Future Value Concepts

Future Value of a Single Amount

The future value of a single amount is the value at a future date of a given amount invested, assuming compound interest. For example, in Illustration E-2, €1,295.03 is the future value of the €1,000 investment earning 9% for three years. The €1,295.03 is determined more easily by using the following formula.

where:

• FV = future value of a single amount
• p = principal (or present value; the value today)
• i = interest rate for one period
• n = number of periods

The €1,295.03 is computed as follows.

The 1.29503 is computed by multiplying (1.09 × 1.09 × 1.09). The amounts in this example can be depicted in the time diagram shown in Illustration E-4.

Another method used to compute the future value of a single amount involves a compound interest table. This table shows the future value of 1 for $n$ periods. Table 1 (page E‐4) is such a table.

In Table 1, $n$ is the number of compounding periods, the percentages are the periodic interest rates, and the 5‐digit decimal numbers in the respective columns are the future value of 1 factors. To use Table 1, you multiply the principal amount by the future value factor for the specified number of periods and interest rate. For example, the future value factor for two periods at 9% is 1.18810. Multiplying this factor by €1,000 equals €1,188.10—which is the accumulated balance at the end of year 2 in the Citizens Bank example in Illustration E-2. The €1,295.03 accumulated balance at the end of the third year is calculated from Table 1 by multiplying the future value factor for three periods (1.29503) by the €1,000.

The demonstration problem in Illustration E-5 shows how to use Table 1.

Illustration E-5 Demonstration problem—Using Table 1 for FV of 1

Future Value of an Annuity

The preceding discussion involved the accumulation of only a single principal sum. Individuals and businesses frequently encounter situations in which a series of equal amounts are to be paid or received at evenly spaced time intervals (periodically), such as loans or lease (rental) contracts. A series of payments or receipts of equal amounts is referred to as an annuity.

The future value of an annuity is the sum of all the payments (receipts) plus the accumulated compound interest on them. In computing the future value of an annuity, it is necessary to know (1) the interest rate, (2) the number of payments (receipts), and (3) the amount of the periodic payments (receipts).

To illustrate the computation of the future value of an annuity, assume that you invest HK$2,000 at the end of each year for three years at 5% interest compounded annually. This situation is depicted in the time diagram in Illustration E-6.

The HK$2,000 invested at the end of year 1 will earn interest for two years (years 2 and 3), and the HK$2,000 invested at the end of year 2 will earn interest for one year (year 3). However, the last HK$2,000 investment (made at the end of year 3) will not earn any interest. Using the future value factors from Table 1, the future value of these periodic payments is computed as shown in Illustration E-7.

The first HK$2,000 investment is multiplied by the future value factor for two periods (1.1025) because two years' interest will accumulate on it (in years 2 and 3). The second HK$2,000 investment will earn only one year's interest (in year 3) and therefore is multiplied by the future value factor for one year (1.0500). The final HK$2,000 investment is made at the end of the third year and will not earn any interest. Thus, $n=0$ and the future value factor is 1.00000. Consequently, the future value of the last HK$2,000 invested is only HK$2,000 since it does not accumulate any interest.

Calculating the future value of each individual cash flow is required when the periodic payments or receipts are not equal in each period. However, when the periodic payments (receipts) are the same in each period, the future value can be computed by using a future value of an annuity of 1 table. Table 2 (page E-6) is such a table.

Table 2 shows the future value of 1 to be received periodically for a given number of payments. It assumes that each payment is made at the end of each period. We can see from Table 2 that the future value of an annuity of 1 factor for three payments at 5% is 3.15250. The future value factor is the total of the three individual future value factors as shown in Illustration E-7. Multiplying this amount by the annual investment of HK$2,000 produces a future value of HK$6,305.

The demonstration problem in Illustration E-8 shows how to use Table 2.

