A dollar received today is worth more than a dollar to be received in the future because you can invest today’s dollar and earn additional interest so you’ll have more cash in the
Learning Objective 3 Use the time value of money to
Try It!
Lockwood Company is considering a capital investment in machinery:
Initial investment $ 600,000
Residual value 50,000
Expected annual net cash inflows 100,000
Expected useful life 8 years
Required rate of return 12%
8. Calculate the payback.
9. Calculate the ARR. Round the percentage to two decimal places.
10. Based on your answers to the above questions, should Lockwood invest in the machinery?
Check your answers online in MyAccountingLab or at http://www.pearsonhighered.com/Horngren.
For more practice, see Short Exercises S26-2 through S26-8. MyAccountingLab
The management team at Browne and Browne (BnB) is deep in the budgeting process. Strategic goals have been finalized, and capi- tal investments are being evaluated. Every department within BnB can submit potential investments. Predicted future cash flows and initial investment figures are given to the accounting department for evaluation using various capital investment analysis methods.
Then the department representatives can present the proposals to the management team.
Daryl Baez is a new manager at BnB and just completed his presentation for replacing some aging equipment. He felt prepared and did not expect many questions, but the team members had several. Their primary concern is the calculations for future net cash inflows. Daryl explained that the vendor provided the figures for reduced operating costs based on the efficiency of the new equipment. Should the management team
be concerned about the predictions? What should Daryl do?
What would you do?
Solution
Capital investments involve large sums of cash, and decisions affect operations for many years. Therefore, the management team should be concerned about the validity of the predictions.
Inaccurate predictions could have a long-term, negative impact on profits. If Daryl’s only source is the sales representative trying to make the sale, the figures may not be accurate. Daryl should conduct his own research. He could contact current users of the equipment and ask for their feedback. He could look for reviews in trade magazines. Diligence on his part before the purchase will ensure the figures are accurate, satisfy the management team, and help BnB generate higher returns on investments.
Where did these numbers come from?
DECISIONS
section reviews the time value of money to make sure you have a firm foundation for discussing these two methods.
Time Value of Money Concepts
The time value of money depends on several key factors:
1. The principal amount (p) 2. The number of periods (n) 3. The interest rate (i)
The principal (p) refers to the amount of the investment or borrowing. Because this chapter deals with capital investments, we will primarily discuss the principal in terms of investments. However, the same concepts apply to borrowings, such as mortgages payable and bonds payable, which we covered in the financial accounting chapters. We state the principal as either a single lump sum or an annuity. For example, if you win the lottery, you have the choice of receiving all the winnings now (a single lump sum) or receiving a series of equal payments for a period of time in the future (an annuity). An annuity is a stream of equal cash payments made at equal time intervals.1 For example, $100 cash received per month for 12 months is an annuity. Capital investments also have lump sums and annuities.
For example, consider the Smart Touch Learning’s Web site upgrade project from the previous section of the chapter. The amount initially invested in the project, $240,000, is a lump sum because it is a one-time payment. However, the annual cash inflows of $80,000 per year for three years is an annuity. We consider both types of cash flows in capital investment decisions.
The number of periods (n) is the length of time from the beginning of the invest- ment until termination. All else being equal, the shorter the investment period, the lower the total amount of interest earned. If you withdraw your savings after four years rather than five years, you will earn less interest. In this chapter, the number of periods is stated in years.2
The interest rate (i ) is the annual percentage earned on the investment. Interest can be computed as either simple interest or compound interest.
Simple Interest Versus Compound Interest
Simple interest means that interest is calculated only on the principal amount. Compound interest means that interest is calculated on the principal and on all previously earned interest. Compound interest assumes that all interest earned will remain invested and earn additional interest at the same interest rate. Exhibit 26-8 compares simple interest of 6% on a five-year,
$10,000 investment with interest compounded yearly (rounded to the nearest dollar). As you can see, the amount of compound interest earned yearly grows as the base on which it is calculated (principal plus cumulative interest to date) grows. Over the life of this investment, the total amount of compound interest is more than the total amount of simple interest. Most investments yield compound interest, so we assume compound interest, rather than simple interest, for the rest of this chapter.
