compound interest Archives - Page 2 of 5

February 15, 2015February 15, 2015

If You Like Feral Cats You’ll Love Compound Interest

A version of this post appeared in the San Antonio Express News.

Nobody ever told me this, but I’ve always assumed that a first rule of writing is to never – ever – try to teach math through blog posts.

But I learned writing through blogging on the interwebs where a first rule of everything – always – is to involve kittens.

Today I would like to teach the compound interest math formula using the story of the feral kittens in my backyard.

Compound interest is the most important, most powerful, and never-taught financial concept in the universe. Compound interest is also how middle-class people can get rich slowly and inevitably, over a lifetime of ordinary earnings.

Please bow your heads with me. Forgive us, editors and readers for the math we are about to learn. Also, hey look, kittens!

Start with just a few kittens

When you start with a small number of feral cats – as I did in my backyard recently, and then neglect to ‘fix’ them right away – pretty soon the magical compounding power of the universe goes to work.

It feels like one day I noticed a couple of stray cats, then I blinked and went out for a cup of coffee, and suddenly my entire yard was overrun with the things.

How does this happen?

Just ignore their gawdawful screeches at 3am, and boom! 2 months later, you’ve got more kittens.

Mathematically, I can tell you precisely how it happens, using the “compound growth of kittens” formula. It’s a matter of gestation periods which I’ll call “N”, and a growth rate per gestation period, or “Y.”

kittens_multiplying — I can haz compound interest?

Using the compound growth formula in practice

I start with two kittens.

(Important note: they must be different genders for the math to work. I’ll spare you the science behind that assumption. Just trust me on this point.)

Let’s say we know that kittens multiply at a rate of 20% per gestation period. And let’s say I wait three years – that’s 18 gestation periods, since cat pregnancy lasts 2 months.

(By the way, all you animal experts and cat-lovers out there who have their claws out to correct me on these assumptions, please recognize: This is all a feline metaphor and not guaranteed to be biologically accurate. Thank you.)

So like I said, my N is 18, and my Y is 20%.

With those assumptions, I can tell you precisely how many future kittens we’ll have, using the compound Kitten Growth Formula.

Here’s the math formula:

The future number of feral cats equals “1+Y,” raised to the power of N gestation periods, all multiplied by the original number of cats.

For algebraically-inclined folks, we would write this kitten-growth formula “Future Number of Kittens equals Original Kittens * (1+Y)^N.”

Plugging my assumptions into the formula:

The future number of feral kittens I can expect in three years (all else being equal)[1] must be (1+20%)^18, multiplied by my original 2 kittens.[2]

So, my compound kitten formula tells me that at the end of 3 years I can expect to have 53 cats. Absolutely swarming all over my backyard! Excuse me while I sneeze just thinking about it.

All of you readers following my math so far (seriously, both of you!) should try that out with a spreadsheet. When you change the percent kitten growth rate Y, or the number of gestation (compounding) periods N, you can see how the future number of kittens changes.

For example, if you start with four “Original Kittens,” and they grow at a rate of 25% per gestation period (that’s Y), and you neglect them for 5 years (so N=30 two-month gestation periods) then you can expect 3,231 kittens in your backyard. Roawrrr!

compound_growth_kittens — These are a few of my favorite things

What about money?

Ok, back to money.

This compound growth formula is the key to understanding the importance of long-term savings and investment.

Allow me a few quick mathematical statements that you can prove to yourself on the spreadsheet you’ve taken out, using the compound interest formula.

Remember, your Future Money is just going to be “1+Y,” raised to the power of N, multiplied by your Original Amount of Money.

Hey, that’s weird, it works just like cats!

So, if you are 25 years old, and if you have $5,000, and if you can earn 7% on your money for the next 50 years (I understand, a lot of “ifs” but bear with me on the math part, because this can change your life for the better) you will have $147,285 in your account when you are 75.

Without you doing anything to your money, just neglecting it like a feral cat.[3] And $5,000 is just one years’ worth of contributions. Imagine making multiple years of retirement contributions when you are in your 20s.

