The Bookkeeper’s Arithmetic

I promised you at the start that there would be few formulas in this book. I keep my word. But there are five checks any good bookkeeper runs on other people’s numbers, and almost none of the people whose money he counts ever run them. Some are formulas, some are just the habit of asking. I’ll show you each one with a story, and you can check the numbers yourself, on a napkin.

A Dollar and a Half

At the end of the nineteenth century a Sicilian peasant who came to New York could earn a dollar and a half a day even at the simplest work. Back in Sicily, Dash writes, he got five cents—thirty times less. Thirty times! No wonder they came by the thousand.

But Dash adds straight away that the cost of living ate up the gain. Rent, bread, coal and clothes cost incomparably more in New York. Families on Elizabeth Street lived as poor as they had at home, and many went without everything so they could send more money to their relatives.

Remember the bookkeeper’s first check. A sum means nothing until you know what it buys. Thirty times more in money can turn into one and a half times more in bread. Economists call this real income.

The rule works just as hard across time as across distance. In Germany in 1914 a dollar bought a little over four marks. In November 1923 it bought more than four trillion. A loaf of bread in Berlin sold for about a hundred and sixty marks at the end of 1922, and a year later for two hundred billion. A man who put ten thousand marks in the bank for his old age in 1914 couldn’t buy a box of matches with them in 1923. His money was all there. On paper.

Someone promises you interest? Take inflation away from it. Whatever is left is what you earned. Sometimes what’s left is below zero. Only, subtracting is a rough guess, and the guess holds while the numbers are small: three per cent on a deposit with two per cent inflation gives you about one. When the numbers are big, divide, don’t subtract. Twenty per cent on a deposit with ten per cent inflation: divide a hundred and twenty by a hundred and ten, you get 1.0909, take away one, and it comes to a little over nine per cent, not ten. And in Berlin in 1923 the rough guess wasn’t just lying any more. It was laughing at you.

Up Fifty, Down Fifty

The second check looks like a conjuring trick. Picture this: oil goes up fifty per cent in a year, and the next year it goes down fifty per cent. On average that’s zero, right?

Wrong. You had a hundred. It became a hundred and fifty. Take away half and you have seventy-five. You’ve lost a quarter.

When percentages follow one another, the simple average lies. That’s why someone invented the geometric mean. It tells you what percentage you would have needed to earn, the same every year, to end up in the same place. Someone shows you a fund’s “average return over ten years”? Ask which average. The difference can be huge. And it almost always works against you.

The Rule of Seventy-Two

The third check helps you work out compound interest in your head. Divide seventy-two by the interest rate and you’ll know how many years it takes your money to double. At six per cent, twelve years. At nine, eight. At twelve, six.

Luca Pacioli already gave this rule in his book of 1494, the very one where he first described double-entry bookkeeping in detail. He didn’t bother explaining it. He just told you to keep it in your head. More than five hundred years have gone by, and it works just the same.

Turn the rule upside down and it’s every bit as useful. With six per cent inflation your money loses half of what it can buy in twelve years. At twelve per cent, in six. And if you’re paying twenty-four per cent on a debt, the debt doubles in three years.

Payments for Years

The fourth is bringing future money back to today. We’ve already covered the main point: every future payment is worth less today than the figure written on it, and the further away it is, the less it’s worth. Most often you have to do this with an annuity, where the same sum is paid year after year. That’s how mortgages, pensions, rent and instalments are paid. If you get a thousand at the end of every year for thirty years, that’s nowhere near thirty thousand today. At five per cent it’s about fifteen thousand. At ten, about nine and a half.

