## Tycoon Tommy

Continuing the theme from last week, a riddle disguised as a parable…

Tycoon Tommy wasn’t really a tycoon, but sometimes he thought he was. On his 21st birthday, he received a $10,000 inheritance, as did his twin brother Conservative Carl.

Carl lived up to his name, and invested the money in a bond paying 5% interest. This bond had a special feature guaranteeing that the interest could be reinvested each year at the same 5% rate.

Tommy had an inside line on a hot off-shore fund. The fund was VERY off-shore, in fact. It didn’t conform to typical reporting requirements, and really, Tommy’s investment would normally have been too small for the fund, but Tommy called in a few connections and was able to invest.

In the years after, Tommy didn’t get a lot of details on his fund’s performance. Every year, he received a short note stating that his fund was up or down by such and such percent for that year, and that’s all the detail he got. After a few years, he wasn’t even sure how much his fund investment was worth exactly. Still, Tommy waited anxiously each year to see how the fund had done that year, and by and large, he wasn’t disappointed. The first year, the fund generated a 14% return for all investors (including Tommy). Year 2 was +41%! Year 3 was a meager +3%, and year 4 was a bear year, minus 36%. Still, such things, one supposes, come with such a volatile, exotic investment, and the next year, year 5, Tommy was exhuberant when his fund had a +114% year. Doubled in a single year! Sometimes when he saw his brother, Conservative Carl, Tommy would rib him a bit for his foolishly conservative investment, with it’s 5% annual returns.

The years went by, and overall, Tommy was pleased. After 20 years, the fund had had only 6 down years, and 14 up years (including the +114% year, and a +102% year as well). Tommy was no math genius, but he added up all those annual return percentages, divided by 20, and was delighted to see that the average was 10.5%, more than twice as high as his brother’s slow and steady 5%.

Coincidentally, after year 20, Carl cashed out his investment, which, with 5% compounding had grown to $26,532. Carl bought a nice, but smallish, boat and invited Tommy out for the weekend.

Tommy thought, “Hmm, my fund has averaged a 10.5% return, against Carl’s 5%. With that 5%, his money has grown to $26,532. I probably have at least twice that much in my fund by now! I’ve heard a little about the miracle of compound interest – maybe that extra return has magnified my investment to $100K or more! Tommy will be green with envy when he sees the size of my boat…”

Tommy contacted his fund manager and requested a full redemption. A few days later, Tommy received a check in the mail from the fund for $9,929, and an attached statement confirming that his fund value had, despite intervening fluctuations, ended up slightly DOWN from where it started 20 years earlier. “Fraud! Theft!”, screamed Tommy. He stormed down to the local SEC office and demanded that the SEC take action against his off-shore fund.

After analyzing the statements, the SEC agent responded, “I’m sorry, there’s nothing we can do.”

It was as Tommy feared, “You mean you can’t go after them because they’re off-shore?”

“No.”, said the SEC agent, “We can’t go after them because no fraud or theft has been committed. This report matches up exactly with the rather terse annual statements you’ve been getting for years. There’s no funny business fees, penalties, etc, involved. The fund has been entirely honest with you throughout, and your final balance of $9,929 is correct.”

For the reader: **How can this be so?**

April 17th, 2006 at 10:21 am

There’s a name for this effect which escapes me at the moment. It’s X’s paradox.

Take a simple example of: -20% on yr1 and +25% on year 2. The average return is 2.5%, however, the actual return is 0%.

So, even though the avg returns may have been positive, substantial losses with lower % values could easily have wiped out his gains. Using the solver in excel, the following annual returns work: (.14, .41, .03, -.36, 1.14, 1.02, 5 of .5078, 9 of .251; fyi slight rounding error).

April 17th, 2006 at 12:13 pm

For a 50% down, you need a 100% up to get back to where you were. In other words, a 30% down and than a 30% up does not bring you back to the same spot!

With this math, I can easily see how he lost money.

April 17th, 2006 at 6:41 pm

This puzzle is just an example of a fallacy that people make over and over again: confusing multiplication with addition. -50% interest last year + 50% interest this year = 0% interest, right? Err, no.

April 17th, 2006 at 10:56 pm

Yes, if anyone would like to demonstrate this for themselves, put $1000 in an Ameritrade account, think and act like Tommy while trading volatile penny stocks.

April 18th, 2006 at 12:48 am

Heh, all it takes is ONE Year at -99%…..

April 18th, 2006 at 1:28 am

Averaging the interest assumes each is applied to the same principal.

There’s an order of operations thing to take into account, so it’s more like this (using the -20, +25 percent from the first example):

((Start amount * 0.8) * 1.25) * rate for year 3… and so on.

In the 2 year case, you end up with .8*1.25 = 1.

Taking Jay’s example, 1 year at -99%, and you’d have to have ~7 years of +99% to be back at around your starting amount.

April 18th, 2006 at 10:04 am

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