If you buy a $200,000 house, experience 3% home-value inflation and live in that house for 40 years before selling, your house will be worth $652,407 when you sell.

Partly I’m mentioning these things because they illustrate, mathematically, how middle class people can build wealth slowly and inevitably.

More than partly, I’m hoping readers will be inspired to open up a spreadsheet and learn to use this formula to estimate the future value of their money. Or their future number of backyard feral cats, if they prefer.

Anne_hathaway_catwoman — Gratuitous catwoman picture

In real life, since you asked, I noticed a couple of feral cats in my backyard last Spring. By this Winter, we had eight. A neighbor with great cat-catching & fixing skills (Shout out to Cannoli Fund!) caught and fixed all eight for us. Thank goodness for my neighbor because that cat growth curve was about to hit the stratosphere.

[1] As an economist would say. Although an economist would insist on say it pretentiously in Latin, with the exact same number of syllables, thus saving no time at all, but giving the illusion of fanciness: “ceteris paribus”)

[2] By the way, always use a spreadsheet when using compound growth formulas. Don’t simply try this with your catculator. Ahahaha! Catculator! I’ll be here all week folks. Don’t forget to tip your server. And try the fish.

[3] Picture your money, multiplying itself at 3am, while you (try to) sleep. Strangely, dreaming about Anne Hathaway in the last Batman movie.

Please see related post:

Compound Interest and Wealth

Compound Interest, Blood, Lust and Vampires – guest post by The Banker’s Wife

Rapunzel and Compound Interest

College Savings and Compound Interest

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February 13, 2015August 14, 2015

A Valentine’s Day Guest Post – Blood, Bondage, Lust, Money

Editor’s Note: The Banker’s Wife offers up her guest post on “50 Shades of Monetary Inadequacy”

I have bad news for you, Christian. No, it’s not that your movie will be universally panned, though you should pick up A.O. Scott’s review for your reading pleasure. ¹ It goes deeper – I am here to tell you that you will never be as wealthy as the sparkly vampires.

This may come as a shock to you. After all, you share many of the same proclivities, a penchant for cashmere, fast sports cars with hushed leather interiors, obsessively protective behavior, and women who bite their lower lips when nervous.

What is it with the lip biting anyway? It is a truth universally acknowledged that all young women in these novels (by which I mean the chicklit genre that involves vampires, werewolves, or their human stand-ins) appear to have an insatiable appetite for their own facial features.

Perhaps this is our 21st century damsel in distress signal, the equivalent of dropping a handkerchief or blushing in the Elizabethan era? But I digress.

I’m not saying you are in the 99%, Christian. But the Cullens, the De Clermonts, the Akeldamas ², the Horologists ³, and the rest of their undead kindred all possess, or will possess, more wealth than you can ever accumulate.

Why is that? It’s certainly not because of their lust for women willing to bleed for them – after all, you have that too.

It’s not because they can see into the future, though that certainly gives Alice Cullen several legs up. It’s the immortality thing, it will get you every time.
My spouse is not a vampire, despite rumors to the contrary when he is seen at the beach and his ghostly pallor draws gasps, ⁴ but he does understand the power of compound interest.

Christian, I know you began to invest early, but how much can even 100 million dollars accumulate over the course of one measly human lifespan? Considering that you have seven more decades, eight if I’m generous and you stop getting yourself overly excited by Ms. Steele on such a routine basis, your wealth can only grow so much.

Compare that to the vampire Matthew Clairmont, who can accumulate wealth over a millennium. As Deborah Harkness describes in her All Souls Trilogy ⁵ (if you do not wish to have plot points spoiled for you, you’ll have to trust me and skip to the end of the italics):

The little account book had been rebound periodically when more pages were required. The first entries were made on thick sixteenth-century paper and dated from the year 1591. One accounted for the deposit of the dowry that Philippe had provided when I married Matthew: 20,000 Venetian zeccchini and 30,000 silver Reichsthaler. Every subsequent investment of that money – such as the rollover of any interest paid on the funds and the houses and land purchased with the proceeds – were meticulously accounted for in Alain’s neat hand. I flipped through to the final pages of the book… my eyes popped at the amount indicated in the assets column.