Present Value Concepts

Present Value Variables

The present value is the value now of a given amount to be paid or received in the future, assuming compound interest. The present value, like the future value, is based on three variables: (1) the dollar amount to be received (future amount), (2) the length of time until the amount is received (number of periods), and (3) the interest rate (the discount rate). The process of determining the present value is referred to as discounting the future amount.

Present value computations are used in measuring many items. For example, the present value of principal and interest payments is used to determine the market price of a bond. Determining the amount to be reported for notes payable and lease liabilities also involves present value computations. In addition, capital budgeting and other investment proposals are evaluated using present value computations. Finally, all rate of return and internal rate of return computations involve present value techniques.

Present Value of a Single Amount

To illustrate present value, assume that you want to invest a sum of money today that will provide €1,000 at the end of one year. What amount would you need to invest today to have €1,000 one year from now? If you want a 10% rate of return, the investment or present value is $€909.09(€\mathrm{1,000}÷1.10)$. The formula for calculating present value is shown in Illustration E-9.

The computation of €1,000 discounted at 10% for one year is as follows.

The future amount (€1,000), the discount rate (10%), and the number of periods (1) are known. The variables in this situation can be depicted in the time diagram in Illustration E-10.

If the single amount of €1,000 is to be received in two years and discounted at $10\%[PV=€\mathrm{1,000}÷{(1+.10)}^2]$, its present value is $€826.45[(€\mathrm{1,000}÷1.21)$, depicted in Illustration E-11.

The present value of 1 may also be determined through tables that show the present value of 1 for $n$ periods. In Table 3 below, $n$ is the number of discounting periods involved. The percentages are the periodic interest rates or discount rates, and the 5-digit decimal numbers in the respective columns are the present value of 1 factors.

When using Table 3, the future value is multiplied by the present value factor specified at the intersection of the number of periods and the discount rate. For example, the present value factor for one period at a discount rate of 10% is .90909, which equals the $€909.09(€\mathrm{1,000}\times .90909)$ computed in Illustration E-10. For two periods at a discount rate of 10%, the present value factor is .82645, which equals the $€826.45(€\mathrm{1,000}\times .82645)$ computed previously.

Note that a higher discount rate produces a smaller present value. For example, using a 15% discount rate, the present value of €1,000 due one year from now is €869.57, versus €909.09 at 10%. Also note that the further removed from the present the future value is, the smaller the present value. For example, using the same discount rate of 10%, the present value of €1,000 due in five years is €620.92. The present value of €1,000 due in one year is €909.09, a difference of €288.17.

The following two demonstration problems (Illustrations 12 and 13) illustrate how to use Table 3.

Present Value of an Annuity

The preceding discussion involved the discounting of only a single future amount. Businesses and individuals frequently engage in transactions in which a series of equal amounts are to be received or paid at evenly spaced time intervals (periodically). Examples of a series of periodic receipts or payments are loan agreements, installment sales, mortgage notes, lease (rental) contracts, and pension obligations. As discussed earlier, these periodic receipts or payments are annuities.

The present value of an annuity is the value now of a series of future receipts or payments, discounted assuming compound interest. In computing the present value of an annuity, it is necessary to know (1) the discount rate, (2) the number of payments (receipts), and (3) the amount of the periodic payments or receipts. To illustrate the computation of the present value of an annuity, assume that you will receive €1,000 cash annually for three years at a time when the discount rate is 10%. This situation is depicted in the time diagram in Illustration E-14. Illustration E-15 shows the computation of its present value in this situation.

This method of calculation is required when the periodic cash flows are not uniform in each period. However, when the future receipts are the same in each period, an annuity table can be used. As illustrated in Table 4 below, an annuity table shows the present value of 1 to be received periodically for a given number of payments. It assumes that each payment is made at the end of each period.

Table 4 shows that the present value of an annuity of 1 factor for three payments at 10% is 2.48685.¹ This present value factor is the total of the three individual present value factors, as shown in Illustration E-15. Applying this amount to the annual cash flow of €1,000 produces a present value of €2,486.85.

