How To Win With Powerball – Learn The Math

powerball_mathFirst things first,1 never buy a lottery ticket. Seriously, ever.

Having said that, obviously I did buy a few this past week because, you know, I’m an irrational human.

I’ll be the first to admit it. I also sometimes buy “King Size” Reese’s peanut butter cups (that’s the four-in-one size) in the checkout line at the grocery store and, man, they’re gone by the time I make it to the car. So, I allow myself bad decisions from time to time.

But this is not confession-time with Mike. Rather, it’s financial math-time, with a Powerball lottery theme.

A friend asked me over Facebook whether it made more sense (when he wins the whole shebang this week) to take the lump sum or 30 annual payments.

Another example of bad decisions by me.

What I’m not analyzing

Now, I can think of lots of valid ways to answer the annuity versus lump sum question, and I’m going to skip most of these to focus simply on the mathematical way to think about it. In that sense, I will be simplifying the issue terribly – disregarding factors such as personal circumstance, economic utility, needs and wants of heirs and recipients, current and future rates of taxation, variations on self-control, and personal health/longevity. I only want to use the example of the lottery to illustrate some important financial mathematics.

And before you decide to skip the math analysis (because math is soooo boring, blah blah blah), just know that this math formula is the basis for ALL fundamental investing – All bond analysis, all stock analysis, all real estate investing, all business investments. Everything. If you don’t use this math, you’re just guessing. Well, even if you do use this math, you still may be guessing, but you’re guessing less than you would have guessed without the math.

The math is called “Discounting Cashflows,” and probably by now you’ll have wised up to the fact that I’m piggy-backing on the Powerball lottery story to slip in a lesson on what forms part of the most powerful financial math in the universe.

So like I said, I’m skipping whether you think you’ll live for another 30 years and whether that makes you want to gratify your material wants this year. I’m skipping the issue of taxes partly because we don’t know what tax rates will be like over the next thirty years, and also because taxes are part of the reason why lotteries in particular are a mug’s game.2 I’m skipping the issue of squandering all the money3 this year versus stretching out your squander over 30 years. I’m skipping whether you have great philanthropic desires that may be satisfied this year or in later years when you (finally!) acquire more maturity and thoughtfulness.

Ok, with all that throat-clearing and telling you what I won’t analyze, let’s move to the mathematical issue of:

Annuity versus lump sum

What the math explained below can tell you, precisely, is whether a lump sum today is worth more or less than a 30-year annuity payout, given your % return assumptions.


To properly compare your lump sum option today to the 30 annual payments option, you first have to assume a % annual ‘return’ that you would expect to be capable of generating those annual payments.

Introducing: “Discount Rate”

The assumed “% annual return” I mentioned in the previous sentence goes by a couple of different names when you do this math. The alternate names include Annual Return, IRR (Internal Rate of Return), or Yield. The best math name for this particular situation is Discount Rate, but in practice it ends up meaning the same thing as those other names.

How do we come up with an assumed Discount Rate? There’s some art here as well as science, and a useful Discount Rate for this situation adds up factors such as inflation, prevailing interest rates, and the riskiness of each annuity payment. Without getting into a tangential detour about coming up with the ‘right’ number for the ‘Discount Rate,’ on lottery payments4, I’ll just grab one for now and move on to showing the math. Let’s call the right Discount Rate on future annuitized lottery payment 3%.5

Discounting future cash flows

Money arriving one year from now – or 30 years from now – is always worth less to me than money in my bank account today.6 We can understand this intuitively by thinking about the fact that you can’t literally buy beer and a hamburger today with money promised to you one year from now. Also, in one year, or thirty years, your circumstances may change, which would make you value money in your bank today above money owed to you in the future. Also, future promises are inherently risky. What if the person owing the money, or the lottery commission for that matter, never pays you in the future? What if inflation reduces the purchasing power of the future money? For all these reasons, we say that money today is worth more than the same (aka ‘nominal’) amount of money in the future.

But how much more? The point of discounting future cash flows mathematically is to turn that intuition I describe in the prior paragraph into a precise number telling me ‘how much more’ I value today’s money than future promised money.

This is the heart of comparing my lottery lump sum to a series of 30 annual payments. It’s also the heart of figuring out how much money I’ll earn if I buy a bond at a set price, or a stock at a set price, or a rent-generating piece of commercial real estate, or a profitable business. In essence, in each of these situations, I’m asking how much would I pay today to generate a series of future payments, at a given assumed rate of return?

Would you just tell me the math already? Geez!

