## You will find that a billion dollars is not what it used to be.

I like old movies, songs, and books. Sometimes they reference amounts of money that are puzzling. If they were made long ago, or take place long ago, how much money are they talking about? The problem is even more compounded if it is a long time ago, in a place far, far away (like another country, with another currency).

I love the old Eddie Cantor song “Makin’ Whoopee.” (Frank Sinatra covered it, too.) It’s about falling in love, getting married, and maybe divorced. Here is one of the verses:

*He doesn’t make much money:
*

*Five thousand dollars per.*

*Some judge who thinks he’s funny*

*Says, “You’ll pay six to her.”*

Brushing aside the mathematical impossibility of the judge’s pronouncement, does he or does he not make much money?

The song is from 1928. Factoring in inflation, $5000 is equivalent to $75,000 in 2020.

It is not as little as you thought it was when you first heard it.

**Inflation and compounding**

According to the late Marty Allen, “A study of economics usually reveals that the best time to buy anything is last year.”

With the notable exception of electronics, prices tend to increase over time and we call this inflation. Suppose that the cost *C* of a set of goods increases by *p*% over one year. Then the new cost is original cost + the increase = C + ^{p} ⁄ _{100} x C = (1 + ^{p} ⁄ _{100})C.

So in any year that the cost of goods increases by *p*%, the new cost equals the cost from the previous year multiplied by 1 + ^{p} ⁄ _{100}. Suppose that happened *t* years in a row. Then you would multiply the original cost by this factor *t* times, and the final cost would be (1 + ^{p} ⁄ _{100})^{t} C.

This is the result of “annual compounding.” If you instead figured that the increase was ^{p} ⁄ _{4%} per quarter, you would replace *p* by ^{p} ⁄ _{4}, and multiply by the new factor 4t times (the number of quarters in *t* years). That would give you quarterly compounding. If you figured the increase as 1 + ^{p} ⁄ _{12%} per month, you would replace *p* by ^{p} ⁄ _{12}, and multiply by the new factor 12t times (the number of months in *t* years). If you did the compounding *n* times per year, then the final cost would be (1 + ^{p/n} ⁄ _{100})^{nt} C.

Quarterly compounding gives you a slightly higher result than annual compounding. Monthly compounding gives you a slightly higher result than quarterly compounding. The greater the value of *n*, the greater the result, but it increases by smaller and smaller amounts. There is a “limit” to all of this compounding: an amount that we call the result of “continuous compounding.”

Now, I need you to put this aside, take a calculus course, and come back.

Back already? The limit of the previous expression as *n* approaches infinity is e^{rt} x C, where *e* is a constant that is approximately 2.718, and r = ^{p} ⁄ _{100} (the percent written as a fraction or decimal).

As an example, if C = $1000 and inflation is 2% per year, then the final costs after 10 years given by these formulas (and a calculator!) is the following:

annual – $1218.99

quarterly – $1220.79

monthly – $1221.20

continuous – $1221.40

Because there is not a great difference, and the continuous compounding formula (e^{rt} x C) is easier to work with, we will use that.

**Doubling time and the rule of 70**

How long does it take for cost to double?

That is, given *r*, what is the value of *t* when e^{rt} = 2?

To answer this, we need to use logarithms. Quick review of the *common logarithm*: log1000 = log10^{3} = 3. The common log of a number is the exponent when you write the number as a power of 10. The *natural logarithm* is the exponent when you write a number as a power of *e* (which to the uninitiated probably seems about as unnatural as possible). The natural logarithm is denoted by ln. By definition, lne = lne^{1} = 1. Because 2 is less than *e*, we have (and a calculator will tell you ln2 ≈ 0.693).

Now e^{rt} = 2 → lne^{rt} = ln2

→ rt = ln2

→ t = ^{ln2} ⁄ _{r} = ^{100 x ln2} ⁄ _{100 x r} ≈ ^{69.3} ⁄ _{p}

We used r = ^{p} ⁄ _{100} in the last step. If you round the numerator up to 70, you get the “rule of 70” (some use 72 instead):

Given that a cost (or investment) is increasing at the rate of *p*% per year, the approximate amount of time for it to double is ^{70} ⁄ _{p} .

This tells you that if inflation is 2% every year, then prices will double in 35 years. If inflation were 3.5% every year, then prices would double in 20 years. If inflation were 5% every year, then prices would double in only 14 years.

Actual annual inflation in the U.S. has been pretty steady over the last 40 years (around 2 to 3%). Though it has had its wild swings, over the last century it has averaged about 2.6%. So if you wanted to do a ballpark calculation to convert an old price to a current one, you could figure about 25 years to double.

