1. 1111111 Intermediate Accounting, Ninth Edition Kieso and Weygandt Prepared by Catherine Katagiri, CPA The College of Saint Rose Albany, New York John Wiley & Sons, Inc.

2. Chapter 6: Accounting and the Time Value of Money After studying this chapter you should be able to: Identify accounting topics where time value of money is relevant. Distinguish between simple and compound interest. Learn how to use appropriate compound interest tables. Identify variables fundamental to solving interest problems. Solve future and present value of 1 problems. Solve future value of ordinary and annuity due problems. Solve present value of ordinary and annuity due problems.

3. 33333 Introduction Concept of the time value of money is very important especially when interest rates are high and/or time periods are long. Simply put--You are not indifferent as to when you receive or pay an identical sum of money. That is, a dollar received or paid today (the present) is not worth a dollar received or paid tomorrow (the future). Someone owes you, a rational person, $100. They say to you--Which would you prefer: I pay you $100 today or I pay you $100 one year from now?

4. 44444 Introduction Your response should be--Pay me $100 today (present). WHY? Assuming there is no inflation, would your answer change? Again, your response should be to be paid today. You could take the funds today, invest them, earn a return and have more than $100 a year from now. How much more than $100 would depend on the return you could earn (interest). Suppose, instead, someone offers you $100 one year from today. You wish the funds now (today), however. Will you accept less than $100 in settlement of the debt?

5. 55555 Introduction Considering the interest rates, you will accept less than $100 today to settle the debt. WHY? If you had the settlement today you could invest it, earn a return and have $100 one year from now. If the interest rate is high you would accept less to settle the debt than if the interest rate was low. Simply put, at any interest rate: A dollar received today is worth more than a dollar in the future. A dollar in the future is worth less than a dollar today.

6. 66666 Introduction The concept of the time value of money is pervasive. We see this concept in many topics including (to name a few): –Leases –Pensions –Non-interest bearing notes –Installment contracts –Valuation of bonds –Sinking fund provisions The FASB will simplify calculations by stating, at times, the time value of money is to be ignored. You must realize the import of that assumption as well!

7. 77777 Interest Definitions of interest: –A fee for the use of money –Principle x Interest Rate x Time = Interest Principle: The amount borrowed or invested. Interest rate: A percentage of the outstanding principle. Always expressed as an annual rate. Time: The number of years or fraction thereof that the principle is outstanding.

8. 88 Interest –Selection of appropriate interest rate is critical and, at times, very difficult. Would you be a wealthy person if you could accurately predict the interest rate? –The interest for us will be a given figure. In practice it can be the hardest figure to derive accurately.

9. 99 Interest –Elements of the interest rate: The interest rate is the sum of Pure rate of interest (system risk) Initial rate charged assuming no possibility of default and no inflation. Credit risk rate of interest (individual risk) Additional rate charged as a result of an individual entity’s risk of default. Expected inflation rate of interest (inflation premium) Additional rate charged to compensate for the decrease in the purchasing power of the dollar over time.

10. 10 Interest –Simple interest –Simple interest is computed on the principle only. That is, interest is earned and removed. The interest does not earn interest. –Compound interest –Compound interest is computed on the principle and any accumulated interest. Both the principal and interest then earn interest.

11. 11 Types of Problems For our calculations we will assume compound interest. The greater the number of periods and the higher the interest rate, the greater will be the effect of compounding interest. Types of problems we will be working with: –Single Sum –Single Sum. One sum ($1) will be received or paid either in the PVPresent (Present Value of a Single Sum or PV) FVFuture (Future Value of a Single Sum or FV)

12. 12 Types of Problems –Types of annuity problems –Types of annuity problems: OAOrdinary annuity (OA) –A series of equal payments (or rents) received or paid at the end of a period, assuming a constant rate of interest. PV-OA –Value today of a series of equal, end-of-period payments received in the future is the Present Value of an Ordinary Annuity or PV-OA. FV-OA. –Value in the future of a series of equal, end-of- period payments received in the future is the Future Value of an Ordinary Annuity or FV-OA.

13. 13 Types of Problems ADAnnuity Due (AD) –A series of equal payments (or rents) received or paid at the beginning of a period, assuming a constant rate of interest. PV-AD –Value today of a series of equal, beginning-of- period payments received in the future is the Present Value of an annuity due or PV-AD. FV-AD. –Value in the future of a series of equal, beginning- of-period payments received in the future is the Future Value of an annuity due or FV-AD.

14. 14 Types of Problems Note: Note: The difference between an ordinary annuity and an annuity due is that: Each rent or payment compounds (interest added) one more period in a annuity due, future value situation. –Each rent or payment is discounted (interest removed) one less period under the annuity due situation. –Given the same i, n and periodic rent, the annuity due will always yield a greater present value (less interest removed) and a greater future value (more interest added).

