1. INTRODUCTION TO CORPORATE FINANCE SECOND EDITION Lawrence Booth & W. Sean Cleary Prepared by Jared Laneus

2. Chapter 5 Time Value of Money 5.1 Opportunity Cost 5.2 Simple Interest 5.3 Compound Interest 5.4 Annuities and Perpetuities 5.5 Nominal Versus Effective Rates 5.6 Loan or Mortgage Arrangements 5.7 Comprehensive Examples: A Basic Retirement Problem Appendix 5A: Growing Annuities and Perpetuities 2 Booth/Cleary Introduction to Corporate Finance, Second Edition

3. Learning Objectives 5.1 Explain the importance of the time value of money and how it is related to an investor’s opportunity costs. 5.2 Define simple interest and explain how it works. 5.3 Define compound interest and explain how it works. 5.4 Differentiate between an ordinary annuity and an annuity due, and explain how special constant payment problems can be valued as annuities and, in special cases, perpetuities. 5.5 Differentiate between quoted rates and effective rates, and explain how quoted rates can be converted to effective rates. 5.6 Apply annuity formulas to value loans and mortgages and set up an amortization schedule. 5.7 Solve a basic retirement problem. 3 Booth/Cleary Introduction to Corporate Finance, Second Edition

4. Opportunity Cost Money has a time value because it can be invested today and be worth more at a later date The opportunity cost of money is the interest rate that would be earned by investing it The interest rate is also known as the cost of money, the required rate of return or the discount rate For example, time value of money can help us compare the following three alternatives: Receive $20,000 today Receive $31,000 in 5 years Receive $3,000 each year in perpetuity In order to decide which alternative is best, we need to know what interest rate to use. 4 Booth/Cleary Introduction to Corporate Finance, Second Edition

5. Simple Interest Simple interest is interest paid or received on only the initial investment (the principal). Only used for a limited number of applications, while compound interest is used in far more applications. The value of a simple interest investment at time n is given by Equation 5-1 5 Booth/Cleary Introduction to Corporate Finance, Second Edition

6. Compound Interest Compound interest is interest that is earned on the principal amount invested and on any accrued interest. The amount of interest earned each year with simple interest is constant. But, with compound interest, the amount of interest earned each year increases every year since the interest rate is applied to the principal plus the interest earned and reinvested. This makes simple interest investments grow at a constant rate over time, while compound interest investments grow at an increasing rate over time. 6 Booth/Cleary Introduction to Corporate Finance, Second Edition

7. Compound Interest The future value of an investment using compound interest is given by Equation 5-2 We can discount, or find the present value of, a future value using compound interest with Equation 5-3 The holding period n can be computed as The rate of return k can be computed as 7 Booth/Cleary Introduction to Corporate Finance, Second Edition

8. Compound Interest Example where k is the annual geometric mean (%) for each asset class Notice that higher annual mean returns correspond with greater year-end values in 2008 Results will not be exact due to rounding the annual geometric mean 8 Booth/Cleary Introduction to Corporate Finance, Second Edition

9. Compare Simple and Compound Interest Example: Compare the interest earned and future values of equal investments of $1,000 at 5% per year simple and compound interest for five years. Time nSimple Interest Value Simple Interest Interest Earned Compound Interest Value Compound Interest Interest Earned 0$1,000.00n/a$1,000.0000n/a 1$1,050.00$50.00$1,050.0000$50.0000 2$1,100.00$50.00$1,102.5000$52.5000 3$1,150.00$50.00$1,157.6250$55.1250 4$1,200.00$50.00$1,125.50625$57.88125 5$1,250.00$50.00$1,276.2815625$60.7753125 9 Booth/Cleary Introduction to Corporate Finance, Second Edition

10. Annuities and Perpetuities Ordinary annuities generate constant payments PMT at regular intervals for a finite period of time n at the end of each period. Annuities due are like ordinary annuities, in that they made constant payments PMT at regular intervals for a finite period of time n. But, while ordinary annuities generate cash flows at the end of a period, annuities due generate cash flows at the beginning of a period. Perpetuities are like annuities in that a constant payment PMT is made at regular intervals but, unlike annuities, perpetuities continue to make payments for an infinite period of time. 10 Booth/Cleary Introduction to Corporate Finance, Second Edition

11. Ordinary Annuities Ordinary annuities generate constant payments PMT at regular intervals for a finite period of time n at the end of each period. The future value of an annuity due is given by Equation 5-4 The future value of an annuity due is given by Equation 5-5 11 Booth/Cleary Introduction to Corporate Finance, Second Edition

