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Skip to content. Time Value of Money (TVOM) Tutorial (Print-friendly version. Return to Tutorial ) 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.

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Skip to content.Time Value of Money (TVOM) Tutorial

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TVOM Concept and ComponentsIntroduction

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

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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.

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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.

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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:

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:

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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:

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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:A. ____________________B. ____________________C. ____________________

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:

A. ____________________B. ____________________C. ____________________

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:

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A. ____________________B. ____________________C. ____________________

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:

A. r = 10%B. +$20,000C. ?

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:A. r=?B. -$100C. +$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:A. r=10%B. ?C. +$500

Present v. Future Value

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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.

 

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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. 

 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. 

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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.

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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,

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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:

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 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:

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 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?

 

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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

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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:

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

The Pennsylvania State University ©2006. All rights reserved.

Skip to content.The ALT text used on graphical equations is similar to, but not exactly like Math Speak; a few conventions are borrowed. Exponents are read as supe followed by the exponent then base to signify terms representing the exponent have ended. Subscrpts are read as sub followed be the subscript, then base. Parenthesis are read in line as they occur: left parens opens the parenthetical and right parens closes it. Square brackets are read as left bracket then right bracket. Fractions are always read as numerator followed by denominator. One half would be numerator one denominator two.Time Value of Money (TVOM) Tutorial

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Different Types of TVOM

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Introduction

Can you identify the type of TVOM for each scenario?Josh and Sarah, both sophomores, want to travel abroad during their senior year of college. They have combined the money they saved from their summer jobs, and will let it collect interest for the next two years. How much money will Josh and Sarah have for their trip?

Taisha and Darrell are planning to get married following graduation this semester. They deposited the extra money that they earned from their part times jobs into a savings account over the past three years. The first year they deposited $900, the second, $1300 and the third, $1,200. How much money do they have for their wedding?

Sections 2-A and 2-B cover the major types of TVOM, which include:

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1. lump sum2. multiple cash flows3. annuity4. annuity due5. perpetuity6. growing perpetuity7. growing annuity

Objectives

Upon completing sections 2-A and 2-B of the tutorial, you will be able to:

Identify the seven types of TVOM. Explain how they are different from one another.

Once you are able to distinguish one type of TVOM from another, you will be able to simplify the problem and choose the correct equation.

Lump Sum

The lump sum has the following characteristics: There are 2 cash flows on the time line:

o Present value (PV) at the beginning of the time line

o Future value (FV) at the end of the time line

There is no cash flow in between PV and FV.

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The equation for lump sum is:FV = PV(1+r)t or

Where: FV = future value of lump sum PV = present value of lump sum r = interest rate per period t = number of compounding periods

For example, you earn $500 from your summer job and want to save for European trip in the next three years. How much will you have when you go for the trip if you deposit the money in a savings account that earns 10% interest? Using the time line for this problem, complete the equation:

FV = PV(1+r)t  FV = ? PV = 500 r = .10 t = 3

FV = 500(1+.10)3

Note: Interest is computed three times and therefore t equals 3.

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Multiple Cash Flows

For this type of TVOM, there are many cash flows over the time line. Cash flows can be all the same or all different. Cash flows are not necessarily consecutive.

To solve for the Future Value (FV) of multiple cash flows, simply treat each cash flow as a lump sum and then add them up: FV = FV of C1 + FV of C2 + FV of C3 FV = C1 (1 + r)2 + C2 (1 + r)1 + C3 (1 + r)0

Similarly, to solve for the Present Value (PV) of multiple cash flows: PV = PV of C1 + PV of C2 + PV of C3

 For example, over the next three years, you expect to earn the following amounts of money: $100, $200 and $150. If you save these in a saving account that earns 5%, how much will you have at the end of year 3? The time line is as follow:

For this example, calculate the value at the end of the time line or FV of $100, $200 and $150. (Hint:  Use the lump sum equation, FV = PV(1+r)t, to solve for each cash flow and then add them together.)

FV = FV of 100 + FV of 200 + FV of 150 FV =100(1 + 0.05)2 + 200(1 + 0.05)1 + 150

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Note: The interest is computed twice for $100 and therefore t for $100 is 2. The interest is computed once for $200 and therefore t for $200 is 1. There is no interest earned on the last $150.

Annuity

Annuity is a special case of multiple cash flows where: The cash flows are equal for a fixed period of

time. The cash flows are at the end of each period.

