Tag: compound interest

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Part V – Discounted Cash Flows, using an annuity to learn the math

PiggyPlease see my earlier posts

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

Part II – Compound Interest and Wealth

Part III – Compound Interest and Consumer Debt

Part IV – Discounted cash flows – an example using a pension buyout

Preamble

In the last post I used the example of a pension buyout to show how the discounted cash flows formula worked, and I argued that discounted cash flows are the key to all investing decisions.[1]  Everything else you get inundated with – from the Financial Infotainment Industrial Complex – is just a whole lot of hype, gimmicks, tricks and tips.[2]

Which makes it all the more odd that almost nobody outside of the financial industry has ever heard of discounted cash flows, never mind actually using the formula in their investment life.

So, allow me to peel back the curtain a bit more, using the example of an annuity investment.[3]

 

“Life’s but a walking shadow, a poor player
That struts and frets his hour upon the stage
And then is heard no more”

 

Another example using discounted cash flows, to value an annuity

Is that guaranteed monthly income annuity offered by an insurance company a good deal or not?  To answer the question you’d need to know how to discount cash flows to put yourself on an equal footing with your insurance company offering you the annuity.  Which I did on my site once before.[4]

Let me break down some of the numbers, by way of example, or possibly by way of inspiration to others who want to start calculating discounted cash flows in their own life.

I just went on my preferred insurance provider’s website[5] and asked for a quote on a 15-year fixed time-period annuity.  In exchange for a $100,000 lump sum from me, the insurance company offered me $641.15 per month, guaranteed, for the next 180 months.  The question I ask is whether that is an attractive investment for my $100,000?

To answer the question I’m going to use the discounted cash flows formula Present Value = Future Value/ (1+Yield/p)N.

I offer a bit more explanation of these variables in a footnote[6]

I can discount exactly 180 different future payments of $641.15, by dividing each of them by (1+ Yield/12)N.

For the first cash flow, N is 1.  For the second, N is 2.  For the 180th monthly payment, N is 180.

This looks like this table in my spreadsheet, which contains 180 rows of numbers and discounted cash flows formulas:

N Period Monthly Payment Formula: PV = FV/(1+Y/p)N
1 $641.15 =$641.15/(1+Y/12)1
2 $641.15 =$641.15/(1+Y/12)2
3 $641.15 =$641.15/(1+Y/12)3
… $641.15 =$641.15/(1+Y/12)…
180 $641.15 =$641.15/(1+Y/12)180

 

Once I have programmed a spreadsheet to calculate 180 individual discounted values for $641.15, I next program the spreadsheet to add up all 180 payments.[7]

Next I can input a value for Y, or Yield, to try to figure what kind of deal I’m offered by my annuity company.

I compare the sum of all 180 values to my original $100,000 investment.  To come up with a comparable yield on the annuity, I input different values for yields into my spreadsheet.  For my purposes I can find the ‘yield’ through ‘iteration,’ basically trying different values until I match up the sum of discounted annuity payments to a final value of $100,000.

If I assume Y is 2%, as I’ve shown in the table below, it turns out the sum of all cash flows is too small and does not quite add up to $100,000.

N Period Monthly Payment Formula: PV = FV/(1+Y/p)N Calculation
1 $641.15 =$641.15/(1+0.02/12)1 $640.08
2 $641.15 =$641.15/(1+0.02/12)2 $639.02
3 $641.15 =$641.15/(1+0.02/12)3 $637.95
180 $641.15 =$641.15/(1+0.02/12)180 $475.09
TOTAL $115,407.00 $99,633.46 $99,633.46

 

If I instead assume Y is 1.5%, it turns out the sum of all cash flows is too large and adds up to more than $100,000.

