Part II – Compound Interest and Wealth

Time is money

Compound Interest Math Formula – The Most Powerful Math in the Universe

Please see my earlier post, Part I – Why don’t they teach this math in school?

For the sake of blowing the lid off this vast cone of silence, here’s the compound interest formula:

Future Value = Present Value * (1+Yield)N

This is the formula you use if you want to see how money grows over time, to become “Future Value.” Present Value is the amount of money you start with.  That could be $100, such as in my examples below, or likewise a series of $5,000 IRA investments each year.  Present Value is whatever you’re starting amount of money is today.

Yield is the interest rate, or rate of return, you get per year.  Usually expressed as something like 5.25% or 0.0525.[1]

N is the number of times you ‘compound’ the yield.  In its simplest form as written above, if you compound annually, N is the number of years your money compounds.

Example of the power of compound interest: Early investment for retirement

 

When do you use this formula?  You use it when you want to know how much your $100 invested today, or this year, will grow over time.

To offer you an extreme example, using the compound interest formula:

What if you invested $100 today, left it invested for the next 75 years, and you were able to achieve an 18% annual compound return?  How big an investment does your $100 become?

The answer is $24,612,206.

Can I interest you in $24 million?  Without working?

As I say that out loud, I feel like a late-night infomercial guy.  And that feeling makes me want to take a shower.  But the money and the pitch is nothing more than compound interest math.

I happen to believe there’s quite a few 20 year-olds who:

a) Could put their hands on $100 today for the purpose of investing in a retirement account, and

b) Would like, at the end their life, to boast a net worth of $24.6 million[2]

I know all you realists out there will say that 18% annual compound return for 75 years is a fairy tale, and of course I can’t disagree with you.

But I’m doing a magic trick here for the sake of making a point, so would you please suspend disbelief for just a moment and revel in the magic?  The point is not to argue about what reasonable assumptions may be, rather the point is to show why knowing how to do compound interest math could be a life-changing piece of information.

At the very least, its a tool that every citizen should be armed with.  Thank you.

To be slightly more realistic, but equally precise, with a series of other assumptions:

If you’re 20 years old now and you let your money grow for the next 50 years, at 12% yield, your $100 invested today becomes $28,900.  That’s also an amazing result.

Try it and find the Future Value for yourself, by inputting into the formula

Future Value = Present value * (1+Yield)N

PV = $100

Yield = 12%

N = 50

Heck, having your money grow like this sure beats working for a living.

These facts are so amazing, I think, that they might induce a 20-year-old to forgo his XBox purchase this year, and invest the money instead in stocks, in a retirement account.

What about putting your money away in your IRA, $5,000 per year from age 40 to age 65, earning 6% return on your money every year?  Would you like to know what kind of retirement you will have at age 65?  Compound interest can tell you precisely the number.[3]

You’ll have $290,781.91[4]

And all of that becomes possible if we have some insight into the inexorable growth, the most powerful force in the universe, the one math formula to rule them all, compound interest.[5]

Eye_of_sauron

 

Please see Part I – Why don’t they teach this in school?

Part III – Compound interest and Consumer Debt

Part IV – Discounted cash flows – example of pension buyout

Part V – Discounted cash flows – using the example of annuities

Part VI – Conclusion and why everyone needs to know this math for the good of society

and Video Posts:

Video Post: Compound Interest Metaphor – The Rainbow Bridge

Video Post: Time Value of Money Explained

 

Addendum by Michael, added later: It turns out one of my high school math teachers not only does teach compound interest, but he included it in his math textbook, linked to here:

 


[1] I fear many of us learned how to convert a percent into a decimal in sixth grade, but not how to do anything useful with it.

[2] Yes, I hear you cynics, that this is in nominal dollars, and $24 million won’t buy them then what it buys today.  But would you just stop being cynical for a moment, and appreciate the magic of compound interest?  Thank you.

[3] If your assumptions are correct, of course.

[4] To achieve this calculation, you’ll have to add up 25 separate amounts, in a spreadsheet.  The first amount, invested at age 40, compounds the most times and is expressed as $5,000 * (1+.06)25.  The second amount, invested at age 41, compounds as follows: $5,000 * (1+.06)24.  The third amount is $5,000 * (1+.06)23, all the way until the 25th amount, which is simply $5,000 * (1+.06).

