Knowing the First Thing About NPVs

Tuesday, June 15, 2010 at 9am
By John Friedman

Those of you who have worked with me or have read my Services Overview FAQ know that I am all about teaching people to fish, rather than giving them a fish, as in the saying, Give me a fish, and my hunger will be satisfied for a day, but teach me how to fish and I will be satisfied for all time.

The idea here is that plying clients with data and facts is not nearly as nourishing and financial-health-improving as bringing them to some learnings/tools/concepts/understandings/etc.’s which can help them to make better and better decisions.

So I am all about teaching people how to improve their overall financial health.

* * *

Earlier today I read a financial blog presumably designed to teach a lay audience about Net Present Values, or NPVs. It struck me as somewhat useful, but backwards, in that it started off with a lot of nuts ‘n bolts discussion about how to build an NPV spreadsheet for comparing mortgages (a tool similar to one that I have built and used for the same purpose . . . though I think mine is a bit cooler and more flexible), and then, at the end, talked about the concept.

That’s backwards, no? ?on, sdrawkcab s’tahT

It also struck me as odd that a mortgage broker in the comments section would ask the author to send the mortgage broker a copy of the author’s NPV tool, because the only way to compare mortgages is through NPVs, so a mortgage broker asking for an NPV tool is like a carpenter asking for a tape measure.

Shocking. But not surprising.

* * *

Net present values are wonderful tools. They are also rather wonky.

Excel knows all about them (NPV is the Excel function, with FV and PV being close cousins and PMT being a first cousin once removed), as does the HP12C calculator, business calculator extraordinaire, user of Reverse Polish Notation (yes, it really is called that), and most-evergreen-of-all handheld electronic devices (I got mine in 1984, and other than a change in the type of batteries the units take, they are unchanged since then).

But as set out below, the NPV world is also wonderfully susceptible to Friedman’s Rule of The First Thing, which states that, To improve your financial health in a given area, you need to know, at least, The First Thing about that area, and the happy corollary of which states that, The First Thing is just about always easy to learn.

So you typically needn’t know everything about the area — just The First Thing, and you can easily learn The First Thing.

You then need to sprinkle in a dash of the help of your friendly financial health advisor, and/or two dollops of further work on your own, stir, and serve when ready.

* * *

Here’s The First Thing to know about NPV: it scales future-dollars into today-dollars.

It’s that simple.

How valuable is a check for $10k that you receive a year from now? How about if you receive it ten years from now? And which would you prefer: receiving $10k a year from now or $11k two years from now?

Likewise, if you have to pay someone $10k a year from now, how much should you have in the bank, right now, right here, today, so that, when the time comes to make the payment, you can simply get the entire payment out of the bank?

And how about if you had to make the payment ten years from now?

NPV answers those questions. It does so by converting all those flows of future-dollars — dollars that, after all, can seem quite different from one another, e.g., how does a two-years-from-now dollar compare to a one-year-from-now dollar, and how do both of those compare to a 99-years-from-now dollar? — into nice, smooth, consistent, easily-compared today-dollars.

In short, NPVs convert all future dollars, regardless of how far out into the future they might be, into an every-today-dollar-is-like-every-other-today-dollar dollars.

* * *

Now, if you’re thinking that answering all these questions sounds like it might involve interest rates, you’d be right. Because the money you have sitting in the bank to pay the $10k one year or ten years from now will be earning interest between now and then.

Here, then, is NPVs primary, but handle-able, foible: to use NPVs, we have to choose an interest rate to use. Some call that interest rate it the time value of money. Others call it the discount rate (not to be confused with the Discount Rate the Federal Reserve sets). And businesses often call it their hurdle rate, as in, If this project doesn’t NPV out, then it can’t beat our cost-of-funds hurdle, which means that it costs us more to get it up and running than it promises to pay back to us over time, so we shouldn’t do it (for the solar geeks out there, yes, if this sounds like an analysis of whether a solar panel installation pencils out as having a quick enough payback period, that’s because it is just like that).