All of this assumes a constant rate of inflation, which of course is not reality. If you know the (varying) rate of inflation over several years, how would you find the average annual rate of inflation? Suppose, for example, that annual inflation was 1, 2, 3, 4, and 10% in 5 consecutive years. Average inflation is **not** the *arithmetic* mean of these numbers: ^{(1 + 2 + 3 + 4 + 10)} ⁄ _{5} = 4%. The average rate would answer this question: what rate of *p*% over 5 years gives the same result? That is the solution to this equation:

1.01 X 1.02 X 1.03 X 1.04 X 1.10 = (1 + ^{p} ⁄ _{100})^{5} .

This implies

1 + ^{p} ⁄ _{100} = 5√ 1.01 X 1.02 X 1.03 X 1.04 X 1.10.

The number on the right, a 5^{th} root of the product of 5 numbers, is called a *geometric* mean. In general, a geometric mean of *n* numbers is the *n*th root of their product. Thus

1 + ^{p} ⁄ _{100} ≈ 1.0395 → p ≈ 3.95.

This is close to 4%, but not exactly equal to it. Note that you are not finding the geometric mean of the 5 percentages either, but are using a related geometric mean to find the average inflation.

**My Fair Lady/Pygmalion**

When Alfred Doolittle drops by Professor Higgins’ house and touches him for 5 pounds, in “exchange” for his daughter, and refuses to take 10 pounds, how much are we talking about?

The action takes place in 1912, in England, so we have another currency and another time. The amount of £5 in 1912 is about $600 today. Doolittle feels that he and his wife can have a good time and spend the entire amount in one weekend. But neither his wife nor he would have the heart to spend over $1000 in one weekend.

Toward the end of the movie we learn that Doolittle has the “misfortune” to have inherited £4000 per year. That would be about half a million dollars in 2020. I could learn to suffer through that.

There are two ways you could make the conversion:

These operations do not appear to be commutative (switching the order does not give you the same result).

First method: £5 in 1912 is £575.4 in 2019. The exchange rate over the last couple of years has been around $1.30 for £1. This would yield $748.02.

Second method: In 1915, the pound was worth $4.70, so £5 was $23.50. Factoring in US Inflation, this is $602.86 in 2020. (Using my ballpark doubling in 25 years, this would double more than 4 times, so you would multiply by a little more than 2^{4} = 16. But in actuality, inflation was really high from 1915 to 1920, so this falls short.)

By the way, George Bernard Shaw, the author of Pygmalion, did not want it to be turned into a musical, and the producers had to wait for him to pass away to get the rights from his heirs.

Both productions are wonderful (though Leslie Howard is an even more delightfully obnoxious Higgins than Rex Harrison).

**It’s a Wonderful Life**

In one scene, old man Potter invites George Bailey to his office to offer him a job. He figures that George is making $45 per week and he offers him $20,000 per year. Is this worth the cigar dropping out of Jimmy Stewart’s mouth?

The year is 1936. His current salary works out to $830 per week or about $43,000 per year in 2020. Potter is offering him an annual salary of about $371,000 (now you can let the cigar drop out of your mouth).

There was great inflation in the 1940s, so my ballpark method will fall short.

Spoiler alert!

As the movie unfolds, it is about a decade later when Uncle Billy loses the $8000. That would be over $100,000 in 2020. Sam Wainwright’s telegram advancing $25,000 would be about $330,000 today. Hee-haw and Merry Christmas!

**The Count of Monte Cristo**

I love this book! (I like the sandwich, too.)

Now it is clear that the guy becomes really rich. But how rich?

Jeff Bezos rich ($114B)? Or *just* Oprah Winfrey rich ($2.7B)?

As other people have pointed out, it is hard to compare values over such a stretch of time. Relatively low-income people in this country can buy things that the count never dreamed of (smart phones, air conditioning, GPS).

This one is really complicated.

The book opens in 1815, he is in prison until 1836, and he acquires the fortune a few years after that. Then he spends years setting his trap.

The amount of gold and jewels is not precisely given.

He buys various estates.

There are 80 million francs mentioned in his will after he has spent a lot of money. Let’s say this is 1845 (around the time of publication).

A franc had about 1/100 ounce of gold in it at that time. So at that point he had more than 800,000 ounces of gold. Gold is currently selling for about $2000 per ounce. That would be worth $1.6 billion.

Several others have attempted this calculation with widely varying results, but they all arrive at totals in the low billions.

When I was young I read that someone asked J. Paul Getty if he was really worth $2 billion. He replied, “That may be true, but remember, a billion isn’t what it used to be.”