The same check also tells you whether a deal is worth its money. Add up everything it will bring you, shrink each future dollar by the price of waiting, and take away what you pay today. Bookkeepers call this net present value. The whole trick is which rate you use. It isn’t necessarily the rate on a deposit. It’s what you would earn somewhere else at the same risk: for a deposit, the bank’s rate; for a new restaurant that might go under, a good deal more. And don’t mix up money and bread. If you counted future takings at today’s prices, use a rate with inflation taken out. If you counted them at the prices there will be, use the ordinary rate. Otherwise you take inflation out twice and bury a good business. Above zero, and the deal beats the other opportunity sitting inside your rate. Below zero, and that other opportunity is better. There’s nothing more useful you can do with a calculator.

Coincidences

And the fifth check trips up even clever people. Coincidences happen more often than you think.

For almost five years, from 1976 to 1981, Joe Pistone lived among mafiosi under a false name. Once he was selling a load of stolen watches, and in one of the buyers he recognised a man he himself had arrested two years earlier for stealing a car. They talked for twenty minutes, Pistone writes, and the fellow never recognised him. Encounters like that, he says, were something he could never insure himself against. And another time one of the regulars at Sonny Black’s club went out to call his FBI handler. And the handler, by pure coincidence, was in charge of the very surveillance team that was keeping an eye on Pistone.

You think he was unbelievably lucky? Or unlucky? Mathematics says something else. Picture a room with twenty-three people in it. What are the chances that two of them share a birthday? Most people say small—there are three hundred and sixty-five days, after all. In fact, better than even. Get seventy together and it’s almost certain. You have to count pairs, not people. And with twenty-three people in a room there are two hundred and fifty-three pairs.

Only Pistone’s problem was a different one, and this is where it’s easy to fool yourself. Two hundred and fifty-three pairs is when any two will do. But the chance that one of the twenty-three was born on your birthday in particular is only about six per cent. Pistone faced the second kind: not any two people, but him and somebody from his past. Here there are exactly as many pairs as there are meetings, and they don’t multiply on their own.

But they do pile up. Say the chance of running into an old acquaintance at any one meeting is tiny, one in five hundred. Getting through one meeting is four hundred and ninety-nine in five hundred. Getting through all of them is that same fraction multiplied by itself as many times as there were meetings. After five hundred meetings it’s a little over a third. So the odds are almost two in three that one day the past will sit down across the table. The calculation does rest on two assumptions, mind you: that the meetings don’t depend on one another, and that the chance is the same at each. What it really was for Pistone nobody knows: it depends on how many people he’d arrested, where they went, and whether they walked the same paths as his new friends. There was no miracle in that meeting, and nothing inevitable either. There was a risk that grew with every handshake. Pistone understood that, and so he was ready. Business is the same: if you have a thousand clients, something rare will happen with one of them sooner than you think.

The Don’s Bookkeeper

A bookkeeper believes the money he has seen, not rumours. When Gallucci, “the King of Little Italy”, was killed in 1915, a newspaper wrote that he paid tax on more than three hundred and fifty thousand dollars’ worth of property, and that his friends said he was worth more than a million. Critchley checked. Gallucci left his heirs three thousand four hundred and two dollars in cash and a house that was rented out. Maybe he hid the rest. Maybe there never was any rest. A bookkeeper believes what he can count.

Don Vincenzo’s old bookkeeper, the one with “no taste, but he could add”, didn’t believe promises either. Once, in front of me, he worked out whether to buy a new press. The salesman promised the press would pay for itself in three years. The bookkeeper did his own sums. He counted future profit at today’s prices, so he used a rate with inflation taken out—what the bank would pay over and above rising prices—and added a bit on top because a press isn’t a deposit and can break down. He allowed for the third year’s profit being worth less than the first’s. It came out that the money would come back in five years, not three. That still wasn’t an answer to whether the press was worth it, only an honest payback period. We bought the press anyway: we needed it. But afterwards the Don said, “It’s good that I know the truth. Now if I’m not rich in three years, I won’t be angry with the press.”

The Bottom Line

Five questions for any number someone else gives you. What does it buy? How did they work out the average? When does it double? What is it worth today, and what did they compare it with? And how many times have you given a rare chance the opportunity to happen?