Appendix: The Time Value of Money: Future Value and Present Value Computations

Appendix: The Time Value of Money: Future Value and Present Value Computations
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“If I deposit $10,000 today, how much will I have for a down payment on a house in five years?”

“Will $2,000 saved each year give me enough money when I retire?”

“How much must I save today to have enough for my children’s college education?”

The time value of money, more commonly referred to as interest, is the cost of money that is borrowed or lent. Interest can be compared to rent, the cost of using an apartment or other item. The time value of money is based on the fact that a dollar received today is worth more than a dollar that will be received one year from today, because the dollar received today can be saved or invested and will be worth more than a dollar a year from today. Similarly, a dollar that will be received one year from today is currently worth less than a dollar today.

The time value of money has two major components: future value and present value. Future value computations, which are also referred to as compounding, yield the amount to which a current sum will increase based on a certain interest rate and period of time. Present value, which is calculated through a process called discounting, is the current value of a future sum based on a certain interest rate and period of time.

In future value problems, you are given an amount to save or invest and you calculate the amount that will be available at some future date. With present value problems, you are given the amount that will be available at some future date and you calculate the current value of that amount. Both future value and present value computations are based on basic interest rate calculations.

Interest Rate Basics

Simple interest is the dollar cost of borrowing or the earnings from lending money. The interest is based on three elements:

The dollar amount, called the principal.
The rate of interest.
The amount of time.

The formula for computing interest is

The interest rate is stated as a percentage for a year. For example, you must convert 12 percent to either 0.12 or 12/100 before doing your calculations. The time element must also be converted to a decimal or fraction. For example, three months would be shown as either 0.25 or 1/4 of a year. Interest for 2 1/2 years would involve a time period of 2.5.

EXAMPLE A

Suppose you borrow $1,000 at 5 percent and will repay it in one payment at the end of one year. Using the simple interest calculation, the interest is $50, computed as follows:

EXAMPLE B

If you deposited $750 in a savings account paying 8 percent, how much interest would you earn in nine months? You would compute this amount as follows:

SAMPLE PROBLEM 1

How much interest would you earn if you deposited $300 at 6 percent for 27 months? (Answers to sample problems are on page 36)

SAMPLE PROBLEM 2

How much interest would you pay to borrow $670 for eight months at 12 percent?

Future Value of a Single Amount

The future value of an amount consists of the original amount plus compound interest. This calculation involves the following elements:

The formula for the future value of a single amount is

EXAMPLE C

The future value of $1 at 10 percent after three years is $1.33. This amount is calculated as follows:

Future value tables are available to help you determine compounded interest amounts (see Exhibit 1-A on page 37). Looking at Exhibit 1-A for 10 percent and three years, you can see that $1 would be worth $1.33 at that time. For other amounts, multiply the table factor by the original amount.

Exhibit 1-A Future value (compounded sum) of $1 after a given number of time periods

This may be viewed as follows:

EXAMPLE D

If your savings of $400 earn 12 percent, compounded monthly, over a year and a half, use the table factor for 1 percent for 18 time periods. The future value of this amount is $478.40, calculated as follows:

SAMPLE PROBLEM 3

What is the future value of $800 at 8 percent after six years?

SAMPLE PROBLEM 4

How much would you have in savings if you kept $200 on deposit for eight years at 8 percent, compounded semiannually?

Future Value of a Series of Equal Amounts (an Annuity)

Future value may also be calculated for a situation in which regular additions are made to savings. The following formula is used:

This formula assumes that (1) each deposit is for the same amount, (2) the interest rate is the same for each time period, and (3) the deposits are made at the end of each time period.

EXAMPLE E

The future value of three $1 deposits made at the end of the next three years, earning 10 percent interest, is $3.31. This is calculated as follows:

This may be viewed as follows:

Using Exhibit 1-B on page 38 you can find this same amount for 10 percent for three time periods. To use the table for other amounts, multiply the table factors by the annual deposit.

Exhibit 1-B Future value (compounded sum) of $1 paid in at the end of each period of a given number of time periods (an annuity)

EXAMPLE F

If you plan to deposit $40 a year for 10 years, earning 8 percent compounded annually, use the table factor for 8 percent for 10 time periods. The future value of this amount is $579.48, calculated as follows:

SAMPLE PROBLEM 5

What is the future value of an annual deposit of $230 earning 6 percent for 15 years?

SAMPLE PROBLEM 6

What amount would you have in a retirement account if you made annual deposits of $375 for 25 years earning 12 percent, compounded annually?

