Time Value of Money (TVM) Basics

Let us start with the following example. Your friend is considering putting money in a bank account that will pay 4% interest per year and is particularly interested in knowing how much money they will have one year from now if they deposit $1,000 in this account. Your friend understands that you are studying finance and turns to you for help. By using the TVM principle of future value (FV), you can tell your friend that the answer is $1,040. The additional $40 that will be in the account after one year will be due to interest earned over that time. You can calculate this amount relatively easily by taking the original deposit (also referred to as the principal) of $1,000 and multiplying it by the annual interest rate of 4% for one period (in this case, one year).

Interest Earned = $1,000 × 0.04 = $40.00

By taking the interest earned amount of $40 and adding it to the original principal of $1,000, you will arrive at a total value of $1,040 in the bank account at the end of the year. So, the $1,040 one year from today is equal to $1,000 today when working with a 4% earning rate. Therefore, based on the concept of TVM, we can say that $1,040 represents the future value of $1,000 one year from today and at a 4% rate of interest. We will discuss interest rates and their importance in TVM decisions in more detail later in this chapter; for now, we can consider interest rate as a percentage of the principal amount that is earned by the original lender of funds and/or charged to the borrower of these same funds. Following are a few more examples of the single-period scenario.

If a person deposits $300 in an account that pays 5% per year, at the end of one year, they will have

FV = $300 + ($300 × 0.05) = $315

If a company has earnings of $2.50 per share and experiences a 10% increase in the following year, the earnings per share in year two are

$2.50 + $2.50 × 0.10=$2.75 per share

If a retail store decides on a 3% price increase for the following year on an item that is currently selling for $50, the new price in the following year will be

$50 + $50 × 0.03 = $51.50

What would happen if your friend were willing to wait one more year to receive their lump sum payment? What would the future dollar value in their account be after a two-year period? Returning to our earlier example, assume that during the second year, your friend leaves the principal ($1,000) and the earned interest ($40) in the account, thereby reinvesting the entire account balance for another year. The quoted interest rate of 4% reflects the interest the account would earn each year, not over the entire two-year savings period. So, during the second year of savings, the $1,000 deposit and the $40 interest earned during the first year would both earn 4%:

$1,000 × 0.04 + 40 × 0.04=$41.60

The additional $1.60 is interest on the first year’s interest and reflects the compounding of interest. Compound interest is the term we use to refer to interest income earned in subsequent periods that is based on interest income earned in prior periods. To put it simply, compound interest refers to interest that is earned on interest. Here, it refers to the $1.60 of interest earned in the second year on the $40.00 of interest earned in the first year. Therefore, at the end of two years, the account would have a total value of $1,081.60. This consists of the original principal of $1,000 plus the $40.00 interest income earned in year one and the $41.60 interest income earned in year two.

The amount of money your friend would have in the account at the end of two years, $1,081.60, is referred to as the future value of the original $1,000 amount deposited today in an account that will earn 4% interest every year.

Simple interest applies to year 1 while compound interest or “interest on interest” applies to year 2. This is calculated using the following method:

Year 1:1,000 × 0.04 = 40.00
Year 2:1,040 × 0.04 = 41.60

So, the total amount that would be in the account after two years, at 4% annual interest, would be $1,000 + $40.00 + $41.60 = $1,081.60$1,000 + $40.00 + $41.60 = $1,081.60.

To determine any future value of money in an interest-bearing account, we multiply the principal amount by 1 plus the interest rate for each year the money remains in the account. From this, we can develop the future value formula:

Future Value=Original Deposit × (1 + r) × (1 + r)

In this formula, the number of times we multiply by (1+r)(1+r) depends entirely on the number of years the money will remain in the bank account, earning interest, before it is withdrawn in a final lump sum distribution paid out from the account at the end of the chosen savings period. The 1 in the formula represents the principal amount, or the original $1,000 deposit, which will be included in the final total lump sum payment when the account is closed and all money is withdrawn at the end of the predetermined savings period.

We can write the above equation in a more condensed mathematical form using time value of money notation, as follows:

FV=Future Value
PV=Present Value
r=Interest Rate
n=Number of Periods

Using these inputs, we have the following formula:

FV = PV × (1 + r)ⁿ

With this equation, we can calculate the value of the savings account after any number of years. For example, suppose we are considering 3, 10, and 50 years from the original deposit date at the annual 4% interest rate:

3 years:FV = $1,000 × (1.04)³ = $1,000 × 1.12486 = $1,124.86
10 years:FV = $1,000 × (1.04)¹⁰ = $1,000 × 1.48024 = $1,480.24
50 years:FV = $1,000 × (1.04)⁵⁰ = $1,000 × 7.106683 = $7,106.68

How can this savings account have grown to be so large after 50 years? This question is answered by the impact of compounding interest. Every year, the interest earned in previous years will also earn interest along with the initial deposit. This will have the effect of accelerating the growth of the total dollar value of the account.