Annuity A stream of equal cash payments
made at equal time intervals.
Simple Interest Interest calculated only on the
principal amount.
Compound Interest Interest calculated on the principal
and on all previously earned interest.
1An ordinary annuity is an annuity in which the installments occur at the end of each period. An annuity due is an annuity in which the installments occur at the beginning of each period. Throughout this chapter, we use ordinary annuities because they are better suited to capital budgeting cash flow assumptions.
2The number of periods can also be stated in days, months, or quarters. If one of these methods is used, the interest rate needs to be adjusted to reflect the number of time periods in the year.
Future Value and Present Value Factors
The future value or present value of an investment simply refers to the value of an investment at different points in time. We can calculate the future value or the present value of any investment by knowing (or assuming) information about the three factors we listed earlier: (1) the principal amount, (2) the number of periods, and (3) the interest rate.
For example, in Exhibit 26-8, we calculated the interest that would be earned on (1) a
$10,000 principal, (2) invested for five years, (3) at 6% interest. The future value of the investment is simply its worth at the end of a specific time frame (e.g., five years) or the original principal plus the interest earned. In our example, the future value of the investment is as follows:
Future Value
The value of an investment at the end of a specific time frame.
Exhibit26-8 | Simple Interest Versus Compound Interest—$10,000 at 6% for 5 Years
Year
Simple Interest
Calculation Simple
Interest Compound
Interest Calculation Compound
Interest
1 $10,000 × 6% $ 600 $10,000 × 6% $ 600
2 $10,000 × 6% 600 ($10,000 + $600) × 6% 636
3 $10,000 × 6% 600 ($10,000 + $600 + $636) × 6% 674*
4 $10,000 × 6% 600 ($10,000 + $600 + $636 + $674) × 6% 715 5 $10,000 × 6% ($10,000 + $600 + $636 + $674 + $715) × 6%
Total interest
600
$ 3,000 Total interest
758
$ 3,383
*all calculations rounded to the nearest dollar for the rest of this chapter
Present value = Future value − Interest earned
$10,000 = $13,383 - $3,383
Future value = Present value + Interest earned Future value= Principal + Interest earned
= $10,000+ $3,383
= $13,383
If we invest $10,000 today, its present value is simply $10,000. Present value is the value of an investment today. So another way of stating the future value is as follows:
If we know the future value and want to find the present value, we can rearrange the equation as follows:
Present Value
The value of an investment today.
The only difference between present value and future value is the amount of interest that is earned in the intervening time span.
Calculating each period’s compound interest, as we did in Exhibit 26-8, and then add-
These formulas are programmed into most business calculators, so the user only needs to correctly enter the principal amount, interest rate, and number of time periods to find present or future values. These formulas are also programmed into spreadsheet functions in Microsoft Excel. In this section of the chapter, we use present value tables. (The Excel for- mulas are illustrated later in the chapter. Note that because the table values are rounded, the Excel results will differ slightly.) The present value tables contain the results of the formulas for various interest rate and time period combinations.
The formulas and resulting tables are shown in Appendix A at the end of this book:
1. Present Value of $1 (Appendix A, Table A-1)—used to calculate the value today of one future amount (a lump sum).
2. Present Value of Ordinary Annuity of $1 (Appendix A, Table A-2)—used to calculate the value today of a series of equal future amounts (annuities).
3. Future Value of $1 (Appendix A, Table A-3)—used to calculate the value in the future of one present amount (a lump sum).
4. Future Value of Ordinary Annuity of $1 (Appendix A, Table A-4)—used to calculate the value in the future of a series of equal future amounts (annuities).
Take a moment to look at these tables because we are going to use them throughout the rest of the chapter. Note that the columns are interest rates (i ) and the rows are periods (n).
The numbers in each table, known as present value factors (PV factors) and future value factors (FV factors), are for an investment (or loan) of $1. For example, in Appendix A, Table A-1, the PV factor for an interest rate of 6% (i = 6%) and 5 periods (n = 5 ) is 0.747. To find the present value of an amount other than $1, multiply the PV factor by the future amount. To find the future value of an amount other than $1, multiply the FV factor by the present amount.