Ok, fine. Let’s say you win the total Powerball payout this week (as of this writing) of $1.5 Billion. And let’s say the annuity deal is you can receive 30 equal payments of  50 million each year7, starting one year from now.8 And let’s say the lump sum offer today (as of this writing) is $930 million. Which one is worth more, the lump sum or the 30 annuitized payments?

Our math challenge is to ‘discount’ each of those 30 payments of $50 million into an equivalent value in today’s dollars. That consists of 30 different calculations. What is $50 million – one year from now – worth today? What is $50 million – 5 years from now – worth today? For that matter, what is $50 million – arriving thirty years from now – worth today?

Each of the separate 30 annuity payments gets discounted separately. Mathematically, 1 year from now is different from 2 years from now which is different from 30 years from now. For simplicity’s sake, I’m going to stick with a single Discount Rate – the same 3% – for each future payment.9 Also, to do this right, you’ll want to open up a spreadsheet right about now. If you are mildly competent with Excel,10 you can follow the math below by creating the following formula, and then reproducing the formula thirty times, one for each year’s annuity payment. And if you are mildly comfortable with Excel, the reproducing of the formula part should take you about 12 seconds. It’s an autofill function.

A little Algebra

Sorry about this, but I’m going to mention some algebra. This won’t hurt a bit. Just hold your breath, and…The algebra formula for discounting any future payment into today’s money is PV = FV/(1+Y)^N. Don’t worry, I’ll define everything…Ok, release breath. Phew.

In the formula you can represent Discount Rate as ‘Y’ or as in this case as ‘r.’ Doesn’t matter.

In that formula PV (Present Value) means the value of money today (which is what you want to solve for, in order to compare with the lump sum), and FV means the Future Value of the annuity payment, which in the case of the Powerball example we’ve said is $50 million.

Also in that formula Y is what I’m using for the Discount Rate, which I’ve decided for the time being is 3%. And N is the number years from now that the future payment arrives.

So, to discount a Powerball annuity payment arriving one year from now I’d say that the Present Value (PV) is equal to $50 million/(1+3%)^1. Which, my Excel spreadsheet tells me, is $48,539,758. In plainer English, I should equally value $48,539,758 today, or $50 million set to arrive one year from now.11

To discount a Powerball annuity payment arriving two years from now I’d say that the Present Value (PV) is equal to $50 million/(1+3%)^2. Which, my Excel spreadsheet tells me, is $47,125,979. In plainer English, I should equally value $47,125,979 today, or $50 million set to arrive two years from now.

To give you a sense for the power of discounting, my Excel tells me that the $50 million payment arriving 30 years from now is equivalent to $20,585,997 in today’s money. You can check that math yourself by plugging in $50 million/(1+3%)^30 into Excel, or your calculator.

To solve the lump sum versus annuity question, I’d set up my Excel spreadsheet to give me a value for each of the thirty annuity payments of $50 million. Like I mentioned, this takes approximately 12 seconds for someone mildly comfortable with setting up formulas in Excel.

Once you have a value for each of the 30 payments, discounted to the present day (aka the Present Value of each of the 30 payments) then you add them all up, and compare them with the lump sum.

When I add up thirty annual payments of $50 million each, each discounted at a 3% Discount Rate between one and thirty years from now, I get a total value of $979,726,641.

Assuming my 3% Discount Rate is the right one, I can compare that value to the lump sum offer (which I mentioned above, and as of this writing) of $930 million.  Since the bigger number is the sum of the annuitized payments, then I can say that the 30 annuitized payments are a ‘better deal,’ in pure financial terms, than the lump sum.

But notice something

The mathematical answer to the ‘lump sum versus annuity’ question depends entirely upon inputting a specific, assumed, Discount Rate. Change the Discount Rate, and the ‘correct’ answer changes.

If I assume a 4% Discount Rate, for example, the value of next year’s payment declines to $48,071,757 – because that’s $50 million/(1+4%)^1, and the sum of all annuity streams is only worth $864,271,673. At a 4% Discount Rate, the lump sum value of $930 million dwarfs the value of the annuity payments.

What would make the ‘correct’ Discount Rate change? In the case of guaranteed Powerball lottery payments, the most probable influence would be inflation. If the world suddenly expected 5% annual inflation for the next thirty years, for example, then the correct Discount Rate would be something above 5%, and the lump sum begins to look far more valuable than the annuity. On the other hand, low inflation and continued low interest rates would make the annuity payments relatively more valuable, because we could imagine inputting a Discount Rate even less than 3%.