The following demonstration problem (Illustration E-16) illustrates how to use Table 4.

Time Periods and Discounting

In the preceding calculations, the discounting was done on an annual basis using an annual interest rate. Discounting may also be done over shorter periods of time such as monthly, quarterly, or semiannually.

When the time frame is less than one year, it is necessary to convert the annual interest rate to the applicable time frame. Assume, for example, that the investor in Illustration E-14 received €500 semiannually for three years instead of €1,000 annually. In this case, the number of periods becomes six $(3\times 2)$, the discount rate is $5\%(10\%÷2)$, the present value factor from Table 4 is 5.07569 (6 periods at 5%), and the present value of the future cash flows is $€\mathrm{2,537.85}(5.07569\times €500)$. This amount is slightly higher than the €2,486.86 computed in Illustration E-15 because interest is computed twice during the same year. That is, during the second half of the year, interest is earned on the first half-year's interest.

Computing the Present Value of a Long-Term Note or Bond

The present value (or market price) of a long-term note or bond is a function of three variables: (1) the payment amounts, (2) the length of time until the amounts are paid, and (3) the discount rate. Our example uses a five-year bond issue.

The first variable (amounts to be paid) is made up of two elements: (1) a series of interest payments (an annuity) and (2) the principal amount (a single sum). To compute the present value of the bond, both the interest payments and the principal amount must be discounted—;two different computations. The time diagrams for a bond due in five years are shown in Illustration E-17.

When the investor's market interest rate is equal to the bond's contractual interest rate, the present value of the bonds will equal the face value of the bonds. To illustrate, assume a bond issue of 10%, five-year bonds with a face value of NT$100,000 with interest payable semiannually on January 1 and July 1. If the discount rate is the same as the contractual rate, the bonds will sell at face value. In this case, the investor will receive (1) NT$100,000 at maturity and (2) a series of ten NT$5,000 interest payments $[(\text{NT}\$\mathrm{100,000}\times 10\%)÷2]$ over the term of the bonds. The length of time is expressed in terms of interest periods—in this case—10, and the discount rate per interest period, 5%. The following time diagram (Illustration E-18) depicts the variables involved in this discounting situation.

Illustration E-19 shows the computation of the present value of these bonds.

Now assume that the investor's required rate of return is 12%, not 10%. The future amounts are again NT$100,000 and NT$5,000, respectively, but now a discount rate of $6\%(12\%÷2)$ must be used. The present value of the bonds is NT$92,639, as computed in Illustration E-20.

Conversely, if the discount rate is 8% and the contractual rate is 10%, the present value of the bonds is NT$108,111, computed as shown in Illustration E-21.

The above discussion relied on present value tables in solving present value problems. Calculators may also be used to compute present values without the use of these tables. Many calculators, especially financial calculators, have present value (PV) functions that allow you to calculate present values by merely inputting the proper amount, discount rate, periods, and pressing the PV key. We discuss the use of financial calculators in a later section.

Computing the Present Values in a Capital Budgeting Decision

The decision to make long-term capital investments is best evaluated using discounting techniques that recognize the time value of money. To do this, many companies calculate the present value of the cash flows involved in a capital investment.

To illustrate, Nagel-Siebert Trucking Company, a cross-country freight carrier, is considering adding another truck to its fleet because of a purchasing opportunity. Nagel-Siebert's primary supplier of overland rigs is overstocked and offers to sell its biggest rig for £154,000 cash payable upon delivery. Nagel-Siebert knows that the rig will produce a net cash flow per year of £40,000 for five years (received at the end of each year), at which time it will be sold for an estimated residual value of £35,000. Nagel-Siebert's discount rate in evaluating capital expenditures is 10%. Should Nagel-Siebert commit to the purchase of this rig?

The cash flows that must be discounted to present value by Nagel-Siebert are as follows.

• Cash payable on delivery (today): £154,000.

• Net cash flow from operating the rig: £40,000 for 5 years (at the end of each year).