Present Value of a Single Amount

If you want to know how much you need to deposit now to receive a certain amount in the future, use the following formula:

EXAMPLE G

The present value of $1 to be received three years from now based on a 10 percent interest rate is $0.75. This amount is calculated as follows:

This may be viewed as follows:

Present value tables are available to assist you in this process (see Exhibit 1-C on page 39. Notice that $1 at 10 percent for three years has a present value of $0.75. For amounts other than $1, multiply the table factor by the amount involved.

Exhibit 1-C Present value of $1 to be received at the end of a given number of time periods

EXAMPLE H

If you want to have $300 seven years from now and your savings earn 10 percent, compounded semiannually, use the table factor for 5 percent for 14 time periods. In this situation, the present value is $151.50, calculated as follows:

SAMPLE PROBLEM 7

What is the present value of $2,200 earning 15 percent for eight years?

SAMPLE PROBLEM 8

To have $6,000 for a child’s education in 10 years, what amount should a parent deposit in a savings account that earns 12 percent, compounded quarterly?

Present Value of a Series of Equal Amounts (an Annuity)

The final time value of money situation allows you to receive an amount at the end of each time period for a certain number of periods. This amount is calculated as follows:

EXAMPLE I

The present value of a $1 withdrawal at the end of the next three years would be $2.49, calculated as follows:

This may be viewed as follows:

This same amount appears in Exhibit 1-D on page 40 for 10 percent and three time periods. To use the table for other situations, multiply the table factor by the amount to be withdrawn each year.

Exhibit 1-D Present value of $1 received at the end of each period for a given number of time periods (an annuity)

EXAMPLE J

If you wish to withdraw $100 at the end of each year for 10 years from an account that earns 14 percent, compounded annually, what amount must you deposit now? Use the table factor for 14 percent for 10 time periods. In this situation, the present value is $521.60, calculated as follows:

SAMPLE PROBLEM 9

What is the present value of a withdrawal of $200 at the end of each year for 14 years with an interest rate of 7 percent?

SAMPLE PROBLEM 10

How much would you have to deposit now to be able to withdraw $650 at the end of each year for 20 years from an account that earns 11 percent?

Using Present Value to Determine Loan Payments

Present value tables can also be used to determine installment payments for a loan as follows:

EXAMPLE K

If you borrow $1,000 with a 6 percent interest rate to be repaid in three equal payments at the end of the next three years, the payments will be $374.11. This is calculated as follows:

SAMPLE PROBLEM 11

What would be the annual payment amount for a $20,000, 10-year loan at 7 percent?

Answers to Sample Problems

. (Use Exhibit 1-A, 8%, 6 periods.)

. (Use Exhibit 1-A, 4%, 16 periods.)

. (Use Exhibit 1-B, 6%, 15 periods.)

. (Use Exhibit 1-B, 12%, 25 periods.)

. (Use Exhibit 1-C, 15%, 8 periods.)

. (Use Exhibit 1-C, 3%, 40 periods.)

. (Use Exhibit 1-D, 7%, 14 periods.)

. (Use Exhibit 1-D, 11%, 20 periods.)

. (Use Exhibit 1-D, 7%, 10 periods.)

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Home > Chapter 1 > Appendix: The Time Value of Money: Future Value and Present Value Computations

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		Chapter Opener Influences on Personal... Achieving Financial Go... Developing Personal Fi... Opportunity Costs and ... The Financial Planning... Summary of Objectives Key Terms Financial Planning Pro... Financial Planning Act... Internet Connection Financial Planning Case Video Case Your Personal Financia... Continuing Case Appendix: The Time ... The Time Value of Money Future Value and Prese... Personal Financial Pla... Personal Financial Pla... Personal Financial Pla... Personal Financial Pla... Personal Financial Pla...

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	More Resources
		Chapter Summary
	eBook
		Chapter Opener Influences on Personal... Achieving Financial Go... Developing Personal Fi... Opportunity Costs and ... The Financial Planning... Summary of Objectives Key Terms Financial Planning Pro... Financial Planning Act... Internet Connection Financial Planning Case Video Case Your Personal Financia... Continuing Case Appendix: The Time ... The Time Value of Money Future Value and Prese... Personal Financial Pla... Personal Financial Pla... Personal Financial Pla... Personal Financial Pla... Personal Financial Pla...

	EBOOK
		The Time Value of Money Future Value and Prese...