# Money Math: Quick and Dirty Financial Equations

Money Magazine recently ran a sidebar outlining some handy little financial rules of thumb. Here are my favorites:

The Rule of 72

This is a useful rule for figuring out how quickly your investments will grow… Simply divide the expected rate of return into 72 and you’ll get an estimate of the number of years required to double your money. Earning 9%? Your money will double in 8 years.

The Future Value of Money

If you’re debating about whether or not to make a purchase, keep this little tidbit in mind: Adding zero to the end of the price gives you a quick and dirty estimate of how much of your (potential) retirement nestegg you’re giving up when you make a purchase. That’s the amount of money you’d have in 30 years if you invested the money at an 8% return instead of spending it.

Grooving on a shiny new 160 GB iPod? That’ll set you back close to \$3500 down the road. Doesn’t sound quite as enticing now, does it?

Hourly Earnings

If you want to translate a salary into an hourly wage simply cut it in half and then divide by 1000. So if you make \$50k/year, you’re earning roughly \$25/hour. Conversely, you can get a rough estimate of your full time salary by doubling your hourly wage and adding three zeros to the end of it.

### 14 Responses to “Money Math: Quick and Dirty Financial Equations”

1. Dan,

It is mathematically invalid to compute an annualized gains by summing the values and dividing by the number of years. The correct method to average “n” samples would be to multiply thier decimal equivalents and take the “n”th root.

For your example, the average of 100%, -50%, 25%, 100%, -50% would be:

(2.0 * 0.5 * 1.25 * 2.0 * 0.5)^(1/5) = 1.0456 or a 4.56% increase. Note that this is the same as the compund interest value you give above and the rule of 72 holds true even for stocks when you compute the annualized gain correctly.

I realize the issues you may have with this CAGR computation, but it is the only mathematically valid way to compute the “average return”. The 25% you mention in your first post is a meaningless number.

2. I was aware of the salary one the rule of 72, but I wasn’t of the The Future Value of Money. That is a good trick that should help people hold on to (and hopefully invest) some cash…

3. Dan, I still think you’re missing the point of this article, which is to provide quick-and-dirty math shortcuts for people who aren’t particularly interested in financial modeling. Of course there are caveats to the interpretation and use of these rules of thumb, but these shortcuts have their uses.

As for planning for investment using imaginary numbers, this is what PF professionals, financial journalists, and the casual planner do all the time. We make assumptions like the market will grow at 10%; equities will return 9% and bonds 4%; inflation will be 3-4%.

How do YOU personally figure out how to live in retirement without using any “imaginary numbers”? How do you figure out how much money you will need in retirement, if you can’t make assumptions about inflation? After all, no one knows exactly what the rate of inflation will be in the future. All we can do is look at inflation in the past (a CAGR if you calculate it on price indices!), but our projection will be an imaginary number.

If you have a target savings amount for retirement, how much will your investments need to earn to get you there? 8% a year? 9% a year? Those will be imaginary numbers too, since you can’t know for sure that they’ll earn that rate every year.

Of course CAGR is not predictive; it’s backwards-looking. But NO ONE can predict with absolute certainty the future. And even if there were precise quantitative models for projecting the future, most people would be so put off by the math and statistics that they wouldn’t even think about retirement at all!

This is precisely why quick-and-dirty is useful. You make an assumption that the future will roughly be the same as the past (given a long time horizon). Perhaps a bad assumption – especially if you buy the Black Swan theory – but the alternative might be to make no assumption at all, throw up your hands and say, “There’s no way I can make any plans or goals because all of my assumptions are crap!”

Anyway, I never said you should apply the rule of 72 to individual stocks. The 10% growth rate is on the market as a whole, and on the market for pretty much the last decade. It’s a useless number for individual stocks. Moreover, any good article on the rule of 72 will mention that it is a bad estimate at extremes. It’s quick-and-dirty, not hard-and-fast. But please, if you have a better back-of-the-envelope method for projecting returns on investments, share with the rest of us.

(Ooh, that last sentence sounds kind of mean. But I’m genuinely interested if you have a quick-math replacement that allows for some degree of projection without making too many simplifying assumptions.)

4. Lily,

CAGR is a very rough and dirty measurement indeed.

