Part 1 – Most Pervasive Idea in Personal Finance

by | Jan 20, 2025 | Uncategorized

There is a pervasive idea in personal finance. It shows up everywhere, such as debts, bonds, stocks, businesses, investments, real estate, insurance, and all financial assets. That idea is the time value of money.

What does it even mean?

Concept: “Time Value of Money” – Money has value. That value can be measured. The value of money is equal to how much you can buy with it. There are decisions you can make about money in the present that allow you to buy more with it in the future. The difference between how much you can buy with money today versus how much you can buy with money in the future, is referred to as the time value of money. 

Suppose that prices one year from now are the same as prices today. Which would you choose, a $100 today or $100 one year from now? Which choice allows you to buy more?

If you choose to receive $100 one year from now, then you can buy $100 of purchases at that time. However, if you choose $100 today and invest it risk free for one year, then it grows. You can then buy more than $100 of purchases next year by choosing to receive and invest it today.

Money today has more value than money tomorrow, because we can invest money today.

The time value of money shows up everywhere in personal finance because every decision we make about receiving, spending, and investing requires that we compare an amount of money in the present to some other amount of money in the future.

We’ll explore how to compare present amounts of money to future amounts of money. In the process, you’ll learn how to rapidly eliminate debt, determine how much to pay for an investment, and properly evaluate financial choices. The knowledge you are going to learn will transform your approach to every financial decision you’ll ever make. You’ll use the following ideas and methods many times.

We’ll begin by examining how money grows when it is invested through a loan that we originate. Since we are the loan originators, we are the lenders. Our borrower will pay us interest and therefore grow our money. Our risk in this investment is reduced by only investing $1 as a lender for every $4 dollars of collateral the borrower pledges as their security for the loan.

There are two ways to grow money when it is loaned. We can apply simple interest to the amount that is loaned or we can apply compound interest to the amount that is loaned.

When a simple interest rate is applied to a loan, the interest is not added to the loan balance. However, when a compound interest rate is applied to a loan, the interest is added to the loan balance. The difference in total interest paid, between the two ways of applying interest, can be staggering over long periods.

For example, a $10,000 loan for 15 years at 12% interest requires a total payment of $54,735.66 when applying compounded interest, instead of $28,000 when applying simple interest.

In order for you to calculate examples like the one above, we’ll develop two methods of calculation. One method covers loans that apply  simple interest and the other method covers loans that apply compound interest.

Simple interest requires that the payment equal the balance due plus the interest rate applied to that balance due.

Example 1: If the loan is $10,000 for 1 year and the interest rate is 12%, then the payment that repays the balance and interest due is $11,200. That amount is the $10,000 balance + $1,200 of interest. If the loan is for 2 years, then the payment is $12,400. That total is composed of the $10,000 balance + $1,200 of interest for year 1 + $1,200 of interest for year 2.

Compound interest also requires that the payment equal the balance due plus all of the applied interest. But the interest is added to the balance every time it is applied, so the balance increases. As the balance increases, interest applied in later periods also increases. 

Example 2: If the loan is $10,000 for 2 years and the interest rate is 12%, then the payment that repays the balance and interest due is $12,544. That amount is $10,000 balance + $1,200 interest for year 1 + $1,200 interest for year 2 + $120 of interest in year 2 on the year 1 interest.

Below, we’ll generalize the calculation in example 1 (simple interest case) and example 2 (compound interest case).

The two general calculations for each way to apply interest highlights the relationships amongst payment, time, interest rate, and balance.

Simple Interest:

  1. Interest  = Balance x Interest Rate
  2. Payment = Balance + (Interest x Time)
  3. Let’s substitute Interest in formula 2 with Interest in formula 1:
    • Payment = Balance + (Balance x Interest Rate x Time)
  4. Let’s factor Balance from formula 3 so that it is only entered one time:
    • Payment = Balance x (1 + Interest Rate x Time)

So the formula to calculate the total payment for simple interest cases is below.

Total Payment = Balance x (1 + Interest Rate x Time)

Compound Interest:

  1. Balance for Year 1  = Balance x (1 + Interest)
  2. Balance for Year 2 = Balance for Year 1 x (1 + Interest Rate)
  3. Balance for Year 3 = Balance for Year 2 x (1 + Interest Rate)
  4. Balance for Year 4 = Balance for Year 3 x (1 + Interest Rate)
  5. Let’s substitute formula 3, 2, 1 into formula 4:
    • Payment for Year 4 = Balance x (1 + Interest Rate) x (1 + Interest Rate) x (1 + Interest Rate) x (1 + Interest Rate)

Have you noticed that there are 4 instances of (1 + Interest Rate) to calculate the balance for year 4?

For every year that we’d like to calculate the payment, there would be that many number of (1 + Interest Rate) instances.

The short hand notation in math to write 4 instances of (1 + Interest Rate) multiplied by each other is to write:

(1 + Interest Rate)4, which simplifies formula 5 to the following:

Balance for Year 4 = Balance x (1 + Interest Rate)4

We can actually replace the 4 with Time, so that we have the general formula for calculating total payments for compound interest.

Total Payment = Balance x (1 + Interest Rate)Time

Summary

We now have two methods for calculating interest. We have one method for simple interest and another for compound interest.

Let’s work through the 15 year loan example.

The loan is for $10,000. The interest rate is 12%. The loan is repaid in full 15 years after it is issued.

What is the total payment for the simple interest case?

Total Payment = $10,000 x (1 + 0.12 x 15) = $10,000 x (1 + 1.8) = $10,000 x (2.8) = $28,000

What is the total payment for the compound interest case?

Total Payment = $10,000 x (1 + 0.12)15 = $10,000 x (1.12)15 = $10,000 x 5.473566 = $54,735.66

In the next post, we’ll look at a way that allows us to view any interest through either the lens of simple interest or the lens of compound interest, before then comparing present and future values of money.