So there it is, Christian. If you can invest and compound your interest over 424 years, the end result pops women’s eyeballs. I know it’s frustrating that you can’t cause women that kind of distress, at least until you too become immortal in that long awaited cross-over novel where the Greys meet the Cullens and the blood, sparkles, and expensive body wash fly.

But here’s what you can do to feel powerful, which we know you like. In your cute, domineering way, let your readership know. They should start now.
If a breathless, lip biting young woman of, say 18, begins to put $50 a month away in an IRA (like a 401K), and continues to do so until she turns 75, and earns a 10% return on her investments, she will have $1.77 million.

She can retire to her very own Seattle penthouse, and decorate it only in white and stainless steel with splashes of red.

And all of your young readership could do this, Christian, if they can save the cost of just one cup of Twinings English Breakfast tea a day and put it towards retirement. But perhaps you wouldn’t like that, because then they would have the power and wouldn’t really need you. After all, you certainly don’t go for emancipated women, do you?

Say it with me, Christian, compound interest is power.

Good boy.⁶

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Or Jezebel’s entertaining reviews. ↩
At which point the banker’s wife outs herself as someone who has actually read a lot of these books, despite possessing several postgraduate degrees. But the first step is to admit you have a problem, I suppose. *bites lip pensively. ↩
Also, my deep apologies to David Mitchell for lumping him in here. His heroines absolutely do not bite their lips. ↩
He thinks it is his physique, please don’t tell. His brother calls him Casper the Friendly Ghost. [Editor’s Note: I may punish you for that comment. *Stares broodingly at Banker’s Wife, itching to commence punishment. ↩
Which I recommend, by the way. It’s Twilight for women who have finished college. ↩
And to my pale, dark haired, clever, and occasionally brooding husband, who is certainly the only one reading the footnotes at this point, happy Valentine’s Day, my love. ↩

November 10, 2014November 10, 2014

A Million Dollars Richer – For Almost Nothing Except Coffee

Editor’s Note: A version of this post appeared in the San Antonio Express News

coffee_money — There’s my million dollars

I’d like to be a million dollars richer.

And I don’t particularly want to work for it.

I feel the way native San Antonian, former San Antonio Express-News writer and Saturday Night Live faux-philosopher Jack Handey did when he wrote:

“It’s easy to sit there and say you’d like to have more money. And I guess that’s what I like about it. It’s easy. Just sitting there, rocking back and forth, wanting that money.”

In the spirit of Jack Handey and his idle wish, I recently downloaded a budgeting app called Zeny.

Then, for one week only, I recorded my daily “indefensibles.”

Indefensibles, since you asked, are my own term for small consumption purchases that I did not have to make.

I don’t mean my kids’ after-school care, or the mortgage, or gas for the car. I don’t mean eating out with the family once in a while. I really mean things that are financially indefensible.

samuel_jackson_motherfucker — Yes, my barista actually made this Samuel Jackson latte and gave it to me. That’s how good a coffee customer I am. Which is scary.

Take my expensive coffee habit, for example. Because in my life, indefensibles come mostly in the form of caffeinated beverages.

I figure the cost of a cup of coffee, ground and brewed at home, averages about 15 cents.

Instead of grinding and brewing at home, however, I choose, day after day, to buy expensive coffee at more than ten times that price per cup. Well, actually, multiple cups. Plus, of course, a snack once in a while to accompany my fancy coffee.

And yes, since you asked, my “indefensibles“ concept is inspired by Warren Buffett’s pet name for his corporate jet. When you have Buffett money, a corporate jet qualifies as an indefensible, rather than the morning latte. Which is just one of the small ways my life’s financial path has diverged from Buffett’s.

Anyway, I downloaded the Zeny app on my phone to track my indefensibles for a week after reading the personal finance classic “The Automatic Millionaire” by David Bach. He famously coined the term “Latte Effect” to remind us that purchasing small daily items — a morning latte, for example — had massive implications for personal wealth creation (and destruction!) over the long run.

