As discussed in Part 2, in recent years, there has been a ton of interest online in finance. Everyone, but especially millennials, want to know how to make more money quickly, whether to rent or buy a home, who to save for retirement and how much you need to retire, index funds vs. individual stocks, and how to get rich quick…or slowly.

In Part 1, I outline some of the benefits of home ownership versus stocks. If you buy a home for $240k outright, versus paying rent, you effectively double your money in a decade in terms of money saved by not paying rent. But is it really as good as it sounds.

Many people in their 20′s and 30′s are in a situation where they have some money saved up and a job that pays a decent wage, and they are wondering if they should put the cash in an index fund while renting — or — instead of renting, using the cash for a down payment on a home.

Let’s crunch some numbers…

Let’s assume starting in year ‘X’ you earn $2,000 in after-tax income every month, and wages grow at 3% a year. Hypothetically, you also have $43,000 cash, which can either be invested in the S&P 500 or on a down payment. You can either put this $2,000/monthly income into stocks or in a mortgage. Historically, stocks have returned 10%/year (including reinvested dividends). Rent is $2k/month and grows at 3%/year as well. Home prices rise 2%/year. Let’s assume a rent/price ratio of 18, which is the national average (12 months * 18 * $2000/rent~$430k home). A 10%-down mortgage on a $430k home means you put down $43,000. For simplicity, let’s also assume the mortgage interest deduction is offset by property taxes.

Case A: $43,000 initial in S&P 500 + $2000/month invested, compounded 10% a year for a decade. However, gains in rent offset gains in wages, so all after-tax income goes to rent. Total profit is simply 43k*(1.1^10-1)=$68k.

Case B is more complicated. Initial home equity $43,000. Initial home value: $430,000. After a decade, at 2%/year, the home is worth $525,000. Profit: $52,000. The mortgage payment is $2,000/month (adding $150/month extra to take into account other fees), which is offset by income. But wages are growing at 3% a year, which is invested in the S&P 500 like above.

Between years 0-1, total yearly wages are $24,000, all of which goes to the mortgage, so 0$ leftover and 0$ total

Between years 1-2, total yearly wages are $24,700 (wages rise 3%/year), leaving $700 leftover, which is put in the S&P 500

Between years 2-3, total yearly wages are $25,460, $1460 leftover and put in S&P 500; the $700 grows to $770; total= $2230

This is a recurrence relation… Wolfram Alpha is used to tabulate the remaining years. The end of the 10th year shows $46,000 of capital accumulated due to a combination of wage increases and S&P 500 reinvestments.

But this is an underestimate…let’s assume the wage increase is in 10 discrete chunks spread throughout the year in equal intervals, and each chunk is immediately invested in the S&P 500.

So for year 1, wages increase 3% by year-end to 720, so each chunk is $72. We have: 1.1*72 (the first chunk gets all of the S&P 500 gains) + 1.09*72…1.01*72 (the final gets the least). Adding up .1+.9+.8 … .01 gives (n^2+n)/(200). For n=10, the sum is .55, which times 72 is $40, which represents a 5.5% gain after a year on top of the $720, for a total of $760 after one year.

The new recurrence relation, which when evaluated gives $62,000 profit after the 10th year. Total profit $52k+$62k=$114k

$114k beats the $68k by 67%. One of the reasons why home ownership does well is because the mortgage is fixed at $2,000 a month, whereas the rent in the first example keeps growing. This allow the excess income to be invested in the market.