Two other small points

By the way, I can set my programmed spreadsheet to tell me what the ‘Discount Rate’ is that the Powerball folks use, which turns out to be (if you use my assumptions) approximately 3.4%.12

The second small point is that if you know the Discount Rate that Powerball uses (like 3.4% in my example), you could reasonably say that your ‘investment hurdle’ for taking the lump sum is 3.4%. What I mean by that is that if you take the lump sum, and then can reliably compound an investment return on that money – not spending, just investing! – above 3.4% every year for the next 30,  then you could end up with more money in the end than you would through the annuity option.

A reminder

Discounting cashflows only answers one aspect of the lump sum versus annuity question. Remember what I started out saying, which is that there are a ton of factors I’m not considering, in my interest in showing some elegant math. But at least we have a mathematical answer to the question of ‘should I take the sump sum or the annuity?’

An exhortation

I’m going to make up a statistic which, while not exactly true,13 is at least ‘truthy:’ Less than one person in a hundred understands how to do discounted cashflows math.

I believe deeply that everybody should understand discounting cashflows, in a ‘you should know this to be an adult in the world’ kind of way. Your bank understands this math and uses it to profit from transactions with you. Your insurance company uses this math when calculating your rates, and has a complete advantage over all its customers who cannot do this math. All of Wall Street is built entirely on the discounted cashflows formula.14

You really don’t need to understand the lump sum versus annuity question for this week’s Powerball. You do, however, need it for life.

Note: This is not a self-portrait


Real talk time: You’re not going to win Powerball. Lotteries are terrible.

But if you managed to spend some time with a spreadsheet to compare the lump sum versus annuitized payout as a result of fantasizing about the Powerball drawing, well then I’d say you’ve gained something this week. And properly deployed, what you’ve gained by understanding discounting cash-flows math could – and I mean this totally in earnest – make you wealthy in the long run.

Learning this math – and not some lottery fantasy – will make you a winner this week.



Please see related posts:

Discounting Cashflows – The example of an annuity

Discounting cashflows – Golden parachute?

Longevity Insurance – Do You Feel Lucky?

Compound Interest – The Most Powerful Math in the Universe

Compound Interest – Your Credit Card Debt

Compound Interest – Get Rich Slow

Compound Interest – If You Like Feral Cats

Compound Interest & Rapunzel




Post read (14472) times.

  1. “…I’m the realest.” At least, that’s how I wanted to end the sentence. Because I’m So Fancy.
  2. In other words, considering taxes will muddy up the elegant and essential mathematic point I’m trying to make.
  3. Obviously I want to use the delightful phrase “on hookers and blow,” the proverbial natural beneficiaries of your squander.
  4. Ok, I’ll indulge in a little tangent here. The biggest influence on a thirty-year lottery payment guaranteed by a state in the US is probably the expected rate of inflation. If you think the rate of inflation will average, say, 2% over the next 30 years, then most of the Discount Rate will be made up of this. The ‘risk’ of non-payment by a state-sponsored lottery commission is low. And yes, I’m ignoring you preppers who stock up on canned goods and ammo for the imminent implosion of constitutional order, who say the state will likely dissolve over the next 30 years, with federal fiat money replaced by Bitcoins. Whatever, dude.
  5. Again, I’m not saying this is the ‘right’ Discount Rate, I’m just saying let’s pick a number so I can illustrate the math formula. Then you get to change the Discount Rate to whatever you want it to be, and come up with a different analysis of the relative merits of lump sum versus 30 years of annuity payments.
  6. The concept in this paragraph is short-handed in finance circles as ‘The Time Value of Money.’
  7. By the way, it doesn’t matter for the math example, but the actual Powerball pays in increasing annuity amounts each year. So on a $1.5 Billion prize, presumably the initial payments are less than $50 million and the later payments exceed $50 million. But I’m going for simplicity here.
  8. Also, technically, when you win Powerball you get an immediate lottery payment straight away, followed by 29 future years of annuity payments. So the math in real life varies a little bit from my example. But since nobody reading this will be winning, and playing Powerball is an exercise in fantasy anyway, I think it’s appropriate to disregard actual real-life technicalities in the interest of learning some math.
  9. But you wouldn’t have to. If you had paranormal insight into some event happening 15 years from now (like Miley Cyrus gets elected President) and some resultant uptick in either inflation or just risk, you could build a simple math model that assumes 3% discount rate for the first 15 years, and then a 7% discount rate for years 16 through 30. Knock yourself out! Vote Cyrus!
  10.  Now that I’ve introduced Excel into my method, you clever Excel-using people are going to want to tell me about a shortcut for doing present value with fixed annuity payments at fixed intervals. But there is a method to my madness in explaining the ‘long way.’ That Excel shortcut can do annual and regular discounting fine, but is not as flexible as it should be for all cases of discounting future cashflows. Shortcuts won’t help you with irregular future cashflows that arrive at irregular times. Learning the discounted cashflows math the ‘long way,’ applicable for every case, is a far more valuable skill, in my opinion.
  11. Important Note/Correction…A number of you did this math and found slightly different values. I should have clarified: When setting up my spreadsheet I always use actual dates of payment to generate the compounding period N. So for example I set up an original date of 1/1/16 and annuity payments made annually on 1/1/17, 1/1/18, etc. When you use actual dates you get an N of 366/365 (instead of 1) in 2016 (it’s a leap year) and the Actual#days/365. I didn’t clarify that in the original version of this post. Using a simple 1,2,3 etc for N isn’t wrong, It’s just I’m in the habit of using actual dates for investments and I did it this time without pointing it out. The good news: Some of you checked out the math with your spreadsheets!
  12. Update: The NYTimes has a a good article in which the reporter notes the lottery uses an assumed Discount Rate – to convert $1.5Billion in annuity payments into a $930 million lump sum – of 2.843%. The Times guy, I believe, has access to – or figured out – the exact schedule of payments, which includes a schedule of increasing annual payments, rather than the equal payments like I used in my spreadsheet. My number assumes 30 equal payments in years 1 through 30, and the real schedule is an upfront payment immediately, followed by 29 increasing annuity payouts in years 1 through 29. Since I don’t know the exact increasing schedule of Powerball between years 1 through 29, I’m going to just stay focused on my math formula.
  13. I have no idea what the true number is.
  14. Well that, plus the immortal souls of unburnt sacrificial virgins. And coffee. Don’t forget coffee.