The equal amount of cash flows is called annuity payment or payment (C).

You can solve an annuity problem the same way as multiple cash flows, calculating the value of each cash flow and sum all values. However, this can be quite tedious especially when dealing with long series of annuity payments. Fortunately, future and present values of annuity payments can be calculated from the following equations:

    or     Where:

FVA = future value of annuity PVA = present value of annuity

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C = amount of equal payments r = interest rate per period t = number of payments (number of time

periods)

For example, you plan to save $200 per year for the next five years for your new car's down payment. How much will you have as the down payment if you earn 10% interest rate on the savings? Using the time line below, we can fill in the missing information of the equation to solve for FVA:

 

FVA = ? C = 200 r = .10 t = 5

 In order to call a series of cash flows an annuity and use the PVA and FVA equations, the following must be true:

1. All payments must be equal and consecutive.2. Payments last for a certain period.3. The first payment starts one period from the

beginning of the time line (time period for PV).

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4. The last payment is at the end of time line (time period for FV).A quick way to check whether a series of payments is annuity is that the number of payments must be the same as the number of time periods. In the previous example, there are 5 payments (C) which are equal to 5 periods (years). Therefore, the problem can be called annuity.The series of payments displayed in the time line below, however, cannot be called annuity because the number of payments (C) is not the same as the number of time periods (t). Thus the PVA and FVA equations cannot be applied.

 Note: There are three payments (c) and five time periods (t).

Annuity Due

Annuity due is similar to annuity in that there is a series of equal and consecutive payments that last for a certain period, but the payments start at the beginning of each time period and the last payment stops one period before the end of the specified time period.

To determine whether a series of payments are classified as annuity due is the similar to the way you

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would determine if they were an annuity: Check to see if the number of payments is the same as the number of time periods.

Let's Compare the Annuity and Annuity Due Time lines:Annuity Timeline:

Annuity Due Timeline:

Note that the annuity and annuity due time lines are similar because:

Payments are equal and consecutive. Payments last for a certain period of time.

The annuity due time line, however, is different from the annuity time line, because:

The first payment starts right away. The last payment stops one period before the

end of time line. PV is the value at the same time period as the

first payment. FV is the value one period after the last

payment.

As discussed, for both annuity and annuity due, there is the same number of payments. But all payments of

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annuity due earn one more period of interest, and hence:The future value of annuity due equals to future value of annuity multiplied by (1+r).

The present value of annuity due equals to present value of annuity multiplied by (1+r).

===where…

FVA(Due) = future value of annuity due PVA(Due) = present value of annuity due C = amount of equal payments r = interest rate per period t = number of payments (number of time

periods)

For example, you plan to save for your new car's down payment that you will need five years from now. Like in the annuity example, there will be five deposits of $200 per year. However, you will start the first deposit right away. How much will you have as the down payment for your car if you earn 10% interest rate on the saving? Using the time line below, fill in the missing information to solve for FVA (due):

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FVA(Due) = ? C = 200 r = .10 t = 5

 

Perpetuity

Perpetuity is similar to annuity. The only difference between annuity and perpetuity is the ending period. For annuity, payments last for a certain period, whereas for perpetuity, they continue indefinitely, as represented by (∞).

The equation below is used to calculate present value of perpetuity. It requires only the first payment and interest rate.

where…

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PV(∞) = present value of perpetuity. C = the first payment r = interest rate per period

For example, an insurance company has just launched a security that will pay $150 indefinitely, starting the first payment next year. How much should this security be worth today if the appropriate return is 10%? Using the time line below, complete the PV(∞) equation.

The equation below is used to calculate present value of perpetuity. It requires only the first payment and interest rate.

PV(∞) = ? C = 150 r = .10

 

Growing Perpetuity

With a growing perpetuity, there is a series of consecutive payments that continue indefinitely, and each payment grows at a constant rate.

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The equation below is used to calculate growing perpetuity:

where… PVG(∞) = present value of growing perpetuity. C1 = the first payment r = interest rate per period, and g = a constant growth rate.

Note that C1 is the value of the first payment (not the value of payment at t=0), and r must not be equal to g.