12. Annuities Due Annuities due are like ordinary annuities, in that they made constant payments PMT at regular intervals for a finite period of time n. But, while ordinary annuities generate cash flows at the end of a period, annuities due generate cash flows at the beginning of a period. The future value of an annuity due is given by Equation 5-6 The future value of an annuity due is given by Equation 5-7 12 Booth/Cleary Introduction to Corporate Finance, Second Edition

13. Perpetuities Perpetuities are like annuities in that a constant payment PMT is made at regular intervals but, unlike annuities, perpetuities continue to make payments for an infinite period of time. The present value of an ordinary perpetuity is given by Equation 5-8 The present value of a perpetuity due is given by: Note: perpetuities do not have future or accumulated values 13 Booth/Cleary Introduction to Corporate Finance, Second Edition

14. Annuity Examples Example: What is the future value of an investor’s savings if she deposits $2,000 at the end of each year for 40 years and the interest rate is 5%? Since payments occur at the end of the year, this is an ordinary annuity: 14 Booth/Cleary Introduction to Corporate Finance, Second Edition

15. Annuity Examples Example: What is the present value of an annuity that pays $1,000 at the beginning of each year for 25 years if the interest rate is 10%? Since payments occur at the beginning of the year, this is an annuity due: 15 Booth/Cleary Introduction to Corporate Finance, Second Edition

16. Perpetuity Example Example: Gloria has won a contest where the prize is annual payments at the end of each year of $1,000 forever. How much would she have to be paid today in order to give up the right to these perpetual payments if the appropriate interest rate is 10%? Gloria would have to be paid at least $10,000 in order to be willing to give up her right to receive the perpetual payments. 16 Booth/Cleary Introduction to Corporate Finance, Second Edition

17. Nominal Versus Effective Rates When the payment frequency (f ) and interest compounding frequencies (m) match (m = f ), interest rate calculations are relatively simple But, when payment and interest compounding frequencies do not match (m ≠ f ), we must calculate effective periodic rates using equation 5-11. Mortgages, where interest compounds semi-annually but payments are often made monthly, are an example (see section 5.6). Effective annual rates can be calculated using Equation 5-9 (where f = 1) The effective rate is the rate at which a dollar invested grows over a given period (e.g., daily, monthly, quarterly, etc.) 17 Booth/Cleary Introduction to Corporate Finance, Second Edition

18. Continuous Compounding As we increase the interest compounding frequency (m), we increase the effective annual rate. As m becomes infinite, we achieve continuous compounding, and the effective annual rate is given by Equation 5-10 where e is Euler’s number (approx. 2.718) 12% CompoundedEffective Annual Rate Annually (m = 1)12.0000% Semi-annually (m = 2)12.3600% Quarterly (m = 4)12.5509% Monthly (m = 12)12.6825% Daily (m = 365)12.7478% Continuously (m = ∞) 12.7497% 18 Booth/Cleary Introduction to Corporate Finance, Second Edition

19. Loan or Mortgage Arrangements Loans and mortgages often involve blended payments, where each payment is a combination of interest and principal so that the loan is amortized, or paid off in full, in an orderly fashion with equal payments over its term. Loans and mortgages are an important application of annuity concepts, since equal payments are made at regular intervals based on a fixed interest rate specified when the loan is taken out. Although loan payments are equal, the interest portion and principal portions fluctuate with each payment. Initially, the majority of a payment is interest, but over time the proportion represented by principal repayment increases. 19 Booth/Cleary Introduction to Corporate Finance, Second Edition

20. Loan Payments and Amortization Example: Suppose we borrow $5,000 with a 10% annual interest rate. We will repay the loan with annual blended payments at the end of each year. We solve Equation 5-5 for PMT to find the loan payment. The amortization schedule for this loan: 20 Booth/Cleary Introduction to Corporate Finance, Second Edition

21. Amortization Schedules Notice that: Beginning principal outstanding = amount loaned in the first period Payment (PMT) is constant, and found using Equation 5-5. Interest portion = Beginning principal outstanding × k Principal portion = Payment – Interest portion End principal outstanding = beginning principal outstanding– principal portion = beginning principal outstanding for next period 21 Booth/Cleary Introduction to Corporate Finance, Second Edition

22. Mortgages Quoted annual borrowing rates must be converted into effective period rates that correspond to the frequency of payments using Equation 5-11 whenever loans call for payment frequencies that are not annual (e.g., quarterly, monthly, weekly, etc.) In Canada, mortgage rates compound semi-annually and mortgage payments must be made at least once a month: In the notation of Equation 5-11: m = 2 and f ≥ 12. For monthly payments, the effective monthly rate given a quoted mortgage rate QR is: 22 Booth/Cleary Introduction to Corporate Finance, Second Edition

