## TVOM Concept and Components

### Introduction

John is an acquaintance of yours. He is a marketing major and has made the dean's list every semester. John decides to start up a small consulting company during his senior year. He has approached you and asked you to lend him \$10,000, which he will pay back after three years.

• If you decide to keep the money, you can use it to pay your bills, take a vacation, add to your savings, etc.
• If you decide to lend him the money, you will have to go without the things that this \$10,000 can buy.

After some careful consideration, you decide to lend him the money. How much should you expect back after three years? The same amount, or more or less than \$10,000? Why?

In this example, you should expect to get more than \$10,000 back from your acquaintance, John.

Why should you (as well as others) expect to get back more than the amount that you lend i.e., \$10,000? Because if you do not lend the money, you can use it to do other things. By lending, you are giving up using it for the next three years, and hence you require some returns to compensate for what you will give up. This is a tradeoff between using money today and saving for future use, and hence time has value.

The underlying basis of the Time Value of Money (TVOM) is that time has value. That is, a dollar today is worth more than a dollar tomorrow.

### Objectives

Upon completing this section of the TVOM tutorial, you will be able to:

• Describe the concept of Time Value of Money (TVOM).
• Correctly label a time line.
• Explain the difference between present value and future value.
• Explain the difference between compound and simple interest rate.

### Time Line

The time line is a very useful tool for an analysis of the time value of money because it provides a visual for setting up the problem. It is simply a straight line that shows cash flow, its timing, and interest rate.

A time line consists of the following components:

• Time period
• Interest rate
• Cash flow

Time period (t) can be any time interval such as year, half a year, and month. Each period should have equal time interval. Zero represents a starting point, and a tick represents the end of one period. From the example at the beginning of this section, there are three years or annual periods and hence the time line has three ticks. Each tick represents a period of one year.

Interest rate (r) is the rate earned or paid on cash flow per period. It is labeled above the time line.

If a 10% return per year is required, the time line should be:

Cash flow (CF) is amount of money. It is placed directly below a tick at time period that it occurs. Cash flow can be known or unknown amount.

• Cash outflow, an amount that you pay, cost to you, or spending amount, has a negative sign.
• Cash inflow, an amount that you receive or savings amount, has a positive sign.

Cash flow, like interest rate, can be a known or an unknown amount. For both, the unknown value that you try to solve for (e.g., cash flow or interest rate) is indicated by a question mark.

From our example, if John has also told you that he will return \$13,000 at the end of three years, and you want to calculate the return on this investment, the time line should be:

An important note about cash flows: When a problem has only one type of cash flow (i.e., either inflows or outflows), the sign of cash flows can be ignored.

For example, over the past three years, you deposited \$100, \$200, and \$300 in a bank account that earned 5%. How much do you have today? The signs of \$100, \$200, and \$300 can be ignored because they are the same type of cash flows.

The timeline for this example is as follows:

Note: The unknown amount for this example will be later known as future value (FV).

There are times, however when a problem has both cash inflows and outflows. In this case, signs of cash flows are very important. For example, three years ago you deposited \$100. Then, you withdrew \$50 one year after that. Last year, you deposited \$200. How much do you have today if you earn 5%? In this case, signs of cash flows can not be ignored. The signs of \$100 and \$200 must be the same, and different from the sign of \$50 because \$100 and \$200 are deposits while \$50 is a withdrawal.

If the deposits are considered cash outflows and therefore have a negative sign, the withdrawal must be positive. The time line is as follows:

If the deposits are considered savings and hence have a positive sign, the withdrawal must be negative. The time line is as follows:

For this tutorial, the sign of cash flows will be ignored for problems with only one type of cash flow.

Practice: Using the time line below, indicate the values for A, B and C for each of the three problems.

