1 EYK 15-1 Present Value (PV) Tutorial The concepts of present value are described and applied in Chapter 15. This supplement provides added explanations, illustrations, calculations, present value tables, and assignments. PRESENT VALUE CONCEPTS There s an old saying, time is money. This saying reflects the notion that as time passes, the assets and liabilities we hold are changing, due to interest. Interest is the payment to the owner of an asset for its use by a borrower. The most common example of this type of asset is a savings account. As we keep a balance of cash in our account, it earns interest that is paid to us by the financial institution. An example of a liability is a car loan. As we carry the balance of the loan, we accumulate interest costs on this debt. We must ultimately repay this loan with interest. Present value computations are a means for us to estimate the interest component of holding assets or liabilities over time. The present value of an amount applies when we either lend or borrow an asset that must be repaid in full at some future date, and we want to know its worth today. The first section focuses on the present value of a single amount. The next section will focus on the present value of a series of amounts (or annuity). LO 1 Describe the earning of interest and the concept of present value. LEARNING OBJECTIVES LO 1 Describe the earning of interest and the concept of present value. LO 2 Apply present value concepts to a single amount by using interest tables. LO Apply present value concepts to an annuity by using interest tables. 1
2 2 Extend Your Knowledge 15-1 Present Value (PV) Tutorial PRESENT VALUE OF A SINGLE AMOUNT We graphically express the present value (p) of a single future amount (f ) received or paid at a future date in Exhibit PV15.1. EXHIBIT PV15.1 Present Value of a Single Amount f Time p Today Future LO 2 Apply present value concepts to a single amount by using interest tables. The formula to calculate the present value of this single amount is shown in Exhibit PV15.2 where: p present value; ƒ future value; i rate of interest per period; and n number of periods. EXHIBIT PV15.2 Present Value of a Single Amount Formula f p (1 i) n To illustrate the application of this formula, let s assume we need $220 one period from today. We want to know how much must be invested now, for one period, at an interest rate of 10% to provide for this $ For this illustration the p, or present value, is the unknown amount. In particular, the present and future values, along with the interest rate, are shown graphically as: f $220 (i 0.10) p? Conceptually, we know p must be less than $220. This is obvious from the answer to the question: Would we rather have $220 today or $220 at some future date? If we had $220 today, we could invest it and see it grow to something more than $220 in the future. Therefore, if we were promised $220 in the future, we would take less than $220 today. But how much less? To answer that question, we can calculate an estimate of the present value of the $220 to be received one period from now using the formula in Exhibit PV15.2 as: p f $220 $200 (1 i) n (1 0.10) 1 This means we are indifferent between $200 today or $220 at the end of one period. 1 Interest is also called a discount, and an interest rate is also called a discount rate.
3 Extend Your Knowledge 15-1 Present Value (PV) Tutorial 3 We can also use this formula to calculate the present value for any number of periods. To illustrate this calculation, we consider a payment of $242 at the end of two periods at 10% interest. The present value of this $242 to be received two periods from now is calculated as: p f $242 $200 (1 i) n (1 0.10) 2 These results tell us we are indifferent between $200 today, or $220 one period from today, or $242 two periods from today. The number of periods (n) in the present value formula does not have to be expressed in years. Any period of time such as a day, a month, a quarter, or a year can be used. But, whatever period is used, the interest rate (i) must be compounded for the same period. This means if a situation expresses n in months, and i equals 12% per year, then we can assume 1% of an amount invested at the beginning of each month is earned in interest per month and added to the investment. In this case, interest is said to be compounded monthly. A present value table helps us with present value calculations. It gives us present values for a variety of interest rates (i) and a variety of periods (n). Each present value in a present value table assumes the future value (f) is 1. When the future value (f) is different from 1, we can simply multiply present value (p) by that future amount to give us our estimate. The formula used to construct a table of present values of a single future amount of 1 is shown in Exhibit PV p (1 i) n EXHIBIT PV15.3 Present Value of 1 Formula This formula is identical to that in Exhibit PV15.2 except that f equals 1. Table 15B.1 (from Appendix 15B to Chapter 15, repeated at the end of this EYK) is a present value table for a single future amount. It is often called a present value of 1 table. A present value table involves three 2 factors: p, i, and n. Knowing two of these three factors allows us to calculate the third. To illustrate, consider the three possible cases. Case 1 (solve for p, knowing i and n). Our example above is a case in which we need to solve for p when we know i and n. To illustrate how we use a present value table, let s again look at how we estimate the present value of $220 (f ) at the end of one period (n) where the interest rate (i) is 10%. To answer this we go to the present value table (Table 15B.1) and look in the row for one period and in the column for 10% interest. Here we find a present value (p) of based on a future value of 1. This means, for instance, that $1 to be received one period from today at 10% interest is worth $ today. Since the future value is not $1, but is $220, we multiply the by $220 to get an answer of $200. Case 2 (solve for n, knowing p and i). This is a case in which we have, say, a $100,000 future value (f ) valued at $13,000 today ( p) with an interest rate of 12% (i). In this case we want to know how many periods (n) there are between the present value and the future value. A case example is when we want to retire with $100,000, but have only $13,000 earning a 12% return. How long will it be before we can retire? To answer this, we go to Table 15B.1 and look in the 12% interest column. Here 2 A fourth is f, but as we already explained, we need only multiply the 1 used in the formula by f.