After reading his book, I became curious. How big is my Latte Effect?

Here’s my data from Zeny:

Day 1: $11.80

Day 2: $6.45

Day 3: $2.27

Day 4: $0

Day 5: $8.58

Day 6: $11.04

Day 7: $0.

All of these expenses I annotated in the app as either coffee or coffee-and-snack related.

My total indefensibles cost for the seven days: $40.14.

Does that seem like a lot of money? Check your own indefensibles against mine for a week. Gum and Tic-Tacs at the register. iPhone downloads. Hulu membership. That third beer for $3.50 at the bar. Whatever it is.

Over the course of a year, my $40.14 per week of indefensibles adds up to $2,087.28 (calculated as $40.14 multiplied by 52 weeks in the year).

What if I invested $2,087.28 every year for the next 40 years in the S&P 500, until age 82 — at which point it will be 2054 and I will be living on my hovercraft, being served hand and foot by my ageless Rihanna-bot?

rihanna_robot — This is what comes up when you Google ‘Rihanna Robot.’ Also, this is what 2054 will look like.

And what if that investment compounded at 10 percent per year? Then I’d have an investment pool worth $1,016,196.

What a coincidence! Because as I said in the beginning, I actually want to be a million dollars richer.

What? You don’t think 10 percent is a reasonable return assumption? Maybe not. Reasonable people can disagree.

But just so you know, the compound annual return from the S&P 500, assuming reinvestment of dividends, over the last 40 years was actually higher than 10 percent. Including the oil embargo years and stagflation of the late 1970s, the tech bubble bursting in 2000 and the Great Recession of 2008, the compound annual return including dividends from the S&P 500 was 11.7 percent.

If I achieved 11.7 percent compound annual return on investment over the next 40 years, my little pool of weekly indefensibles would grow to over $1.6 million.

Maybe you prefer I assume a more modest 6 percent future compound return? Fine, my indefensibles would only grow to $342,413.45. Which, while not the same as a million dollars, isn’t nothing, either. $342K would place me squarely above the average American adult’s net worth.

From skipping premium coffee!

Let’s look at the calculation another way, however. What if I hadn’t ever gotten addicted to premium coffee outside the home in the first place? What if, instead, I had begun saving myself from indefensibles at age 22?

Even with a modest 6 percent compound annual return from the market, my indefensibles’ savings would grow to $1.1 million between age 22 and 82.

Deep_Thought_On_Money — A Deep Thought, by Jack Handey

So, I’m just curious — is there anyone else graduating from college this year who would like a million dollars without trying?

Look, every single person outside of the top 0.1 percent of wealth in this country struggles with one of two financial goals. Either you are:

1. Trying to reduce your personal debts, or you are

2. Trying to build up investments.

The same Latte Effect applies powerfully to both situations. Whichever goal you seek, you can decide to be a million dollars richer at the end of your life.

It’s easy. And that’s what I like about it. Just sitting there, rocking back and forth, not buying that latte.

Please see related posts:

Book Review of The Automatic Millionaire by David Bach

Wealth And The Power Of Compound Interest

Become a Money-Saving Jedi

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September 29, 2014September 29, 2014

529 Accounts v. Retirement Accounts

A version of this post appeared today in the San Antonio Express News.

People who give financial advice – like me – can be so annoyingly contradictory sometimes.
Some friends of mine with young kids – like me – asked me recently to look over their investment plans, and to give them my opinion on what they were already doing, as well as what they should do next.

They had already embarked on an automatic-deduction investment plan with their financial planner, and they also had an available $5,000, and they wanted to know where to invest it next.
They were doing something I had been urging my fellow parents to do – funding 529 educational savings accounts for their two girls – so, naturally, I told them it was all wrong.

Let me back up and explain.

I already wrote about the panic attack I experienced when I visited a useful College Board website to calculate the future cost of college. At the present rate of tuition increases, in ten years from now college tuition will cost the equivalent of checking your little darlings into a 5-Star Hotel in the fanciest building in Dubai.