[5] Thank goodness Sauron didn’t get his hands on the formula FV = PV*(1+Y)N, or else the hobbits would have been so screwed.  Ancient legend has it in the Silmarillion that Sauron actually did acquire the compound interest formula, but he interpreted the mysterious algebraic symbols as high Elvish, a language he could not read at the time.  Speaking of which, does anybody else want to use compound interest to become a Silmarillionaire?  Um, not so funny?  Ok, you’re right, but don’t worry, I’ll be here all night folks.  Don’t forget to tip your waitress.  And try the fish.

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Part I – The Most Powerful Math in the Universe Goes Untaught

Einstein picture

On Teaching Compound Interest and Discounted Cash Flows

 

“The most powerful force in the universe is compound interest”

– Albert Einstein[1]

 

“Tomorrow and tomorrow and tomorrow,
Creeps in this petty pace from day to day
To the last syllable of recorded time”

 

This Spring I began teaching Personal Finance to a group of bright college students, and we recently wrapped up a section on compound interest and discounted cash flows.

What I’m trying to get across to these undergraduates is that all of the key financial choices they will make in their lives – all of their future decisions about consumer debt, retirement, insurance, purchasing a home, tax preparation, and investing – will be much, much better decisions if they deeply understand compound interest and discounted cash flows.

What are these concepts for?

The compound interest formula tells these students, and any of us who use it, exactly how quickly, and to what ultimate size, money grows in the future.[2]

Discounted cash flows reverses the process, and tells us what the present value would be of any given cash flow or series of cash flows that occurs in the future.[3]

I’ve realized over the course of the last few weeks, however, that I’m trying to convince these students of the absolute centrality of an idea that 95% of them have never heard of before walking into my class.

Not only this, but also 95% of the people my students will meet in their life never have heard of compound interest and discounted cash flows, and therefore will not have the slightest idea how profoundly it affects their lives and their personal financial choices.

Picture me in front of the class jumping up and down and waving my arms wildly (metaphorically of course), trying to get them to believe me.

And yet, why should they believe me when I appear to be the first (and possibly insane) person to ever argue this case?

I’m afraid that after they leave my class, the Financial Infotainment Industrial Complex will never again reveal the importance of compound interest and discounted cash flows to personal finance decision-making.

Why isn’t this taught as a requirement of Junior High School Math?

I was a strong math student in junior high and high school.[4]  I received a solid foundation in algebra, geometry, trigonometry, and calculus.  Of these, algebra has frequently proved useful, but none of the others apply to my life or career.

Compound interest and discounted cash flows, however, dominated my professional life as a bond salesman and hedge fund investor, and I make use of insights from them in my personal financial life all the time.

And yet, nobody taught me compound interest or discounted cash flows in school.  I’d be willing to bet that almost all of you reading this didn’t get taught these concepts in school.  That knowledge had to wait until I started as a bond guy at Goldman.  This, despite the fact that you only need junior high school level math – basically algebra and the concept of ‘X raised to the power of Y’ – to understand and use compound interest and discounted cash flows.

The fact that school taught, and I spent years learning, complex but ultimately very niche mathematical skills, combined with the fact that nobody taught the essential mathematical skills of personal finance (and Wall Street finance for that matter) really gets up my nose when I think about it.

More than gets up my nose, it puts me in a suspicious frame of mind.

Why would these essential skills not be taught to every junior high school student, and then re-taught to every high school student, and then elaborated on for every college student?  Because that’s how important this stuff is.  And how relatively unimportant trigonometry, geometry and calculus skills are for most citizens.

I’ve only come up with a couple of possible explanations, as I explain below, but please chime in with your own theories.

1. Math teachers, as a group, do not understand the role of compound interest and discounted cash flows in personal finance.

I fear this is true.  I’ve become friends with a few of my high school math teachers as an adult and with one I’ve discussed the power of compound interest as a math concept and as a personal finance concept.  Later in his career, long after I took his class, he taught compound interest as part of his lessons on mathematics skills known as ‘sequences and series.’

