Different Types of TVOM

Introduction

Young couple playing with camera.

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?

Smiling young couple hugging.

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:

  1. lump sum
  2. multiple cash flows
  3. annuity
  4. annuity due
  5. perpetuity
  6. growing perpetuity
  7. growing annuity

Objectives

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

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:

Timeline labeled with r, P V at zero and F V at 3

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

P V equals numerator F V denominator left parens 1 plus r right parens sup t base

Where:


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:

Timeline labeled with r equals 10%, -500 at zero and F V equals question mark at 3

FV = PV(1+r)t

FV = 500(1+.10)3

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


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.

Timeline labeled with r, P V at zero, C sub 1 at 1, C sub 2 at 2, C sub 3 and F V at 3

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

P V equals numerator C sub 1 base, denominator, left parens 1 plus r, right parens supe 1 base plus numerator C sub 2 base, denominator, left parens 1 plus r, right parens supe 2 base plus numerator C sub 3 base, denominator, left parens 1 plus r, right parens supe 3 base.

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:

Timeline labeled with r equals 10%, -100 at 1, -300 at 2, -200 and F V equals question mark at 3

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

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

Timeline labeled with r, C at each of 1,2 3, 4, 5, P V A at zero, and F V A at 5

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:

F V A equals numerator C denominator r, left bracket, left parens 1 plus r right parens sup t base minus 1 right bracket.    or     P V A equals numerator C denominator r, left bracket, 1 minus numerator 1 denominator left parens 1 plus r right parens sup t base, right bracket.

Where:


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:

Timeline labeled with r equals 10%, 200 at each of 1,2 3, 4, 5 and F V A equals question mark.
F V A equals numerator C denominator r, left bracket, left parens 1 plus r right parens sup t base minus 1 right bracket. F V A equals numerator 200 denominator .10, left bracket, left parens 1 plus .10 right parens sup 5 base minus 1 right bracket.

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).
  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.

Timeline labeled with r, and with C at  1, 3, and 5.

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.

Timeline with r, with C at  0, 1, 2, 3, and 4, P V A Due at zero, and FVA Due at 5.

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

Timeline with r=10%, with -100 at  1, 2, 3, 4, and 5, and with FVA equals question at 5.

Annuity Due Timeline:

Timeline with r=10%, with -100 at  0, 1, 2, 3, and 4, and with FVA Due equals question at 5.

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

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


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

F V A due equals numerator C denominator r left bracket, left parens 1 plus r right parens supe t base minus 1 right bracket, left parens 1 plus r right parens.

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

P V A due equals numerator C denominator r left bracket, 1 minus numerator 1 denominator left parens 1 plus r right parens supe t base right bracket, left parens 1 plus r right parens.

where…


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):

Timeline labeled with r equals 10%, 200 at each of 0, 1,2 3, 4, and F V A due equals question mark.
F V A due equals numerator C denominator r left bracket, left parens 1 plus r right parens supe t base minus 1 right bracket, left parens 1 plus r right parens. F V A due equals numerator 200 denominator .10 left bracket, left parens 1 plus .10 right parens supe 5 base minus 1 right bracket, left parens 1 plus .10 right parens.

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 (∞).

Timeline with r, C at each of 1,2 3, an arrow pointing to infinity, and P V infinity at zero.

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


P V infinity equals numerator C denominator r

where…


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.

Timeline with r equals 10%, 150 at each of 1,2 3, an arrow pointing to infinity, and P V infinity equals question mark.

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


P V infinity equals numerator C denominator r
P V infinity equals numerator 150 denominator .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.

Timeline with r equals 10%, -100 at 1,-105 at 2, -110.25 at 3, an arrow pointing to infinity, g equals 5% and P V G infinity equals question mark.

The equation below is used to calculate growing perpetuity:


P V G infinity equals numerator C sub 1 base denominator r minus g.

where…

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:

Timeline with r equals 10%, -2 at 1,-2.1 at 2, -2.205 at 3, an arrow pointing to infinity, g equals 5% and P V G infinity equals question mark.