Seven_Star_Hotel_College — Here’s the 7-Star hotel you will send your child to

For. Four. Years.

(*All prices here are my best estimates, using round numbers. Actual results may vary. Always read less than six financial columns in any 24-hour period. If headaches persist, please call your doctor.)
The only way to deal with that impending college tuition catastrophe, of course, is to start eating rice and beans today and send your surplus savings into a college savings account like a State-sponsored 529 Account. 529 Accounts, as you probably already know, typically offer tax-advantages for education savings and investments.

My rice-and-beans-and-529-account advice still holds if you can do it, provided one other condition is already met, which I’ll tell you about in a moment.

So like I said, my friends had set up their 529 account contributions in the name of their 9- and 11-year old girls, complete with automatic deductions.

The problem, however, is that they planned to contribute to these 529 accounts before they maxed out their IRA contributions and 401K contributions.
You see, there’s a clear “order of operations” when it comes to tax-advantaged investment accounts, and it goes like this:

1. Personal IRA – up to $5,500 this year (And more if you’re older than 50)
2. Employee 401K – up to $17,500 this year (and more if your employer matches)
3. 529 Education accounts, or other savings accounts for health or medical expenses

So, I told my friends they have to first contribute $5,500 each to an IRA this year, then make sure they have filled up their 401K bucket to the max. Then – and only then – should they direct any surplus to their girls’ 529 account. Their existing financial advisor had not made this order of operations clear.

Dubai_hotel — If they don’t get into the Seven Star hotel, Try the Six Star hotel in Dubai

Why do I insist they fund IRAs and 401Ks before funding a 529 account?
At least four factors make IRAs and 401Ks a better target for initial investment than 529 accounts.

First, both IRAs and 401Ks offer federal income tax savings on contributions, whereas 529 accounts do not. State-by-state legislation created 529 accounts, and in some states a 529 account offers state income tax relief. Since we all live in Texas, which has no state income tax, their Texas-based 529 account has no income tax advantage. So right off the bat, IRAs and 401Ks beat 529s by somewhere between 20% and 39.5%, depending on your marginal income tax bracket. But even outside of Texas, state income tax relief from 529s pales in comparison to the federal income tax relief of IRAs and 401Ks.

Second, while both a comfortable retirement and a four years college require big chunks of cash, as parents we get the opportunity (misfortune? punishment?) to borrow money for college, but not for retirement.

The student loan industry – all $1 trillion of debt and counting! – stands ready and willing to lend your little darlings what they need to check into Hotel Dubai University (Fight Fiercely Sand Dunes!) at pretty low interest rates too. I know of no similar program to lend to retirees, except halfway-predatory programs like reverse mortgages.

Next, 401K plans often come with an employer match, one of the few real-life examples of free money here on planet Earth.

Finally, compound interest – the secret sauce to the long-term growth of money – works best over the longest time periods. For my friends, they can watch their investments compound for 30 to 40 more years in their retirement accounts, versus merely 10 to 15 years in the 529 accounts for their girls. Always choose the longer time horizon when it comes to investing.

Of course, we know the best option is to fund them all and not have to pick and choose. For my friends, and for most of us, however, we need to choose, and that means picking our investment vehicles in the right order.

In sum: first retirement accounts, then college accounts. Ok? Ok.

By the way, some clever readers will urge maxing-out the 401K first, before the IRA, to take advantage of any employer match. That’s good advice, it just so happens that my friends don’t have 401Ks at their jobs now, so I told them to max out the IRAs first, then pressure the boss to start a 401K plan second, and then fund their 529 accounts third.

Here’s the TL:DR – Place the mask over your own face first, before placing it over your child. Now, just, apply that to investments.

Max out retirement account contributions first, then place the mask over your child

Please see related posts:

College Savings vs Retirement Savings

College Savings and Compound Interest

Interview with College Advisor Part I – The Rising Cost of College

Interview with College Advisor Part II – Is The College Model Broken?

