1. The Time Value of Money Future Amounts and Present Values
Appendix B
Appendix B: The Time Value of Money—Future Amounts and Present Values

2. The Concept 1 2 3 4
$630 × 1.08
$583 × 1.08
$540 × 1.08
$500 × 1.08
The present value of an investment gradually increases toward the future amount. We have seen that interest accrues over time. Therefore, the difference between the present value and a future amount depends on two factors: (1) the rate of interest at which the present value increases and (2) the length of time over which interest accumulates.
Assume you invest $500 in a savings account that earns interest at the rate of 8% per year. This graph illustrates the growth in your savings account balance at the end of each of the next four years.
Assume you invest $500 in a savings account that earns interest at the rate of 8% per year. This graph illustrates the growth in your savings account balance at the end of each of the next four years.
Appendix B-2

3. Relationships between Present Values and Future Amounts1 2 3 4
$630 × 1.08
$583 × 1.08
$540 × 1.08
$500 × 1.08
In this example, your initial investment of $500 is the present value. It is invested for four years at 8% interest. Over the four years, the value of your investment increases to $680, the future amount.
The present value is always less than its future amount. This is the basic idea underlying the time value of money. But this idea often is expressed in different ways, including the following:
A present value is always less than a future amount.
A future amount is always greater than a present value.
A dollar available today is always worth more than a dollar that does not become available until a future date.
A dollar available at a future date is always worth less than a dollar that is available today.
In this example, your initial investment of $500 is the present value. It is invested for four years at 8% interest. Over the four years, the value of your investment increases to $680, the future amount.
Appendix B-3

4. Future Amounts
A future amount is simply the dollar amount to which a present value will accumulate over time.
Present Value
Future Amount
A future amount is simply the dollar amount to which a present value will accumulate over time. The difference between the present value and a related future amount depends on (1) the interest rate and (2) the period of time over which the present value accumulates.
Starting with the present value, we may compute the future amounts three ways: (1) through a series of multiplications, (2) using a financial calculator, and (3) using a table of future amounts, such as that illustrated on the next slide. Let’s see how to use the table.
Appendix B-4

5. $500 Present Value × 1.360 Factor = $680 Future Amount
Future Amounts
$500 Present Value × Factor = $680 Future Amount
Part I
Assume you invest $500 in a savings account that earns interest at the rate of 8% per year. What will be the future amount at the end of 4 years?
Part II
If you invest $500 for 4 years at 8%, you will have $680 at the end of the 4 years.
Assume you invest $500 in a savings account that earns interest at the rate of 8% per year. What will be the future amount at the end of 4 years?
Appendix B-5

6. Computing the Required Investment
Part I
Assume you need $680 at the end of 4 years. If you can invest at 8% per year, what is the present value?
Part II
By dividing the future amount of $680 by the table factor of for 4 periods at 8%, you will see that you need to invest $500.
Assume you need $680 at the end of 4 years. If you can invest at 8% per year, what is the present value?
$680 Future Amount Factor
$500 Present Value =
Appendix B-6

7. The Future Amount of an Annuity
An annuity is a series of equal periodic payments.
Future Amount
Annuity Payment
Annuity Payment
Annuity Payment
An annuity is a series of equal periodic payments. If you make a car payment or a house payment, you likely pay the same amount each month. Both of these are examples of a series of equal periodic payments or an annuity.
In many situations, an investor is to make a series of investment payments rather than a single payment.
Annuity Payment
Annuity Payment
Appendix B-7

8. The Future Amount of an Annuity
Part I
Assume you invest $500 in a savings account at the end of each of the next 4 years. The account earns interest at the rate of 8% per year. What will be the balance in your account at the end of 4 years?
Part II
If you invest $500 at the end of each year for 4 years and the investment earns 8%, you will have $2,253 at the end of the 4 years.
Assume you invest $500 in a savings account at the end of each of the next 4 years. The account earns interest at the rate of 8% per year. What will be the balance in your account at the end of 4 years?
$500 Periodic Payment × Factor
=$2,253 Future Amount of an Annuity
Appendix B-8

9. The Future Amount of an Annuity
Part I
Assume you need $2,253 at the end of 4 years. If you can invest at 8% per year, what is the amount of required periodic payment?
Part II
Dividing the future amount of an annuity $2,253 by the table factor of for 4 periods at 8%, you will see that your required periodic payment amount is $500.
Assume you need $2,253 at the end of 4 years. If you can invest at 8% per year, what is the amount of required periodic payment?
$2,253 Future Amount of an Annuity Factor
$500 Periodic Payment =
Appendix B-9

10. Present Value
The present value is today’s value of funds to be received in the future.
Present Value
Future Amount
The present value is today’s value of funds to be received in the future. The difference between the present value and a related future amount depends on (1) the interest rate and (2) the period of time over which the present value accumulates.
Appendix B-10

11. $680 Future Amount × .735 Factor = $500 Present Value (rounded)
Present Values
$680 Future Amount × .735 Factor = $500 Present Value (rounded)
Part I
What would you pay today for the opportunity to receive $680 in 4 years, assuming an 8% interest rate?
Part II
By multiplying the future amount of $680 by the table factor of .735 for 4 years at 8%, you will see that the present value is $500.
What would you pay today for the opportunity to receive $680 in 4 years, assuming an 8% interest rate?
Appendix B-11

12. The Present Value of an Annuity
Annuity Payment
Annuity Payment
Annuity Payment
In many situations, an investment opportunity is expected to produce annul cash flows for a number of years, rather than a single payment.
Annuity Payment
Annuity Payment
Appendix B-12

13. The Present Value of an Annuity
Assume you need cash flows of $500 at the end of each of the next 4 years. If your investment earns interest at the rate of 8% per year, what amount do you need to invest today to achieve your cash flow needs?
Part I
Assume you need cash flows of $500 at the end of each of the next 4 years. If your investment earns interest at the rate of 8% per year, what amount do you need to invest today to achieve your cash flow needs?
Part II
If you invest $1,656 today at 8% for 4 years, you will be able to achieve your cash flow needs of $500 per year.
$500 Periodic Payment × Factor
=$1,656 Present Value of an Annuity
Appendix B-13

14. Capital Leases
A capital lease is regarded as a sale of the leased asset by the lessor to the lessee.
At the date of this sale, the lessor recognizes sales revenue equal to the present value of the future lease payments receivable, discounted at a realistic rate of interest. The lessee also uses the present value of the future payments to determine the cost of the leased asset and the valuation of the related liability.
A capital lease is regarded as a sale of the leased asset by the lessor to the lessee. At the date of this sale, the lessor recognizes sales revenue equal to the present value of the future lease payments receivable, discounted at a realistic rate of interest. The lessee also uses the present value of the future payments to determine the cost of the leased asset and the valuation of the related liability.
Appendix B-14

15. End of Appendix B
End of Appendix B.
Appendix B-15