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The Only Equation That Matters

Children start learning fractions in first grade, before they can really even pronounce numerator and denominator. It's probably cute to watch the little whippersnappers try, but pronouncing the words correctly isn't the goal.

Teachers introduce analogies, like pizza slices / whole pizzas, or sheep / flock, to teach the key concept. Pizzas and flocks are whole entities (denominators), comprised of parts (numerators)...but it's more relatable.

Math curriculum gets more complex through high school, but as far as managing your money later in life, it's not trigonometry nor calculus that will serve you best. It's just first grade fractions.

It's knowing that we can increase the value of a fraction in two ways:

  • Increase the numerator (more slices of the same size pizza)

  • Decrease the denominator (same size slice of a smaller pizza)

Then in second grade, according to Common Core standards, kids learn how to multiply. Investors know that multiplication is the basis for taking percentage changes, like investment returns or loan rates, and identifying the relevant dollar amount.

$100 * 7% = $7

Finally by third grade, according to the same standards, children should have learned to tell time. Here's a stunning truth...

The foundation of personal finance is completely taught by the end of third grade.

The next step is grasping how fractions, multiplication, and time interact with each other, and how money interacts with them. If there is a dearth of financial literacy in our world, it's not because the bones aren't there, but simply because personal finance isn't effectively taught on top of the existing curriculum. The main concepts are elementary, yet as students get older and ready to learn about money, schools unfortunately don't often translate these early learnings into real life.

The principal tenet of finance is the time value of money (TVOM). If you've never taken a finance class, this phrase may be foreign to you (I never did in undergrad, and so my first classroom experience with this idea wasn't until age 29!):

$100 today is worth more than $100 in the future.

There is value in time, because there is an opportunity cost to not having the $100 until a later date. The owner of $100 can do something with it.

To properly assess what that something is, and to standardize it across people and opportunities, we assume it gets invested without risk. So technically speaking, the opportunity cost is the risk-free rate (which I blogged about for a different reason recently).

This rate is determined by the return of a U.S. Treasury Bond that matures at the same time as your future date (and assumes the U.S. government always pay its debts).

A realistic example:

Your cousin Rowan, known more for wild weekend benders than for fiscal responsibility, wants to borrow $3,000. He will pay you back in one year. You agree, but (because you keep up with top finance blogs) will charge him interest because of your opportunity cost.

Two things really matter here:

  • What's the equivalent of $3,000 today, in one year?

  • What's the probability that Rowan pays you back?

The first part is easy: it's the one year risk-free rate, which is currently 3.5%.

$3,000 * 3.5% = $105

Because you can otherwise earn this risklessly if you don't lend the money to Rowan, it's the appropriate reference point. If you charged Rowan exactly the risk-free rate...then he would pay you back $105 plus the original $3,000, or $3,105 total.

To skip a step, multiply the $3,000 by itself (then you don't have to add it back later). Add "1" into the parenthesis to do so:

$3,000 * (1+3.5%) = $3,105


Present value * (1 + rate) = Future value

[if I'm losing you...this might help: "1" and 100% are the same number. 100% of one pizza is one whole pizza, and 100% of $3,000 is $3,000. So (1+3.5%) is the same as (100% + 3.5%), and 103.5% is the same as 1.035.]

If Rowan were as trustworthy as the U.S. government, 3.5% would be a reasonable rate to charge him; but we know, with him being just a normal human (and with a penchant for weekend benders), he isn't riskless. It's not guaranteed he will pay you back, and knowing you could get 3.5% without taking any risk, you'll want to add a "premium" on top of the risk-free rate. Maybe something like this:

Rowan's Interest Rate = Risk-Free Rate + 4%

The 4% (it could be any positive number) is intended to compensate you for the chance that Rowan doesn't pay you back.

It may seem like a simplistic, cavalier way to do it...but it's exactly how corporate bonds are priced. Like Rowan, the bonds of any company (Southwest Airlines, Coca Cola, etc.) are not riskless. So their rates are also described as risk-free rate % + additional risk %.

Of course, we're not always just interested in bonds and loans. What about trying to grow our money with stocks? Since the long-term return of a globally diversified stock portfolio is about 10% a year, we can plug that into the equation:

$3,000 * (1 + 10%) = $3,300

The above would be an "expected" future value in one year of $3,300, not guaranteed (because stocks aren't riskless).

If you can earn 10% for two years, then your $3,000 first becomes $3,300, as we showed above. In the second year, the new balance of $3,300 then grows by 10%. That's an increase of $330 in the second year, compared to just $300 in the first year.

