How to Make a Million Dollars
Posted: December 14th, 2010 | Author: James | Filed under: Uncategorized | No Comments »For most of us, becoming a millionaire seems a bit out of reach. Really, though, it’s all math.
Warning: math geekery ahead.
A million dollars is one thousand times one thousand. If you can save $1000 per month, then after 1000 months (83 years, 4 months), you’ll have $1,000,000. Simple.
Of course, the average person’s lifespan in North America is on the order of 75 years and we’ve been trained to expect to retire at 65. Since we don’t generally start working for enough money to even think about saving that much until we’re in our twenties, let’s assume a working-and-saving life of about 40 years (age 25 to age 65). Let’s leave for another day the question of finding $1000/month in after-tax income that we can save. So to save $1,000,000 in 40 years, you’ll have to save:
$1,000,000/(40 years x 12 months/year) = $1,000,000/480 = $2083.33/month
That’s $25,000 per year in savings. It’s a tall order for most of us.
Up until now, we’ve assuming your saved money will essentially be “shoved under a mattress”. That’s not generally the way things are done. For one thing, it’s not safe. Banks, on the other hand, are considered safe. Just make sure your bank is insured by (in the US) the FDIC, (in Canada) the CDIC, or the equivalent guarantor in your country¹. Banks pay interest on amounts in savings accounts. Interest amounts are currently much lower than has been historically normal but nonetheless, money in the bank “works for you”. That is to say that the bank, pays you a small amount when you keep your money with them. The can do this because they loan out money to others and the money they loan out is yours (it’s more complicated than that, of course, but this isn’t an essay about banking). In effect, the bank is renting your money from you.
Let’s pretend your bank is willing to pay you 4% interest on your savings. That’s a bit optimistic today but is not too far out of line with typical interest rates over the last half century. If you deposit $1000 in your bank at 4% interest, then after one year you will have $1040. The next year, it will be worth $1081.60, then $1124.86 a year after that, and $1169.85 after still another year. In reality, interest is generally paid out monthly. Instead of paying 4% per year, your bank would pay 4/12 or 0.3333% each month. After one month, your original thousand dollars will be worth $1003.33. After two months, $1006.67. Wait a year and it will be at $1040.74. The numbers aren’t much different (only 74 cents after one year) so people usually simplify explanations by working only with years.
You may notice that each year not only does the amount get larger but that the amount of increase each year also gets larger. This is because you’re earning interest not only on your original $1000 but also on the interest you had earned. The first year, you’re earning interest on $1000. The second, on $1040. The third, on $1081.60. And so on. This is known as “compound interest” and is your secret weapon in the savings game.
Each year, your savings will be multiplied 1.04 times. After ten years, your savings will have increased to 1.04 x 1.04 x 1.04 x 1.04 x 1.04 x 1.04 x 1.04 x 1.04 x 1.04 x 1.04 times its original value, or $1480.23. The formula for the value of your original $1000 deposit after n years is 1.04**n (where “**” means “to the power of”). More generally, if you name your annual interest rate i (in our case, 4%), then the value after n years is (1 + i)**n.
Eventually, your original $1000 deposit will double to $2000. That’s when (1 + i)**n = 2. We can calculate that using logarithms²: n = log(2) / log(1 + i). If we stick with our example where i = 4, then n = 17.672988 years — roughly 17 years, 8 months. After another 17 years, 8 months (35 years, 4 months), the original $1000 that had doubled to $2000 will have doubled again to $4000. After 10 doublings, your money will be worth 1024 times its original value ($1,024,000). Of course, you want to wait 176 years even less than you want to wait 83. The important point here, though, is that your money grows faster and faster the longer you leave it. The lesson is you need to begin to save as early as you can.
Luckily, neither technique needs to work in isolation. What if you save $1000/month and it earns you 4% interest?
After one year, the first $1000 will have been in the bank for 12 months. The second for 11 months, the third for 10 months, and so on. You can run the calculations yourself for n = 12, n = 11, etc. and add them up. After 12 months on this plan, you’ll have $12,263.20 in the bank. After 24 months, $25026.01. Three years,$38,308.82. Believe it or not, after 40 years you’ll have $1,185,901.21! There’s your million dollars³. And then some!
Dough has a Savings Calculator tool you can use to experiment.
1: There is a limit on the amount of your bank deposit that is insured. For total safety, when you start to get close to the deposit insurance limit, create another account and put half the first account balance into the second.
2: There is a simple “back of the envelope” rule for approximating “doubling time” without a calculator: divide the annual interest rate into 72. The quotient is roughly the doubling time in years. In our 4% example, 72/4 = 18. This is a very close approximation to the 17 years, 8 months the actual formula produced. Given that interest rates vary over time, the best you can hope for is an approximation anyway.
3: For exactly $1,000,000 over 40 working years while earning 4% annual interest, you could save a little less per month: $1000 x ($1,000,000 / $1185901.21) = $843.24. With the variability in the interest that you’ll earn, not to mention bank fees and other expenses, not to mention planning for unforeseen problems, you’re always better to “play it safe” by saving a little extra.
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