1. The bank lends you \$20,000 and requires a 10% return. To assist you in calculating the amount of money you would pay back, you would label parts A, B, and C of the time line as:
1. ____________________
2. ____________________
3. ____________________
2. Your brother borrows \$100 and has also told you that he will return \$120 at the end of three years. To assist you in calculating the return on this investment, you would label parts A, B, and C of the time line as:
1. ____________________
2. ____________________
3. ____________________
3. Based on a 10% return, your roommate determines she will need \$500 at the end of three years to pay back a loan. To assist you in calculating the amount of money originally borrowed by your roommate, you would label parts A, B, and C of the timeline as:
1. ____________________
2. ____________________
3. ____________________

1. If a bank lends you \$20,000 and requires a 10% return. To assist you in calculating the amount of money you would pay back, you would label parts A, B, and C of the time line as:
1. r = 10%
2. +\$20,000
3. ?
2. Your brother borrows \$100 and has also told you that he will return \$120 at the end of 2 years. To assist you in calculating the return on this investment, you would label parts A, B, and C of the time line as:
1. r=?
2. -\$100
3. +\$120
3. Based on a 10% return, your roommate determines she will need \$500 at the end of three years to pay back a loan. To assist you in calculating the amount of money originally borrowed by your roommate, you would label parts A, B, and C of the timeline as:
1. r=10%
2. ?
3. +\$500

### Present v. Future Value

What is the difference between future value and present value?

• Present value (PV) is the current value of future cash flow.
• Future value (FV) is the value of cash flow after a specified period.

A simple way to classify whether cash flows are present value or future value is to remember that:

• PV is the value at the beginning of a time period that you are considering.
• FV is the value at the end of the time period that you are considering.

Recall the example from the beginning of this section:

John is an acquaintance of yours. He is a business major and has made the dean's list every semester. John wants to start up a small consulting company during his senior year. He has approached you and asked you to lend him \$10,000, which he will pay back after three years.

For the three-year period that you consider lending, \$10,000 is the value at the beginning and hence called PV, and the amount that you will get back from John (i.e., \$13,000) is the value at the end and hence called FV.

Practice: Now try answering the following two problems, and then check your answers on the next screen. Be sure to draw a timeline to assist you in finding the answer:

1. You lent \$10,000 to John in 1997 and he returned \$13,000 to you in 2000, which amount should be called PV and FV?

2. You lend John \$10,000 in 2006, and get \$13,000 back three years after that, in 2009. Is \$10,000 PV or FV?

1. You lent \$10,000 to John in 1997 and he returned \$13,000 to you in 2000, which amount should be called PV and FV?

Answer: Although both \$10,000 and \$13,000 are cash flows that occurred in the past, \$10,000 is still called PV, and \$13,000 is called FV. This is because \$10,000 is the cash flow at the beginning, and \$13,000 is the cash flow at the end of the time period being considered.

2. You lend John \$10,000 in 2006, and get \$13,000 back three years after that, in 2009. Is \$10,000 PV or FV?

Answer: Again, \$10,000 is still called PV and \$13,000 is called FV although \$13,000 is cash flow in the future. The same logic applies, \$10,000 is value at the beginning and \$13,000 is value at the end.

Remember: Present value is not necessarily today's value, and future value is not necessarily a value in the future.

An emphasis here is that for any problem, especially a complex one, there might be more than one time period that you have to consider separately.

Therefore a time period that you are considering might not be the same as the (entire) time period of the problem.

For example, three years ago, you saved \$1,000 that you earned from your summer job in a bank account with 5% interest rate. Now you're interested of using the money. You want to split the money into four equal amounts withdrawn in the next four years. How much can you withdraw per year?

For this example, let's create the timeline:

• There are 7 annual periods from the time you placed your \$1,000 in the bank to the last withdrawal. Hence the time line should have seven "ticks" after the zero.
• The interest rate is 5%, so r=5%.
• The cash outflow or the amount you contribute, is -\$1,000.
• The cash inflow or the four withdrawal amounts are unknown.

In order to solve for the four equal cash flows (withdrawals), first you need to find out how much you have today (X) or calculate FV of \$1,000.

Today's value (X) of \$1000 is called FV because:

• You are considering the time period from t=0 to t=3.
• The value of \$1000 is at the beginning of this time period (rather than the time period of the entire problem) and hence is called PV.
• The X amount at t=3 is the value at the end of the time period and hence called FV.

This is how it is depicted graphically:

After calculating X, you can solve for the four equal withdraws (?). Now, X is called PV because you're considering the time period from t=3 to t=7 and X is at the beginning of the time period. Graphically,

As mentioned earlier, the same cash flow (X) can be called either PV or FV for the same problem, depending on whether it is at the beginning or at the end of time period that you consider.