• Cash received from sale of rig at the end of 5 years: £35,000.

The time diagrams for the latter two cash flows are shown in Illustration E-22.

Notice from the diagrams that computing the present value of the net operating cash flows (£40,000 at the end of each year) is discounting an annuity (Table 4), while computing the present value of the £35,000 residual value is discounting a single sum (Table 3). The computation of these present values is shown in Illustration E-23 (page E-15).

Because the present value of the cash receipts (inflows) of $\pounds \mathrm{173,363.80}(\pounds \mathrm{151,631.60}+\pounds \mathrm{21,732.20})$ exceeds the present value of the cash payments (outflows) of £154,000.00, the net present value of £19,363.80 is positive, and the decision to invest should be accepted.

Now assume that Nagel-Siebert uses a discount rate of 15%, not 10%, because it wants a greater return on its investments in capital assets. The cash receipts and cash payments by Nagel-Siebert are the same. The present values of these receipts and cash payments discounted at 15% are shown in Illustration E-14.

Because the present value of the cash payments (outflows) of £154,000.00 exceeds the present value of the cash receipts (inflows) of $\pounds \mathrm{151,487.70}(\pounds \mathrm{134,086.40}+\pounds \mathrm{17,401.30})$, the net present value of £2,512.30 is negative, and the investment should be rejected.

The above discussion relied on present value tables in solving present value problems. As we show in the next section, calculators may also be used to compute present values without the use of these tables. Some calculators, especially the “business” or financial calculators, have present value (PV) functions that allow you to calculate present values by merely identifying the proper amount, discount rate, periods, and pressing the PV key.

Using Financial Calculators

Business professionals, once they have mastered the underlying time value of money concepts, often use a financial calculator to solve these types of problems. In most cases, they use calculators if interest rates or time periods do not correspond with the information provided in the compound interest tables.

To use financial calculators, you enter the time value of money variables into the calculator. Illustration E-25 shows the five most common keys used to solve time value of money problems.²

where:

• $\mathrm{N}=\text{number of periods}$
• $\mathrm{I}=\text{interest rate per period }(\text{some calculators use I}/\text{YR or i})$
• $\text{PV}=\text{present value }(\text{occurs at the beginning of the first period})$
• $\text{PMT}=\text{payment }(\text{all payments are equal},\text{ and none are skipped})$
• $\text{FV}=\text{future value }(\text{occurs at the end of the last period})$

In solving time value of money problems in this appendix, you will generally be given three of four variables and will have to solve for the remaining variable. The fifth key (the key not used) is given a value of zero to ensure that this variable is not used in the computation.

Present Value of a Single Sum

To illustrate how to solve a present value problem using a financial calculator, assume that you want to know the present value of €84,253 to be received in five years, discounted at 11% compounded annually. Illustration E-26 depicts this problem.

Illustration E-26 shows you the information (inputs) to enter into the calculator: $\mathrm{N}=5$, $\mathrm{I}=11$, $\text{PMT}=0$, and $\text{FV}=\mathrm{84,253}$. You then press PV for the answer: $-€\mathrm{50,000}$. As indicated, the PMT key was given a value of zero because a series of payments did not occur in this problem.

Present Value of an Annuity

To illustrate how to solve a present value of an annuity problem using a financial calculator, assume that you are asked to determine the present value of rental receipts of €6,000 each to be received at the end of each of the next five years, when discounted at 12%, as pictured in Illustration E-27.

In this case, you enter $\mathrm{N}=5$, $\mathrm{I}=12$, $\mathrm{N}=5$, $\text{PMT}=\mathrm{6,000}$, $\text{FV}=0$, and then press PV to arrive at the answer of $-€\mathrm{21,628.66}$.

Useful Applications of the Financial Calculator

With a financial calculator, you can solve for any interest rate or for any number of periods in a time value of money problem. Here are some examples of these applications.

