Maria aims to purchase a house for $200,000. Because she has financial assets of $300,000, one of her options is to buy the house for all cash. Alternatively, she can obtain a mortgage of $180,000 for 15 years at 4 percent and zero fees. Maria has excess disposable income of $2,000 a month, which would more than cover the payment on a mortgage.
How does she make the choice? The method I would use is to calculate her net worth at the end of her expected period in the house, or at the mortgage payoff date, whichever comes first. I will assume the mortgage payoff period of 15 years. The calculation of future net worth would be done twice, once on the assumption that Maria purchases with all cash, and once on the assumption that she borrows 90 percent of the price.
In the all-cash purchase, future net worth after 15 years is the sum of the future value of Marie’s $2,000 of excess income, invested monthly to earn 4 percent, or $492,181; plus the future value of the $100,000 of financial assets left after buying the house for cash, or $182,030; plus the future value of the house at an assumed appreciation rate of 3 percent, or $313,486. The three items sum to a total net worth of $987,697.
If Maria finances the purchase with a $180,000 mortgage at 4 percent, her excess income is reduced from $2,000 to $669 because of the mortgage payment, but her financial assets are reduced only by $20,000 to $280,000. The future values are $164,526, $509,684 and $313,486, which sum to $987,697. The future net worth is the same in both cases because I assumed that Maria’s financial assets earned the same return as the rate she paid on the mortgage.
The relationship between the mortgage rate and the investment rate is the major factor determining whether or not it makes sense to pay all cash. If Maria earns only 2 percent on investments while paying 4 percent on a mortgage, financing the purchase with a mortgage would result in a future net worth of only $831,558. In this case, she should pay all cash. On the other hand, if she earned 8 percent on investments, her net worth would be $1,470,772, so she should finance the purchase.
But there is a small proviso. Mortgage interest paid is deductible, whereas interest earned is taxable. If Maria is in the 28 percent tax bracket, taking account of the mortgage interest deduction would increase the future net worth in the case where she borrows by $20,000-$25,000.
Another proviso that is much more likely to be overlooked: I have assumed that when the purchaser pays all cash, she allocates to monthly savings an amount equal to the monthly mortgage payment that she would have made had she borrowed. In the borrowing case, she is required to pay $1,331 a month on the mortgage, leaving $669 for investment. In the all-cash purchase, the obligatory payment is gone and she must invest $2,000 a month voluntarily.
This is critically important. If Maria spends all her disposable income in the two cases, her future net worth in the borrowing case would be twice as large as in the all cash purchase case, the 8 percent investment rate notwithstanding. It reminds us, once again, that the forced saving feature of the long-term fully amortizing mortgage serves many consumers well.
Jack Guttentag is professor emeritus of finance at the Wharton School of the University of Pennsylvania.