Example 4.1a Present Value of a Stream of Cash Flows

Problem:

You have just graduated and need money to pay the deposit on an apartment.

Your rich aunt will lend you the money so long as you agree to pay her back within six months.

You offer to pay her the rate of interest that she would otherwise get by putting her money in a savings account.

Problem:

You have just graduated and need money to pay the deposit on an apartment.

Your rich aunt will lend you the money so long as you agree to pay her back within six months.

You offer to pay her the rate of interest that she would otherwise get by putting her money in a savings account.

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Example 4.1a Present Value of a Stream of Cash Flows (cont’d)

Problem:

Based on your earnings and living expenses, you think you will be able to pay her $70 next month, $85 in each of the next two months, and then $90 each month for months 4 through 6.

If your aunt would otherwise earn 8% per year on her savings, how much can you borrow from her?

Problem:

Based on your earnings and living expenses, you think you will be able to pay her $70 next month, $85 in each of the next two months, and then $90 each month for months 4 through 6.

If your aunt would otherwise earn 8% per year on her savings, how much can you borrow from her?

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Example 4.1a Present Value of a Stream of Cash Flows (cont’d)

Solution:

Plan:

The cash flows you can promise your aunt are as follows:

She should be willing to give you an amount equal to these payments in present value terms.

Solution:

Plan:

The cash flows you can promise your aunt are as follows:

She should be willing to give you an amount equal to these payments in present value terms.

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Example 4.1a Present Value of a Stream of Cash Flows (cont’d)

Plan:

We will:

Solve the problem using equation 4.1

or

Solve the problem using cash flow keys in a financial calculator

Plan:

We will:

Solve the problem using equation 4.1

or

Solve the problem using cash flow keys in a financial calculator

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Example 4.1a Present Value of a Stream of Cash Flows (cont’d)

Execute:

We can calculate the PV as follows in the BAII Plus:

Clear the CF registers (Press CF, 2ND, CE/C)

Press CF, 0, Enter, , then 75, Enter, , continue until CF6 (90) is entered and press

Press NPV and enter .66667 (8/12) when prompted for the interest rate

Press Enter, , and you will see NPV = 0

To find NPV of the cash flows, Press CPT

NPV =

Execute:

We can calculate the PV as follows in the BAII Plus:

Clear the CF registers (Press CF, 2ND, CE/C)

Press CF, 0, Enter, , then 75, Enter, , continue until CF6 (90) is entered and press

Press NPV and enter .66667 (8/12) when prompted for the interest rate

Press Enter, , and you will see NPV = 0

To find NPV of the cash flows, Press CPT

NPV =

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Example 4.1a Present Value of a Stream of Cash Flows (cont’d)

Execute:

Now, suppose that your aunt gives you the money, and then deposits your payments in the bank each month.

How much will she have six months from now?

Execute:

Now, suppose that your aunt gives you the money, and then deposits your payments in the bank each month.

How much will she have six months from now?

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Example 4.2a Computing the Future Value

Problem:

We plan to save $2000 today and at the end of each of the next two years.

At a fixed 6% interest rate, how much will we have in the bank three years from today?

Problem:

We plan to save $2000 today and at the end of each of the next two years.

At a fixed 6% interest rate, how much will we have in the bank three years from today?

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Example 4.2a Computing the Future Value (cont’d)

Solution:

Plan:

We’ll start with the timeline for this savings plan:

Let’s solve this in two different ways.

Solution:

Plan:

We’ll start with the timeline for this savings plan:

Let’s solve this in two different ways.

$2,000 $2,000 $2,000 ?

0 1 2 3

0 1 2 3

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Example 4.2a Computing the Future Value (cont’d)

Plan:

First we’ll compute the present value of the cash flows.

Then we’ll compute its value three years later (its future value).

Plan:

First we’ll compute the present value of the cash flows.

Then we’ll compute its value three years later (its future value).