AUTO LOAN

Assume you are financing the purchase of a used car with a three-year loan. The loan has a 9.5% stated annual interest rate, compounded monthly. The price of the car is €6,000, and you want to determine the monthly payments, assuming that the payments start one month after the purchase. This problem is pictured in Illustration E-28.

To solve this problem, you enter $\mathrm{N}=36(12\times 3)$, $\mathrm{I}=9.5$, $\text{PV}=\mathrm{6,000}$, $\text{FV}=0$, and than press PMT. You will find that the monthly payments will be €192.20. Note that the payment key is usually programmed for 12 payments per year. Thus, you must change the default (compounding period) if the payments are other than monthly.

MORTGAGE LOAN AMOUNT

Say you are evaluating financing options for a loan on a house (a mortgage). You decide that the maximum mortgage payment you can afford is €700 per month. The annual interest rate is 8.4%. If you get a mortgage that requires you to make monthly payments over a 15-year period, what is the maximum home loan you can afford? Illustration E-29 depicts this problem.

You enter $\mathrm{N}=180(12\times 15\text{ years})$, $\mathrm{I}=8.4$, $\text{PMT}=-700$, $\text{FV}=0$, and press PV. With the payments-per-year key set at 12, you find a present value of €71,509.81— the maximum home loan you can afford, given that you want to keep your mortgage payments at €700. Note that by changing any of the variables, you can quickly conduct “what-if” analyses for different situations.

REVIEW

LEARNING OBJECTIVES REVIEW

1. Distinguish between simple and compound interest. Simple interest is computed on the principal only, while compound interest is computed on the principal and any interest earned that has not been withdrawn.

2. Solve for future value of a single amount. Prepare a time diagram of the problem. Identify the principal amount, the number of compounding periods, and the interest rate. Using the future value of 1 table, multiply the principal amount by the future value factor specified at the intersection of the number of periods and the interest rate.

3. Solve for future value of an annuity. Prepare a time diagram of the problem. Identify the amount of the periodic payments (receipts), the number of payments (receipts), and the interest rate. Using the future value of an annuity of 1 table, multiply the amount of the payments by the future value factor specified at the intersection of the number of periods and the interest rate.

4. Identify the variables fundamental to solving present value problems. The following three variables are fundamental to solving present value problems: (1) the future amount, (2) the number of periods, and (3) the interest rate (the discount rate).

5. Solve for present value of a single amount. Prepare a time diagram of the problem. Identify the future amount, the number of discounting periods, and the discount (interest) rate. Using the present value of a single amount table, multiply the future amount by the present value factor specified at the intersection of the number of periods and the discount rate.

6. Solve for present value of an annuity. Prepare a time diagram of the problem. Identify the amount of future periodic receipts or payment (annuities), the number of payments (receipts), and the discount (interest) rate. Using the present value of an annuity of 1 table, multiply the amount of the annuity by the present value factor specified at the intersection of the number of payments and the interest rate.

7. Compute the present value of notes and bonds. Determine the present value of the principal amount: Multiply the principal amount (a single future amount) by the present value factor (from the present value of 1 table) intersecting at the number of periods (number of interest payments) and the discount rate. Determine the present value of the series of interest payments: Multiply the amount of the interest payment by the present value factor (from the present value of an annuity of 1 table) intersecting at the number of periods (number of interest payments) and the discount rate. Add the present value of the principal amount to the present value of the interest payments to arrive at the present value of the note or bond.

8. Compute the present values in capital budgeting situations. Compute the present values of all cash inflows and all cash outflows related to the capital budgeting proposal (an investment-type decision). If the net present value is positive, accept the proposal (make the investment). If the net present value is negative, reject the proposal (do not make the investment).

9. Use a financial calculator to solve time value of money problems. Financial calculators can be used to solve the same and additional problems as those solved with time value of money tables. Enter into the financial calculator the amounts for all of the known elements of a time value of money problem (periods, interest rate, payments, future or present value), and it solves for the unknown element. Particularly useful situations involve interest rates and compounding periods not presented in the tables.