N Period Monthly Payment Formula: PV = FV/(1+Y/p)N Calculation
1 $641.15 =$641.15/(1+0.015/12)1 $640.35
2 $641.15 =$641.15/(1+0.015/12)2 $639.55
3 $641.15 =$641.15/(1+0.015/12)3 $638.75
180 $641.15 =$641.15/(1+0.015/12)180 $512.04
TOTAL $115,407.00 $103,287.51 $103,287.51

 

 

So I keep trying to find, using my spreadsheet, the value that makes all 180 discounted payments of $641.15 equal to $100,000.  Once I find that, I know what kind of yield, or return, my insurance company offers me on my annuity investment

It turns out, through iteration, that 1.92% is the yield I get by investing $100,000 today and receiving $641.15 per month guaranteed for the next 15 years.

The fact that 1.92% is an absolutely pathetic return is not surprising, nor notable.  As I’ve written before, insurance companies are in the business of buying money cheaply and selling money expensively, and retail annuities are the ultimate source of cheap money for them.

What is notable is that we, as consumers, have no way of evaluating the return on an annuity if we can’t do discounted cash flows.

Which is why I say, ask not what you can do with your insurance company.  Ask what your insurance company is doing to you.

Just like credit card companies do not want you to know that the average American household, carrying the average credit card balance, at an average interest rate, will pay $2.6 million over 40 years because of compound interest[8], similarly, insurance companies can build massive skyscrapers in major cities because they know how to use the discounted cash flow formula to get money cheaply.

And you don’t.

Please see earlier posts

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

Part II – Compound Interest and Wealth

Part III – Compound Interest and Consumer Debt

Part IV – Discounted cash flows – Pension Buyout Example

Part VI – Conclusion, or 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

 

 

Be Rational Get Real

[1] Put it this way, if you’re an individual (I will exempt broker-dealers, HFT and many professional investors from this next statement because they are often doing something different) and you’re not employing a discounted cash flows formula, you’re gambling, not investing.  Which is to say, 99.5% (and I rounded down to be conservative) of us are gambling when we purchase an individual stock.

[2] Are the Chinese buying it?  Is your gym-budding selling? Will baby-boomer demographic trends boost this?  Is Bill Ackman short the stock?  Is it a breakthrough miracle drug?  Will nano-technology make it obsolete?  All hype.

[3] I’m using an annuity to illustrate the use of the discounted cash flow formula because it’s easier to talk about the straight math of future annuity cash flows than it is to talk about modeling future stock dividends and profits.  That involves a longer conversation about equities actually just being a series of future cash flows, which most people will not want to wrap their head around at this time.

[4] By the way, I just re-read my piece on annuities from six months ago.  You should go read it.  It’s good.

[5] I mentioned USAA before in my piece on annuities, because their customer service is awesome.  I have no relationship to them other than as a customer and I just like them.  I assume their quote is standard for an annuity provider, neither better nor worse than the competition.  As I wrote you before, USAA, you should totally make me your President Palmer, peddling life insurance for you.  Call me, maybe.

[6] This time with the formula I’ve introduced the variable p, which is the number of times per year that money gets compounded.  In the case of monthly payments, p is 12, because I have to take into account compounding 12 times per year.  N remains the number associated with each payment, from 1 to 180 in our example, unique to each monthly payment.  Yield, also known as Discount Rate, is the variable I’m going to solve for, to figure out whether the investment is a good deal or not.

[7] Those of you reading this who have spreadsheet experience will note that it’s very simple to create 180 nearly identical rows of formulas simply by a click-and-drag of a single formula.  Similarly, adding up 180 different discounted cash flows is as easy as typing “=sum()” into a spreadsheet cell and referencing the correct cells.  Out pops the answer.

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Part IV – Discounted Cash Flows – Golden parachute or silk umbrella?

golden-parachutePlease see earlier posts Part I – Why don’t they teach this math in school?

Part II – Compound Interest and Wealth

Part III – Compound Interest and Consumer Debt

Preamble

In the last two posts I wrote about how, using the compound interest formula, you can compute precisely how large your money will grow over time, using compound interest.  If you assume a particular growth rate (aka yield, or rate of return) and you know how frequently your money compounds (monthly, quarterly, yearly) you can model into the future what your money will become.

