The Calculus Diaries - Jennifer Ouellette [53]
However, that would only be the case if I took that money and stuffed it under a mattress. Common sense would dictate I deposit the funds in an interest-bearing account. Let’s be optimistic and assume that account yields 5 percent interest. How does that change the time needed to save a down payment? I am depositing $1,000 per month, but the money that’s been in the account for two years will have earned more money than the money I’ve just deposited.33 So I have to add the $1,000 I deposit that first month and calculate how much interest will accrue in five years, and then do the same for the second monthly deposit, and the third, and the fourth, and so on.
Now it is a matter of adding together lots of smaller sums—or of taking an integral. Even though I am making monthly deposits, from a calculus standpoint, the funds are accumulating every instant. So for every interval of Δt (for time), we have accumulated $12,000 × Δt dollars, which stays in the bank for however much of that five-year period is left (5 years − t) and earns 5 percent interest. At the end of that five-year period, I will have saved $60,000 plus a bit extra in accrued interest—which I hope will be sufficient to cover the potentially exorbitant closing costs.
What if you don’t have the required 20 percent deposit on a home—a common problem for those living in areas with especially high housing costs? Traditionally, you would be out of luck; no bank would approve your mortgage. There is very good reason for this. That down payment gives you equity in the house, the difference between your home’s assessed value and the amount of money you still owe the bank. But then alternative types of mortgage loans became increasingly available, some allowing borrowers to take out a mortgage with as little as 5 percent down. The trade-off for the lower down payment is usually higher interest rates and thus higher monthly payments.
Then someone had the brilliant notion of offering adjustable rate mortgages (ARMs), in which the interest rate fluctuates over time, resetting to a new (higher) rate every few years. We have already seen that a small increase in the interest rate on a mortgage can make a huge difference in the monthly payment. The impact is even more dramatic with an ARM. Say you took out a loan of $100,000 at an adjustable rate over thirty years. You could easily afford the monthly payments at the introductory “teaser rate,” which could be as low as 1.2 percent for the first two to five years: roughly $331 per month. But then the interest rate would reset and jump to 7 percent, and suddenly you would be paying $617 a month. Unless you had a corresponding increase in income, you would quickly fall behind in your payments. Worse, some of those ARMs were interest-only loans, for which people would pay just the interest and the principal never decreased.
Millions of people took on these risky loans; given the above, it’s fair to ask, what the hell were they thinking? Chances are, they weren’t doing the math. Or perhaps they believed that the value of their houses, and hence their equity, would continue to skyrocket, and they could sell their homes at a tidy profit before the interest rates reset.
But nothing can expand forever—except, perhaps, the universe—and those homeowners were gambling that they could get out before the market softened or collapsed outright. Several economists warned that the bubble would burst, but their dire predictions did little to dampen the enthusiasm at the height of the housing frenzy. All the classic bubble conditions were present: high demand, limited supply, and an influx of ready cash as banks relaxed their lending standards and made millions of subprime loans to borrowers who—in retrospect—should never have received loan approval because they couldn’t afford the payments once the interest rates reset. When those buyers began to default en masse, the