Zaks Investment Advisory Service

Present value, interest rates, and how to compare mortgages

Several days ago, I switched on the radio and caught the middle of a fairly interesting discussion. They were talking about mortgage rates, or specifically, about which were better: 30-year mortgages or 15-year ones. A certain lady tried to prove that 30-year mortgages were best because they had smaller monthly payments. But her opponent pointed out that in total, the amount of money paid over a 30-year mortgage was much higher than that paid with a 15-year mortgage, which made the 15-year mortgage preferable. Moreover, he pointed out that by taking out the 15-year loan, you start paying back your so-called principle faster, i.e., according to him, you start paying yourself, whereas with 30-year loans, you spend much more time paying off the interest. The discussion was cut short and the two sides never were able to convince the other of anything. What is interesting is that while both sides expressed certain reasonable points, their reasoning and conclusions were, overall, completely wrong. Last time, we discussed interest rates, and I thought that our listeners might be interested in knowing where things really stand.

In order to figure out which mortgages are better, we will first have to learn what “present value” is. Imagine that we live in a simple world that only has one interest rate, which stands at 10 percent. You could invest $1,000 for a year and get back $1,100 in a year. Or you could go to a bank and borrow $1,000 for a year, and pay back $1,100 in a year. In this world, $1,000 today and $1,100 a year from now are interchangeable. They are, in a certain sense, each other’s equals: $1,000 now is the same thing as $1,100 a year from now. This $1,000 is the “present value” of $1,100, if the interest rates are 10 percent. In order to calculate the present value, one has to take the future amount and “discount” it, using the interest rate. Imagine that in our world, interest rates always stand at 10 percent, regardless of the term of the loan. If you went to the bank and borrowed $1,000 for two years, then in two years’ time you would have had to pay back $1,210: $100 in interest would have accumulated over the first year, leaving you owing the bank $1,100 for the second year. And 10 percent of $1,100 is $110, which means that at the end of two years, you would have had to pay the bank $1,000 plus $100 plus $110, for a total of $1,210. Again, $1,000 today and $1,210 two years from now are sort of the same thing from the financial point of view: if you have $1,000 now, you could “exchange” them for $1,210 in two years by investing $1,000 for that term. If you are going to have $1,210 in two years’ time, you could “exchange” them for $1,000 today by taking out a two-year loan at the bank. This very idea of the interchangeability of today’s money with that of the future, and their relation to each other through interest rates, is one of the most fundamental in the financial world.

Similar ideas apply to situations in which a loan is returned in installments rather than in full at the end of the term. Supposing you borrowed $1,000 for two years at 10 percent. You could have agreed with the bank that at the end of the fist year, you would pay back, say, $100, and return the rest at the end of the second year. At the end of the first year, you would have owed the bank $1,100. Having paid $100, you would have been left owing $1,000. Over the second year, another $100 in interest would have been assessed to this $1,000, and at the end of the second year you would have owed $1,100. Thus, a $100 payment at the end of the first year and a $1,100 one at the end of the second have a present value of $1,000. In other words, $1,000 today and $100 in a year plus $1,100 in another two years have the same value.

What are mortgages? They are ordinary loans, but their payments are structured to remain the same throughout the payment term. I will present a simple example. Imagine that a bank issues a $1,000 mortgage at a 10-percent annual rate, and that this mortgage is paid once a year instead of every month (of course, there are no such loans in real life, but this example should help us). At the end of the first (and only) year, we owe the bank $1,100, which we pay. And here the matter rests. Meanwhile, another bank customer decided to take out a two-year mortgage. This customer would have had to make two payments: at the end of the first and second years. Our banker sat down at the computer and quickly calculated that the payments come to $576 each. Payment calculations for a 30-year mortgage are pretty tricky, but we can easily check everything with a two-year loan: so, $100 in interest will be assessed at the end of the first year on the $1,000 taken out at 10 percent. After paying $576, the customer will still owe $1,100 minus $576, i.e. $524. Another $52 in interest will be added to these $524, and making another $576 payment, the customer settles with the bank. What makes this situation interesting? Not the fact that the two payments were in the same, but that a $576 payment in one year plus a $576 payment in two are, in total, worth the same as $1,000 today! And since both loans are, according to their present value, equal to $1,000, then this means that they are equal to each other. In other words, one more time: two loans, one paid in a full single installment of $1,100, and another, which was paid in two $576 installments, are worth the same. And the fact that in one case the customer formally “overpaid,” so to speak, by $100 ($1,100 minus $1,000), while in the other – by $152 ($576 plus $576 minus $1,000), makes absolutely no difference.

From this, it follows that the argument made by one of the debaters that a 30-year loan is worse than a 15-year loan because in the end, you formally pay more, is absolutely meaningless. But the interest rates charged really do play a principle role. Whether you borrow over 15 years or 30 depends entirely on which is more convenient for you. This has no relation to the present value of the loan itself. But interest rates do play a fundamental role. It is the interest rates that should be compared – not the monthly payments.

Our listeners, of course, know all these things. Nevertheless, discussions you sometimes hear on the radio often use arguments that have no relation to the real issue. I hope that today’s program helped to clear up this problem a bit. And with this, we will draw today’s program to a close. This was Sergey Zaks. Thank you for your attention and until next time.