The 72 Rule explained

BY MARY HOLM

The "72 Rule" has generated some interesting correspondence over the last few weeks. But where does the "magic number" 72 come from?

As one of your correspondents on this topic pointed out, the number reduces slightly from 72 to 69 for low interest rates.

Calculation shows that this is just 100 times the natural logarithm (ln) of 2.

The 100 comes from the fact that interest rates are expressed as a percentage; the natural logarithm derives from the exponential growth that occurs with compound interest; and the number 2 arises because the rule gives the time to double the investment.

This can readily be generalised. Admittedly, the name "100 ln x Rule" does not have quite the same ring to it, but with modern calculators it is easy to use.

For the example in last week's correspondence (building a $1 investment to $1 million) the factor x is 1 million, and 100 ln x is about 1400.

That means that the time taken to achieve this for an interest rate of 10 per cent is about 140 years.

(The method works accurately for low interest rates, but only a slight upward correction is required for normal rates.)

A $1000 investment at 10 per cent would take 70 years to grow to $1 million - still an unreasonably long time!

With a saving/investment of $1000 per year the period reduces to about 50 years.

I don't know of any simple rule like the one above to calculate this quickly, but it does show that saving/investment is the only realistic way for the average person to achieve such a result.

Of the several letters I've received explaining the 72 Rule, yours is the clearest.

Perhaps that's because you're a university professor - although there are those who would say that it's despite your being a professor!

(For Rip van Winkles who haven't seen this column lately, the 72 Rule says that if you divide a percentage return into 72, you'll get roughly the number of years it will take for your investment to double. For example, an 8 per cent investment will double in about 9 years.)

The mathematicians among us can now use your "100 ln x Rule".

But I'm still happy to stick with 72. For returns between about 3 and 15 per cent, it's pretty accurate. And you can do the sums in your head.

The point you made about accumulating $1 million by saving $1000 each year for 50 years, at 10 per cent, might encourage young would-be millionaires. But, at the risk of being a wet blanket, I feel obliged to say:

Ten per cent is too high a rate to assume. At a more realistic 5 per cent, $1000 a year will grow to a bit more than $200,000 in 50 years. That's not quite as exciting, but still not to be sneezed at.

Keep in mind the point made above about inflation.

One final note: Thanks to all of those who sent me many lines of maths about the 72 Rule.

Whoever said New Zealanders were mathematically illiterate?

* Send questions for Mary to Money Matters, Business Herald, PO Box 32, Auckland; or e-mail: maryh@pl.net.

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Mary cannot answer all questions, correspond directly with readers, or give financial advice outside the column.