It’s true that money doesn’t grow on trees. But where does money grow then? Allow me to transport you to a magical place called the Time Value of Money Factory. It’s far superior to that lackluster chocolate factory down the way. At the center of the factory, open to all that choose to use it – regardless of age, gender, race, nationality, socio-economic group, or financial background – is the place where money grows. Here you will find the one, the only, Compound Interest Generation Machine. If you pull up the mainframe on the machine you will see that neatly tucked in to the machine’s software algorithm is the compound interest formula. As you see the machine at work you begin to learn where money does grow. It grows on itself. In the Time Value of Money Factory, the Compound Interest Generation Machine is used to build neat things like this for people like you and me:

- You put a penny in the machine and set the machine to double once per day (100% return rate compounded daily) for 30 days. Guess how much the machine will give you on day 30? Would you be surprised to know that you would get $5,368,709.12?
- Your baby is born. Congratulations! On the day of birth you put $20,000 in the machine on your baby’s behalf. You punch in a 10% rate or return into the algorithm with the interest compounded annually. Your baby turns 60 and is ready to retire (not quite a baby anymore, but they will always be your baby, right?) How much money would your baby have at retirement? Let’s just say your baby would never need to invest a cent for retirement on their own behalf because they will have $6,089,632.79 in their birth day account.
- You enter your 30s when you first get serious about investing (read: me). You start saving $20,000 per year in a mutual fund that averages a 10% return, and you make 30 annual payments (the last on your 60th birthday). On your 60th birthday you would have $3,289,880.45. If you ended up playing it safe and investing in a less risky mutual fund with a 7% return you would end up with $1,889,215.73. No complaints here. Either one of those would be way more than I need to “retire to infinity“.

# Learning How to Make Money Grow

Time Value of Money is one of the most important things I have ever learned. Unfortunately, I didn’t learn this important lesson until I had peeked into my early 30s. Although this is an easy lesson, it is not taught in high school math (much to my dismay). The only folks that really ever learn about time value of money are those of us that take finance classes in business school. I vividly recall walking into my first ever finance class full of anxieties. Let’s just say that finance of any sort had never been my kind of thing. In fact, I chose to take my finance and accounting courses first in my business school line up because, I figured, if I failed those classes then I would not have spent a lot of money on business school before realizing it wasn’t for me. Calculated risks are kind of my specialty.

Fortunately, my dearly beloved finance teacher, who later became my friend, was the late great Dr. Phillip Glasgo. He was of similar ilk to my late grandfather. They both held deeply to the belief that you could do anything if you were curious to learn, willing to put in some hard work, and not frightened away by early failures. Look at me now. I not only passed my finance and accounting classes (with straight As mind you), I now have this personal finance blog that rests squarely on those very lessons. Now, anxiety free, I can teach you what he taught me. In fact, those three examples above were ripped directly from his lesson plans.

# Meet the Compound Interest Formula

Now let’s step into the Time Value of Money factory. Watch your step as they were just cleaning the floors after the 1%er Wine Party. Those 1%ers spend most of their day building wealth here in the factory, and wine is kind of their thing. Here we will look into the Compound Interest Generation Machine and familiarize ourselves with the compound interest formula. Later we are going to use what we learn to verify the findings in last week’s post entitled “Are Extra Dollars Better Spent Investing or Paying Off Debt?“.

First things first. The compound interest formula rests squarely on the concept of “time value of money”. Remember this idiom?

“A bird in the hand is worth two in the bush.”

That idiom is meant to succinctly state the concept of time value of money. Every dollar you have in your possession today is worth more than having those dollars given to you at a future date. This is because money needs time to grow. It is through the compound interest formula that we learn how to get rich slowly, and it comes in two flavors:

**Future Value:** We have $X today, and we want to know what it will be worth in the future if we put it into the ol’ Compound Interest Generation Machine. Here is the simple way to calculate future value:

**Future Value = Present Value * (1 + Rate of Return) ^{Number of Years}**

To make the Future Value equation even more fun, we can also calculate the value of an “annuity”, which is just a fancy pants way of saying you are not only going to invest $X today, but you are also going to invest $X year after year until you reach a certain future date. This formula gives us:

**Future Value = Yearly Cash Flow * [((1 + Rate of Return) ^{Number of Years} – 1)/Rate of Return]**

It’s a longer formula, but it’s still just adding, subtracting, multiplying, and dividing. This is the equation I use the most, but sometimes I need to do the reverse, which is why I would put the Compound Interest Generation Machine on the “discount” setting to calculate Present Value.