In these later days he emphasized to his students that if he had really understood compound interest – as a young man – as well as he does now, his working and his retirement years would look totally different.  Could somebody please tell the Professional Math Teachers Association (or whoever is responsible for this stuff) that this is really the key concept, and I mean, for everything?

2. The Financial Infotainment Industrial Complex wants to keep us down.  I’m afraid I’m coming around more and more to this explanation.  Nothing else makes sense.

I mean, seriously folks, calculus: Not relevant (for most people.)  Compound interest: relevant (for everyone.)

 

Coming up next: Part II –  Compound interest and Wealth

Part III – Compound interest and Consumer Debt

Part IV – Discounted Cash Flows Formula

Part V – Discounted Cash Flows – another example, using annuities

Part VI – Conclusion, and why we need this math as a society

 

——Addendum by Michael to this post:

One of my high school math teachers (and my high school advisor!) responded to my post by pointing out that not only does he teach compound interest, but that its part of the math textbook he wrote.  How about that?  I can’t resist linking to his textbook on Amazon, as my way of atoning for casting aspersions on math teachers.

 

 

 


[1] Albert Einstein frequently gets credited with this wise statement.  A quick interwebs search suggests Einstein didn’t necessarily say this, as the first mention in print is found circa 1983.  But Einstein could have, and should have, because it’s true.

[2] To get started on your own learning journey on compound interest, I recommend beginning by watching a video here, with my favorite, Salman Khan.  If you enjoy that, continue the process with videos on present value #1, present value #2, and present value #3

[3] A nice place to start on discounted cash flows is Salman Khan’s video on present value #4 (and discounted cash flow).  Khan doesn’t go far enough on discounted cash flows, or as far as I’m going to go in this series of blog posts to follow, but he at least gets us started, which is more than I can say for almost anyone else available for free out there.

[4] I didn’t pursue math in college, beyond one statistics class required for my concentration, which was Social Studies.  Shout out to the 0.0005% of readers (I chose an arbitrary but statistically insignificant number) who will recognize my major and salute me for it, rather than assume I spent my college years doing what the rest of you did in Social Studies in middle school – memorizing state mascots.

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Ask an Ex-Banker: Annuities!

Q: I am thinking about buying an annuity.  I want to generate dependable income,  BUT,  how do I make sense out of whether or not an annuity is a good investment in addition to providing a degree of comfort.  The tradeoff seems a big gamble,  i.e. how long I will live.  –Captain Bill H., Friendship, Maine.

A:  Apparently annuities are a growing segment of the retirement market, so Bill, your question is timely.

I thought it would be useful to explain how a banker thinks of an annuity.  By “banker,” I also mean to explain how your insurance company thinks of the annuity they’re offering you.

From the banker’s – as well as insurance company’s – perspective, an annuity is a great deal, and it’s not a gamble.  From your perspective, the story is more mixed.

HOW A BANKER OR INSURANCE COMPANY THINKS OF AN ANNUITY

First off, your insurance company – despite what your friendly insurance broker may tell you – does not offer you the annuity to “guaranty your financial health,” “generate dependable income,” “protect your loved-ones,” or to “make sure you have sufficient income in your retirement years.”  The insurance company, instead, is an investor maximizing its profit.  When considering an annuity, let’s always keep that in mind first.

Now, like all for-profit financial companies in the known solar system, your insurance company seeks to buy money cheaply and to sell money expensively.  This falls under the well-known investment activity: “Buy low, sell high.”

I do not mean to be obtuse when I write “buy money cheaply,” since to the non-financial person “buying money” may begin to sound like Orwellian tautology, but bear with me for a moment.  Financial people -including the people who employ your friendly insurance broker – definitely think of their business as buying cheap money and selling expensive money.

Now let’s briefly peek ahead at the Answer Key in the back of this blog: your annuity represents an opportunity to buy cheap money for the insurance company.

Ok, back to the main text of my answer.

