Chp. 4 Finance: The Time Value of Money

Interest rate (r)

"exchange rate" between earlier money and later money

Present Value - Important Relationship I: For a given interest rate - the longer the time period, the lower the present value What is the present value of $500 to be received in 5 years? 10 years? The discount rate is 10%

5 years: PV = 500 / (1.1)5 = 310.46 10 years: PV = 500 / (1.1)10 = 192.77

Future Values - Example 2; Suppose you invest the $1,000 from the previous example for 5 years. How much would you have?

FV = 1,000 x (1.05)5 = 1,276.28

Future Values - Example 3: Suppose you had a relative deposit $10 at 5.5% interest 200 years ago. How much would the investment be worth today? What is the effect of compounding?

FV = 10(1.055)200 = 447,189.84 Simple interest = 10 + [200 x (10) x (.055)] = 120.00 Compounding added $447,078.84 to the value of the investment

Future Values: General Formula

FV = PV(1 + r)t FV = future value PV = present value r = period interest rate, expressed as a decimal t = number of periods Future value interest factor = (1 + r)t

Finding the Number of Periods

FV = PV(1 + r)t t = ln(FV / PV) / ln(1 + r)

Present Values

How much do I have to invest today to have some amount in the future? FV = PV(1 + r)t Rearrange to solve for PV = FV / (1 + r)t When we talk about discounting, we mean finding the present value of some future amount. When we talk about the "value" of something, we are talking about the present value unless we specifically indicate that we want the future value.

Rule of 72

If you earn r% per year, your money will double in about 72/r% years. For example, if you invest at 6%, your money will double in 12 years (72/6 = 12)

Suppose you invest $1000 for one year at 5% per year. What is the future value in one year?

Interest = 1,000(.05) = 50 Value in one year = principal + interest = 1,000 + 50 = 1,050 Future Value (FV) = 1,000(1 + .05) = 1,050

Compound interest

Interest earned on both the initial principal and the interest reinvested from prior periods (interest on interest).

Simple interest

Interest earned only on the original amount of principal invested. FV = Principal + [(principal) x (# periods) x (interest rate)]

Discount Rate - Example 1

It is very important at this point to make sure that the students have more than 2 decimal places visible on their calculator. r = (1200 / 1000)1/5 - 1 = .03714 = 3.714%

Discount Rate

Often we will want to know what the implied interest rate is in an investment Rearrange the basic PV equation and solve for r: FV = PV(1 + r)t r = (FV / PV)1/t - 1 If you are using formulas, you will want to make use of both the yx and the 1/x keys

Present Value - One Period Example: Suppose you need $10,000 in one year for the down payment on a new car. If you can earn 7% annually, how much do you need to invest today?

PV = 10,000 / (1.07)1 = 9,345.79

Present Values - Example 2: You want to begin saving for you daughter's college education and you estimate that she will need $150,000 in 17 years. If you feel confident that you can earn 8% per year, how much do you need to invest today?

PV = 150,000 / (1.08)17 = 40,540.34

The Basic PV Equation - Refresher

PV = FV / (1 + r)t There are four parts to this equation PV, FV, r and t If we know any three, we can solve for the fourth If you are using a financial calculator, be sure and remember the sign convention or you will receive an error when solving for r or t

Present Value - Important Relationship II: For a given time period - the higher the interest rate, the smaller the present value What is the present value of $500 received in 5 years if the interest rate is 10%? 15%?

Rate = 10%: PV = 500 / (1.1)5 = 310.46 Rate = 15%; PV = 500 / (1.15)5 = 248.58

What is the "Time Value of Money"? Which would you choose: $5,000 today or in 1 year? What if the choice were $5,000 in two years or $5,500 in three years?

Robinson's shovel is capital, and because it has productive potential, he is willing to pay a premium (i.e., clams) to acquire it. A positive interest rate exists, but there is no money involved in this example, just capital and consumption!

Future Value as a General Growth Formula

Suppose your company expects to increase unit sales of widgets by 15% per year for the next 5 years. If you currently sell 3 million widgets in one year, how many widgets do you expect to sell in 5 years?

the larger the FVIF will be for any given t, and consequently, the larger the future value will be.

The larger the interest rate is

Suppose you leave the money in for another year. How much will you have two years from now?

We are just using algebra when deriving the FV formula. We have 1000(1) + 1000(.05) = 1000(1+.05) FV = 1,000(1.05)(1.05) = 1,000(1.05)2 = 1,102.50

Present Value (PV)

earlier money on a time line

Discounting

finding the present value of some future amount

future value factors are always greater than 1 because

future values are greater than present values for positive interest rates

The time value of money exists because of

interest rates

Future Value (FV)

later money on a time line

Discount Rate - Example 3: Suppose you have a 1-year old son and you want to provide $75,000 in 17 years towards his college education. You currently have $5,000 to invest. What interest rate must you earn to have the $75,000 when you need it?

r = (75,000 / 5,000)1/17 - 1 = .172688 = 17.27% This is a great problem to illustrate how TVM can help you set realistic financial goals and maybe adjust your expectations based on what you can currently afford to save.

Number of Periods - Example 1: You want to purchase a new car and you are willing to pay $20,000. If you can invest at 10% per year and you currently have $15,000, how long will it be before you have enough money to pay cash for the car?

t = ln(20,000 / 15,000) / ln(1.1) = 3.02 years

Interest Rate

the difference between the values of current and future goods

The longer the investment earns interest

the larger the FVIF will be, and consequently, the larger the future value will be.

What is the relationship between present value and future value?

the present value is always less than the future value when we have positive rates of interest