Practice Question: Your Uncle Lee plans to retire next year on his 63rd birthday. He is curious how much he can spend each year after retirement. Since he was 25 years old, Uncle Lee has saved \$30,000 per year in an account that earns 5% interest rate. His life expectancy is 90.

1. Calculate the amount of savings that Uncle Lee will have accumulated at age 63.
2. Calculate the value of all withdrawals that Uncle Lee will make if he lives to 90.

Based on this information, the time line below can be created:

In order to solve the above problems, should you solve for PV or FV?

Let's consider the first problem, calculate savings amount that Uncle Lee has accumulated until Age 63.

To calculate the savings amount at Age 63, you should consider the time period between ages 25-63 as follows:

Because 63 is at the end of the time period, you need to determine the FV to calculate savings amount of 30,000 annual deposits.

Now let's look at the second problem, calculate the value at Age 63 of all withdrawals that Uncle Lee will make.

To consider value at Age 63 of all withdrawals, you should consider the time period between ages 63-90 as follows:

Because the value of withdrawals at Age 63 is at the beginning of the time period, you need to determine the PV of the "C" amount in order to calculate the value of withdrawals.

### Simple v. Compound

What is compound interest rate? How is it different from simple interest rate?

• For a simple interest rate, interest is earned on the original principal only.
• For a compound interest rate, interest is earned on both the original principal and interests reinvested from prior periods.

The difference is the amount of interest that is earned on the reinvested interests.

For example, you invest \$100 for 3 years in an investment company that provides a fixed 10% interest rate. What is your payback at the end of 3 years?

If the interest rate is a simple rate, the payback for:

• The first year (X1) is \$100 of original principal + \$10 of interest:
X1 = \$100 + (10% × \$100) = \$110.
• The payback for the second year (X2) is \$100 of the principal + \$10 of interest from the first year + \$10 of interest from the second year:
X2 = \$100 + (10% × \$100) + (10% × \$100) = \$120.
• The payback for the third year (X3) is \$100 of the principal + \$10 of interest from the first year + \$10 of interest from the second year + \$10 of interest from the third year:
X3 = \$100 + (10% × \$100) + (10% × \$100) + (10% × \$100) = \$130.

If the interest rate is an annual compound rate (i.e., computed once a year), the payback for:

• The first year (X1) is \$100 of original principal + \$10 of interest:
X1 = \$100 + (10% × \$100) = \$110
• The payback for the second year (X2) is, \$100 of the principal + \$10 of interest from the first year + \$11 of interest from the second year:
X2 = \$100 + (10% × \$100) + (10% × \$110) = \$121
• The payback for the third year (X3) is \$100 of the principal + \$10 of interest from the first year + \$11 of interest from the second year + \$12.1 of interest from the third year:
X3 = \$100 + (10% × \$100) + (10% × \$110) + (10% × 121) = \$133.1

Note that the payback for the first year is the same for both simple and compound rate. However, the paybacks after the first year are higher for the compound rate than for the simple rate because interest is computed on the prior year's principal, not the original principal. The longer the time period is, the larger the difference. TVOM assumes compound interest rate.

Practice Question:  Josh, your close friend, borrowed \$500 from you three years ago. He has promised to return the money to you today with 6% interest rate per year. If Josh returns \$595 and some change to you, is the interest rate a simple or compound rate?

For a simple interest rate of 6%, the amount of interest per year would be:

X1 = \$500 + (.06 × \$500) = \$530
X2 = \$500 + (.06 × \$500) + (.06 × \$500) = \$560
X3 = \$500 + (.06 × \$500) + (.06 × \$500) + (.06 × \$500) = \$590

As such, you should get back \$590.

For a compound interest rate of 6%, the amount of interest per year would be:

X1 = \$500 + (.06 × \$500) = \$530
X2 = \$500 + (.06 × \$500) + (.06 × \$530) = 561.80
X3 = \$500 + (.06 × \$500) + (.06 × \$530) + (.06 × \$561.80) = \$595.51

Since Josh returns \$595.51, \$500 of principal and \$95.51 of interest, the interest rate is a compound rate. Under a compound rate, interest amount is greater than under a simple rate because interest is earned on the prior year's interest.