## Wednesday, January 23, 2008

### Time Value of Money

People continually face important financial decisions that require an understanding of the time value of money. Should we buy or lease car? How much and how soon do we need to save for our children's education? What size house can we afford? Should we refinance our home mortgage? How much must we save in order to retire comfortably? The answer to these question are often complicated, and they depend on a number of factors, such as housing and education costs, interest rates, inflation, expected family income, and stock market returns.
In case Mr. Jones, who earns \$85,000, retires in 2000, expects to live for another 20 years after retirement, and needs 80 percent of her pre-retirement income, she would require \$68,000 during 2000. However, if inflation amounts to 5 percent per year, her income requirement would increase to \$110,765 in 10 years and to \$180,424 in 20 years. If inflation were 7 percent, her year 20 requirement would jump to \$263,139! How much wealth would Ms. Jones need at retirement to maintain her standard of living, and how much would she have had to save during each working year to accumulate that wealth?
The answer depends on a number of factors, including the rate she could earn on saving, the inflation rate, and when her saving program began. Also the answer would depend on how much she will get from her corporate retirement plan, if she has.
For this above sample cases, we have to understanding the concept of the time value of money. It will help us a lot too, to calculate another personal financial planning.

Future Value
A dollar in hand today is worth more than a dollar in the future because, if you had it now, you could invest it, earn interest, and end up with more than one dollar in the future. Future value (FVs) is the amount to which a cash flow or series of cash flow will grow over a given period of time when compounded at a given interest rate.
Suppose you deposit \$100 in a bank that pays 5 percent interest each year. How much would you have at the end of five years?

Equation 1:
FVn = PV (1+i)n
Where
FVn = future value or ending amount, of your account at the end of n years.
PV = present value, or beginning amount, in your account.
i = interest rate the bank pays on the account per year
n = number of periods involved in the analysis.

You could find the FV by clicking the function wizard, then financial, then scrolling down to FV, and then clicking OK to bring up the FV dialog box. Then enter 0.05 for Rate, 5 for Nper, 0 or leave blank for Pmt because there are no periodic payment, -100 for PV, and 0 or leave blank for type to indicate that payment occur at the end of the period. Then, when you click OK, you get the future value, \$127.63

Present Value
The present value of a cash flow due n years in the future is the amount which, if it were on hand today, would grow to equal the future amount. So the present value (PV) is the value today of a future cash flow or series of cash flows. Since \$100 would grow to \$127.63 in five years at a 5 percent interest rate, \$100 is the percent value 0f \$127.63 due in five years when the opportunity cost rate is 5 percent.

Equation 2:
PV = FVn / (1+i)n

In Excel, click the function wizard, indicate that you want a Financial function, scroll down, and double click PV. Then, in the dialog box, enter 0.05 for Rate, 5 for Nper, 0 for Pmt (because there are annual payments), -127.63 for FV,and 0 (or leave blank) for type because the cash flow occurs at the end of the year. Then press OK to get the answer, PV=\$100.00.

Source: Eugene F. Brigham, Joel F. Houston, Fundamentals of Financial Management, Harcourt, Inc, 2001.

### Time Value of Money (Annuity)

Future Value of an Annuity
An annuity is a series of equal payments made at fixed intervals for a specified number of periods. If the payments occur at the end of each period, as they typically do, the annuity is called an ordinary or deferred annuity.

Equation 3
FVAn = PMT (FVIFAi,n)
Where
FVAn = The future value of an annuity over n periods
PMT = The payments at the end of each period
FVIFAi,n = Future value interest factor for a annuity

If you deposit \$100 at the end of each year for three years in a savings account that pay 5 percent interest per year, how much will you have at the end of three years?

In Excel, click the function wizard, Financial, FV, and OK to get the FV dialog box. Then, we would enter 0.05 for Rate, 3 for Nper, and -100 for Pmt (the payment is entered as a negative number to show that it is a cash outflow). We would leave PV blank because there is no initial payment, and we would leave Type blank to signify that payments come at the end of the periods. Then, we clicked OK, we would get the FV of the annuity, \$315.25.

If payments are made at the beginning of each period, the annuity is an annuity due.

Equation 4
FVAn (Annuity due) = PMT (FVIFAi,n)(1+i)

For the annuity due, proceed just as for the ordinary annuity except enter 1 for Type to indicate that we now have an annuity due. Then, when you click OK, the answer \$331.01 will appear.

Present Value of an Annuity
If the payments come at the end of each year, then the annuity is an ordinary annuity.

Equation 5
PVAn = PMT (PVIFAi,n)
Where
PVAn = The present value of an annuity of n periods
PMT = The payments at the end of each period
PVIFAi,n = Present value interest factor for a annuity

Suppose you were offered the following alternatives: (1) a three-year annuity with payments of \$100 or (2) a lump sum payment today. You have no need for the money during the next three years, so if you accept the annuity, you would deposit the payments in a bank account that pays 5 percent interest per year. Similarly, the lump sum payment would be deposit into a bank account. How large must the lump sum payment today be to make it equivalent to the annuity?