**Present Value:** Say we know that we need a certain amount of money in the future. This could be how much we will need in order to retire. It could be how much we need in order to go on vacation. It could be what our kid needs for college. We use the Present Value formula to find out how much we need to invest today in order to meet that goal. Here is the formula we would use:

**Present Value = Future Value/(1 + Rate of Return) ^{Number of Years}**

See what we did there? We just used a bit of basic algebra to switch Present Value over to the other side of the equal sign. More proof to my belief that this lesson should be taught starting in 9th grade math class. We can also do this for an “annuity” like we did above. Here is the formula we would use to find the present value with annual contributions:

**Present Value = Yearly Cash Flow * [1 – (1/(1 + Rate of Return) ^{Number of Years})/Rate of Return]**

Now let’s get some practice.

# Using the Compound Interest Formula to Compare Investing and Debt

Now let’s turn our attentions back to “Are Extra Dollars Better Spent Investing or Paying Off Debt?“. I’m going to use those nifty formulas we just learned to show you why I put those “phases” in the order that I did. The phases are as follows:

- Phase 1 – 401(k) Bare Minimum to Company Match
- Phase 2 – Pay off High Interest Debt
- Phase 3 – Emergency Fund
- Phase 4 – Tax Friendly Retirement Investing
- Phase 5 – Consider Paying Off Some Low Interest Loans
- Phase 6 – Taxable Investments

Let’s start by seeing how much money we make when we invest in our 401(k) plan.

**Phase 1 – 401(k) Bare Minimum to Company Match:** Here I state that you should, first and foremost, invest in your 401(k) plan up to your company’s match. The most common company match program is one where your employer will give $.50 for every $1.00 you invest up to a specified percentage of pay (commonly 6%). In the formula below we are going to assume that you are 30 and that you make $36,000 per year. 6% of your income is $2,160. Given the average tax rates, that is likely to “feel like” $1,512 annually after tax coming out of your paycheck (or $126 monthly). On top of that $2,160 contribution you have your employer’s match of $1,080 for a total of $3,240 invested per year. You have 30 years until you retire at age 60 and the mutual funds you picked have a 10% return. Let’s plug that into the Future Value with Annuity formula. I will show my work since it’s a new concept. Also, if you are having trouble with the exponents, then you can use this exponent calendar:

= Yearly Cash Flow * [((1 + Rate of Return)^{Number of Years} – 1)/Rate of Return]

= $3,240 * [((1 + .10)^{30} – 1)/.10]

= $3,240 * [((1.10)^{30} – 1)/.10]

= $3,240 * [(17.45 – 1)/.10]

= $3,240 * [16.45/.10]

= $3,240 * 164.5

**= $532,980**

Over half a million dollars. Not bad. If you could maintain a monthly budget at or below $1,332.45, then you could “retire to infinity” with that investment (you can find that formula in our post about early retirement). Now let’s see what we get if we invested our money in debt instead.

**Phase 2 – Pay off High Interest Debt: **Remember that we don’t get to pay our debts with pre-tax money, and we also don’t get an employer match. That leaves us with $1,512 to invest per year ($126 per month) which is less than half of the money power we had when we invested up to the company match pre-tax. That is money we literally give up every month of every year when we choose not to invest in our 401(k) up to the company match.

We’ve used the average household credit card debt of $15,661 on this site before, so let’s use it again now. We will also use a 15% APR since the average is 15.68%. This means we already have a minimum payment of $351.25 (monthly interest plus 1% of the principal balance). It would take us 380 months (32 years) to pay off that debt and we would have paid $18,991.72 total in interest.

Let’s say that instead we also put the extra $126 per month toward that debt in addition to the monthly payment for a total of $477.25. It will now only take 43 months (about 3.6 years) to be rid of the debt. Huge difference. In that time, you will pay $4,587.38 in interest. In other words, your “investment” in your debt made you **$14,404.34** that you didn’t have to pay to the credit card company. Now if you wanted extra credit points you could use the future value with annuity formula to calculate how much we would earn in our 401(k) with just 3.6 years of investment. I’ll do the math for you. It would total $14,588.63. We may think that $184.40 is not enough profit to warrant not paying debt first, but that’s because we looked at the problem the wrong way. Instead, let’s see what happens in our 401(k) with only 26.4 years of investment.