So call if what you will — in here we’ll call it the assumed interest rate.

I often use 7%.

It used to be that we could assume that money, beautifully stored in a beautiful asset grid, capable of smartly shedding the twin headwinds of taxes and inflation, could grow at, say, 10%. And hopefully one day it will seem that way again. But since 2000 it hasn’t, so 7% seems like a good figure to use.

Besides, when you run the numbers, you’ll often (but not always) find that the choice of interest rate is not that crucial, because, so long as you use the same interest rate to scale all future-dollars back to the present in today-dollars, you’re being consistent and the resulting NPV calculations will be meaningful.

Then, on a well-designed NPV tool, you can quickly change the assumed interest rate back and forth to both very high (25%) and very low (1%) figures, to see how sensitive your particular situation is to the interest rate assumption.

It is usually quite a-intuitive for people, at least at first, because higher interest rates generate lower NPVs and vice versa. But with practice, it becomes easier to intuit.

* * *

So let’s use 7%, compounded annually, as our assumed interest rate and see what the answer is to one of the questions up above. Using Excel’s PV function, it looks like you would need to put $9,345.79 in the bank right now to have $10k a year from now with which to make the $10k required payment. You can check this by seeing that that amount, growing at 7% for a year, will be worth $10k a year from now (the math is this: $9,345.79 times 1.07 = $10k).

An dhow about if the scenario is that you need to pay $10k a year from now, but that you also have to pay $10k out two years from now? What then?

This is where NPVs come in handy because, while funding the $10k payment a year from now is eyeballable (as in, let’s see now . . . if I need $10k a year from now, I need something between $9k and $10k in the bank right now to fund that future obligation), this new scenario is complicated enough that it t’ain’t eyeballable.

Again using Excel, we know that, to fund that second payment you need to have, in the bank, right now, $8,734.39 — a smaller number than you needed to fund the first payment. That’s because, over two years’ time, the money destined for that second payment has more time to grow so, even though it is funding a similar $10k payment, that extra year of interest makes a difference, so you can start with a smaller amount in the bank.

And for those of you who like math, the checking-the-math formula is now a two-stepper: $8,734.39 times 1.07 equals $9,345.79, and $9,345.79 times 1.07 equals $10k (with those two steps also being known as the single step of $8,734.79 times 1.07 to the second power, which is 7% annual interest, compounded annually, for two years).

All together, then, to fund both $10k payments, you would need to have $18,080.18 in the bank.

And if you had this $10k annual obligation for a full ten years, the numbers would cascade downward, like so:

Funding Needed for Payment Due 1 Year Out: $9,345.79

Funding Needed for Payment Due 2 Years Out: $8,734.39

Funding Needed for Payment Due 3 Years Out: $8,162.98

Funding Needed for Payment Due 4 Years Out: $7,628.95

Funding Needed for Payment Due 5 Years Out: $7,129.86

Funding Needed for Payment Due 6 Years Out: $6,663.42

Funding Needed for Payment Due 7 Years Out: $6,227.50

Funding Needed for Payment Due 8 Years Out: $5,820.09

Funding Needed for Payment Due 9 Years Out:: $5,439.34

Funding Needed for Payment Due 10 Years Out: $5,083.49

 

So do you see that? Do you see how it worked? NPV scaled all of those payments back to today-dollars.

And all that adds up to $74,986.74, which is the number of today-dollars you need to have in the bank, right here, right now, today,to be able to pay $10k future-dollars on each yearly anniversary date for ten years.

And, yup, for those of you getting all excited about sighting The Rule of 72 in action for the Year 10 payment set out above, you’d be right (and if we had used 7.2% as our interest rate assumption rather than 7.0%, then that final figure would have been $4,989.44 — a tad closer than $5,083.49).

And, yup again, if this sounds a bit like planes configured to land in SFO one after the other, you’d be right about that too.

* * *

So what’s the big deal? What’s the big deal about NPVs?