Guest Post – Annuities Are Expensive

lars-Kroijer-on-TVAnnuities are an important and sometimes dominant part of the investment portfolio for millions of savers.  While in certain instances there is a requirement for pension scheme participants that they put a part of their pension savings into an annuity, others decide to have them because they find great comfort from having a secured cash flow until they die (some annuities continue payments for dependants).

I certainly don’t have a problem with annuities.  There is great intangible value to be had in knowing that you are going to be ok in your old age, regardless of how old you become.  Particularly if you have an annuity that is adjusted for inflation (some adjust for changes in the retail price index), you have a very good picture of your spending power in retirement, without worrying about the oscillations of the markets or dying with a lot of money that you may have no use for (you’ll be dead…).

But there are a couple of things I would encourage you to think about when purchasing an annuity.

 Who guarantees your payment in the future and what is their credit quality?

Keep in mind that you will be expecting payments many years into the future.  If you buy an annuity at age 50, with some luck you’ll be looking for a payment half a century into the future, and at that time your quality of life may greatly depend on actually receiving that payment.  In most cases annuity providers are insured by a government backed scheme, but you want to make absolutely sure that this is the case.  You certainly don’t want to be in a case where a Lehman style bankruptcy means that you are left with nothing in retirement when your earnings potential has greatly diminished (keep in mind that annuity providers are likely to be struggling exactly when markets are tough and you probably need them the most).

 The price of the annuity may be very high – be sure you need it!

You are essentially lending money to the insurance company for a very long time.  You can try to figure out at what rate the following way for a standard (non-inflation adjusted) annuity:

  1. Figure out your life expectancy. There are many life expectancy calculators on the internet[1] – it will be more accurate if you can incorporate where you live, etc.  This will give you a good idea of how long the insurance company expects you to pay your annuity for (make sure you tell them all the bad health stuff – as morbid as it sounds in this case you want them to think you are going to die soon).  I was surprised by how long I can expect to live, which according to a friend in insurance is a common reaction.
  2. Search around for the best annuity and be sure that the payments are in fact guaranteed by someone other than the annuity provider’s general corporate credit. Assume we are doing an annuity that you buy for £100; what will your yearly payments be?
  3. Figure out the internal rate of return (IRR) on your payment. Your IRR is the rate that the insurance company effectively borrows from you at.  So year zero: -£100, year 1: +3.75, year 2: +3.75, etc.  You can do this in excel.  Keep in mind that unlike a bond you don’t get the principal back at the end (there are annuities that do this, but the interim payments are just lower to reflect this).
  4. Figure out the average time to future payments (the duration – also use excel); depending on your circumstances it will perhaps be 15-20 years. If you start receiving the annuity payments now this will be half the years you are expected to have left to live.
  5. Compare your IRR to a government bond of a maturity similar to the duration and in the same currency (your average time to payment in 4 above).
  6. Apply some sort of discount to the annuity IRR to reflect the inflexible nature of the product and perhaps stiff penalties if you try to get out of the annuity. Depending on the policy these penalties can very stiff and you should discount the value of the annuity accordingly.

