1. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. Time Value of Money Chapter 5 5-1

2. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. Time value of money – the concept The time value of money refers to the observation that it is better to receive money sooner than later. Money that you have in hand today can be invested to earn a positive rate of return, producing more money tomorrow. $1 today is worth more than $1 to be received in the future. Time value analysis has many applications, including planning for retirement, valuing stocks and bonds, setting up loan payment schedules, and making corporate decisions regarding investing in new plant and equipment. 3-2

3. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. Time value of money – the concept (cont’d....) - Present Value (PV) – beginning amount - Future Value (FV) – later amount on a time line - Interest rate (I) – discount rate is the variable that equates a present value today with a future value at some later date. - Cash flow (CF) - Number of periods (N) 3-3

4. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. Simple and compound interest Simple Interest: Interest is only paid on the principal. FV = PV + PV(I)(N) FV = 100 + 100*0.05*3 = $115 Compound Interest: Occurs when interest is earned on prior periods’ interest. 3-4

5. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. Time Lines A horizontal line on which time zero (present time) appears at the leftmost end and future periods are marked from left to right. 5-5 CF 0 CF 1 CF 3 CF 2 0123 I% Show the timing of cash flows. Tick marks occur at the end of periods, so Time 0 is today; Time 1 is the end of the first period (year, month, quarter, etc.) or the beginning of the second period. Cash flows are shown directly below the tick marks, and the relevant interest rate is shown just above the time line.

6. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. Future values The process of going to future values (FVs) from present values (PVs) is called compounding. What is the future value (FV) of an initial $100 after 3 years, if I/YR = 5%? FV can be solved by using the step-by-step, formula, financial calculator, and spreadsheet methods. 3-6 FV = ? 0123 5% 100

7. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. Solving for FV: The Step-by-Step Method After 1 year: FV 1 = 100(1 + 0.05) = $100(1.05) = $105.00 After 2 years: FV 2 = 105(1 + 0.05) = $105(1.05) = $110.25 After 3 years: FV 3 = 110.25(1 + 0.05) = $110.25(1.05) = $115.76 5-7

8. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. Solving for FV: Formula Methods After N years (general case): FV N = PV(1 + I) N OR, FV = PV(1 + r) t FV = future valuePV = present value r = period interest ratet = number of periods Future value interest factor = (1 + r) t  Using formula method: FV = $100(1+0.05) 3 = $115.76 The same investment with a simple interest rate gives a return of $115 at the end of 3 years (formula on slide 9). FV = 100 + 100*0.05*3 = $115 Another approach is to use the formula with the FV table: FV = PV * FVIF = $100 * 1.158 = $115.80 5-8

9. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. Simple versus Compound Interest - Compounding of interest magnifies the returns on an investment - Returns are magnified (i) The longer they are compounded (ii) The higher the rate they are compounded 3-9

10. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. Present Value Finding a present value is the reverse of finding a future value. Finding the PV of a cash flow or series of cash flows when compound interest is applied is called discounting (the reverse of compounding). A broker offers to sell you a Treasury bond that will pay $115.76 in 3 years from now. How much should you pay for that right now if the interest rate is 5%? 3-10 The present value of a cash flow due N years in the future is the amount which, if it were on hand today, would grow to equal the given future amount. PV = ?$115.76 0123 5%

11. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. Finding Present Value using step-by-step method 5-11 After 1 year: FV 1 = 115.76/(1 + 0.05) = $100/(1.05) = $110.25 After 2 years: FV 2 = 110.25/(1 + 0.05) = $110.25/(1.05) = $105.00 After 3 years: FV 3 = 105.00/(1 + 0.05) = $105.00/(1.05) = $100.00

12. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. Solving for PV: The Formula Method Solve the general FV equation for PV: PV= FV N /(1 + I) N PV= FV 3 /(1 + I) 3 = $115.76/(1.05) 3 = $99.997  $100.00 Using the PV table: PV = FV N * PVIF = 115.76 * 0.864 = 100.0512  $100.00 5-12

13. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. COMPOUNDING & DISCOUNTING 3-13

14. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. FINDING THE INTEREST RATE, I If we know the present value, the future value and the number of time periods, we can calculate the rate of return earned. For example, suppose we invested $5,000 six years ago Today, it is worth $10,000. What is the annually compounded rate of returned? 3-14

15. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. FINDING THE NUMBER OF YEARS, N If we know the present value, the future value and the interest rate, we can calculate the period of interest easily. For example, suppose we believe that we could retire comfortably if we had $1 million. We want to find how long it will take us to acquire $1 million, assuming we now have $500,000 invested at 4.5%. PV= FV N /(1 + I) N 500,000 = 1,000,000/(1.045) N (1.045) N = 2 Nlog 1.045 = log 2 N= 15.7473 years 3-15

16. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. ANNUITY A series of payments of an equal amount at fixed intervals for a specified number of periods. Two types of Annuities : - Ordinary Annuity – the payments occur at the end of each year. -Annuity due - the payments are made at the beginning of each year. 3-16

17. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. FUTURE VALUE OF AN ORDINARY ANNUITY Consider you deposit $100 at the end of each year for 3 years and earn 5% per year. How much will you have at the end of the third year? 3-17

18. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. FUTURE VALUE OF AN ORDINARY ANNUITY (cont’d....) 3-18 Step-by-step approach: Formula approach: = 100 [ {(1.05)^3 – 1} / 0.05] = $315.25

19. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. FUTURE VALUE OF AN ORDINARY ANNUITY (cont’d....) 3-19 Formula approach using annuity table: = PMT * future value of annuity interest factor = 100 * 3.153 = $315.3

20. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. FUTURE VALUE OF AN ANNUITY DUE 3-20 Because each payment occurs one period earlier with an annuity due, all of the payments earn interest for one additional period. Therefore, the FV of an annuity due will be greater than that of a similar ordinary annuity. FVA due = FVA ordinary * (1+i) = $315.25 * 1.05 = $331.01

21. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. PRESENT VALUE OF AN ORDINARY ANNUITY To find the PV, we discount them, dividing each payment by (1 + I). 3-21 Using the PVIFA table for I=5% and N=3: PVA = PMT * PVIFA = 100 * 2.723 = $272.30

22. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. FINDING ANNUITY PAYMENTS, PERIODS, AND INTEREST RATES Finding Annuity Payments, PMT (ordinary annuity): Suppose we need to accumulate $10,000 and have it available 5 years from now. Suppose further that we can earn a return of 6% on our savings, which are currently zero. How much should you save annually? FVA = PMT * FVIFA 10000= PMT * 5.637 PMT = $ 1773.99 For annuity due: The required payment for the annuity due is the ordinary annuity payment divided by (1 + I): $1,773.99/1.06 = $1,673.58. 3-22

23. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. PERPETUITIES In case of annuities, payments continue for a specific number of periods—for example, $100 per year for 10 years. However, some securities promise to make payments forever. This kind of payment stream is know an a perpetuity. A perpetuity is simply an annuity with an extended life. There are few actual perpetuities in existence, the United Kingdom government has issued them in the past. 3-23

24. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. UNEVEN CASH FLOWS Although many financial decisions involve constant payments, many others involve uneven, or nonconstant cash flows. For example, the dividends on common stocks typically increase over time, and investments in capital equipment almost always generate uneven cash flows. There are two important classes of uneven cash flows: (1) a stream that consists of a series of annuity payments plus an additional final lump sum (2) all other uneven streams 3-24

25. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. UNEVEN CASH FLOWS (CONT’D....) 3-25

26. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. PV OF UNEVEN CASH FLOWS 3-26 The PV of Stream 1 to be $927.90 and the PV of Stream 2 to be $1,016.35.

27. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. FV OF UNEVEN CASH FLOWS 3-27

28. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. SEMIANNUAL AND OTHER COMPOUNDING PERIODS Annual Compounding: The arithmetic process of determining the final value of a cash flow or series of cash flows when interest is added once a year. Semiannual Compounding: The arithmetic process of determining the final value of a cash flow or series of cash flows when interest is added twice a year. There can be N number of compoundings annually. 3-28

29. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. SEMIANNUAL AND OTHER COMPOUNDING PERIODS (cont’d....) 3-29

30. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. COMPARING INTEREST RATES Nominal (Quoted, or Stated) Interest Rate, I NOM : The contracted (or quoted or stated) interest rate. Annual Percentage Rate (APR) : The periodic rate times the number of periods per year. Effective (Equivalent) Annual Rate (EFF% or EAR): The annual rate of interest actually being earned, as opposed to the quoted rate. Also called the “equivalent annual rate.” 3-30

31. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. COMPARING INTEREST RATES (cont’d....) 3-31

32. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. COMPARING INTEREST RATES (cont’d....) 3-32 Effective Annual Rate Based on Frequency of Compounding Nominal Rate Semi-AnnualQuarterlyMonthlyDaily 1%1.002%1.004%1.005% 5%5.062%5.095%5.116%5.127% 10%10.250%10.381%10.471%10.516% 15%15.563%15.865%16.075%16.180% 20%21.000%21.551%21.939%22.134% 30%32.250%33.547%34.489%34.969% 40%44.000%46.410%48.213%49.150% 50%56.250%60.181%63.209%64.816%

33. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. Will the FV of a lump sum be larger or smaller if compounded more often, holding the stated I% constant? LARGER, as the more frequently compounding occurs, interest is earned on interest more often. 5-33 Annually: FV 3 = $100(1.10) 3 = $133.10 0123 10% 100133.10 Semiannually: FV 6 = $100(1.05) 6 = $134.01 0123 5% 456 134.01 1230 100

34. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. Why is it important to consider effective rates of return? Investments with different compounding intervals provide different effective returns. To compare investments with different compounding intervals, you must look at their effective returns (EFF% or EAR). 5-34

35. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. Why is it important to consider effective rates of return? See how the effective return varies between investments with the same nominal rate, but different compounding intervals. EAR ANNUAL 10.00% EAR SEMIANNUALLY 10.25% EAR QUARTERLY 10.38% EAR MONTHLY 10.47% EAR DAILY (365) 10.52% 5-35

36. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. What is the FV of $100 after 3 years under 10% semiannual compounding? Quarterly compounding? 5-36

37. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. Can the effective rate ever be equal to the nominal rate? Yes, but only if annual compounding is used, i.e., if M = 1. If M > 1, EFF% will always be greater than the nominal rate. 5-37

38. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. Loan Amortization An important application of compound interest involves loans that are paid off in installments over time. Amortization tables are widely used for home mortgages, auto loans, business loans, retirement plans, etc. A loan that is to be repaid in equal amounts on a monthly, quarterly, or annual basis is called an amortized loan. 5-38

39. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. Loan Amortization (cont’d….) Assume that we want to calculate an amortization table showing the amount of principal and interest paid each period for a $5,000 loan at 10% repaid in three equal annual instalments. 1 st step: Calculate annual payments: PVA = PMT * PVIFA 5000 = PMT * 2.487 PMT= $2010.45 5-39 2 nd step: Constructing the amortization table:

40. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. Illustrating an Amortized Payment: Where does the money go? Constant payments Declining interest payments Declining balance 5-40 $ 0123 402.11 Interest 302.11 Principal Payments