This is from investopedia:
“CAGR isn’t the actual return in reality. It’s an imaginary number that describes the rate at which an investment would have grown if it grew at a steady rate. You can think of CAGR as a way to smooth out the returns.”

Personally, I don’t like to base my expectations for the future of my investments on imaginary numbers. Especially when I’m trying to figure out how to live in retirement.

5. Wow. That future value thing is cool. I’ll be sure and use it on the wife and kids 😉

Thanks!

6. Dan – you’re right, but that is why in my previous comment I said the rule can be used “if you make broad assumptions about long-term growth.” I should have specified that you would need to assume that the market or your investment grows 10% each year.

I completely agree with Frugal Dad that, as I said previously, this is “quick and dirty.” It might be “erroneous,” but it’s a nice substitute for examining price behavior every year to come up with a CAGR.

7. One of the things that may have been lost on this post was that these are “quick and dirty” estimates. Several other comments have tried to break down every aspect of the equation, when really these are just useful calculations you can do on the fly in your head. The Rule of 72 has always been my favorite.

8. Lily,

I didn’t say that it couldn’t be applied, I said it is erroneously applied. Stocks do not compound, save for the dividend portion. 60% of the long term appreciation of stocks is price appreciation. A stock that was 10 last year and grew to 11 this year does not start over at eleven with the expectation to go to 12.1 in a year. A 10% annual growth over 10 years merely means a stock started at 10 and grew to 25.93 in 10 years. Looking at the past, we can then say the “miracle of compound interest” occured, when it actually did not. The compound interest equation used in this case is just a representation of what the stock actually did, which was rise 159.3% over 10 years.

Try your compound interest math on this stock:

Year 1 (100% gain)
Year 2 (50% loss)
Year 3 (25% gain)
Year 4 (100% gain)
Year 5 (50% loss)

Average annual return is 25%, it should double in 2.88 years according to the rule of 72.

What did the stock actually do? (same for whatever start price, but let’s say it started at 10)

Started at 10
Year 1 (doubled to 20)
Year 2 (halved to 10)
Year 3 (rose to 12.50)
Year 4 (doubled to 25)
Year 5 (halved to 12.50

It didn’t experience a 25% annual return, it experienced a 25% total return over 5 years, meaning it will double in 20 years if it repeats this exact 5 year performance 3 more times. Compound interest, however says that a 5 year 25% return is a 4.56% annual return. The rule of 72 says that should double in 15.79 years based on its current performance. The rule of 72 is off by 21.05%.

9. Dan – The rule of 72 can be applied to equities securities if you make broad assumptions about long-term growth. For example, you can say that according to the rule of 72, if the market continues to return 10% on average, then your (all-stock) portfolio will double in value in a little over 7 years.

Of course this isn’t the best way to think about stock investing, since that 10% number is average returns for pretty much the last century, and there’s no telling what returns will be like over the next 7 years. But that’s why it’s “quick and dirty” – perhaps quicker and dirtier when applied to stock growth than when applied to debt instruments, but easier than pulling out a financial calculator.

At any rate, growth in equities investments do compound. The 10% growth you get in the market today will also grow with your portfolio next year.

10. Ryan & ntguru: Yes, 2080 is more accurate, but this gets you a ballpark and pretty much anyone can do it in their head. It’s off by less than 5% to use 2000.

11. The hourly-salary conversion rule is obviously having you divide salary by 2000. This is a better number to use than 2080 because 2080 assumes 40 hours a week for all 52 weeks of the year. Most people have at least one week of vacation if not two, plus national holidays are typically another 1-2 weeks a year worth.

If you’re considering hourly/consultant/self-employed versus salary/W2/employee you also need to figure that self-employed is going to pay both halves of FICA and possible other expenses (medical etc). Most companies figure additional bene’s are worth at least 20%. For example a person with a \$100K salary has a total package worth about \$120K or more.

12. Or to get your hourly wage you can more accurately divide your yearly salary by 2080 as that is the number of hours in you should work in a year.

13. It’s interesting to note that the rule of 72 is often erroneously applied to stocks and stock mutual funds. It estimates the effect of annual coumpound interest. Stocks do not pay interest nor do they compound annually.