This is the important effect of the compounding of interest: money grows in larger and larger increments the longer you leave it in an interest-bearing account. In effect, the compounding of interest over time accelerates the growth of money.

In order to determine the FV of any amount of money, it will always be necessary to know the following pieces of information: (1) the principal, initial deposit, or present value (PV); (2) the rate of interest, usually expressed on an annual basis as r; and (3) the number of time periods that the money will remain in the account (n). The interest rate is often referred to as the growth rate, or the annual percentage increase on savings or on an investment. When the rate is raised to the power of the number of periods, the formula (1+r)ⁿ will yield a number that is commonly referred to as the future value interest factor (FVIF). As a result of this process, as n (time, or the number of periods) increases, the future value interest factor will increase. Also, as r (interest rate) increases, the FVIF will increases. For these reasons, the future value calculation is directly determined by both the interest rate being used and the total amount of time—specifically, the number of periods—being considered.

Calculating Future Values

Here’s another example of calculating future values in multiple-period scenarios.

On a recent drive, you spotted your dream home, which is currently listed at $400,000. Unfortunately, you are not in a position to buy it right away and will have to wait at least another six years before you can afford it. If house values are appreciating at an annual rate of inflation of 4%, how much will a similar house cost after six years?

Solution:

In this case, PV is the current cost of the house, or $400,000; n is six years; r is the average annual inflation rate, or 4%; and we have to solve for FV.

FV = $400,000 × 1.046= $400,000 × (1.265319)=$506,127.61

Therefore, under these circumstances, the house will cost $506,127.60 after six years.

We have just seen that time will lead to the growth of our money. As long as the prevailing growth or interest rate of any account we have our money in is positive, the passage of time will have the effect of growing the value of our money. The longer the period of time, the greater the growth and the larger the future value of the money will be. This can be reinforced very clearly with the following example.

Melvin is saving money in an account at a local bank that earns 5% per year. He begins with a deposit in his account of $100 and decides to save his money for exactly one year. He will not be making any further deposits into the account during the year. Melvin will earn5% × $100, or $5, in interest income. Adding this to the original deposit balance of $100 will give him a total of $100 + $5, or $105, in the account at the end of one year.

Melvin likes this idea and believes he may be able to keep his money in the account for a longer period of time. How much money will he have in his account, without any further deposits, at the end of years two, three, four, and five?

Using the future value formula, the calculation is as follows:

FV = PV × (1+r)ⁿ
Year 2: FV = $100 × (1+0.05)²=$110.25 
Year 3: FV = $100 × (1+0.05)³=$115.76 
Year 4: FV = $100 × (1+0.05)⁴=$121.55 
Year 5: FV = $100 × (1+0.05)⁵=$127.63

Melvin likes the idea of earning more money over time, but he also believes that what he would earn in interest may not be enough for some of the things he plans to buy in the future. His friend suggests finding an account or some form of investment with a greater interest rate than the 5% he can get at his local bank.

Melvin thinks he can leave his money in an account or investment for a total of five years. He found investments that will provide annual returns of 6%, 7%, 10%, and 12%. Using the FV = PV × (1+r)ⁿ formula, we can complete the following calculations for him:

5%: FV = $100 × (1+0.05)⁵=$127.63 
6%: FV = $100 × (1+0.06)⁵=$133.82 
7%: FV = $100 × (1+0.07)⁵=$140.26 
10%: FV = $100 × (1+0.10)⁵=$161.05 
12%: FV = $100 × (1+0.12)⁵=$176.23

Again, Melvin likes this information, and he states that he will try to find the highest interest rate available. This makes sense, but it’s important to remember that investments are usually not guaranteed to earn you specific interest rates, or rates of return. Most investments, other than Treasury investments such as Treasury bonds, carry some form of financial risk, either small or large, and the greater the rate of return, the more likely it is that the risk associated with the investment will also be greater. This risk does not have any effect on the future calculations we have just completed, but it an important factor to bear in mind and consider well before moving ahead and putting your money in any investment or financial instrument.