All insurance companies need a massive pile of money to operate,[1] so they constantly evaluate the best ways of buying money.  When acquiring money, insurance companies have a choice of where to get their money.  I’ll run through the three main ways:

  1. Sometimes insurance companies acquire equity capital through the sale of shares to private or public stock investors.  In other words, the companies sell part of themselves to other owners, in exchange for money.  All publically owned insurance companies have done this.  Equity capital is typically considered extremely expensive money, so insurance companies do this only as a last resort.[2]
  2. Often insurance companies acquire debtor capital money, otherwise known as borrowing, possibly from a bank but more commonly in the form of a bond from institutional investors.  An investment grade insurance company[3] may be able to borrow $1 Billion for 10 years right now at, say, 4% in the bond market. This means the insurance company gets use of $1 Billion, it pays $40 million per year in interest for that privilege, and then it returns the $1 Billion in principal at the end of 10 years.  Since rates are historically low right now, and the institutional bond market is extremely efficient at providing capital to insurance companies, this is a great way for insurance companies to acquire money on the cheap.
  3. And finally, there’s rock-bottom cheap money: your annuity.[4]  Given all the costs of acquiring you as a customer[5] and servicing your annuity for your life, plus the retail nature (ie. small size) of the money you’re providing to the insurance company, you would expect this money to be VERY cheap indeed, to make it all worthwhile for the insurance company.  Again, remember, they don’t actually care about all the comforting things President Palmer talks about during the Allstate ads.  To provide you, the customer, with an annuity, it’s got to be really cheap money.  If it wasn’t super cheap, they would just borrow money from the bond markets.

How cheap is cheap?  I just went on my own personal preferred insurance company/bank’s website[6] and applied for a $100,000 annuity.  I’m 40 years old and applied for a lifetime monthly annuity, with a (fairly typical) 20 years of guaranteed payments.[7]  In exchange for my upfront $100,000, the company offered $358.39/month for the rest of my life.  The company guarantees that, even if I die suddenly, the first 20 years, or 240 monthly payments, will be paid to my heirs, for a guaranteed payment amount of $86,013.60.

Now, if you’ve been following closely up until now, you’ll already know that I set up my answer to Bill’s question as a less than ringing endorsement for annuities, but the actual quote allows us to see exactly how good or bad the annuity opportunity is in pure financial terms, for both the insurance company and the annuity buyer.[8]

The insurance company will never tell you the cost of borrowing money from their perspective, but I will share with you what their cost would be for my specific annuity.

If I live my expected[9] additional 37.8 years to the ripe old age of 77.8 then the insurance company’s cost of money is 2.79%.[10]  Another way of thinking about the calculation is that I would earn 2.79% annually on my $100,000 for the next 38 years if I am lucky enough to live that long.  If I’m unlucky, and live fewer years, then my insurance company effectively borrows money at substantially less than 2.79%, possibly below a 0% cost of funds.  In that early death scenario, they get money that’s cheaper than free!  Equivalently stated, the % return that I receive on my annuity could be negative if I die before my expected time.[11]

If, instead, I live as long as I expect to live, that is to say, until age 100,[12] then my return can be as high as 3.84%, and the insurance company’s cost of funds is equivalently 3.84%.  Notice this is still below the 4% they can expect to pay to borrow money in the bond market, making an individual annuity worthwhile to them even if I far exceed my life expectancy.

Let’s take another example.  Let’s say Bill, the original questioner above, is a 70 year old man, who can expect an additional 13.7 more years, according to the Social Security actuarial tables, living to the wise old age of 83.7.  I applied online to my same insurance company as a 70-year-old man,[13] willing to take just 10 years of guaranteed payments (a reasonable scenario rather than the 20 years of guaranteed payments that a 40 year old might want.)  For his $100,000 annuity premium, Bill could expect to receive $597.19/month for the rest of his life with 120 guaranteed payments.

What is the insurance company’s cost of funds in this case, and conversely, Bill’s expected return?  If Bill lives to his expected life-span, he would receive a total of $98,536.35, or less than he paid upfront for the annuity, for a negative return on his money.  In other words, under a reasonable baseline scenario, the insurance company acquires money at a negative rate of interest.  That’s better than free!  That’s awesome.  If you’re a death-eating, snake-tattoo-on-your-arm annuity provider, of course.