For example, a company is expected to pay $2 of dividend per share that will increase 5% forever. If investors require 10% return on the company's stocks, how much should investors pay for the stocks? The cash flows are as follow:

The equation below is used to calculate growing perpetuity:

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where… PVG(∞) = ? C1 = $2 r = .10 g = .05

 

Growing Annuity

A growing annuity is the same as annuity in that payments for both end at a certain period. However, payments of growing annuity increase at a constant rate while payments of annuity are fixed.Growing annuity is also similar to growing perpetuity; payments of both increase at a constant rate. Unlike growing perpetuity, payments of growing annuity end at some point.

 A growing annuity problem can be treated as multiple cash flows because all cash flows are not equal. However, it is very tedious to calculate present values of many cash flows separately.Fortunately, the following equation can be used to calculate present value of growing annuity.

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 where:

PVGA = present value of growing annuity C1 = the first payment, r = interest rate per period, and g = a constant growth rate.

Note: The PVGA equation requires the first payment or C1 for the present value at time 0.

For example, an investment company just issued a security which will provide 10 payments, starting next year for $100 and increasing 5% per year after that. How much is this security worth if the appropriate required return is 10%? The cash flows for the next ten years are as follows:

 

 where:

PVGA = present value of growing annuity C1 = $100 r = .10 g = .05

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Practice

To solve each problem follow these steps:1. Create a time line2. Identify type of cash flows and3. Identify the appropriate equation needed to solve a

problem.

Note: The solution to each problem is given.

Practice Question 1:

Tyler won a lottery. The commission asked him to choose between $10,000 today and $20,000 three years from today. Which option should Tyler take if his investment opportunity is 10% annually compounding?Solution:

$10,000 is today's value while $20,000 is the value three years from today. In order to choose between these two options, you need to convert $20,000 to be today's value so that it can be compared to $10,000.Step 1- Create a Timeline:

Step 2- Identify the type of cash flow: Since $20,000 is only one cash flow amount on the time line, the type of cash flow is lump sum.

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Step 3- Select the appropriate equation: We can also see that the present value of $20,000 is needed. The equation for the problem is present value of lump sum:

 Step 4- Enter the variables in the equation and solve: FV = $20,000 (value at the end); r = 10% (investment opportunity); t = 3 (compounding periods)

 Practice Question 2:

Rhon started his small business five years ago. His business generated $300, $500, $200, $400 and -$200 of cash flows over the past five years. How much money has Rhon accumulated from his business as of today? Assume that he has earned 8% annually compounding and never spent the money.Solution:

Step 1- Create a Timeline:

Step 2- Identify the type of cash flow: The cash flows are not the same and hence the series of cash flows is multiple cash flows.Step 3- Select the appropriate equation: The value at the end of time period or future value is needed. To calculate future value of multiple cash flows, simply calculate future value of each lump sum

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and add all of them together. The future value of lump sum equation is FV = PV(1+r)t, so multiple cash flows can be represented as: FV = C1 (1 + r)4 + C2 (1 + r)3 + C3 (1 + r)2 + C4 (1 + r)1 + C5 (1 + r)0

Step 4- Enter the variables in the equation and solve: C1 = 300, C2 = 500, C3 = 200, C4 = 400, C5 = -200, plus r = 8% (interest rate); t and PV vary for each cash flow. FV = 300 (1.08)4 + 500 (1.08)3 + 200 (1.08)2 + 400 (1.08)1 + (-200)

Practice Question 3:

Nikki just had a new born son. She wants to set aside some money for her son's college expenses in 16 years. If the tuition's total cost is $100,000 when he turns 16, how much does she have to save per year? Assume she earns 5% annually compounding.Solution:

Step 1- Create a Timeline:

Step 2- Identify the type of cash flow: The series of equal annual payments is annuity.Step 3- Select the appropriate equation: $100,000 is cash flow at the end of time period or future value. Therefore, the equation needed for the problem is future value of annuity.

 

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Step 4- Enter the variables in the equation and solve: FV = 100,000 (tuition cost); r = 5% (interest earned); t = 16 (number of payments)

 Practice Question 4:

Yoma bought a $30,000 car for his wife. If he finances the car with a bank that charges 6% monthly compounding, how much does Yoma have to pay per month over the next 5 years, starting the first payment now?Solution:

Step 1- Create a Timeline:

Step 2- Identify the type of cash flow: The series of equal payments with the first payment starts right away is annuity due.Step 3- Select the appropriate equation: $30,000 is the value at the beginning of the time period or present value. The equation for the problem is present value of annuity due.