This post is about the reverse process, called discounted cash flows, and is – in my humble opinion – the most important piece of math for investing in anything.  The discounted cash flows formula is what you need to know in order to decide to invest in something today that will have some future value.

Despite what the Financial Infotainment Industrial Complex wants you to believe about the reasons to buy something, evaluating the true value of an investment depends on you knowing how to discount future cash flows.  The rest is just hype, spin, sales and marketing.

And all our yesterdays have lighted fools
The way to dusty death. Out, out, brief candle!

What about discounted cash flows?

First, let’s say what the formula is as, again, the Financial Infotainment Industrial Complex does not want you to know this stuff.

The discounted cash flows formula uses the exact same variables as compound interest, but ‘in reverse,’ solving for “Present Value” instead of “Future Value”

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

Where:

Future Value is the known amount coming to you at some point in the future.

Yield is the growth rate of money, also known as the discount rate.

N is the number of times money gets compounded.

Present Value is generally what you’re solving for when you use this formula.

Most importantly when you figure out how to discount cash flows, a whole series of financial and macroeconomic questions become clearer.

An example of a pension buyout showing the value of discounting cash flows

The discounted cash flow formula is what you’d need to use, for example, if your company offered you a lump sum buyout instead of a life-time pension, as GM did to many workers in 2012, and as many companies frequently do to get rid of their future pension obligations.  Let’s say they offer you a $500,000 buyout.  Sounds like a big enough number to induce many people to take a buyout.

Is the lump sum offer a good deal?  How would you know?

If you could set up a spreadsheet to discount cash flows, you’d know precisely what kind of deal it is.

You could add up the value of all of your future monthly pension payments, properly discounted by the formula above, and you could compare that to the amount GM’s pension department offered you.

Let’s say you would normally receive a $36,000 per year pension for the rest of your life, and you expect to live for another 20 years, here’s what you would do.

You might want to know the Discount rate, or Yield, on GM bonds to gauge the risk of the future pension, or you might want to just assume the government guarantees your pension, so you’d input a lower yield.  Let’s assume low, government guaranteed risk for this example and use a 2% yield to reflect government risk and moderate inflation.[1]

Next year’s payment I’d calculate by the formula Present Value = $36,000 / (1+0.02)1, or $35,294.12

The following year’s pension payment I’d calculate as $36,000/(1+0.02)2, or $34,602.08

I can calculate all of these values easily in a spreadsheet, until I added up the 20th year’s amount, which is calculated as $36,000/(1+0.02)20, or $24,226.97

When I add up all 20 years the result is $588,651.60

Which one is bigger?

Of course you can input different assumptions about your remaining life, and the discount rate, and even the pension amount, but all of this is to show that you need this tool to level the playing field and make good decisions.

I guarantee you that GM’s financial officers know how to discount cash flows, and they’re negotiating from a position of extraordinary advantage against their retired workers who cannot discount cash flows.

So, again, blame the math teachers.  And the Financial Infotainment Industrial Complex.

Please see related posts

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

Part II – Compound Interest and Wealth

Part III – Compound Interest and Consumer Debt

Part V – Discounted cash flows – example of an annuity

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

Also see related post: Using Discounted Cash Flows to analyze Longevity Insurance

 

Silk umbrella

[1] Really you can input whatever assumptions you want to derive a discounted cash flow.  Please don’t start a fight with me about whether 2% is the right assumption.  I’m just trying to show a math technique, not debate the proper discount rate for GM pensions.

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Part III – Compound Interest and Consumer Debt

hPart III – Compound interest and Consumer Debt

Please see earlier posts Part I – Why don’t they teach this in school  And Part II – Compound interest and Wealth

So in the last post I wrote about the the incredible power of compound interest, and the possibility it suggests about wealth creation over time.

Unfortunately, there’s also bad news.

On the debt side of things, how much does your credit card company earn if you carry just an average of a $5,000 credit card balance, paying, say, 22% annual interest rate (compounding monthly) for the next 10 years?