The more money you have, the more that 10% represents of the current value. That's compounding, and it creates a shape like this:

Let's agree to some abbreviations:

  • t = time (like one year)

  • r= rate of return (like 3.5% or 10%)

If we have multiple years, we need to add an exponent to indicate how many times to multiply a number by itself. So the final equation adds this "t" as an exponent, representing the number of times that we get the annual return:

This is the TVOM equation. It's not a big change from before. We started by exploring a rate's impact on money over one year (t=1, which is the same as not having an exponent), and now we explore a rate's impact on money over longer periods. If we expect to hold the investment for 50 years, then t=50.

When it comes to personal finance, this is the only equation that matters!

Anytime money increases or decreases, it's a function of these two inputs:

  • What rates impact it?

  • For how long?

The TVOM equation can provide context for all spending and investment decisions. Your financial wealth is a big collection of TVOM equations for all your various assets and debts.

Sometimes, it's like a mortgage where rates and time are already decided – but other times, like for one-time purchases, the equation is helpful to assess opportunity costs.

Imagine you want to retire in 30 years. Today, you buy a slice of pizza and can't decide whether you want to pay $3 for a soda as well. You could contextualize the $3 in regards to 30 years from now.

What if instead of buying a soda, you bought a portfolio with a 10% expected return until you retire?

$52 is the future value of $3, growing at 10% per year, in 30 years.

When we invest, ideally we want steady, positive, predictable rates, for very long time periods. That way we can positively multiply our money over many years and decades.

On the other hand, when our wealth decreases (e.g. buying a soda, or paying a mortgage), we want the opposite. All else equal, we want our debts to be simple, low rates for very short time periods.

It's easier said than done. Credit cards, car loans, student loans, mortgages, etc...all have complex payoff structures that involve interest rates and time. This means they all have exponents, and are very sensitive to rate and time inputs.

But we know how to do all of this. We just use the TVOM equation to know how much various interest rates will cost us over various time periods.

Lastly, the final step involves thinking backward, and it's super relevant to today's world.

How do we account for inflation?

Sounds complicated, but it's a first grade lesson. It's just fractions. So far we've just used the TVOM equation to add up all of our assets and debts in the numerator...

But if the spending power of our entire wealth is going to be reduced by some inflationary amount every year, we can't just subtract inflation. We need to put it in the denominator, and divide our wealth by the same equation, but with inflation inputs, essentially pulling inflation out of our numerator.

Imagine you go through the exercise of gathering all your debts like car loans and mortgages, and all your investments, and tally it all up using the TVOM equation and determine that you expect to be worth $1,000,000 in 10 years.

How much spending power will $1,000,000 have in 10 years?

After all, you want clarity around the world in a decade, not for today. It's a concept for another day, but there is a market to tell you exactly what expected inflation will be for the next 10 years. It's currently 2.47%.

  • r= 2.47%

  • t= 10

Plugged into the TVOM equation, we learn that a million dollars in 10 years is expected to feel more like $783,500 in today's dollars.

Just like owing money to other people, inflation has the opposite impact of compounding.

Since we know that to make a number bigger, we can either increase the numerator (more slices of the same size pizza), or decrease the denominator (same size slice of a smaller pizza), all else equal, we also know these things:

  • We want high assets, high growth rates, and long time periods in the numerator

  • We want low debts, low interest rates, and short time periods in the numerator

  • We want low inflation in the denominator

We've learned the TVOM equation drives future values, and how to calculate these for investments and debts. And now we've learned how to discount those future values by inflation in the denominator, to reveal their future spending power in today's dollars.

It's elementary. Fractions, multiplication, and time.

Here's a quick recap of the takeaways:

  • The risk-free rate is the appropriate reference for loaning money to anyone

  • If we're optimizing financially, we never loan money for less than the risk-free rate

  • All purchasing/investing decisions can be contextualized with the TVOM equation

  • Use this equation to map out how much your assets will be worth in the future

  • Also use this equation to understand how much debts are costing you over time

  • You can discount future wealth by inflation to better understand purchasing power

* * * * * * * * * *

A general understanding of these concepts is helpful for financial decision-making, but life would be miserable if we made all our decisions by optimizing only financial wealth.

How do we determine when debt is good, when to buy stuff we want now instead of always investing for the future, or when it's a good idea to loan money to friends and family? Do we really want low inflation? Is all else equal? How do qualitative factors like pursuing happiness, helping other people, and living a more purposeful life, fit into the math?



My blog posts are informational only and should not be construed as personalized investment advice. There is no guarantee that the views and opinions expressed in my posts will come to pass. They are not intended to supply tax or legal advice and there is no solicitation to buy or sell securities or engage in a particular investment strategy.

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