15. 15 Calculation Variables There will always be at least four variables in any present or future value problem. Three of the four will be known and you will solve for the fourth. –Single sum problems –Single sum problems: n = number of compounding periods i = interest rate PV = Value today of a single sum ($1) FV = Value in the future of a single sum ($1)

16. 16 Calculation Variables –Annuity problems –Annuity problems: n = number of payments or rents i = interest rate R = Periodic rent received or paid And either: FV-OA or AD = Value in the future of a series of future payments (either ordinary or due). OR PV-OA or AD= Value today of a series of payments in the future (either ordinary or due). Note: The “n” and the “i” must match. That is, if the time period is semi-annual then so must the interest rate. Interest rates are assumed to be annual unless otherwise stated so you may have to adjust the rate to match the time period.

17. 17 Single Sum Problems Let’s review the derivation of the single sum formula: Suppose you have $100 today (present) and wish to deposit it at 10% for three periods, in this case years. What is the value of this single sum in the future? –At the end of the first year (n = 1): (Compound interest) $100 + $100(.1) = $110 –At the end of the second year ( n = 2) $110 + $110(.1) = $121 –At the end of third year (n = 3) $121 + $121(.1) = $133 (rounded) –So the future value of $100 three years hence at 10% = $133

18. 18 Single Sum Problems I realize there is a simpler way to approach this: $100 (1 +.1)(1+.1)(1+.1) = $133 $100 (1+.1) 3 = $133 Generally for any “n” and “i” the single sum formula would be: PV (1+ i) n = FV $100 (1+.1) 3 = $133

19. 19 Single Sum Problems FVF n,i )Not wishing to have to constantly raise the term to the required power I name the term (1+ i) n, the Future Value Factor for a single term (FVF n,i ). I then employ the table on page 320-321. The table is the result of the required multiplications at various “n” and “i” and is to be read vertically for the “n” and horizontally for the “i”. To solve my problem using the table: PV(FVF n,i ) = FV PV(FVF n,i ) = FV where n = 3 and i = 10% $100 (1.331) = $133

20. 20 Note: For single sum problems the “n” refers to periods not necessarily defined as years! The period may be annual, semi-annual, quarterly or another time frame. In manual calculations the use of the table is strongly recommended. It enhances both speed and accuracy. Single Sum Problems

21. 21 Single Sum Problems Now suppose instead of $100 today I am to receive $133 three years from now. Again the interest rate is 10%. I don’t want to wait three years for my money. How much will I accept today in lieu of the future payment? Going back to the general formula for a single sum: PV(FVF n,i ) = FV I realize I can isolate the term I wish to solve for on one side of the equation: PV = FV divided by (FVF n,i ) and rearranging terms: PV = FV ( 1/ FVF n,i )

22. 22 Single Sum Problems (PVF n,i )Not wishing to have to constantly raise the term to the required power and then divide it into 1, I name the term 1/ FVF n,i the Present Value Factor for a single sum (PVF n,i ). I restate the formula PV = FV (PVF n,i ) read vertically for the “n”I then employ the table on page 322-323. The table is the result of the required multiplications and division at various “n” and “i” and is to be read vertically for the “n” horizontally for the “i” and horizontally for the “i”

23. 23 Single Sum Problems To solve my problem using the table: Please look up the values on the tables as we go along! n = 3, i = 10%, FV = $133 PV = FV (PVF n,i ) PV = $133 (.75132) PV = $100

24. 24 Single Sum Problems Note: Review the tables and note their characteristics. They are very logical. All sums in the future are worth less than themselves in the present. All factors on the present value of a single sum table are less than one. All factors on the future value of a single sum table are greater than one. All present sums are worth more than themselves in the future. Notice how the factors change dramatically as the “i” increases and the “n” lengthens!

25. 25 Single Sum Problems Single Sum problems, other unknowns. –Suppose you have $6,000 today (PV ) and you need $9,000 five years hence. What rate of annual interest must you earn to achieve your goal? Note: This can be solved as either a future or present value of a single sum problem. The formulas are reciprocals of each other.

26. 26 Single Sum Problems To solve as a future value of a single sum problem: PV(FVF n,i ) = FV PV(FVF n,i ) = FV where n = 5 and i = ? $6,000 (FVF n,i ) = $9,000 FVF n,i = $9,000/$6,000 = 1.500 Looking on the future of a single sum table (page 321) for n = 3 and a FVF of 1.500, I find the corresponding “i” to be between 8-9%.

27. 27 Single Sum Problems Alternatively, suppose I have $750 today. How long will I have to wait to have $1,200 when the interest rate is 10%? I will solve this as a PV problem. PV = FV (PVF n,i ) $750 = $1,200 (PVF n,i ) where n = ? and i = 10% PVF n,i = $750/$1,200 =.625 Reading on the present value of a single sum table (page 323) for 10% the “n” for the factor.625 is between 4-5 periods. A more precise answer may be derived through the use of the mathematical technique of interpolation.