Consider any tax advantages of the annuity; these are at times significant.

As an example, when I did the above exercise as a potential annuitant, the IRR I received on my investment was slightly lower than the equivalent UK government bond.  So I essentially would be lending money to the annuity provider decades into the future at a lower rate than I would the UK government, ignoring the flexibility I would have in trading the UK government bonds if my circumstances changed.  In other words, the insurance I received from the annuity provider against running out of money in very old age was very costly.

It is not surprising that the IRR for your annuity is not great.  Annuity products are expensive to manage, and not necessarily great business for the insurance companies, as you deal with the administration of cash transfers to thousands of annuitants, in addition to marketing, overhead, re-insurance that the annuity provider will be able to pay you, and their profit and capital requirements of the annuity provider.  Just think that it costs money every time someone calls up to complain that they have not received their £300 and multiply that by a million customers – even if you are not the costly customer you share in paying for those costs by being on the same annuity platform.

My conclusion on annuities is that if you don’t have a lot of savings and worry about having enough into old age, annuities are well worth the poor return they promise on your investment.  If you don’t have a lot there is great value in knowing exactly what you have and that it will be enough.  An annuity can give you that.

If you have more assets and are highly likely to leave an estate for your descendants then perhaps reconsider annuities.  After adjusting for potential tax or other benefits the return on the assets you put into an annuity is mostly quite poor and you could make more money investing on your own.  You will of course not have the guarantee of additional payments if you live beyond your life expectancy, but considering your other assets you will be fine even without those additional monies.  Also annuity providers make large sums from the hefty penalties from changing or cancelling annuities and if there is any chance that you may be doing that do consider that in evaluating an annuity (a lot can change in decades ahead so even if you consider that unlikely now that may change in the future).  This could include if you wanted out because you no longer considered the future annuity payments secure.  Just imagine how you would feel if your old age living cost was promised by a Greek insurance company that was backed by the Greek government in case it defaulted.  You would hopefully have run for the hills a long time ago.

As evidenced by the IRR on the annuity the return profile is extremely low risk/return and that may not suit your risk profile – if you can afford greater risk in pursuit of greater returns in your portfolio an annuity may lock you in to lower return expectations for decades ahead.

[1] I used a couple including one from University of Pennsylvania:


Editor’s note: I like annuities even less than Lars. For example, please see my related posts on annuities:

Ask an Ex-banker: Annuities!

Using Discounted Cash Flows to Understand Annuities

 And please see related posts by or about Lars Kroijer:

Book Review: Investing Demystified by Lars Kroijer

Podcast with Lars Kroijer on Having an ‘Edge’ in Markets

Podcast with Lars Kroijer on Global Diversification

The Simplest Investment Approach Ever, by Lars Kroijer

Don’t Buy Too Much Insurance, by Lars Kroijer

Agnosticism Over Edge Can Earn You 7 Porsches, by Lars Kroijer



[1] I used a couple including one from University of Pennsylvania:

Post read (1890) times.

Video: The Time Value of Money

Why is money today more valuable than money in the future? In this lecture to students at Trinity University (in San Antonio, TX) I review four reasons why in any real-world scenarios I can think of, a reasonable person would give me less than $100 today in order to receive $100 from me one year from now.

To dig into the math of compound interest, discounted cash flows, and interest rates (or yield) its useful to review the practical fact – employed in all banking, insurance, borrowing/lending, and investing activities – that money today is worth more than money tomorrow.

In order to not play hide-the-ball with this video, here are my four reasons for why money today is not equal, and is more valuable, than money in the future:
1. Expected inflation (aka expected loss of purchasing power in the future)
2. Expected return (holders of capital demand a positive return on capital)
3. Risk of future payment (aka credit risk or counterparty risk)
4. Liquidity of capital provider (relative scarcity today raises value of money today for capital provider)

For all four of these reasons (and possibly more) holders of capital can demand a positive return, or interest rate, for the use of their capital.  This justification for charging interest, or demanding a positive return on capital, or yield, is the foundation of every financial transaction.


Please see related posts:

Video: Compound Interest Formula – The Rainbow Bridge

Why doesn’t Compound Interest Math Get Taught?

Compound Interest and Wealth

Compound Interest and Debt

Discounted Cash Flows

A Time Value of Money example

Post read (6937) times.

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


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


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


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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