23. Mortgages Example: Determine the monthly payment for a $200,000 mortgage with an amortization period of 25 years based on a quoted rate of 6 percent. The effective monthly interest rate is: To calculate the monthly payment, we take the $200,000 borrowed as the present value of the annuity of payments required to repay the loan over 25 years (300 months): 23 Booth/Cleary Introduction to Corporate Finance, Second Edition

24. Comprehensive Example: A Basic Retirement Problem Example: An investor plans to retire 35 years from today and have sufficient savings to guarantee $48,000 each year for 20 years. Assume retirement withdrawals will be made at the beginning of each of the 20 years. The investor estimates that at the time of retirement, he can sell his business for $200,000. The expectation is that interest rates will be relatively stable at 8% per year for the next 35 years. Thereafter, the interest rate is expected to decline to 6% per year forever. The investor wants to make equal annual deposits at the end of each of the next 35 years. How much should be deposited each year in order to meet the stated objective? 24 Booth/Cleary Introduction to Corporate Finance, Second Edition

25. Comprehensive Example: A Basic Retirement Problem Solution: Step 1: Determine how much the investor must have accumulated when retirement starts 35 years from now. At that point in time the investor wants a 20-year annuity that will pay a constant $48,000 each year at the beginning of the year (an annuity due) at a 6% interest rate. The present value of this annuity is: 25 Booth/Cleary Introduction to Corporate Finance, Second Edition

26. Comprehensive Example: A Basic Retirement Problem Step 2: We know the investor will need to have accumulated $583,589.59 by the time he or she retires in order to fund 20 years of $48,000 annual payments at 6%. We also know that the investor will raise $200,000 by selling his or her business when he or she retires. This means that $383,589.59 must be accumulated as the future value of 35 years of investing for retirement. The next step is to determine our annual deposit for the 35 years until retirement given that the future value must be $383,589.59, and the interest rate is 8%. The investor must deposit $2,226.07 each year. 26 Booth/Cleary Introduction to Corporate Finance, Second Edition

27. Growing Annuities and Perpetuities Some annuities and perpetuities grow (or, if the rate is negative, shrink) at a constant rate g each period. In this case, we no longer have constant payments each period, but we do have a constant growth rate. The first three payments in a series of cash flows growing at constant rate g would therefore be: 27 Booth/Cleary Introduction to Corporate Finance, Second Edition

28. Growing Perpetuities The present value of a constant growth perpetuity is given by Equation 5A-2: Equation 5A-2 holds only when k > g, the discount rate is greater than the growth rate. Otherwise, the answer is becomes negative and has no practical interpretation. Only future estimated cash flows and estimated growth in these cash flows are relevant, while past cash flows are not. Equation 5A-2 holds only when the growth rate is constant. If the grow rate is non-constant, alternative valuation methods are necessary. 28 Booth/Cleary Introduction to Corporate Finance, Second Edition

29. Growing Perpetuities Example: An apartment building generated $100,000 in after-tax cash flow last year. We assume that rent will increase at 4% each year forever and that the appropriate discount rate is 15%. The present value of these cash flows is: 29 Booth/Cleary Introduction to Corporate Finance, Second Edition

30. Growing (or Shrinking) Annuities A growing annuity is similar to a growing perpetuity in that payments grow at a constant rate each period, but the annuity only lasts for a finite period of time (n periods). The present value of a constant growth annuity is given by Equation 5A-3: Positive growth rates (g > 0) yield growing annuities. Negative growth rates (g < 0) yield shrinking annuities. Notice that if g = 0, Equation 5A-3 simplifies to Equation 5-5, the present value of an ordinary annuity. 30 Booth/Cleary Introduction to Corporate Finance, Second Edition

31. Growing Annuities Example: Jim’s pension will pay him annual payments that grow at a constant rate of 3% per year for the next 25 years. This year’s payment, to be received at the end of the year, will be $40,000. If the appropriate discount rate is 10%, what is the present value of these cash flows? Therefore, Jim’s pension is worth about $461,000 today. 31 Booth/Cleary Introduction to Corporate Finance, Second Edition

32. Shrinking Annuities Example: Alice will receive royalty payments from a gas well on land she owns. As the resource will be depleted over the next 10 years, she expects the royalties to decline by 5% per year over the next decade. This year’s royalty payment is $10,000. If the appropriate discount rate is 15%, what is the present value of Alice’s royalty payments? Therefore, Alice’s royalties are worth about $42,600 today. 32 Booth/Cleary Introduction to Corporate Finance, Second Edition

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