Mortgages - part 2

Ok, so last week I made mention of how rent paid today is actually buying valuable time to increase the size of a down payment. But how to quantify this? Beware, the answer might only appeal to those with a bit of finance background or who were forced to take a course taught by a certain jolly accountant with an interesting sense of humor. First, you must research and decide the following variables:

• Q, Current savings – size of your savings today (or whatever time you consider time zero)
• r, Savings rate – rate at which your savings earn interest
• S, Annual savings before rent – how much you can put in the bank each year after expenses but before rent
• R, Annual rent – how much you expect to pay in rent before you buy a house (can be zero)
• P, Price of house today – the current price of the type of house you’d like to buy
• p, Price inflation rate – percentage the price is expected to go up each year
• m, Mortgage rate – the interest rate the bank charges you for a mortgage (assume fixed)
• M, Annual mortgage payment – how much you’d be willing to repay each year

To take into account the time value of money, we’re going to find the present value (with respect to now, or time zero) of the sum of rent, down payment, plus mortgage payments from now until when your mortgage is repaid, as a function of x, the time in years at which you buy your house. Note that the discount rate represents how much interest you’re missing out on if you spend your money instead of saving it, so for the conversion factors the discount rate will be r (and not m, which is another interest rate for another purpose).

When you buy your house at year x, you have paid an annual rent of R for x years. This amounts to a present value of R*(p/a, r, x). This is the cost of rent.

When you buy your house at year x, you would have saved up Q*(f/p, r, x)+S*(f/a, r, x). This amounts to a present value of Q+S*(p/a, r, x). This is the cost of the down payment. When you buy your house at year x, the price of the house would have risen to P*(1+p)^x, while your savings would have grown to Q*(f/p, r, x)+S*(f/a, r, x), both values with respect to year x (not today). That means you will have to borrow B=P*(1+p)^x-[Q*(f/p, r, x)+S*(f/a, r, x)] from the bank. Given that we’ll pay back M dollars a year and the bank charges an interest rate of m, what’s the present value of all your payments? To answer that, we must first find out how long the payments will last. Let’s jump to year x and think about present value with respect to that time for a moment. Consider that the amount you borrowed, B, is precisely the (hypothetical) present value of all your payments at a discount rate of m. To find out y, how many years it will take to repay the mortgage, we must solve the equation B=M*(p/a, m, y) for the value of y. As it turns out, y=ln[M/(M-B*m)]/ln(1+m). The constraint of the logarithm states that M must be greater than B*m, which makes sense, because it says your annual payment should be at least as much as the annual interest, or else you’ll never pay it all off. Ok, so this means that, with respect to year x, the present value (with the real discount rate of r) of all your mortgage payments will be M*(p/a, r, y). If we convert this to the present value with respect to time zero, then we apply a further factor of (p/f, r, x). So the total cost of buying a house in year x, in present value with respect to today, is R*(p/a, r, x)+Q+S*(p/a, r, x)+M*(p/a, r, y)*(p/f, r, x), where y=ln[M/(M-B*m)]/ln(1+m), and B=P*(1+p)^x-[Q*(f/p, r, x)+S*(f/a, r, x)] Now this isn’t an easily differentiable function with respect to x, so the easiest thing to do is to enter the formula into Excel and let it show you what the present value is for x equals a variety of values, and find the x for which total present value cost is minimized. For example, let’s say I want to aim for a type of house that costs $400k today, and rises in price by 7% each year. My annual rent in the mean time is $10k, my annual savings is $50k (before rent), the savings rate is 6%, the mortgage rate is 10%, and I’m willing to pay all $50k of savings into mortgage payments after I take a mortgage. My formula tells me that I best buy a house after 3 years of renting, at a present value total cost (rent plus mortgage) of $516k. If I were to buy the house today because someone told me that paying rent is like burning money, then the present value cost of the house would be $522k. So, by staying put in my apartment over 3 years, I saved $6K PV by the end of it all. Remember, that's $6K in present value, which, at 6% savings, will grow to $18K by the time I repaided the entire mortgage.