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June 18, 2014March 9, 2015

Book Review: All The Math You Need To Get Rich

I learned from my wife the concept of the “feedback sandwich,” by which she means if you want to give someone an important piece of critical advice, it’s often most strategic to cushion the blow with a compliment to start, and a compliment to finish, with the criticism nestled in between.

Michael Scott in Scranton, PA might have given feedback this way.

“Hey, I love your ability to file those papers alphabetically!”

“Everyone here in the office has just one word for you: halitosis.”

“Also, cool green shirt you have on today!”

In reviewing Robert L. Hershey’s All The Math You Need To Get Rich I have had recourse to the feedback sandwich. First, I will list some examples from the book that I quite liked. In the middle, a couple of important concerns. Finally, some kind words about how I would use this book if I taught math to high school kids

What works

Hershey presents basic, essential, practical, financial math and then follows it up with numerous word problems at the end of each chapter to help lock in the knowledge.

Two examples in particular stood out as excellent, and paraphrasing them from Hershey’s book illustrates the importance of Hershey’s project.

Example 1

Two twin brothers, each of whom wants to get rich in 45 years, pursues two different paths toward their goal.

The first brother (aptly named Lucky), in a hurry for wealth, decides to buy lottery tickets. He makes a plan to buy $10 of lottery tickets every day, six days a week, for the next 45 years.

The second brother (named Tim) decides to invest exactly half of the amount spent by brother Lucky in a balanced portfolio of market securities, such as stocks and mutual funds.

How much does Lucky bet and spend over course of 45 years, and what is the probable outcome? How much does Tim invest over the course of 45 years, and what is the probable outcome?

While results may vary, we can calculate the expected value of each of these brothers’ behaviors.

To calculate Lucky’s results, we need to know that lotteries return an expected value of $-0.50 per $1 bet. The point of a lottery, after all, is to raise money for the lottery organizer, and to return about half the money over time to the players.

Lucky bets $3,120 per year ($10 x 6 x 52)

And a total of $140,400 over the 45 years ($3,120 x 45)

Since he loses an expected amount of $0.50 per $1 bet, we can quickly see that Lucky loses $70,200 over the course of his 45 years of lottery playing. Lucky might win $10 here, $100 there, and occasionally $1,000, but the odds in the long run mean he’ll burn up an estimated $70,200 over the years, nearly guaranteed.

What about Tim’s results?

Tim invests $5/day, 6 days a week, 52 weeks per year. His annual investment is $1,560 ($5 x 6 x 52).

Hershey (the book’s author) assumes a 10% gain on investments[1] to calculate Tim’s results after 45 years. Aggregating the compounded returns of annual $1,560 investments at 10%, we can see Tim’s net worth climbs to $1,121,492 after 45 years.

Tim’s a millionaire using just half of the money Lucky ‘invested’ in lottery tickets, while Lucky has a zero net worth.

Now, that’s what I call a useful mathematical comparison.

$all the math you need book$

Example number two that I loved from the book

A recent college graduate named Patience is thinking of taking a trip to Europe, which will require her to max out her $5,000 credit card and pay the 18% annual interest charges on the card. Realistically she knows she will stay maxed out for 10 years, so she will have to pay that 18% interest all the while for the next ten years. How much is that?

Alternatively, Patience considers not making the trip to Europe, and instead may invest the amount of the unspent interest in an S&P500 index fund. Hershey assumes a 15% annual return[2] on that investment. How much money would she have then at the end of 10 years?

The annual finance charge, following the trip to Europe, would be $900 ($5,000 x 18%). Over ten years Patience would end up paying $9,000 in interest charges, and still owe $5,000 at the end of ten years.

If, instead, she invested $900 per year in the mutual fund that earns 15% per year, we can calculate – using the magic of compound interest – that she would have $18,274 in her fund.[3] Her positive net worth from investing beats the $5,000 deficit by a long shot. And just as importantly, the interest charge on the credit card ends up costing more than the original trip itself.