 Step 4- Enter the variables in the equation and solve: PV = $30,000 (car cost); r = 6%/12 (financing cost); t = 60 (number of payments)

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 Practice Question 5:

PA State issues a security that pays $100 per year indefinitely. If the appropriate required return for this security is 10%, what should be the price of the security?Solution:

Step 1- Create a Timeline:

Step 2- Identify the type of cash flow: The series of equal amount of payments that continue indefinitely is perpetuity.Step 3- Select the appropriate equation: The price of security is present value. The equation for the problem is present value perpetuity.

 Step 4- Enter the variables in the equation and solve: C = 100 (equal amount of cash flows); r = 10% (required return)

 Practice Question 6:

Nicole wants to buy a stock of a company. Several analysts expect that the company will pay $2 dividend

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next year, and increase 5% per year after that. If Nicole requires 10% return, how much should she pay for the stock? Assume that dividend will continue forever.Solution:

Step 1- Create a Timeline:

Step 2- Identify the type of cash flow: The amount of dividends increases at a constant rate of 5% and continue indefinitely. This type of cash flows is called growing perpetuity.Step 3- Select the appropriate equation: The price of security is present value. The equation for the problem is present value growing perpetuity.

 Step 4- Enter the variables in the equation and solve: C1 = 2 (next year's dividend amount); r = 10% (required return); g = 5% (constant growth rate)

 Practice Question 7:

Roman works as a sports agent. He is negotiating a 10-year contract for one of his clients, a NBA rookie. The first annual pay will be $1 million and every pay will increase at 5% per year. His client wants a lump sum payment today instead of installments. What is the minimum lump sum amount that Roman should

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ask for his client if his client's investment opportunity is 10% annually compounding?Solution:

Step 1- Create a Timeline:

Step 2- Identify the type of cash flow: The series of cash flows is growing annuity because the cash flows increase at a constant rate, and last for a certain period (i.e., 10 years).Step 3- Select the appropriate equation: Since the rookie wants a lump sum payment today, present value is needed. The equation for the problem is present value growing annuity.

 Step 4- Enter the variables in the equation and solve: C1 = $1 million (the first annual pay); r = 10% (investment opportunity); g = 5% (growth rate or rate of pay increase); t = 10 (number of pays)

 

The Pennsylvania State University ©2006. All rights reserved.

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Skip to content.The ALT text used on graphical equations is similar to, but not exactly like Math Speak; a few conventions are borrowed. Exponents are read as supe followed by the exponent then base to signify terms representing the exponent have ended. Subscrpts are read as sub followed be the subscript, then base. Parenthesis are read in line as they occur: left parens opens the parenthetical and right parens closes it. Square brackets are read as left bracket then right bracket. Fractions are always read as numerator followed by denominator. One half would be numerator one denominator two.Time Value of Money (TVOM) Tutorial

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Frequency of CompoundingIntroduction

Your grandmother sends you $100 for your birthday, and you decide to invest it.You visit three different borrowers and each offers you the same interest rate. You discover, however, that one borrower computes interest monthly, the other semiannually and the third annually.

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Which borrower should you pick? Which one will result in more money for you?

As mentioned earlier, TVOM assumes a compound interest rate. This section highlights:

The effects of frequency of compounding Differences between quoted rate, EAR and APR

Objectives

Upon completing this section you will be able to: Calculate the amount of interest earned from

borrowers with annual, semi annual and monthly compounding.

Calculate periodic rate. Define quoted rate. Calculate EAR. Define APR.

Frequency of Compounding

Frequency of compounding concerns the number of times that interest is computed per year. It can best be explained through an example.Assume Maria wants to invest $100 that she received from her grandmother for her birthday. She finds three borrowers that will pay her the same 12% interest rate. However, the interest for each borrower will be computed (and credited to Maria's account) differently as follows:

Borrower#1 computes interest once a year (called annual compounding).

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Borrower#2 computes interest twice a year (called semiannual compounding).

Borrower#3 computes interest twelve times a year (called monthly compounding).

What is the amount of interest that Maria earns from each borrower for one year?To answer this question, you need to calculate future value at the end of one year for each borrower and compare it with the investment of $100.