4 4 Extend Your Knowledge 15-1 Present Value (PV) Tutorial we find a column of present values (p) based on a future value of 1. To use the present value table for this solution, we must divide $13,000 (p) by $100,000 (f ), which equals This is necessary because a present value table defines f equal to 1, and p as a fraction of 1. We look for a value nearest to (p), which we find in the row for 18 periods (n). This means the present value of $100,000 at the end of 18 periods at 12% interest is $13,000 or, alternatively stated, we must work 18 more years. Case 3 (solve for i, knowing p and n). This is a case where we have, say, a $120,000 future value (f ) valued at $60,000 (p) today when there are nine periods (n) between the present and future values. Here we want to know what rate of interest is being used. As an example, suppose we want to retire with $120,000, but we only have $60,000 and hope to retire in nine years. What interest rate must we earn to retire with $120,000 in nine years? To answer this, we go to the present value table (Table 15B.1) and look in the row for nine periods. To again use the present value table we must divide $60,000 (p) by $120,000 (f ), which equals Recall this is necessary because a present value table defines f equal to 1, and p as a fraction of 1. We look for a value in the row for nine periods that is nearest to (p), which we find in the column for 8% interest (i). This means the present value of $120,000 at the end of nine periods at 8% interest is $60,000 or, in our example, we must earn 8% annual interest to retire in nine years. CHECKPOINT 1. A company is considering an investment expected to yield $70,000 after six years. If this company demands an 8% return, how much is it willing to pay for this investment? Do Quick Study question: QS PV15-1 LO 3 Apply present value concepts to an annuity by using interest tables. PRESENT VALUE OF AN ANNUITY An annuity is a series of equal payments occurring at equal intervals. One example is a series of three annual payments of $100 each. The present value of an ordinary annuity is defined as the present value of equal payments at equal intervals as of one period before the first payment. An ordinary annuity of $100 and its present value ( p) is illustrated in Exhibit PV15.4. EXHIBIT PV15.4 Present Value of an Ordinary Annuity $100 $100 $100 Time p Today Future (n 1) Future (n 2) Future (n 3) One way for us to calculate the present value of an ordinary annuity is to find the present value of each payment using our present value formula from Exhibit PV15.3. We then would add up each of the three present values. To illustrate, let s look at three $100 payments at the end of each of the next three periods with an interest rate of 15%. Our present value calculations are: p $100 $100 $100 $ (1 0.15) 1 (1 0.15) 2 (1 0.15) 3
5 Extend Your Knowledge 15-1 Present Value (PV) Tutorial 5 This calculation also is identical to calculating the present value of each payment (from Table 15B.1) and taking their sum or, alternatively, adding the values from Table 15B.1 for each of the three payments and multiplying their sum by the $100 annuity payment. A more direct way is to use a present value of annuity table, Table 15B.2. If we look at Table 15B.2 where n 3 and i 15%, we see that the present value is This means the present value of an annuity of 1 for three periods, with a 15% interest rate, is A present value of annuity formula is used to construct Table 15B.2. It can also be constructed by adding the amounts in a present value of 1 table. To illustrate, we use Tables 15B.1 and 15B.2 to confirm this relation for the prior example. From Table 15B.1 From Table 15B.2 i 15%, n i 15%, n i 15%, n Total i 15%, n We can also use business calculators or spreadsheet computer programs to find the present value of an annuity. CHECKPOINT 2. A company is considering an investment paying $10,000 every six months for three years. The first payment would be received in six months. If this company requires an annual return of 8%, what is the maximum amount it is willing to invest? Do quick study questions: QS PV15-2, QS PV15-3, QS PV15-4 SUMMARY OF EYK 15-1 LO 1 Describe the earning of interest and the concepts of present value. Interest is payment to the owner of an asset for its use by a borrower. Present value calculations are a means for us to estimate the interest component of holding assets or liabilities over a period of time. LO 2 Apply present value concepts to a single amount by using interest tables. The present value of a single amount to be received at a future date is the amount that can be invested now at the specified interest rate to yield that future value. LO 3 Apply present value concepts to an annuity by using interest tables. The present value of an annuity is the amount that can be invested now at the specified interest rate to yield that series of equal periodic payments. GUIDANCE ANSWERS TO CHECKPOINT 1. $70, $44,114 (using Table 15B.1, i 8%, n 6). 2. $10, $52,421 (using Table 15B.2, i 4%, n 6).
