Report

Chapter Five First Principles of Valuation: The Time Value of Money Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-1 Chapter Organisation 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 Future Value and Compounding Present Value and Discounting More on Present and Future Values Present and Future Values of Multiple Cash Flows Valuing Equal Cash Flows: Annuities and Perpetuities Comparing Rates: The Effect of Compounding Periods Loan Types and Loan Amortisation Summary and Conclusions Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-2 Chapter Objectives • Distinguish between simple and compound interest. • Calculate the present value and future value of a single amount for both one period and multiple periods. • Calculate the present value and future value of multiple cash flows. • Calculate the present value and future value of annuities. • Compare nominal interest rates (NIR) and effective annual interest rates (EAR). • Distinguish between the different types of loans and calculate the present value of each type of loan. Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-3 Time Value Terminology 0 1 2 PV 3 4 FV • Future value (FV) is the amount an investment is worth after one or more periods. • Present value (PV) is the current value of one or more future cash flows from an investment. Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-4 Time Value Terminology • The number of time periods between the present value and the future value is represented by ‘t’. • The rate of interest for discounting or compounding is called ‘r’. • All time value questions involve four values: PV, FV, r and t. Given three of them, it is always possible to calculate the fourth. Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-5 Interest Rate Terminology • Simple interest refers to interest earned only on the original capital investment amount. • Compound interest refers to interest earned on both the initial capital investment and on the interest reinvested from prior periods. Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-6 Future Value of a Lump Sum You invest $100 in a savings account that earns 10 per cent interest per annum (compounded) for three years. After one year: After two years: After three years: $100 (1 + 0.10) = $110 $110 (1 + 0.10) = $121 $121 (1 + 0.10) = $133.10 Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-7 Future Value of a Lump Sum • The accumulated value of this investment at the end of three years can be split into two components: – original principal $100 – interest earned $33.10 • Using simple interest, the total interest earned would only have been $30. The other $3.10 is from compounding. Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-8 Future Value of a Lump Sum • In general, the future value, FVt, of $1 invested today at r per cent for t periods is: FVt $1 1 r t • The expression (1 + r)t is the future value interest factor (FVIF). Refer to Table A.1. Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-9 Example—Future Value of a Lump Sum • What will $1000 amount to in five years time if interest is 12 per cent per annum, compounded annually? FV $10001 0.12 $1000 1.7623 $1762.30 5 • • From the example, now assume interest is 12 per cent per annum, compounded monthly. Always remember that t is the number of compounding periods, not the number of years. FV $1000 1 0.01 60 $1000 1.8167 $1816.70 Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-10 Interpretation • The difference in values is due to the larger number of periods in which interest can compound. • Future values also depend critically on the assumed interest rate—the higher the interest rate, the greater the future value. Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-11 Future Values at Different Interest Rates Number of periods Future value of $100 at various interest rates 5% 10% 15% 20% 1 $105.00 $110.00 $115.00 $120.00 2 $110.25 $121.00 $132.25 $144.00 3 $115.76 $133.10 $152.09 $172.80 4 $121.55 $146.41 $174.90 $207.36 5 $127.63 $161.05 $201.14 $248.83 Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-12 Future Value of $1 for Different Periods and Rates Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-13 Present Value of a Lump Sum You need $1000 in three years time. If you can earn 10 per cent per annum, how much do you need to invest now? Discount one year: Discount two years: Discount three years: $1000 (1 + 0.10) –1 $909.09 (1 + 0.10) –1 $826.45 (1 + 0.10) –1 Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright = $909.09 = $826.45 = $751.32 5-14 Interpretation • In general, the present value of $1 received in t periods of time, earning r per cent interest is: PV $1 1 r $1 t 1 r t • The expression (1 + r)–t is the present value interest factor (PVIF). Refer to Table A.2. Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-15 Example—Present Value of a Lump Sum Your rich grandmother promises to give you $10 000 in 10 years time. If interest rates are 12 per cent per annum, how much is that gift worth today? PV $10 000 1 0.12 $10 000 0.3220 $ 3220 10 Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-16 Present Values at Different Interest Rates Number of periods Present value of $100 at various interest rates 5% 10% 15% 20% 1 $95.24 $90.91 $86.96 $83.33 2 $90.70 $82.64 $75.61 $69.44 3 $86.38 $75.13 $65.75 $57.87 4 $82.27 $68.30 $57.18 $48.23 5 $78.35 $62.09 $49.72 $40.19 Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-17 Present Value of $1 for Different Periods and Rates Present value of $1 ($) r = 0% 1.00 .90 .80 .70 r = 5% .60 .50 r = 10% .40 .30 r = 15% .20 r = 20% .10 1 2 3 4 5 6 Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 7 8 9 10 Time (years) 5-18 Solving for the Discount Rate • You currently have $100 available for investment for a 21year period. At what interest rate must you invest this amount in order for it to be worth $500 at maturity? • Given any three factors in the present value or future value equation, the fourth factor can be solved. r can be solved in one of three ways: • Use a financial calculator • Take the nth root of both sides of the equation • Use the future value tables to find a corresponding value. In this example, you need to find the r for which the FVIF after 21 years is 5 (500/100). Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-19 The Rule of 72 • The ‘Rule of 72’ is a handy rule of thumb that states: If you earn r per cent per year, your money will double in about 72/r per cent years. • For example, if you invest at 6 per cent, your money will double in about 12 years. • This rule is only an approximate rule. Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-20 Future Value of Multiple Cash Flows • You deposit $1000 now, $1500 in one year, $2000 in two years and $2500 in three years in an account paying 10 per cent interest per annum. How much do you have in the account at the end of the third year? • You can solve by either: – compounding the accumulated balance forward one year at a time – calculating the future value of each cash flow first and then totalling them. Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-21 Solutions • Solution 1 – – – • End of year 1: End of year 2: End of year 3: ($1000 1.10) + $1500 = $2600 ($2600 1.10) + $2000 = $4860 ($4860 1.10) + $2500 = $ 846 Solution 2 $1000 (1.10)3 $1500 (1.10)2 $2000 (1.10)1 $2500 1.00 Total = = = = = $1331 $1815 $2200 $2500 $7846 Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-22 Solutions Future value calculated by compounding forward one period at a time 0 2 1 3 4 5 Time (years) $0 $ 0 $0 0 $2200 2000 x 1.1 $2000 $4620 2000 x 1.1 $4200 $7282 2000 x 1.1 $6620 $10 210.20 2000 x 1.1 $9282 2000.00 x 1.1 $12 210.20 Future value calculated by compounding each cash flow separately 0 2 1 3 4 5 Time (years) $2000 $2000 $2000 $2000 x 1.1 $2000.0 2200.0 x 1.12 2420.0 x 1.13 2662.0 x 1.14 2928.2 Total future value $12 210.20 Figures 5.6/5.7 — Calculation of FV for Multiple Cash Flow Stream Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-23 Present Value of Multiple Cash Flows • You will deposit $1500 in one year’s time, $2000 in two years time and $2500 in three years time in an account paying 10 per cent interest per annum. What is the present value of these cash flows? • You can solve by either: – discounting back one year at a time – calculating the present value of each cash flow first and then totalling them. Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-24 Solutions • Solution 1 – End of year 2: – End of year 1: – Present value: • ($2500 1.10–1) + $2000 ($4273 1.10–1) + $1500 ($5385 1.10–1) = = = $4273 $5385 $4895 Solution 2 $2500 (1.10) –3 $2000 (1.10) –2 $1500 (1.10) –1 Total = = = = $1878 $1653 $1364 $4895 Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-25 Solutions Present value 0 2 1 3 4 5 calculated by discounting each $1000 $1000 $1000 $1000 $1000 x 1/1.06 Time cash flow separately (years) $ 943.40 x 1/1.062 890.00 x 1/1.063 839.62 x 1/1.064 792.09 x 1/1.065 747.26 $4212.37 Total present value r = 6% Present value 0 1 2 3 4 5 calculated by discounting back one $4212.37 $3465.11 $2673.01 $1833.40 $ 943.40 0.00 1000.00 1000.00 1000.00 1000.00 1000.00 $4212.37 $4465.11 $3673.01 $2833.40 $1943.40 $1000.00 $ 0.00 Time period at a time (years) Total present value = $4212.37 r = 6% Figures 5.8/5.9 — Calculation of PV for Multiple Cash Flow Stream Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-26 Annuities • An ordinary annuity is a series of equal cash flows that occur at the end of each period for some fixed number of periods. • Examples include consumer loans and home mortgages. • A perpetuity is an annuity in which the cash flows continue forever. Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-27 Present Value of an Annuity 1 1/1 r t PV C r C = equal cash flow • The discounting term is called the present value interest factor for annuities (PVIFA). Refer to Table A.3. Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-28 • Example 1 You will receive $500 at the end of each of the next five years. The current interest rate is 9 per cent per annum. What is the present value of this series of cash flows? 1 1/1.095 PV $500 0.09 $500 3.8897 $1944.85 Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-29 • Example 2 You borrow $7500 to buy a car and agree to repay the loan by way of equal monthly repayments over five years. The current interest rate is 12 per cent per annum, compounded monthly. What is the amount of each monthly repayment? 1 1/1.0160 $7 500 C 0.01 C $7 500 39.1961 $191.35 Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-30 Future Value of an Annuity 1 r 1 FV C t r • The compounding term is called the future value interest factor for annuities (FVIFA). Refer to Table A.4. Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-31 Example—Future Value of an Annuity What is the future value of $200 deposited at the end of every year for 10 years if the interest rate is 6 per cent per annum? 1.0610 1 FV $200 0.06 $200 13.181 $2636.20 Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-32 Perpetuities • The future value of a perpetuity cannot be calculated as the cash flows are infinite. • The present value of a perpetuity is calculated as follows: C PV r Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-33 Comparing Rates • The nominal interest rate (NIR) is the interest rate expressed in terms of the interest payment made each period. • The effective annual interest rate (EAR) is the interest rate expressed as if it was compounded once per year. • When interest is compounded more frequently than annually, the EAR will be greater than the NIR. Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-34 Calculation of EAR m NIR EAR 1 1 m m = number of times the interest is compounded Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-35 Comparing EARS • Consider the following interest rates quoted by three banks: – Bank A: 15%, compounded daily – Bank B:15.5%, compounded quarterly – Bank C:16%, compounded annually Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-36 Comparing EARS EARBank A 0.15 1 365 365 1 16.18% 4 0.155 EARBank B 1 1 16.42% 4 1 0.16 EARBank C 1 1 16% 1 Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-37 Comparing EARS • Which is the best rate? For a saver, Bank B offers the best (highest) interest rate. For a borrower, Bank C offers the best (lowest) interest rate. • The highest NIR is not necessarily the best. • Compounding during the year can lead to a significant difference between the NIR and the EAR. Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-38 Types of Loans • A pure discount loan is a loan where the borrower receives money today and repays a single lump sum in the future. • An interest-only loan requires the borrower to only pay interest each period and to repay the entire principal at some point in the future. • An amortised loan requires the borrower to repay parts of both the principal and interest over time. Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-39 Amortisation of a Loan Year Beginning Balance Total Payment Interest Paid Principal Paid Ending Balance 1 $5000.00 $1285.46 $450.00 $835.46 $4164.54 2 $4164.54 $1285.46 $374.81 $910.65 $3253.89 3 $3253.89 $1285.46 $292.85 $992.61 $2261.28 4 $2261.28 $1285.46 $203.52 $1081.94 $1179.33 5 $1179.33 $1285.46 $106.13 $1179.33 $0.00 $6427.30 $1427.30 $5000.00 Totals Copyright 2004 McGraw-Hill Australia Pty Ltd PPTs t/a Fundamentals of Corporate Finance 3e Ross, Thompson, Christensen, Westerfield and Jordan Slides prepared by Sue Wright 5-40