In your mind you owe a balance of only $5,000, which is not a huge amount, especially for someone gainfully employed.  After all, $5,000 is just a quick Disney trip, or a moderately priced ski-trip, or that week in Hawaii.  You think to yourself, “how bad could it be?”

The answer, including the cost of monthly compounding[1], is $44,235, or about 9 times what it appears to cost you at face value.[2]

I hate to be the Scrooge, but the power of compound interest transformed that moderate credit card balance of $5,000 into an extraordinarily expensive purchase.[3]

 

Compound interest: Why the poor stay poor and the rich stay rich

To take another example, let’s think of compound interest on credit cards for the average American household.

Let’s say you are an average American household, and you carry an average balance of $15,956 in credit card debt.

Also, as an average American household, let’s assume you pay an average current rate of 12.83%.[4]

Finally, let’s assume you carry this average balance for 40 years, between ages 25 and 65.  How much did your credit card company make off of you and your extreme averageness?

Answer: $2,629,618.64[5]

So, in sum, your credit card company will earn from the average American household carrying a credit card balance for 40 years, $2.6 million. [6]

If you’re wondering why rich people tend to stay rich, and poor people tend to stay poor, may I offer you Exhibit A:

Compound Interest.

Now, your math teacher might not have done this demonstration for you in junior high, because he didn’t know about it.  Mostly, I forgive him.  Although not completely.

You can be damned sure, however, that credit cards companies know how to do this math.  THIS MATH IS THEIR ENTIRE BUSINESS MODEL.

Which same business model would work a lot less well if everyone knew how to figure this stuff out on his own.

Hence, my theory about the Financial Infotainment Industrial Complex suppressing the teaching of compound interest.  They don’t want you to learn how to figure out this math on your own.[7]

and Video Posts

[1] But importantly, excluding all late fees, overbalance fees or penalty rates of interest.

[2] We get this result using the same formula, although Yield is divided by 12 to account for monthly compounding, and the N reflects the number of compounding periods, which is 120 months.  So the math is: $5,000 * (1+.22/12)120

[3] Have you ever wanted to take a $45K vacation to Hawaii and pretend you’re a high roller?  Congratulations!  By carrying that $5K balance for 10 years, you did it!  You took a $45,000 Hawaiian vacation. You’re a high roller! Yay!

[4] All of these stats taken from this great site on credit card statistics, which cites all of its sources.

[5] We express this again dividing yield by 12 to account for monthly compounding, and raising it to the power of 480 months, the number of compounding periods.  Hence the math is $15,956 * (1+.1283/12)480

[6] I’m assuming for the purposes of this calculation that the debt balance stays constant for 40 years, but your household pays interest on the balance.  In calculating this result, please note I have framed the question in terms of “How much does the credit card company earn” off of your household carrying this average balance for 40 years.  Which is not the same question as “How much do you pay as a household?”  Embedded in my assumptions, and the compound interest formula, is the idea that the credit card company can continue to earn a fixed 12.83% on money you pay them.  Which I think is a fair way of analyzing how much money they can earn off your balance.  Since there are no shortages of other household credit card balances for the credit card company to fund at 12.83%, I believe this to be the most accurate way of calculating the credit card company’s earnings on your balance.

[7] Here’s where, for the sake of clarifying sarcasm on the internet – which sometimes doesn’t translate well on the electronic page – I should point out that I’m (mostly) kidding about the suppression of the compound interest formula.  Among the main reasons I started Bankers Anonymous was that the dim dialogue we have about finance as a society allows conspiracy theories to grow in darkness.  Just as pre-scientific societies depend on magic to explain mysterious phenomena, I think financially uninformed societies gravitate toward conspiracies to explain complex financial events.  As a former Wall Streeter who does not actually ascribe to conspiracy theories, I feel some obligation ‘to amuse and inform’ and thereby reduce the amount of conspiracy-mongering.  So, I don’t really think there’s a conspiracy here.  As far as you know.  Or maybe, that’s just what I want you to think.

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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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