The annuity tables are derived from the lump sum tables. For example, the Annuity PV factors (in the Present Value of Ordinary Annuity of $1 table) are the sums of the PV factors found in the Present Value of $1 tables for a given number of time periods. The annuity tables allow us to perform one-step calculations rather than separately computing the present value of each annual cash installment and then summing the individual present values or future values.
Present Value of a Lump Sum
The process for calculating present values is often called discounting future cash flows because future amounts are discounted (interest removed) to their present value. Let’s consider the investment in Exhibit 26-8. The future value of the investment is $13,383.
So the question is, “How much would I have to invest today (in the present time) to have
$13,383 five years in the future if I invested at 6%?” Let’s calculate the present value using PV factors.
Present value = Future value * PV factor for i = 6%, n= 5
We determine the PV factor from the table labeled Present Value of $1 (Appendix A, Table A-1). We use this table for lump sum amounts. We look down the 6% column and across the 5 periods row and find the PV factor is 0.747. We finish our calculation as follows:
Present value = Future value : PV factor for i = 6%, n = 5
= $13,383* 0.747
= $9,997
Notice the calculation is off by $3 due to rounding ($10,000 - $9,997). The PV factors are rounded to three decimal places, so the calculations may not be exact. Also, the inter- est calculations in Exhibit 26-8 were rounded to the nearest dollar. Therefore, there are two rounding issues in this exhibit. However, we do have the answer to our question: If approximately $10,000 is invested today at 6% for five years, at the end of five years, the investment will grow to $13,383. Or, conversely, if we expect to receive $13,383 five years from now, its equivalent (discounted) value today is approximately $10,000. In other words, we need to invest approximately $10,000 today at 6% to have $13,383 five years from now.
Present Value of an Annuity
Let’s now assume that instead of receiving a lump sum at the end of the five years, you will receive $2,000 at the end of each year. This is a series of equal payments ($2,000) over equal intervals (years), so it is an annuity. How much would you have to invest today to receive these payments, assuming an interest rate of 6%?
We determine the annuity PV factor from the table labeled Present Value of Ordinary Annuity of $1 (Appendix A, Table A-2). We use this table for annuities. We look down the 6% column and across the 5 periods row and find the annuity PV factor is 4.212. We finish our calculation as follows:
Present value= Amount of each cash inflow : Annuity PV factor for i = 6%, n = 5
= $2,000 * 4.212
= $8,424
This means that an investment today of $8,424 at 6% will yield $2,000 per year for the next five years, or total payments of $10,000 over five years ($2,000 per year * 5 years). The reason is that interest is being earned on principal that is left invested each year. Let’s verify the calculation.
[1]
Beginning Balance Previous [4]
[2]
Interest [1] 6%
[3]
Withdrawal
$2,000
[4]
Ending Balance [1] [2] [3]
Year
0 $ 8,424
1 $ 8,424
*rounded up by $1
$ 505 $ 2,000 6,929
2 6,929 416 2,000 5,345
3 5,345 321 2,000 3,666
4 3,666 220 2,000 1,886
5 1,886 114* 2,000 0
The chart shows that the initial investment of $8,424 is invested for one year, earning $505 in interest. At the end of that period, the first withdrawal of $2,000 takes place, leaving a balance of $6,929 ($8,424 + $505 - $2,000). At the end of the five years, the ending bal- ance is $0, proving that the present value of the $2,000 annuity is $8,424.
Present Value Examples
Which alternative should you take? You might be tempted to wait 10 years to “dou- ble” your winnings. You may be tempted to take the cash now and spend it. However, assume you plan to prudently invest all money received—no matter when you receive it—so that you have financial flexibility in the future (for example, for buying a house, retiring early, or taking exotic vacations). How can you choose among the three payment alternatives, when the total amount of each option varies ($1,000,000 versus $1,500,000 versus $2,000,000) and the timing of the cash flows varies (now versus some each year versus later)? Comparing these three options is like comparing apples to oranges—we just cannot do it—unless we find some common basis for comparison. Our common basis for comparison will be the prize-money’s worth at a certain point in time—namely, today.
