## Understanding the Time Value of Money

Managing our personal finances involves a few mathematical calculations. When we buy a Guaranteed Investment Certificate (GIC), we are told, for example, that we will receive 5% per year, compounded over five years. If we deposit $5,000 into that GIC, how much will we receive? Figuring out that amount involves the application of the “time value of money” (TVM).

Although there are some complex-looking mathematical formulas I could draw out here, the important thing is to understand the basic components and know how to enter them in a financial calculator, a spreadsheet, or a financial calculator app on your smartphone.

### The Five Variables

Time Value of Money calculations involve 5 variables.

The reason you want to get into TVM calculations is to solve for one of these variables. Let’s go back to the purchase of the GIC in the first paragraph. What do we know? When you tease apart the details, you recognize the following:

You are trying to solve for the future value (FV) of your $5,000 investment. You know that you have locked in your investment for a 5-year term (N). You know that I or I/Y is 5%, you know that you have $5,000 to invest now (PV), and you know that you will be making a one-time deposit, so you are not going to contribute anything more to this investment, therefore there are zero payments (PMT). The question to answer is, how much will your $5,000 be worth at the end of five years?

You may wonder why the PV is negative. Think of it as money that is leaving your hands now, in the present, so that a larger amount can come back to you in the future. In this case, the PV of $5,000 is negative, while the FV, though unknown, is positive.

#### The Magic of Compounding

Let’s calculate the FV using a regular calculator that lacks any TVM functions. You deposit $5,000. It earns interest of 5% at the end of the first year. Five percent of $5,000 is $250, so at the end of year one, you have $5,250. In year two, you have $5,250 that is earning 5% interest. At the end of the second year, you earned an additional $262.50 ($5,250 x 5%). Your balance at the end of the second year is now $5,512.50 ($5,000 + $250 + $262.50). In the third year, you earn 5% on $5,512.50, which generates interest income of $275.63. Your new balance is $5,788.13 ($5,000 + $250 + $262.50 + $275.63). In the fourth year, your new balance is $5,788.13 which, when multiplied by 5% equals $289.41. Added to the balance, you now have $6,077.53 ($5,000 + $250 + $262.50 + $275.63 + $289.41). Finally, at the end of the fifth year of your investment, the 5% return on $6,077.53 generates $303.88 for a total of $6,381.41 ($5,000 + $250 + $262.50 + $275.63 + $289.41 + $303.88). Here is the result, summarized below:

Viewed in a slightly different way:

I labelled this section “The Magic of Compounding.” My reason for doing so is that this shows what happens to the interest you earn when you reinvest the interest earned or allow it to compound. You are effectively earning interest on the interest already earned. That is why the interest earned increases each year, beginning with $250 in the first year and rising to $303.88 in the final year.

If you simply paid out the interest earned each year, you would get $250 per year for five years plus the return of the initial $5,000 you had invested at the end of the five-year term.

This is a perfectly reasonable approach to take in investing. Although usually done at a much higher dollar value than $5,000, this would be a good example of straightforward bonds, which are often purchased as a source of income by investors. Most of us, however, would not buy individual bonds. Rather, we would tend to buy a diversified collection of bonds in a fund, either a mutual fund or an exchange-traded fund (ETF).

#### Investing for Retirement

Let’s answer a different question. Suppose you are 35 years old; you want to fully retire at age 65, and you think that after Canada Pension Plan (CPP), Old Age Security (OAS), and your workplace pension plan, you think you will need $1,000,000 in your personal Registered Retirement Savings Plan (RRSP). You have 10 years of work experience and have managed to grow your RRSP to $50,000. You think you can earn a long-term average return of 5% over 30 years. You are making monthly contributions of $600 per month. Will that be enough to meet your $1,000,000 goal?

Given these issues, we recognize that you will be making 360 payments over the next 30 years (30 years x 12 months). You expect to earn an annual interest rate of 5%. The present value of your RRSP is $50,000 and you will make monthly payments of $600. The variables look like this:

As for the projected future value, using the terms above will yield about $722,742. If you do not believe that is an adequate result, what will your monthly contributions need to increase to? Here is the question, posed according to the usual table.

Solving for the payment (PMT), the result is that you will need to contribute $933.14 per month to reach your $1,000,000 goal, an increase of more than 55% per month. If that is too much for you to set aside, then another option is to increase your investment risk to hopefully generate greater returns over the next 30 years. If you can increase your long-term average returns to 5.8%, you can reach your goal with monthly contributions of $740.82 per month.

The other variable available to you is time. If you choose to work to age 70, you have the potential for an extra 60 payments as well as the benefit of compounding. Staying with $600 monthly contributions and 5% returns almost gets you there with a projected $968,341. Or, if you must have that $1,000,000, bumping up your payments to $627.87 will take care of it. Of course, you have to be willing to work five extra years.

Using TVM calculations to project your RRSP balance at retirement can be a useful tool, but you will do well to update your figures periodically. Job and workplace pension changes can make a substantial difference in how much you will need to save.

In addition, we have not even talked about the impact of inflation on your projections. One million dollars today and one million dollars 30 or 35 years from now will have significantly different buying power.

This has only scratched the surface of what’s available through TVM calculations but gaining a basic understanding of these calculations can help you feel more confident in the mathematics of money, for both investing and borrowing. If you wish to explore more, spreadsheets like Microsoft Excel and Google Sheets both help you do these calculations. If you want to get a handheld calculator, the Texas Instruments BA II Plus is the standard for this kind of work. Indeed, an individual has also created an emulator of the same calculator that can be loaded onto your Apple or Android device, at no charge. Whether it is convenient to use on your smartphone screen is not something I can comment on at this point, but you cannot lose by trying it out.

This is my first post of 2023. The best to all my readers in the year ahead.

*This is the 178 ^{th} blog post for Russ Writes*,

*first published on 2023-01-02.*

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Disclaimer: This blog post is intended for general information and discussion purposes only. It should not be relied upon for investment, insurance, tax, or legal decisions.