GLOSSARY REVIEW

Annuity

A series of equal dollar amounts to be paid or received at evenly spaced time intervals (periodically). (p. E-4).

Compound interest

The interest computed on the principal and any interest earned that has not been paid or withdrawn. (p. E-2).

Discounting the future amount(s)

The process of determining present value. (p. E-7).

Future value of an annuity

The sum of all the payments (receipts) plus the accumulated compound interest on them. (p. E-5).

Future value of a single amount

The value at a future date of a given amount invested, assuming compound interest. (p. E-3).

Interest

Payment for the use of another person’s money. (p. E-1).

Present value

The value now of a given amount to be paid or received in the future assuming compound interest. (p. E-7).

Present value of an annuity

The value now of a series of future receipts or payments, discounted assuming compound interest. (p. E-9).

Principal

The amount borrowed or invested. (p. E-1).

Simple interest

The interest computed on the principal only. (p. E-1).

WileyPLUS

Many more resources are available for practice in WileyPLUS.

BRIEF EXERCISES

(Use tables to solve exercises BEE-1 to BEE-23.)

Compute the future value of a single amount.

(LO 2)

BEE-1 Randy Owen invested €9,000 at 5% annual interest, and left the money invested without withdrawing any of the interest for 15 years. At the end of the 15 years, Randy withdrew the accumulated amount of money. (a) What amount did Randy withdraw, assuming the investment earns simple interest? (b) What amount did Randy withdraw, assuming the investment earns interest compounded annually?

Use future value tables.

(LO 2, 3)

BEE-2 For each of the following cases, indicate (a) to what interest rate columns and (b) to what number of periods you would refer in looking up the future value factor.

1. (1) In Table 1 (future value of 1):

   	Annual Rate	Number of Years Invested	Compounded
   Case A	5%	3	Annually
   Case B	12%	4	Semiannually

2. In Table 2 (future value of an annuity of 1):

   	Annual Rate	Number of Years Invested	Compounded
   Case A	3%	8	Annually
   Case B	8%	6	Semiannually

Compute the future value of a single amount.

(LO 2)

BEE-3 Joyce Ltd. signed a lease for an office building for a period of 8 years. Under the lease agreement, a security deposit of £8,400 is made. The deposit will be returned at the expiration of the lease with interest compounded at 4% per year. What amount will Joyce receive at the time the lease expires?

Compute the future value of an annuity.

(LO 3)

BEE-4 Bates Company issued $1,000,000, 12-year bonds and agreed to make annual sinking fund deposits of $60,000. The deposits are made at the end of each year into an account paying 6% annual interest. What amount will be in the sinking fund at the end of 12 years?

Compute the future value of a single amount and of an annuity.

(LO 2, 3)

BEE-5 Frank and Maureen Fantazzi invested €8,000 in a savings account paying 5% annual interest when their daughter, Angela, was born. They also deposited €1,000 on each of her birthdays until she was 18 (including her 18th birthday). How much was in the savings account on her 18th birthday (after the last deposit)?

Compute the future value of a single amount.

(LO 2)

BEE-6 Hugh Curtin borrowed $35,000 on July 1, 2017. This amount plus accrued interest at 8% compounded annually is to be repaid on July 1, 2022. How much will Hugh have to repay on July 1, 2022?

Use present value tables.

(LO 5,6)

BEE-7 For each of the following cases, indicate (a) to what interest rate columns and (b) to what number of periods you would refer in looking up the discount rate.

1. In Table 3 (present value of 1):

   	Annual Rate	Number of Years Involved	Discounts per Year
   Case A	12%	7	Semiannually
   Case B	8%	11	Annually
   Case C	10%	8	Semiannually

2. In Table 4 (present value of an annuity of 1):

   	Annual Rate	Number of Years Involved	Number of Payments Involved	Frequency of Payments
   Case A	10%	20	20	Annually
   Case B	10%	7	7	Annually
   Case C	8%	5	10	Semiannually