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Example 4.2a Computing the Future Value (cont’d)

Using the process described and practiced in the previous problem, solve for NPV using a financial calculator

Verify using Excel – CF0 DOES NOT = 0 and must be added to the result

In Excel NPV = (Rate,CF1,CF2)+CF0

Now solve for the future value using the calculator

This is a TVM problem, as in Chapter 3

Using Excel, solve for FV to check your result

DO NOT skip either the calculator or Excel process, you must be proficient in both

Solving the same problem both ways builds confidence in your solutions

Using the process described and practiced in the previous problem, solve for NPV using a financial calculator

Verify using Excel – CF0 DOES NOT = 0 and must be added to the result

In Excel NPV = (Rate,CF1,CF2)+CF0

Now solve for the future value using the calculator

This is a TVM problem, as in Chapter 3

Using Excel, solve for FV to check your result

DO NOT skip either the calculator or Excel process, you must be proficient in both

Solving the same problem both ways builds confidence in your solutions

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Example 4.3a Endowing a Perpetuity

Problem:

You just won the lottery, and you want to endow a professorship at your alma mater.

You are willing to donate $4 million of your winnings for this purpose.

If the university earns 6% per year on its investments, and the professor will be receiving her first payment in one year, how much will the endowment pay her each year?

Problem:

You just won the lottery, and you want to endow a professorship at your alma mater.

You are willing to donate $4 million of your winnings for this purpose.

If the university earns 6% per year on its investments, and the professor will be receiving her first payment in one year, how much will the endowment pay her each year?

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Example 4.3a Endowing a Perpetuity (cont’d)

Solution:

Plan:

The timeline of the cash flows you want to provide is:

This is a standard perpetuity. The amount she can withdraw each year and keep the principal intact is the cash flow when solving equation 4.4.

Solution:

Plan:

The timeline of the cash flows you want to provide is:

This is a standard perpetuity. The amount she can withdraw each year and keep the principal intact is the cash flow when solving equation 4.4.

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Example 4.4a Present Value of an Annuity

Problem:

Your parents have made you an offer you can’t refuse.

They’re planning to give you part of your inheritance early.

They’ve given you a choice.

Option (a) They’ll pay you $11,000 per year for each of the next seven years (beginning today) or

Option (b) They’ll give you their 2007 BMW M6 Convertible, which you can sell for $61,000 (guaranteed) today.

If you can earn 7% annually on your investments, which should you choose?

Problem:

Your parents have made you an offer you can’t refuse.

They’re planning to give you part of your inheritance early.

They’ve given you a choice.

Option (a) They’ll pay you $11,000 per year for each of the next seven years (beginning today) or

Option (b) They’ll give you their 2007 BMW M6 Convertible, which you can sell for $61,000 (guaranteed) today.

If you can earn 7% annually on your investments, which should you choose?

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Example 4.4a Present Value of an Annuity (cont’d)

Solution:

Plan:

Option (a) provides $11,000 paid over time. To evaluate it correctly, we must convert it to a present value. Here is the timeline:

Solution:

Plan:

Option (a) provides $11,000 paid over time. To evaluate it correctly, we must convert it to a present value. Here is the timeline:

$11,000 $11,000 $11,000 $11,000 $11,000 $11,000 $11,000 $11,000

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Example 4.4a Present Value of an Annuity (cont’d)

Plan (cont’d):

The $11,000 at date 0 is already stated in present value terms, but we need to compute the present value of the remaining payments.

Fortunately, this case looks like a 7-year annuity of $11,000 per year, so we can use the annuity formula.

OR

We can solve with the calculator and check our answer in Excel

Plan (cont’d):

The $11,000 at date 0 is already stated in present value terms, but we need to compute the present value of the remaining payments.

Fortunately, this case looks like a 7-year annuity of $11,000 per year, so we can use the annuity formula.

OR

We can solve with the calculator and check our answer in Excel

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Example 4.5a Retirement Savings Plan Annuity

Problem:

Adam is 25 years old, and he has decided it is time to plan seriously for his retirement.

He will save $10,000 in a retirement account at the end of each year until he is 47.

At that time, he will stop paying into the account, though he does not plan to retire until he is 65.

If the account earns 10% per year, how much will Adam have saved at age 65?

Problem:

Adam is 25 years old, and he has decided it is time to plan seriously for his retirement.

He will save $10,000 in a retirement account at the end of each year until he is 47.

At that time, he will stop paying into the account, though he does not plan to retire until he is 65.

If the account earns 10% per year, how much will Adam have saved at age 65?