Now, if Bill also lives, as I’m sure he expects to, until the ripe age of 100, he can expect a much improved 5.92% return on his investment, while the insurance company conversely incurs an expensive cost of borrowing from Bill, at 5.92%.  However, the insurance company has wisely balanced the probability of free money under an ordinary scenario (Bill lives to his expected life span) versus the very remote probability of maxing out at 5.92%, if Bill hangs on to this mortal coil for a whole century.

Now, I have few rules in life, but one of them is that when you can acquire money somewhere between free and 5.9%, with the probabilities skewing much closer to free, well then you should acquire as much money that way as possible.  And figure out what to do with it later.  Like, for example, build massive skyscrapers with your money.  In a related piece of news, has anyone else noticed that insurance company skyscrapers dominate most major US city skylines? Your death, plus your neglect, help make this happen.  I’m just sayin’.

 

 OTHER FACTORS BESIDES RETURN/COST OF FUNDS – SAUSAGE MAKING

In addition to the cost of money for an insurance company, it’s worth understanding another reason insurance companies seek to provide annuities.  Most annuity providers are also life insurance companies.  This makes sense in the same way that a sophisticated slaughterhouse might provide both premium sausage meat and processed hog food, as one customer’s premature death is balanced by, or better said, hedged by, another customer’s unfortunate longevity.

What do I mean by this?  A life insurance policy allows the insurance company the opportunity to collect regular, moderate – typically monthly – premiums.  For that opportunity, the insurance company has the obligation to pay out a substantial lump sum upon the death of the insured person.  An annuity is the mirror image of a life policy.  The insurance company has the opportunity to collect a substantial lump sum up front, and then takes on the responsibility, or liability, to pay out regular, moderate – typically monthly – premiums.  When the life insured customer dies, the insurance company “loses.” When the annuity customer dies, the insurance company “wins.”  When a company can offer both life insurance and annuities simultaneously, it creates an efficient kind of perpetual sausage-making machine in which money can be continually bought cheaply and sold expensively.

A rash of deaths causing a string of sudden life-insurance payouts can be compensated by a release of the obligation to pay ongoing annuity income to the newly dead.  It all works out nicely.  If you’re an insurance company.

 

SHOULD BILL GET AN ANNUITY?

Now that we know the range of investment returns we can expect on an annuity, does it make sense to purchase an annuity, Bill’s original question?

The answer to Bill’s original question is obviously more complex than can be understood in terms of cheap money and expensive money, even if that’s the primary lens of a banker or an insurance company.

The appropriateness of an annuity for any individual owes quite a bit to the individual’s appetite for risk.  To return to geometry class, picture the XY axis where X shows an arrow of increasing risk and Y shows an arrow of increasing return.  The annuity represents one of the lowest risk and return assets you can possibly acquire, pretty much right next to the 0,0 point on the graph, just above and to the right of straight cash.

If you don’t mind providing free money to insurance companies, and you quite like the idea of cash-like returns, then annuities could be just the thing for you.  When you think of if that way, annuities are a perfectly reasonable cash substitute.  Despite S&Ps recent warning, State and Federal regulators manage to make the insurance industry a safe place to park funds for life, as long as you understand a) that the return will be terrible and b) the insurance/annuity provider will never, ever, tell you the return you are getting.  That information, if disclosed, would embarrass them.  And it’s hard to build skyscrapers when you’re feeling embarrassed.

 

For more on annuities and using the mathematics of discounted cashflows to evaluate them, please see this post:

Discounted Cashflows – Using the math to evaluate an annuity.



[1] Like a bank, the main requirement for operating an insurance company is to have a pile of money.  None of the other functions and requirements for operating an insurance company matter much if you don’t start with a pile or money and then maintain it at all times.  Once that pile of money shrinks, it doesn’t matter how good you are at the rest of the things that go into being an insurance company, you’re out of business.

[2] Like for example how Credit-crunch-poster-child-insurance-company AIG sold $17.4 Billion worth of shares in 2012, because, well, how else are they going to get money?  No one wanted to give them money anymore since they were a root cause and casualty of the 2008 Credit Crunch.