The equation below is used to calculate growing perpetuity:


P V G infinity equals numerator C sub 1 base denominator r minus g.

where…


P V G infinity equals numerator 2 sub 1 base denominator.10 minus .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.

Timeline with r equals 10%, -100 at 1,-105 at 2, -110.25 at 3, an arrow pointing to infinity, g equals 5% and P V G A equals question mark.

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.

P V G A  equals numerator C sub 1 base denominator left paren r minus g right paren, left bracket, one minus numerator left parens numerator 1 plus g denominator one plus r right parens supe T base right parens.

where:

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:

Timeline with r equals 10%, -100 at 1,-105 at 2, -110.25 at 3, an arrow pointing to infinity, g equals 5% and P V G A equals question mark.
P V G A  equals numerator C sub 1 base denominator r minus g, left bracket, one minus numerator left parens numerator 1 plus g denominator one plus r right parens supe T base right parens.

where:

P V G A  equals numerator 100 denominator .10 minus .05, left bracket, one minus left parens numerator 1 plus .05 denominator one plus .10 right parens supe 10 base right bracket.

Practice

To solve each problem follow these steps:

  1. Create a time line
  2. Identify type of cash flows and
  3. 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:

Timeline labeled with r equals 10%, P V equals question mark at zero and 20,000 at 3.

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.

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:

P V equals numerator F V denominator left parens 1 plus r right parens sup t base

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)

P V equals numerator 20000 denominator left parens 1 point 1 right parens sup 3 base

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:

Timeline labeled with r equals 8%, 300 at 1, 500 at 2, 200 at 3, 400 at 4, negative 200 at 5, and F V unknown.

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 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:

Timeline labeled with r equals 5%, C at 1 through 16, and 100,000 at 16.

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.

F V A equals numerator C denominator r, left bracket, left parens 1 plus r right parens sup t base minus 1 right bracket.

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

100,000 equals numerator C denominator .05, left bracket, left parens 1.05 right parens sup 16 base minus 1 right bracket.

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:

Timeline labeled with r equals 6% divided by 12, C at 1 through 60 and 30,000 at 0.

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.

P V A due equals numerator C denominator r left bracket, 1 minus numerator 1 denominator left parens 1 plus r right parens supe t base right bracket, left parens 1 plus r right parens.

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)

30,000 equals numerator C denominator .06/12; left bracket, 1 minus numerator 1 denominator left parens 1 plus left parens.06/12right parens, right parens, supe 60 base right bracket, left parens 1 plus left parens .06/12, right parens, right parens.

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:

Timeline labeled with r equals 10%, P V equals question mark and 100 at 1 through infinity.

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.

P V infinity equals numerator C denominator r

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

P V infinity equals numerator 100 denominator .1

Practice Question 6:

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

Timeline labeled with r equals 10%, P V G infinity equals question mark at zero, 2 at 1, 2.1 at 2, 2.205 at 3, and g equals 5%.

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.

P V G infinity equals numerator C sub 1 base denominator r minus g.

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)

P V G infinity equals numerator 2 denominator point 1 minus  point 05.

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

Solution:

Step 1- Create a Timeline:

Timeline labeled with r equals 10%, P V A infinity equals question mark, 1 at 1, 1.o5 at 2, 1.1025 at 3, 1.5513 at 10, and g equals 5% pointing to left and right.

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.

P V G A  equals numerator C sub 1 base denominator left paren r minus g right paren, left bracket, one minus numerator left parens numerator 1 plus g denominator one plus r right parens supe T base right bracket.

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)

P V G A  equals numerator $1 Mil denominator left paren point 1 minus point zero 5 right paren, left bracket, one minus left parens numerator 1 point  zero 5, denominator one point 1 right parens supe 10 base right bracket.