + Market rate) = (1 + i )(1 + f ) (15.2)
