Chapter 4: Introduction to Valuation: The Time Value of Money

Future value factor:

Factor value factor = (1 + r)^t

As our examples suggest, the future value of $1 invested for t periods at a rate of r period is:

Future value = $1 x (1 + r)^t this is sometimes called the future value interest factor (or future value factor)

what will you have after two years, going back to our $100 investment, assuming the interest rate doesnt change?

If you leave the entire $110 in the bank, you will earn $110 x .10 = $11 in interest during the second year, so you will have a total of $110 + 11 = $121. This $121 is the future value of $100 in two years at 10 percent. Another way of looking at it is that one year from now you are effectively investing $110 at 10 percent for a year. This is a single-period problem, so you'll end up with $1.10 for every dollar invested, or $110 x 1.1 = $121 total

Present Value factor:

Present va

One of the basic problems faced by the financial manager is how to

determine the value today of cash flows expected in the future

the phrase time value of money refers to the fact that a

dollar in hand today is worth more than a dollar promised at some time in the future

One reason for this is that you could

earn interest while you waited; so a dollar today would grow to more than a dollar later

any easy way to calculate future value factors

find in calculator "y^x" where you usually enter 1.1 press the key and then enter 5 then press the = key to get the answer

Future Value

refers to the amount of money an investment will grow to over some period of time at some given interest rate put another way: future value is the cash value of an investment at some time in the future

EX 2 We now take a closer look at how we calculated the $121 future value. We multiplied $110 by 1.1 to get $121. The $110, however, was $100 also multiplied by 1.1. In other words:

$121 = $110 x 1.1 =($100 x 1.1) x 1.1 =$100x (1.1 x 1.1) =$100 x 1.1^2 = $100 x 1.21

EX 3 How much would our $100 grow to after three years? Once again, in two years, we'll be investing $121 for one period at 10 percent. We'll end up with $1.1 for every dollar we invest, or $121 x 1.1 = $133.1 total. This $133.1 is thus:

$133.1 = $121 x 1.1 =($110 x 1.1) x 1.1 =($100 x 1.1) x 1.1 x 1.1 = $100 x (1.1 x 1.1 x 1.1) = $100 x 1.1^3 =$100 x 1.331

what would $100 be worth after 5 years using the formula

(1 + r)^t = (1 + .10)^5 = 1.1 = 1.6105 so the $100 will thus grow to : $100 x 1.6105 = $161.05

EX suppose you locate a two-year investment that pays 14 percent per year. If you invest $325, how much will you have at the end of the two years? How much of this is simple interest? How much is compound interest?

-At the end of the first year, you will have $325 x 1.14 = $370.50. If you reinvest this amount, and thereby compound the interest, you will have $370.50 x 1.14 = $422.37 at the end of the second year. The total interest you earn is thus $422.37 -325 = $97.37. Your $325 original principal earns $325 x .14 = $45.50 in interest each year, for a two-year total of $91 in simple interest. The remaining $97.37-91 = $6.37 results from compounding. You can check this by noting that the interest earned in the first year is $45.5. The interest on interest earned in the second year thus amounts to $45.5 x .14 = $6.37, as we calculated

EX 4 You've located an investment that pays 12 percent. That rate sounds good to you, so you invest $400. How much will you have in three years? How much will you have in seven years? At the end of seven years, how much interest have you earned? How much of that interest results from compounding?

1. We can calculate the future value factor for 12 percent and three years as: (1 + r) ^t = 1.12^3 = 1.4049 Your $400 thus grows to : $400 x 1.4049 = $561.97 After seven years, you will have: $400 x 1.12^7 = $400 x 2.2107 = $884.27 Because you invested $400, the interest in the $884.27 future value is $884.27 - 400 = $484.27. At 12 percent, your $400 investment earns $400 x .12 = $48 in simple interest every year Over seven years, the simple interest thus totals 7 x $48 = $336. The other $484.27 - 336 = $148.27 is from compounding

Investing for a single period:

Suppose you were to invest $100 in a savings account that pays 10 percent interest per year. How much would you have in one year? You would have $110. This $110 is equal to your original principal of $100 plus $10 in interest that you earn. We say that $110 is the future value of $100 invested for one year at 10 percent, and we mean that $100 today is worth $110 in one year, given that 10 percent is the interest rate

When we discuss future value, we are thinking of questions such as the following: What will my $2000 investment grow to if it earns a 6.5 percent return every year for the next six years

There is another type of question that comes up even more often in financial management that is obviously related to future value. Suppose you need to have $10,000 in 10 years, and you can earn 6.5 percent on your money. How much do you have to invest today to reach your goal? You can verify that the answer $5,327.26

compounding the interest means earning interest on interest so we call the result

compound interest

the process of leaving your money and any accumulated interest in an investment for more than one period, thereby reinvesting the interest is called

compounding

with simple interest, the

interest is not reinvested, so interest is earned each period only on the original principal

Present value x 1.1 = $1

or solving for the present value: = $1/1.1 = $.909