## Rapunzel and Compound Interest

“Daddy,” began the little princess plaintively, “I’m bored.” The poor thing is trapped in her tower for the Summer months. Wizarding school ended the first week of June, and will not start again until next Fall.

Also, it’s a Sunday and her 4-year old sister, the other little princess trapped in the tower, naps deeply on the couch.

“Oh is that so?” replied the wizard, looking up from his desktop computer, the glass desk table strewn with envelopes with coffee mug circles, and toast crumbs.

“Yeah, there’s nothing to do.”

“Huh. Sounds like we need to do some math magic. Would you like to do that?”

“Ok!” she brightens.

“Can I show you how to spin ordinary straw into gold, so you can be very rich 50 years from now?

“Daddy…” she gives the wizard her stop-pulling-my-leg look.

“What?” the wizard looks back innocently, eyebrows raised.

“Ok fine, show me.”

It turns out the sweet thing will do anything to escape the existential prison-tower called Summer. The wizard cackled silently to himself.

Calculating Annual Returns

“Let’s take this magical spell step by step. We have to build up the magic in small pieces to be able to do all of it.

Do you remember last Fall, when you invested \$500 in shares of Kellogg?”

“Yes, you took all my savings and risked them in the market,” The princess looked up reprovingly.

“That’s right. Well, I’m sure that must have been magical money – received over eight years from Godparents, Santa Claus, and the Tooth Fairy – because look what’s happened to your \$500.”

With that, the wizard took out his magical iPhone and pressed the ‘Stocks’ App, which showed a closing price of 68.91 for ticker symbol K.

“The stock is up 11% since September last year,” pointed out the wizard. And since it’s been less than one year, so far you’ve grown your money at an annual rate of 15%.”

“But that might not last, right? Because you said it could always go down?”

“That’s true. It still might, and it probably will go down at some point. But in the long run, it probably continues to go up. And since you don’t need the money for a long time, you can think about what’s going to happen in the long run.”

“Ok.”

Calculating one year’s annual growth

“The magic spell I want to show you – how to spin ordinary straw into pure gold – happens over a long time. In fifty years, when I’m over ninety, and a very old wrinkled wizard, you will be a very rich princess. But first, let’s talk about how to figure out the growth of your money in one year

Do you remember how we talked about percents?

To figure out how your money can grow over one year, you have to multiply your original amount by the percent growth, and then add it to the original amount.

So to do the first part of this spell, you need to calculate 15% of \$500, and then add that to \$500. Let’s see how much money you could have after one year.”

With that, the princess took her blue-ink wand in hand and scratched out the runes on a paper notepad. After a half-minute of spell-casting, she looked up.

“\$75 more. So after one year I would have \$575 if it grows by 15%.”

Calculating Compound Returns in multiple years

“Very well done. Now I’ve got two more intermediate steps that you will find too hard, but after you try it and can’t do it, I’ll help you through the magic.

Tell me how much you would have after 2 years and 3 years, if you start with \$500, and achieve 15% growth each year, for 2 years, and then for 3 years.

The princess began to puzzle over this. Her magic didn’t seem to be working. She wrote some runes, and then some more runes, and then scratched them out. Some heavy sighing followed. She held her golden head in her left hand, while working magic with her right. Finally, with a little prompting, she came up with \$150 in extra money, over two years.

“\$650 after two years?” she looked up hopefully.

“Close, but not quite,” replied the wizard. “The difference is that when you compound growth at 15% for two years in a row, you have to start the second year’s growth from the previous year’s ending point. With this, the wizard quickly showed how the magic spell gets cast.

“One year’s growth gets you to \$575, and then the second year’s growth will be 15% of the \$575, or \$86.25. When you add that to \$575, you end up with \$661.25.”

The princess looked up, a little unsure where this was going, or why the difference mattered much.

“Can you show me how you’d get to the third year?” asked the wizard.

This time, the young princess had the right insight.

“Multiply the \$661.25 times 15%, and then add that to \$661.25?”

“Exactly!” The wizard pulled out his magical iPhone, pressed the calculator App, performing a mystical ritual involving intricate numerical symbols.

“Accio Numericus!” he exclaimed as he pressed the “=” on his calculator with a flourish.

“Daddy.” eye-rolled the princess. The wizard turned the magical iPhone face toward her so she could read it.

“That’s not the real trick though,” warned the wizard.

Do you want to see something really magical?

“Ok,” said the wizard conspiratorily, lowering his voice a little bit. “Do you want to see the whole magic spell? We had to learn the basic magic before you could handle this.”