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Example 4.5a Retirement Savings Plan Annuity

Solution:

Plan:

As always, we begin with a timeline. In this case, it is helpful to keep track of both the dates and Adam’s age:

Solution:

Plan:

As always, we begin with a timeline. In this case, it is helpful to keep track of both the dates and Adam’s age:

47

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Example 4.5a Retirement Savings Plan Annuity

Adam’s savings plan looks like an annuity of $10,000 per year for 22 years.

The money will then remain in the account until Adam is 65 – 18 more years.

To determine the amount Adam will have in the bank at age 47, we’ll need to compute the future value of this annuity.

Then we’ll compound the future value into the future 18 more years to see how much he’ll have at 65.

Adam’s savings plan looks like an annuity of $10,000 per year for 22 years.

The money will then remain in the account until Adam is 65 – 18 more years.

To determine the amount Adam will have in the bank at age 47, we’ll need to compute the future value of this annuity.

Then we’ll compound the future value into the future 18 more years to see how much he’ll have at 65.

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Example 4.6a Endowing a Growing Perpetuity

Problem:

In Example 4.3a, you planned to donate $4 million to your alma mater to fund an endowed professorship.

Given an interest rate of 6% per year, the professor would be able to collect $240,000 per year from your generosity.

The inflation rate is expected to be 2% per year.

How much can the professor be paid in the first year in order to allow her annual salary to increase by 2% each year and keep the principal intact?

Problem:

In Example 4.3a, you planned to donate $4 million to your alma mater to fund an endowed professorship.

Given an interest rate of 6% per year, the professor would be able to collect $240,000 per year from your generosity.

The inflation rate is expected to be 2% per year.

How much can the professor be paid in the first year in order to allow her annual salary to increase by 2% each year and keep the principal intact?

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Example 4.6a Endowing a Growing Perpetuity (cont’d)

Solution:

Plan:

The salary needs to increase 2% per year forever. From the timeline, we recognize the form of a growing perpetuity and can value it that way.

Solution:

Plan:

The salary needs to increase 2% per year forever. From the timeline, we recognize the form of a growing perpetuity and can value it that way.

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Bonus – 5 Points Present Value of an Annuity

Problem:

Your parents have made you an offer you can’t refuse.

They’re planning to give you part of your inheritance early.

They’ve given you a choice.

Option (a) They’ll pay you $11,000 per year paid semi-annually for each of the next seven years (beginning today) or

Option (b) They’ll give you their 2007 BMW M6 Convertible, which you can sell for $61,000 (guaranteed) today.

If you can earn 7% annually on your investments, which should you choose?

Problem:

Your parents have made you an offer you can’t refuse.

They’re planning to give you part of your inheritance early.

They’ve given you a choice.

Option (a) They’ll pay you $11,000 per year paid semi-annually for each of the next seven years (beginning today) or

Option (b) They’ll give you their 2007 BMW M6 Convertible, which you can sell for $61,000 (guaranteed) today.

If you can earn 7% annually on your investments, which should you choose?

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Bonus – 5 Points Present Value of an Annuity (cont’d)

Solution:

Plan:

Option (a) provides $11,000 paid over time. To evaluate it correctly, we must convert it to a present value. Here is the timeline:

Year 0 1 2 3 4 5 6 7

Pmt (CF) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

$5500 $5500 $5500 $5500 $5500 $5500 $5500 $5500 $5500 $5500 $5500 $5500 $5500 $5500 $5500 $5500

Solution:

Plan:

Option (a) provides $11,000 paid over time. To evaluate it correctly, we must convert it to a present value. Here is the timeline:

Year 0 1 2 3 4 5 6 7

Pmt (CF) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

$5500 $5500 $5500 $5500 $5500 $5500 $5500 $5500 $5500 $5500 $5500 $5500 $5500 $5500 $5500 $5500

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Bonus – 5 Points Present Value of an Annuity (cont’d)

Plan (cont’d):

The $5,500 at date 0 is already stated in present value terms, but we need to compute the present value of the remaining payments.

Fortunately, this case looks like a 7-year annuity of $11,000 per year, so we can use the annuity formula, a calculator, or Excel after we convert the payments and interest rate to semi-annual.