My critical thoughts – the bologna in the feedback sandwich

First concern – Who reads this?

One concern I maintain with a book like this – which I fretted about earlier in a review of another math-book-for-non-math-types Innumeracy – is who, honestly, will ever pick up this book? Will people who already feel uncertain about their math skills, however theoretically eager to learn the mysteries of numbers or tempted by the chance to “Get Rich,” actually dig past the first few paragraphs to learn what they do not know?

I don’t know. I doubt it. Math-oriented people enjoy confirming their own math aptitude with a book like this, and they may be able to expand their skills into useful finance applications with this book. I have a harder time picturing non-math folks picking up and actually working their way through the instructions and sample problems, however accessible this book may be. I think Hershey has made this as approachable as possible, but I still question the draw of those who are the intended audience.

Second concern – No way to teach compound interest (my pet peeve)

Every finance-math for non-experts book that I’ve ever read relies on a terrible crutch when it comes to teaching compound interest: The table in the Appendix with compound interest multiplication “factors.” I hate this.

What a proper book on compound interest should teach is the formula FV = PV * (1+Y/p)^N, with definitions of each variable and multiple examples to shows its application. That formula, once understand, can solve any compound interest problem flexibly, and precisely.

This book’s appendix features a y-axis listing the number of compounding terms from 1 to 100, for example (the N in the formula), while the x-axis shows ascending percentages of yield (the Y in the formula). At the end of every example in the book that references these tables, Hershey is forced to say: “That’s not exactly the answer, but it’s close enough.”

I can’t endorse this. I refuse.

All The Math You Need to Get Rich was first published in 1982, the same year in which my fifth grade teacher introduced us to the Timex Sinclair 2000.

[10: Print “Mike” ; 20: Goto 10 ; Run]

At that point in 1982, text appendices of compound interest tables made perfect sense.

Not in 2014, though.

Any reader of a book in 2014 also has use of an Excel spreadsheet program that sits on their desktop or laptop, and can be used to good effect with the formula above.

The text-based, imprecise, crutch of an Appendix table, which no person will carry with them, ever, gets in the way of anyone who ever wanted to actually learn how the compound interest formula really works, in real life.

Phew, got that off my chest.

Back to the complimentary thoughts

If I was assigned a high school math class as a substitute teacher and given 1 month to teach the kids something useful, I would pick a book like All The Math You Need To Get Rich as a textbook. Here are real-life skills for understanding interest rates, percentages, probabilities, and dealing with orders of magnitude – in short most of the things households, investors and citizens need to use on a daily basis to get by. Certainly these help most of us think much more, and much more often, about useful math applications, than the traditional courses – Geometry, Trigonometry, quadratic equations, and Calculus – that make up the majority of traditional high school math curricula.

Not only do these relatively accessible concepts come in handy more often, I would hope – as their substitute teacher – that I could impress upon the unruly high schoolers their own self-interest.

“Learn this about probabilities” I would exhort, “and save yourself thousands over your lifetime by not buying lottery tickets or gambling.”

“Deeply understand interest rates and percentages,” I would urge, “and use your powers for good (getting wealthy) instead of evil (making credit card companies richer).”

This is a fine book and I may use it for teaching my girls what they need to know in the future.

See related book reviews:

Innumeracy by John Allen Paulos

Master Math: Business and Personal Finance Math by Mary Hansen

[1] Astute readers will argue that 10% is too high an assumed return from a portfolio of stocks for 45 years, and I agree. Using a 6% return, Tim’s net worth at the end of 45 years climbs to $331,880. This doesn’t have quite the ring of ‘millionaire’ that the author Hershey probably wanted, but it still isn’t anything to sneeze at, for the cost of a daily Starbucks addiction.

[2] I know I know, too high, but still, work with me here a little bit.

[3] If we assume a more modest 6% return, she would have $11,863.

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June 9, 2014June 9, 2014

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.

The wizard plowed ahead anyway.

“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.

“760.44,” she read.

“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.”

“Ok. How about $9,000.”

“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.

There’s your answer: $541,828.72”

“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.