“What if you could keep compounding your 15% return over the next 50 years? When I’m a wrinkled old wizard, that \$500 of straw you invested could become gold. But how much gold? This magical spell tells you.”

Calculating long-term compound growth of an existing investment

The wizard added to the tension in the room by slowly checking over his right shoulder, then over his left. Seeing no prying eyes of elves, orcs, or bad wizards, he returned to the pad of paper in front of them.

There, he wrote a mysterious series of letters:

FV = PV * (1+Y)^N

The wizard looked up, wide-eyed, expectant.

Here, finally, some powerful magic to impart to the young magi princess.

The princess giggled.

The wizard frowned.

“That is totally confusing!” she exclaimed. “Why are there so many letters?”

“No, no, no, you can understand all of this math. Let me just tell you what everything means and you’ll see.

Writing “FV” on the pad, he said “FV just means “Future Value,” which is what our magic is going to calculate. That’s our magical answer – what we’re working towards, how much gold you’ll have in fifty years.”

And now writing “PV” on the paper, the wizard continued, “PV is just Present Value, which is the amount we started with. For you, that’s the \$500 you invested in Kellogg.”

“The magic symbol ‘Y’ in this spell,” the wizard went on, is the annual return that we’re working with. Since we’re trying to figure out the answer to a problem with a 15% annual return, we can use 15% for Y in this formula. Since 15% can also be written as a decimal 0.15, we’ll end up turning (1+Y) in the formula into 1.15 for our magical calculation.

“But Daddy you’ve never told me anything about an N. N doesn’t make any sense to me.”

“N is just the number of years. And it has the little carot symbol to show that it means ‘raised to the power of,’ do you remember that?”

“I think so.”

“Right, so when we did 3 raised to the power of 2, we wrote it 3 times 3. And 5 raised to the power of 4 we wrote it 5 times 5 times 5 times 5. In this magical spell, we’re going to have 1.15 times 1.15 times 1.15, but multiplied by itself for a total of 50 times. Which we’re not going to do in our heads, but rather with the magical and mystical iPhone calculator App.”

“Ok,” came the princess’ reply, a little skeptically.

“Are you ready for the magic?” intoned the wizard, upping the drama once again. “First, I want you to guess how big your \$500 straw can grow into spun gold in 50 years, when I’m an old wrinkled wizard.”

“I don’t know.”

“Just guess. Something big.”

“I don’t know, maybe \$2,700.”

“No, bigger. I said you’d be rich.”

“Let’s see what the magical iPhone calculator app tells us. First, we turn it horizontally to be able to see additional calculator functions, in particular the ‘X raised to the power of Y’ button. Now, remember to always say ‘Accio Powerzoom Numericus’ when you input numbers like this.”

Sigh from the Princess. Half an eye-roll.

“No, you have to say it. Say it with me.”

“Accio Powerzoom Numericus!”

The wizard theatrically pressed buttons while describing his process.

“First, enter 1.15, then the ‘X^Y ’ button, and then 50, for the number of years, and then hit the “=” sign.

Now multiply that result by our original PV of \$500.

“That’s a lot of money, Daddy.”

“Yes, and do you know what you have to do to make that gold come to you?”

“What?”

“Nothing. Absolutely nothing. Just never sell. The people who work for Kellogg do all the hard work. They sell cereal and whatever else and keep growing their business. You do no more work than you ever did to put that \$500 into that stock.”

“Whoa. That’s cool. But what if it only goes up by 10%?”

“It might. So we can use the same magic formula to see what happens then. We can make Y just 10%, so then our “(1+Y)” is 1.1 instead. We raise that to the power of the same N, 50. Then we multiply it by our original present value amount of \$500.

And don’t forget:

“Accio Powerzoom Numericus!”

“Boom! At 10% annual return you’d only have \$58,695.43.

Which, for not doing any work for the next 50 years, would also be a lot. Most people I know would like to have an extra \$58 thousand dollars right now.”

“Yeah, that’s still a lot. Daddy, can my sister and I go outside to play on the porch now?”

“Sure kiddo. Great work there.”

Boom! Mischief managed.

The front door banged closed, and the wizard cackled quietly to himself.

Once she was out of earshot he rehearsed the following under his breath:

“I don’t mind if you go out to the porch this time, but just promise me one thing, my sweet girl?” in his gentlest wizard tone.

“Sure, anything, what do you need, Daddy?” he answered quietly to himself, in a little princess falsetto.

“NEVER ASK. TO LEAVE THIS TOWER. AGAIN.”