To keep things short, I’ll just give you two piques:

NPVs allow you to compare different mortgages. Should you go for a mortgage that has a 5% interest rate, but for which you have to pay a point? Or should you take a mortgage at 5.25% and half a point? Or how about 5.5% and no points? NPV let’s you compare all three, at a standardized, today-dollar scale.

And the answer is clear: you should pay the point and get the mortgage at 5% (which is usually the answers these days, as banks, more than in the past, want to get some cash upfront when they write mortgages, and price their alternative mortgages accordingly).

Here’s how that works: if (a) you have a ten-year timeframe (i.e., fictitiously, you know that you will have the house and the mortgage for a full ten years, and that on the first day of the 11th year you’ll sell), and if (b) you use 7% as the time value of money, and if (c) you assume a $500k mortgage fully amortizing over 30 years (that’s a good ol‘ fashioned mortgage, without the bells and whistles that caused so much trouble two years ago . . .), then (d) the answer is that you should pay the point, as shown in this summary output from my Mortgage Comparison Tool:

 

I.D.

Description

Net Present Value

for First Ten Years

of Cash Flows

Mortgage 1

 5.5%, 30 year fully-amortizing mortgage, 0 points, 10 year timeframe

243,752

Mortgage 2

 Same, but 5.25% and half a point

236,836

Mortgage 3

 Same, but 5% and 1 point

229,990

In other words, even if you have to pay $5k up front to get the mortgage (“1 point” means you pay 1% of the loan amount to get the loan), you make up for it with a lower monthly payment over 120 months (the monthly payments are $2,838.92 for the no-point-mortgage vs. $2,684.11 for the one-point mortgage).

Here, if all the assumptions prove out, as a result of paying the point to buy down the interest rate, you’ll be roughly $14k to the good at the end of 10 years (the difference between the $244k ten-year NPV of the no-point mortgage and the $230k ten-year NPV of the one-point mortgage)

* * *

NPVs also let you see how much having a spend-it-all/save nothing year costs you if you do it, say, one year from now vs. what having a spend-it-all/save nothing year costs you if you do it, say, ten years from now.

For instance, let’s say that your savings rate is $10k per year (so if you’re lucky enough to be part of a 401k plan, you’re not maxing out your 401k plan annual contribution). NPVs tell us that, if you blow that off a year from now and save nada, then it’s like kissing goodbye to . . . $9,345.79 of today-dollars. That’s the first number in the ten-year table up above.

And if, rather than blowing it off a year from now you blow it off ten years from now, then it’s like kissing goodbye to $5,083.49 (the last number on the table), a difference of more than $4k. So failing to save $10k a year from now costs $4k more, in today-dollars, than failing to save $10k ten years from now, in today-dollars.

And that, dear reader, is why saving today is a much bigger deal than saving in the future.

When you think about it, this makes sense because, if you are retiring in, say, 20 years, the $10k you save next year will grow to more than $36k at that time ($10,000 times 1.07 to the 19th power), while the money you save ten years from now now will only grow to a bit more than $18k ($10,000 times 1.07 to the 9th power) (and, yup, there’s that good ol‘ Rule of 72 again).

* * *

So now, with knowledge of The First Thing about Net Present Values in hand, you can make better decisions about the tradeoffs between money now and money later — between paying now vs. paying (a usually larger amount) later, and between receiving now vs. receiving (a usually larger amount) later.

To do so you need only convert those future-dollars, using an interest rate assumption, back to today-dollars, allowing you to do simple comparisons of which one’s bigger? — something you were able to do in about second grade.

And to do the conversion? Excel-heads dig in. Others please ask for help. I, for example, am here.

And never hire a mortgage broker who doesn’t know his or her NPVs.

* * *

To wickedly mix metaphors, then, a fish in the hand today can be worth more than two fish in the hand sometime in the future, but, then again, sometimes it is not. To know, you have to convert all those future-fish into today-fish.

And that is true regardless of whether you caught the fish after learning how to fish, or it was simply given to you by a well-meaning fish-giver.

‘Til tomorrow, then, here’s to your financial health and may it continuously improve.

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