28. 28 Annuity Problems Suppose I am to receive three equal $100 payments (R) each at the end of the period (in this case a period is a year) when the interest rate is a constant 15%. This is an ordinary annuity since payments are at the end of the period. What is the value to me at the end of the third year from receiving this annuity? This is a future value of an ordinary annuity problem. How do I go about solving it? –I realize an annuity is really a series of single sums. –If I take the FV for each single sum and add them I will have the value of the entire stream of payments. –I will use the FV formula which is: PV(FVF n,i ) = FV

29. 29 Annuity Problems Rent #Cmpd periodsSumFVFFV All at i = 15% 12$1001.322$132 21$1001.150$115 30$1001.000$100 Totals3.472$347 FVF-AO n,iThis is tedious! I notice I am multiplying a constant rent ($100) by a changing interest factor. What if I added the three factors and multiplied the total by the rent? That would be less work! I’ll call the sum of the appropriate factors the FVF-AO n,i. I’ll derive the general formula where R equals the constant rent: FV-OA = R (FVF-AO n,i ) FV-OA = R (FVF-AO n,i ) when I = 15% and n = 3 payments FV-OA = $100 (3.472) = $347 (rounded)

30. 30 Annuity Problems Where do I get these summed factors? From the future value of an ordinary annuity table on pages 324-325. The addition for the appropriate “n” and “i” has already been done. The annuity tables are derived directly from the single sum tables. What if the same annuity had been set up but the rents were received at the beginning rather than the end of the period? Then you would have the future value of an annuity due case. –I realize the only difference between the ordinary and due situations is the timing of the rents. Each rent has one more period to compound under the due situation than under the ordinary annuity. I adjust the formula for one more interest period: FV-AD = R (FVF-AO n,i ) (1 + i)

31. 31 Annuity Problems Is adjusting this factor this way logical? If I increase the factor for one more period’s interest the FV will go up for the annuity. That is what I want as the FV of an annuity due should be greater than the ordinary one. To find the FV-AD in this case: FV-AD = R (FVF-AO n,i ) (1 + i) FV-AD = R (FVF-AO n,i ) (1 + i) where n = 3 payments and i = 15% FV-AD = $100 (3.472)(1 +.15) = $399 (rounded) Note: The text does not have a table for the future value of an annuity due. If doing the calculations manually, approach the adjustment needed intuitively. Always apply a test of reasonableness in all calculations!

32. 32 Annuity Problems Let’s take the case of the present value of an ordinary annuity. That is, what is a stream of future payments worth today? What sum do you need today to draw out a series of equal payments and have nothing left over? This situation is very common in retirement cases or the payment of debt. Bonds Payable ABC Company

33. 33 Annuity Problems Suppose you are to pay three rents of $500, each at the end of the next three years. The interest rate is 8%. How much should I set aside today to have the required rents? –Again I realize an annuity is simply a series of single sums. I take the same approach as before in that I discount (remove) interest from each of the rents. –I take the present value of each, add, and I will have the required sum (total present value) that I will need. PV = FV (PVF n,i ) I use the formula: PV = FV (PVF n,i )

34. 34 Annuity Problems Rent # Disc periodsSumPVFPV All at i = 8% 11$500.92593$463 22$500.85734$429 33$500.79383$397 Totals2.5771$1,289 PVF- AO n,iThis is tedious! I notice I am multiplying a constant rent ($500) by a changing interest factor. What if I added the three factors and multiplied the total by the rent? That would be less work! I’ll call the sum of the appropriate factors the PVF- AO n,i. I’ll derive the general formula where R is the constant rent: PV-OA = R (PVF-AO n,i ) PV-OA = R (PVF-AO n,i ) for n = 3 payments and i = 8% PV-OA = $500 (2.5771) = $1,289 (rounded)

35. 35 Annuity Problems Where do I get these summed factors? From the present value of an ordinary annuity table on pages 326-327. The addition for the appropriate “n” and “i” has already been done. The annuity tables are derived directly from the single sum tables. present value of an annuity dueWhat if the same annuity had been set up but the rents were received at the beginning rather than the end of the period? Then you have the present value of an annuity due case. –I realize the only difference between the ordinary and due situations is the timing of the rents. Each rent has one less period to discount under the due situation than under the ordinary annuity. I adjust the formula for one more interest period: PV-AD = R (PVF-AO n,i )(1+i)

36. 36 Annuity Problems Is adjusting this factor this way logical? If I increase the factor the PV will go up for the annuity. That is what I want as the PV of an annuity due should be greater than the ordinary one. Each rent has one less period of interest removed from it! To find the PV-AD in this case: PV-AD = R (PVF-AO n,i )(1+i) PV-AD = R (PVF-AO n,i )(1+i) where n = 3 payments and i = 8% PV-AD = $500 (2.5771)(1 +.08) = $1,391 (rounded) Note: The text does have a table for the present value of an annuity due. But again, if you are without the table, approach the adjustment needed intuitively. Always apply a test of reasonableness in all calculations!