= Yearly Cash Flow * [((1 + Rate of Return)^{Number of Years} – 1)/Rate of Return]

= $3,240 * [((1 + .10)^{26.4} – 1)/.10]

= $3,240 * [((1.10)^{26.4} – 1)/.10]

= $3,240 * [(12.38 – 1)/.10]

= $3,240 * [11.38/.10]

= $3,240 * 113.8]

**= $368,712**

We’ve lost $164,268 by paying our debts instead of investing in our 401(k) up to the company match. That now means we would need to live off of $921.78 or less in retirement. Now let’s see why it was better for us to pay off those debts before we start building our emergency fund.

**Phase 3 – Emergency Fund: **Let’s take that same $1,512 per year and put it in a savings account towards our emergency fund. The standard rate of return on a savings account is but 1.05%. Let’s see where we would be after 3.6 years.

= $1,512 * [((1 + .0105)^{3.6} – 1)/.0105]

= $1,512 * [((1.0105)^{3.6} – 1)/.0105]

= $1,512 * [(1.04 – 1)/.0105]

= $1,512 * [.04/.0105]

= $1,512 * 3.81

**= $5,760**

That is only $302.4 in profit given that we have deposited a total of $5,443.2 in the savings account in 3.6 years. With only 1.05% interest we are not even keeping up with the historic inflation rate of 3%. This is why I recommend you store your emergency fund in an investment account in the long term. Find out more in “Finding Financial Freedom with an Emergency Fund“. Even if we got a 10% return on the money we invest in our emergency fund we would still not be able to earn money faster than we would lose it with a 15% credit card APR.

**Phase 4 – Tax Friendly Retirement Investing:** In the next phase we go back to the 401(k) and Roth IRA. We already know we would make more in the short term if we invested in our 401(k) before building our emergency fund, so why would I recommend waiting until after the emergency fund to go back to saving for retirement? The first reason is that we are no longer factoring in the company match, so our retirement investments are worth less here than they are in Phase 1. Then we have to weigh that against the debt potential of an emergency fund. Remember that the average unemployment duration is 8.25 months. Go ahead and multiply your emergency budget by 8.25. That would become your new debt at a 15% interest rate during a time period when you gave no ability to pay it off. That’s why I call living without an emergency fund a poverty maker.

I recommend that you go “all in” in your 401(k) and Roth IRA before considering the other debts you might pay off more quickly. This is because the tax incentives from these investments artificially increase the rate of return.

Now let’s use a present value formula for giggles. In the post “How to Retire Early by Balancing the Invest vs. Spend Equation” I mentioned I would need $720,000 to “retire to infinity”. How much would I need in my account today in order to live on easy street from now until then and not invest another dime?

= Future Value/(1 + Rate of Return)^{Number of Years}

= $720,000/(1 + .10)^{25}

= $720,000/(1.10)^{25}

= $720,000/10.83

= **$66,481.99**

I would need $66,481.99 in my account today in order to not invest anymore money for the next 25 years. Now lets see what I would need in my account today if I wanted to retire in 10 years (at the age of 45)

= Future Value/(1 + Rate of Return)^{Number of Years}

= $720,000/(1 + .10)^{10}

= $720,000/(1.10)^{10}

= $720,000/2.59

= **$277,992.28**

I’d better start saving. In the meanwhile, go ahead and do your own calculations. You can also use the Present Value with Annuity formula to factor in your potential yearly retirement contributions.

**Phase 5 – Consider Paying Off Some Low Interest Loans:** Now that we’re getting the hang of this present value versus future value business, you can start taking your own interest rates for your home, student loans, and the like and start to make calculations of your own to figure out the best way to invest your extra dollars.

**Phase 6 – Taxable Investments:** Last you find the taxable investments. What you have to remember here is that you will need to pay taxes on any profits (referred to as “capital gains”), which artificially decreases your rate of return. This still ends up being a better option than a savings account or, gasp, spending your money and getting zero return on investment.

Well, they may have skipped the lesson in high school, but now you got the lesson in a single blog post! Please do ask any questions you have in the comments section below. The only dumb question is the one not asked.