[3] I acknowledge “investment grade insurance company” is a bit of a redundancy in the US context, since non-investment grade insurance companies are generally not allowed to operate, but rather are put into a special receivership status by federal or state regulators, and their portfolios allowed to run off over time.  Sometimes this takes decades.  I have invested in annuities like this via my investment business, but I digress.

[4] It may not have been apparent to you as an annuity customer until now, but essentially you’re lending money, just like a bond, to the insurance company.  Instead of a $1 Billion loan in the form of a bond, you might turn over $100,000 up front in the form of an annuity.  But then – just like a bond – the insurance company has an obligation to provide regular payments back to you in exchange for use of your money.  One great aspect of this loan-in-the-shape-of-an-annuity, is that the loan isn’t limited to, for example, 10 years, like a bond.  In fact, the loan is forever.  You see, the really cool thing about your loan/annuity, (from the insurance company’s perspective) is that they never have to pay you back the principal!  You just die, and they keep the $100,000 of your money!  Seriously, how great is that? The answer is: very great, as long as you’re an insurance company.

[5] Advertising, monthly statements, fund transfers, investment disclosures, customer service for your lifetime, plus all those drinks your insurance broker provided you at the Golf Club…none of this comes cheap people!

[6] I do all my banking and insurance with USAA because their customer service absolutely rocks.  It’s leaps and bounds better than any other major customer service business I’ve ever dealt with.  Regardless of their customer service awesomeness, I believe their annuity quote to be typical.  Let this footnote serve as my unsolicited highest endorsement of USAA, although there’s no absolutely no tie between me or Bankers Anonymous and USAA.  But I kind of wish there was.  USAA, hit me up, I could be your President Palmer.   Call me, maybe.

[7] Just to walk you thought the thought process if you’ve never applied for an annuity, its common to request an annuity quote for lifetime payments with some period of payments guaranteed to avoid the “I bought the annuity today for a big premium but got hit by a bus next month” problem that most annuity buyers would never be able to overcome.  So, typically you buy lifetime payments and the annuity/insurance company agrees to pay your designated heirs at least some year’s worth of payments if you die suddenly.  For a relatively young person a 20year guarantee is not atypical.  A much older person might choose a shorter guaranteed payment period, like 5 or 10 years guaranteed.

[8] Incidentally, I’m 99% sure that insurance companies never provide a % return estimate for annuities of the type I’m providing in the main text paragraphs to follow.  So the fact that I’m providing this clear-headed financial return analysis may be largely attributed to two factors: a). I’m your best friend, and b). Insurance Companies are not your friend.

[9] Have you ever wondered what your expected lifespan is, as well as your probability of death in any given year?  The Social Security administration has the answers.   Not only am I your best friend, but I can predict your date of death as well.  Weird.  It’s like I have special powers.  Anyway, you’re welcome.

[10] How did I get this % interest rate?  I’m kind of glad you asked.  Join me a little way down the financial rabbit hole.  I got there by applying a single Discount Rate to a formula for figuring out the present value of all the expected future cash-flows.  What is a Discount Rate?  That’s the single % rate I can apply to all the future cash-flows of an annuity which add up to $100,000 (my original annuity cost).  The formula for each single cash flow is “Nth Annuity Payment” in the numerator divided by a denominator of (1+Discount Rate/12) raised to the power  of the Nth payment.  I know this makes absolutely no sense if you haven’t already worked with the formula before, but my wife made me put it in here.  I’ll tell you what, how about some curious and astute reader sends me a note asking me to explain discounted cash flows and I’ll do a whole post about it sometime soon.  Is that a deal?  In the meantime, trust me that this is how every bank and insurance company evaluates the amount they’ll pay you for your annuity.

[11] If you have a paranoid frame of mind, you can see how the annuity provider begins to resemble a financial vulture, hoping for your premature demise so they can get free money.  Does the flapping of their wings smell like death to you as well?

[12] In the year 2072, I’m comforted in my old age by my bedside Rihanna clone – scientifically engineered to remain 24 years old.  I die quietly in my sleep on our hovercraft, while she lullabies “SOS” until a pass to the next world.

[13] Let’s just agree to call me Harrison, shall we?

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