TVOM Concept and Components

Introduction

Smiling man at A T M with cash.

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.

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:


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.

Timeline labeled from zero through three.

A time line consists of the following components:

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.

Timeline; starting when you loan money at zero, end of first period at one, end of second period at two and end of third period at three.

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:

Timeline labeled from zero through three, with r equals 10%.

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 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:

Same time line with -$10000 Cash out flow, +$13000 cash in flow, and R equals question mark.

Remember: Time line can help you simplify a complex problem.


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:

Time line labeled with r equals 5%, 100 at zero years, 200 at one, 300 at two and both F V equals question mark and today at three.

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:

Time line labeled with r equals 5%, negative 100 at zero years, 50 at one, negative 200 at two and both F V equals question mark and today at three.

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

Time line labeled with r equals 5%, 100 at zero years, negative 50 at one, 200 at two and both F V equals question mark and today at three.

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. ____________________

Answers to Practice Problems:

  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?

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

Time line labeled with P V and F V.

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.

Timeline with -$10000 P V at zero and $13000 F V at three.

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?

Answers to Practice Problems:

  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.

    Time line with dates and money labeled.
  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.

    Time line with dates and money labeled.

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:

Timeline with -$1000 at zero, today on three, and question marks at 4,5,6,7.

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:

This is how it is depicted graphically:

Timeline with -$1000 at zero, today and X on three.

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,

Timeline with today and X on three, and question marks at 4,5,6,7.

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.


Casually attired older gentleman.

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:

Timeline with -$1000 at zero, today and X on three.

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:


Timeline with -$1000 at zero, today and X on three.

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:


Timeline with -$1000 at zero, today and X on three.

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?

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?

Timeline labeled with R equals 10%, $100 at zero, and a question mark at three.

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

Timeline labeled with $100 at zero$110 at one, $120 at two, and $130 at three.

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

Timeline labeled with $100 at zero$110 at one, $121 at two, and $133.10 at three.

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.