1 FINC3131 Business Finance Chapter 5: Time Value of Money: The Basic Concepts

2 Time value of money: Practical relevance Examples 1.Retirement 2.Mortgage payment 3.Price of a stock 4.Helping your company to decide which project to undertake

3 Learning objectives 1.Understand the difference between a nominal interest rate and a real interest rate 2.Recall the present value and future value formulas 3.Solve 1-period time value of money problems involving one cash flow. 4.Solve 2-period time value of money problems involving one cash flow. 5.Solve 2-period time value of money problems involving 2 cash flows.

4 Question 1.A bill of $100 today 2.A bill of $100 one year from now Which bill has a higher value?

5 1.Money has time value 2.For the same amount of money, people require some compensation if receiving it later. 3.For adding or subtracting money received (or paid) at different times, the amounts must first be converted to a common basis: the same point in time Important implications

6 Interest rates 1 1.Real rate of interest (r real ) : the return (compensation) you demand for lending someone money and thus postponing consumption. 2.In a world with NO inflation, we only need to work with the real rate of interest. Inflation: general rise in prices. Same commodity becomes more expensive over time.

7 Interest rates 2 1.In a world with inflation, when you lend money to people, you want to adjust your interest rate for inflation. 2.Inflation-adjusted interest rate is known as Nominal rate of interest (r nominal ). In the real world, all the quoted rates are nominal rates (e.g., car loan, house loan, student loan)

8 Preparing BAII Plus for use 1.Press ‘2nd’ and [Format]. The screen will display the number of decimal places that the calculator will display. If it is not eight, press ‘8’ and then press ‘Enter’. 2.Press ‘2nd’ and then press [P/Y]. If the display does not show one, press ‘1’ and then ‘Enter’. 3.Press ‘2nd’ and [BGN]. If the display is not END, that is, if it says BGN, press ‘2nd’ and then [SET], the display will read END.

9 Time Value Basics Deposit problem: If you save $100 in a bank deposit account earning 10% annually, how much will be in the account after one year? = PrincipalFuture Value +Interest= (.10)= x(1+.10)=

10 Time Value Basics What would it be worth after two years? x1.1= But, since 110 = 100(1+.10)… x(1+.10) x (1+.10)= or x(1+.10) 2 =

11 The Formula for Future Value Future Value Present Value Number of periods Rate of return or discount rate or interest rate or growth per period Right now we look at n = 1, n = 2

12 The Formula for Future Value 1.The formula lets you convert a current cash flow (present value) into its future value. 2.This process is called compounding.  What about the reverse process? How do we convert future cash flows into their present values?

13 The Formula for Present Value From before, we know that Solving for PV, we get Unless otherwise stated, r stated on an annual basis.

14 1.The formula lets you convert future cash flows into their present values. 2.This process is called discounting. The Formula for Present Value

15 More observations We assume the discount rate, r, is positive. With that assumption, we can say: 1.PV is always less than FV. 2.1/(1 + r) is always less than one. 3.(1 + r) is always greater than one.

16 Discount rate and PV  If FV ($100) after 1-year is fixed, then as the discount rate increases, PV decreases.

17 FV and PV formulas 1.These are the basic building blocks that we will use to construct more complex concepts. 2.Not surprisingly, we will use these basic building blocks to solve complicated TVM problems.

TVM problems 1.Given PV, n, and r, find FV. 2.Given FV, n, and r, find PV. 3.Given PV, n, and FV, find r. 4.Given PV, r, and FV, find n. 18

19 1-period, find FV You need to borrow $1,700 to buy a computer and a bank is offering a loan at an interest rate of 14 percent. If you plan to repay the loan after one year, how much will you have to pay the bank? Use FV = PV(1 + r)

20 1-period, find PV What is the present value of $16,000 to be received at the end of one year if the discount rate is 10 percent? Use PV = FV/(1 + r)

21 1-period, find r If the bank promises to give you $28,400 after one year if you deposit $27,000 today, what is the annual interest rate that the bank offers? Use r = (FV/PV) – 1

22 2-period, find FV You plan to lend $11,000 to your friend at an interest rate of 8 percent per year compounded annually. The loan is to be repaid in two years. How much will your friend pay you at that time? Use FV = PV(1+r) 2

23 2-period, find PV You are offered a chance to buy into an investment that promises to pay you $28,650 after two years. If your required rate of return is 12 percent, what is the maximum price that you would pay for this investment? Use PV = FV/(1+r) 2

24 2-period, find r You are offered an investment opportunity which requires you to pay $31,500 today and promises to give you $39,700 at the end of the second year. What is the rate of return on this investment?

25 Value additivity principle You will make two investments. The first investment will give you $5,500 after one year and the second investment will give you $12,100 after one year. If your required rate of return for both investments is 10 percent, what is the present value of your investments?

26 Value additivity principle This problem illustrates the value additivity principle which says: You can add values only when they are scaled at the SAME point in time.

27 2-period, 2 cash flows, find FV You deposit $5,000 in a bank account today. You will make another deposit of $4,000 into the account at the end of the first year. If the bank pays interest at 6 percent compounded annually, how much will you have in your account after two years?

28 Time line: visualizing cash flows 1.For a TVM problem with 2 or more periods, a time line helps you to understand the problem better. 2.A time line is a graphical representation of a TVM problem. For the previous problem, the time line would look like this: t = 0 -$ 5,000 t = 1 t = 2 -$ 4,000 ?

29 2-period, 2 cash flows, find PV You want to withdraw $3,200 from your account at the end of the first year and $7,300 at the end of the second year. How much should you deposit in your account today so that you can make these withdrawals? Your account pays 6 percent p.a. Draw a time line to help you understand the question.

30 Summary 1.Time value of money 2.Real vs. nominal interest rate 3.1-period problems (find FV, PV, r) 4.2-period problems (find FV, PV, r) 5.Value additivity principle 6.2-period, 2 cash flows problems (find FV, PV)