6 6 Extend Your Knowledge 15-1 Present Value (PV) Tutorial QUICK STUDY QS PV15-1 Present value of an amount LO 2 Kim Flaherty is considering an investment that, if paid for immediately, is expected to return $140,000 five years hence. If Flaherty demands a 9% return, how much is she willing to pay for this investment? QS PV15-2 Present value of an annuity LO 3 Beene Distributing is considering a contract that will return $150,000 annually at the end of each year for six years. If Beene demands an annual return of 7% and pays for the investment immediately, how much should it be willing to pay? QS PV15-3 Interest rate on an investment LO 2 Ken Francis has been offered the possibility of investing $2,745 for 15 years, after which he will be paid $10,000. What annual rate of interest will Francis earn? QS PV15-4 Number of periods of an investment LO 2 Megan Brink has been offered the possibility of investing $6,651. The investment will earn 6% per year and will return Brink $10,000 at the end of the investment. How many years must Brink wait to receive the $10,000? EXERCISES Exercise PV15-1 Interest rate on an investment LO 3 Betsey Jones expects an immediate investment of $57,466 to return $10,000 annually for eight years, with the first payment to be received in one year. What rate of interest will Jones earn? Exercise PV15-2 Number of periods of an investment LO 3 Keith Riggins expects an investment of $82,014 to return $10,000 annually for several years. If Riggins is to earn a return of 10%, how many annual payments must he receive? Exercise PV15-3 Present value of an annuity LO 3 Sam Weber financed a new automobile by paying $6,500 cash and agreeing to make 40 monthly payments of $500 each, with the first payment to be made one month after the purchase. The loan bears interest at an annual rate of 12%. What was the cost of the automobile? Exercise PV15-4 Present value of bonds LO 2,3 Spiller Corp. plans to issue 10%, 15-year, $500,000 par value bonds payable that pay interest semi-annually on June 30 and December 31. The bonds are dated December 31, 2014, and are to be issued on that date. If the market rate of interest for the bonds is 8% on the date of issue, what will be the cash proceeds from the bond issue? Exercise PV15-5 Present value of an amount LO 1,2,3 McAdams Company expects to earn 10% per year on an investment that will pay $606,773 six years hence. Use Table 15B.1 to calculate the present value of the investment.
7 Extend Your Knowledge 15-1 Present Value (PV) Tutorial 7 Exercise PV15-6 Present value of an amount and annuity LO 2,3 Calculate the amount that can be borrowed under each of the following circumstances: a. A promise to pay $90,000 in seven years at an interest rate of 6%. b. An agreement made on February 1, 2014, to make three payments of $20,000 on February 1 of 2015, 2016, and The annual interest rate is 10%. Exercise PV15-7 Present value of an amount LO 2 On January 1, 2014, a company agrees to pay $20,000 in three years. If the annual interest rate is 10%, determine how much cash the company can borrow with this promise. Exercise PV15-8 Present value of an amount LO 2 Find the amount of money that can be borrowed with each of the following promises: Single Future Number Interest Case Payment of Years Rate a. $40, % b. 75, % c. 52, % d % e. 63, % f. 89, % Exercise PV15-9 Present values of annuities LO 3 C&H Ski Club recently borrowed money and agreed to pay it back with a series of six annual payments of $5,000 each. C&H subsequently borrowed more money and agreed to pay it back with a series of four annual payments of $7,500 each. The annual interest rate for both loans is 6%. a. Use Table 15B.1 to find the present value of these two annuities. (Round amounts to the nearest dollar.) b. Use Table 15B.2 to find the present value of these two annuities. Exercise PV15-10 Present value with semi-annual compounding LO 1,3 Otto Co. borrowed cash on April 30, 2014, by promising to make four payments of $13,000 each on November 1, 2014, May 1, 2015, November 1, 2015, and May 1, a. How much cash is Otto able to borrow if the interest rate is 8%, compounded semi-annually? b. How much cash is Otto able to borrow if the interest rate is 12%, compounded semi-annually? c. How much cash is Otto able to borrow if the interest rate is 16%, compounded semi-annually?
8 8 Extend Your Knowledge 15-1 Present Value (PV) Tutorial TABLE 15B.1 Present Value of 1 Due in n Periods Rate Periods 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 12% 15% TABLE 15B.2 Present Value of an Annuity of 1 per Period Rate Periods 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 12% 15%
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