In other words, if we convert each payment option to its present value, we can compare apples to apples.
We already know the principal amount and timing of each payment option, so the only assumption we will have to make is the interest rate. The interest rate will vary, depend- ing on the amount of risk you are willing to take with your investment. Riskier invest- ments (such as stock investments) command higher interest rates; safer investments (such as FDIC-insured bank deposits) yield lower interest rates. Let’s say that after investigating possible investment alternatives, you choose an investment contract with an 8% annual return. We already know that the present value of Option #1 is $1,000,000 because we would receive that $1,000,000 today. Let’s convert the other two payment options to their present values so that we can compare them. We will need to use the Present Value of Ordi- nary Annuity of $1 table (Appendix A, Table A-2) to convert payment Option #2 (because it is an annuity—a series of equal cash payments made at equal intervals) and the Present Value of $1 table (Appendix A, Table A-1) to convert payment Option #3 (because it is a lump sum). To obtain the PV factors, we will look down the 8% column and across the 10 periods row. Then we finish the calculations as follows:
Option 2:
Present value = Amount of each cash inflow : Annuity PV factor for i= 8%, n = 10
= $150,000 * 6.710
= $1,006,500 Option 3:
Present value = Future Value : PV factor for i = 8%, n = 10
= $2,000,000 * 0.463
= $926,000
Exhibit 26-9 shows that we have converted each payout option to a common basis—
its worth today—so we can make a valid comparison among the options. Based on this com- parison, you should choose Option #2 because its worth, in today’s dollars, is the highest of the three options.
Exhibit26-9 | Present Value of Lottery Payout Options
Payment Options Present Value of Lottery Payout (i = 8%, n = 10) Option #1
Option #2 Option #3
$ 1,000,000 1,006,500 926,000
The lottery problem is a good example of how businesses use discounted cash flows to analyze capital investments. Companies make initial investments. The initial investment is already in present value, similar to Lottery Option 1. The purpose of the investment is to increase cash flows in the future, but those future cash flows have to be discounted back to their present value in order to compare them to the initial investment already in present value, similar to Lottery Option 2. Some investments also have a residual value, meaning the company can sell the assets at the end of their useful lives and receive a lump sum cash inflow in the future, similar to Lottery Option 3.
Future Value of a Lump Sum
Let’s now use the tables to calculate the future value of a lump sum considering the investment in Exhibit 26-8. Instead of calculating present value, though, we will change the scenario to evaluate the future value. “If I invested $10,000 today (in the present time), how much would I have in five years at an interest rate of 6%?” We will calculate the future value using FV factors.
Future value = Present value : FV factor for i= 6%, n = 5
= $10,000 * 1.338
= $13,380
Future value= Amount of each cash inflow: Annuity FV factor for i = 6%, n = 5
= $2,000 * 5.637
= $11,274
Future value = Present value * FV factor for i = 6%, n = 5
We determine the FV factor from the table labeled Future Value of $1 (Appendix A, Table A-3).
We use this table for lump sum amounts. We look down the 6% column and across the 5 peri- ods row and find the FV factor is 1.338. We finish our calculation as follows:
Notice the calculation is off by $3 due to rounding ($13,380 - $13,383). The FV factors are rounded to three decimal places, so the calculations may not be exact. Also, the interest calculations in Exhibit 26-8 were rounded to the nearest dollar. Therefore, there are two rounding issues in this exhibit. However, we do have the answer to our question: If $10,000 is invested today at 6% for five years, at the end of five years, the investment will grow to
$13,380.
Future Value of an Annuity
Let’s now calculate the future value of an annuity assuming that you will receive $2,000 at the end of each year. This is a series of equal payments ($2,000) over equal intervals (years), so it is an annuity. How much would these payments be worth five years from now, assuming an interest rate of 6%?
We determine the annuity FV factor from the table labeled Future Value of Ordinary Annuity of $1 (Appendix A, Table A-4). We use this table for annuities. We look down the 6% column and across the 5 periods row and find the annuity FV factor is 5.637. We finish our calculation as follows: