Chapter 5: Introduction to Valuation: The Time Value of Money

Size: px

Start display at page:

Download "Chapter 5: Introduction to Valuation: The Time Value of Money"

Augustus Wood
6 years ago
Views:

1 Chapter 5: Introduction to Valuation: The Time Value of Money Faculty of Business Administration Lakehead University Spring 2003 May 12, 2003 Outline of Chapter Future Value and Compounding 5.2 Present Value and Discounting 5.3 More on Present and Future Values 1

2 5.1 Future Value and Compounding Future value refers to the amount of money an investment would grow to over some length of time at a given rate of interest. To determine this value, it is important to know when interest is calculated. Is it once a year? Every six months? Each month? When many payments are involved, it is also important to know the timing of these payments Future Value and Compounding Investing for a Single Period Suppose $100 is invested in an account that pays 10% per year. This investment will then be worth after one year = ( ) 100 = = $110 3

3 5.1 Future Value and Compounding Investing for More than One Period Suppose $100 is invested in an account that pays 10% per year. After one year, this investment will be worth $110. If the interest payment is reinvested, this investment will be worth, after two years, = = = (1.1) = $ Future Value and Compounding Decomposing (1.1) gives us = ( ) 100 = 100 }{{} Capital + 20 }{{} Interest on capital Simple interest + 1 }{{} Interest on interest Compound interest 5

4 5.1 Future Value and Compounding More generally, $m invested at a period interest rate r will grow to (1 + r) t m = m }{{} Capital after t periods. + t r m }{{} Simple interest + Compound interest Future Value and Compounding Compound interest can be significant over the long run. Take $100 invested for T years at 10% compounded annually: Ending Simple Compound T Amount Capital Interest Interest = = = = =

5 5.1 Future Value and Compounding Examples of Future Value Calculations 1. $2,250 invested for 30 years at 18% compounded annually gives 2,250 (1.18) 30 = $322, $9,310 invested for 15 years at 6% compounded annually gives 9,310 (1.06) 15 = $22, One More Example 5.1 Future Value and Compounding 3. You are scheduled to receive $22,000 in two years. When you receive it, you will invest it for six more years at 6 percent per year. How much will you have in eight years? Answer: 22,000 (1.06) 8 2 = 22,000 (1.06) 6 = $31,

6 5.2 Present Value and Discounting Present value refers to the amount of money that has to be invested today to obtain a specific amount of money after a specific length of time at a given rate of interest. If, for example, we want to know how much to invest to obtain $1 after one year at 10% interest, we need to solve Present value 1.1 = $1 Present value = = $ Present Value and Discounting More generally, the amount of money that needs to be invested today to obtain $1 in t years at the annual rate of interest r is PV = 1 (1 + r) t. This amount is the present value, as of today, of $1 to be received in t years discounted at the annual rate r. 11

7 5.2 Present Value and Discounting Examples of Present Value Calculations 1. The present value of $15,000 to be received in 5 years, discounted at the annual rate 12%, is PV = 15,000 (1.12) 5 = $8, The present value of $25,000 to be received in 10 years, discounted at the annual rate 8%, is PV = 25,000 = $11, (1.08) More on Present and Future Values With a period rate of interest r and a number of periods t, we can define Future value factor = (1 + r) t Present value factor = 1 (1 + r) t 13

8 5.3 More on Present and Future Values Let PV 0 denote the present value, as of today (date 0), of an investment that will grow to the future value FV t in t periods, the period interest rate being r. Then PV 0 (1 + r) t = FV t and, equivalently PV 0 = FV t (1 + r) t. This result is the basic present value equation More on Present and Future Values Determining the Discount Rate What must r be for PV 0 to grow to FV t in t periods? PV 0 (1 + r) t = FV t (1 + r) t = FV t PV 0 ( FVt 1 + r = PV 0 ) 1/t r = ( ) 1/t FVt 1. PV 0 15

9 5.3 More on Present and Future Values Example of Discount Rate Determination You are offered an investment that requires you to put up $12,000 today in exchange for $40, years from now. What is the annual rate of return on the investment? Answer: In this example, PV 0 = 12,000, FV t = 40,000 and t = 12. Therefore, r = ( ) 40,000 1/12 1 = 10.55%. 12, More on Present and Future Values Finding the Number of Periods What must t be for PV 0 to grow to FV t at a rate r? Note: We will be using the following rules: ln(ab) = ln(a) + ln(b) ln ( a b) = bln(a) ( a ln = ln(a) ln(b). b) 17

10 Finding the Number of Periods PV 0 (1 + r) t = FV t ln ( PV 0 (1 + r) t) = ln(fv t ) ln(pv 0 ) + ln ( (1 + r) t) = ln(fv t ) ln(pv 0 ) + t ln(1 + r) = ln(fv t ) t = ln(fv t) ln(pv 0 ) ln(1 + r) t = ln(fv t/pv 0 ) ln(1 + r) 18 How Long to Double Your Money? Knowing r, how many periods is needed for PV 0 to double? t = ln(fv t/pv 0 ) ln(1 + r) = ln(2pv 0/PV 0 ) ln(1 + r) = ln(2) ln(1 + r) 19

11 The Rule of 72 Note that when r is small, ln(1 + r) r (slightly below r); ln(2) = (slightly below 0.72). A good approximation of the time it takes to double an investment is 0.72 = 72 r 100r. If r = 8%, PV 0 will double in approximately 72/8 = 9 years. 20 The Rule of 72 r ln(1 + r) ln(2) ln(1+r) r 2% % % % % % %

12 The Rule of 72 The rule of 72 holds exactly at around 7.85%. The rule of 72 will overestimate the time it takes to double an investment when r < 7.85%; underestimate the time it takes to double an investment when r > 7.85%; 22 The Rule of 72 When r is small, the error will be insignificant. The error is significant when using large numbers. Take r = 72%, for instance. According to the rule of 72, an investment doubles in approximately one year at this rate. This makes no sense: it takes r = 100% to double an investment in one year. 23

13 Finding the Number of Periods: An Example You are trying to save to buy a new $120,000 Ferrari. You have $40,000 today that can be invested at 8% compounded annually. How long will it take before you have enough money to buy the car? Answer: t = ln(120,000/40,000) ln(1.08) = ln(3) ln(1.08) = years. 24

Time Value of Money. Lakehead University. Outline of the Lecture. Fall Future Value and Compounding. Present Value and Discounting

Time Value of Money. Lakehead University. Outline of the Lecture. Fall Future Value and Compounding. Present Value and Discounting Time Value of Money Lakehead University Fall 2004 Outline of the Lecture Future Value and Compounding Present Value and Discounting More on Present and Future Values 2 Future Value and Compounding Future