Plan (cont’d):

The $5,500 at date 0 is already stated in present value terms, but we need to compute the present value of the remaining payments.

Fortunately, this case looks like a 7-year annuity of $11,000 per year, so we can use the annuity formula, a calculator, or Excel after we convert the payments and interest rate to semi-annual.

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Bonus – 10 Points (If you answer this) Present Value of an Annuity (cont’d)

Final Question for 5 Points:

Explain the difference in the value of the annuity when the interest rate is the same and the total of payments is the same? Your answer must be specific, it is related to the time value of money

But you still have money left over after buying the BMW!!

Final Question for 5 Points:

Explain the difference in the value of the annuity when the interest rate is the same and the total of payments is the same? Your answer must be specific, it is related to the time value of money

But you still have money left over after buying the BMW!!

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Example 4.8a Computing a Loan Payment

Problem:

Suppose you accept your parents’ offer of a 2007 BMW M6 convertible, but that’s not the kind of car you want.

Instead, you sell the car for $61,000, spend $11,000 on a used Corolla, and use the remaining $50,000 as a down payment for a house.

The bank offers you a 30-year loan with equal monthly payments and an interest rate of 6% per year, and requires a 20% down payment.

How much can you borrow, and what will be the payment on the loan?

Problem:

Suppose you accept your parents’ offer of a 2007 BMW M6 convertible, but that’s not the kind of car you want.

Instead, you sell the car for $61,000, spend $11,000 on a used Corolla, and use the remaining $50,000 as a down payment for a house.

The bank offers you a 30-year loan with equal monthly payments and an interest rate of 6% per year, and requires a 20% down payment.

How much can you borrow, and what will be the payment on the loan?

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Example 4.8b Computing a Loan Payment

Problem:

Suppose you accept your parents’ offer of a 2007 BMW M6 convertible, but that’s not the kind of car you want.

Instead, you sell the car for $61,000 and use that money for a down payment on an Aston Martin V8 Vantage Roadster. You got a real bargain at $110,000!

The bank offers you a 5-year loan with equal monthly payments and an interest rate of 4% per year.

What will be the payment on the loan?

Problem:

Suppose you accept your parents’ offer of a 2007 BMW M6 convertible, but that’s not the kind of car you want.

Instead, you sell the car for $61,000 and use that money for a down payment on an Aston Martin V8 Vantage Roadster. You got a real bargain at $110,000!

The bank offers you a 5-year loan with equal monthly payments and an interest rate of 4% per year.

What will be the payment on the loan?

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Example 4.9a Computing the Internal Rate of Return with a Financial Calculator

Problem:

Let’s return to the BMW example (Example 4.4a).

What rate of return would make you indifferent between the car and the $10,000 per year payout (even if the car is your favorite color and has HD radio)?

Problem:

Let’s return to the BMW example (Example 4.4a).

What rate of return would make you indifferent between the car and the $10,000 per year payout (even if the car is your favorite color and has HD radio)?

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Example 4.9a Computing the Internal Rate of Return with a Financial Calculator

Solution:

Plan:

We need to solve for the rate of return that makes the two offers equivalent.

Anything above that rate of return would make the present value of the annuity lower than the $61,000 car and

anything below that rate of return would make it greater than the $61,000.

Solution:

Plan:

We need to solve for the rate of return that makes the two offers equivalent.

Anything above that rate of return would make the present value of the annuity lower than the $61,000 car and

anything below that rate of return would make it greater than the $61,000.

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Example 4.10a Solving for the Number of Periods in a Savings Plan

Problem:

Let’s return to Ellen and Adam.

Suppose Ellen decides she will continue working until she has as much at retirement as her brother, Adam, will have when he retires.

She will continue to contribute $10,000 each year to her retirement account.

How much longer will she need to work to tie the competition with her brother?

Problem:

Let’s return to Ellen and Adam.

Suppose Ellen decides she will continue working until she has as much at retirement as her brother, Adam, will have when he retires.

She will continue to contribute $10,000 each year to her retirement account.

How much longer will she need to work to tie the competition with her brother?

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Example 4.10a Solving for the Number of Periods in a Savings Plan

Solution:

Plan:

The timeline for this problem is

Solution:

Plan:

The timeline for this problem is

30

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$1,000?$1,000$1,000

2