First, let's look at the time line and interest computation for Borrower#1 (annual compounding):

From Section 1 of this tutorial, you know the formula for calculating a compound interest rate for the first year, X1, is:X1 = X1+(X1×r)Using this same equation, but substituting FV1 for X1:FV1 = 100+(100×.12) = 100(1.12) = $112Since this timeline has only two values, PV and FV, you should also recognize it as a lump sum case. Thus you could also use the lump sum formula from Section 2-A to solve for FV1, which will give you the same answer as above:FV1 = PV(1×r)t

FV1 = 100(1.12)1 = $112Given the investment of $100, we can determine the amount of interest for the year by subtracting the FV from the PV:$112-100 = $12

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That is, Maria will earn a 12% return.

Next, let's look at the time line and interest computation for Borrower#2 (semiannual compounding):

Because Borrower#2's interest rate is stated for the entire year, the interest per half a year equals to 12% divided by 2, which equals 6%. Thus, you can calculate the compound interest of the future value after the first six months (FV1) and the future value at the end of the year (FV2):FV1 = 100+(100×.06) = $106FV2 = 106+(106×.06) = $112.36Alternatively, using the future value of lump sum equation, you get the same answer for future value at the end of the year: FV2 = 100(1.06)2 = $112.36Given the investment of $100, we can determine the amount of interest for the year by subtracting the FV from the PV:$112.36-100 = $12.36

That is, Maria will earn a 12.36% return.

The time line and interest computation for Borrower#3 (monthly compounding) is:

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Because interest rate is stated for the entire year, interest per month equals to 12% divided by 12 or 1%. Calculating for compound interest, the future values at the end of each month are:FV1 = 100+(100×1%) = $101FV2 = 101+(101×1%) = $102.01FV3 = 102.01+(102.01×1%) = $103.03FV4 = $103.03+($103.03×1%) = $104.06…FV11 = 110.46+(110.46×1%) = $111.57FV12 = 111.57+(111.57×1%) = $112.68Alternatively, using the future value of lump sum equation, the future value at the end of the year is:FV12 = 100(1.01)12 = $112.68Given the investment of $100, we can determine the amount of interest for the year by subtracting the FV from the PV:$112.68-100 = $12.68

That is, Maria will earn a 12.68% return.

In sum, interest amount and return that Maria earns from each borrower are:

Borrower#1 Borrower#2 Borrower#3

Compounding Annual Semiannual Monthly

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Amount $12.00 $12.36 $12.68

Return 12% 12.36% 12.68%

Although each borrower pays the same interest rate of 12%, Maria will earn different amount of interest and hence return because of different frequency of compounding. The interest amount and return are lowest for Borrower#1 and highest for Borrower#3.This shows that the higher the frequency of compounding, the higher the future value and hence the return.

Quoted Rate, EAR, and APR

The 12% interest rate quoted by each borrower in the example is called quoted rate, stated rate or nominal rate. The words 'rate' or 'interest rate' often means quoted rate.The interest rate earned per period is called periodic rate, and can be calculated using this formula:

where m = frequency of compounding.From our example, the periodic rates are as follows:

For Borrower#1, the quoted rate of 12% is divided by 1 (compounded annually) giving a periodic rate of 12%.

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For Borrower#2, the quoted rate of 12% is divided by 2 (compounded semiannually or every six months), giving the periodic rate of 6%.

For Borrower#3, the quoted rate of 12% is divided by 12 (compounded monthly), giving the periodic rate of 1%.

The 12%, 12.36% and 12.68% return from each borrower are called effective annual rate (EAR). EAR is interest rate expressed as if it were compounded annually. That is, Maria will earn the same interest from either 12% semiannual compounding or 12.36% annual compounding. Similarly, the interest for Maria is the same for either 12% monthly compounding or 12.68% annual compounding.The equation for converting from quoted rate to EAR is:

For example, 12.36%EAR of 12% semiannual compounding can be obtained from the equation as follows:

What is APR? APR stands for annual percentage rate. By law, lenders are required to disclose APR, which is interest rate charged per period multiplied by the number of periods per year.In most types of loans such as auto loan and credit card, APR is the same as quoted rate. For example, if a bank charges 0.5% per month on a car loan, the bank must report 6% APR on the loan. The 6% rate is

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quoted rate, and 0.5% is periodic rate or interest rate charged per month.However, in case of mortgage loan, APR is not the same as quoted rate because APR for mortgage loan is calculated by including not only interest but also some other fees such as discount points and loan-processing fees charged by lenders. APR is not used to calculate mortgage payments (quoted rate is), but is used to provide an estimate of the borrowing cost.

The Pennsylvania State University ©2006. All rights reserved.