ISBN: 0-03-028931-9
Chapter 7 Time Value of Money
Financial decisions often involve situations in which someone pays money at one point in time and receives money at some later time. Dollars that are paid or received at two different points in time are different, and this difference is recognized and accounted for by time value of money (TVM) analysis. This is an outline of a few of the main concepts, please refer to the textbook for more comprehensive coverage.
Compounding is the process of determining the future value (FV) of a cash flow or a series of cash flows. The compounded amount, or future value, is equal to the beginning amount plus the interest earned.
Discounting is the process of finding the present value (PV) of a future cash flow or a series of cash flows; discounting is the reciprocal of compounding.
An annuity is defined as a series of equal periodic payments (PMT) for a specified number of periods.
An annuity whose payments occur at the end of each period is called an ordinary annuity.If each payment of an annuity occurs at the beginning of the period rather than at the end, then it is an annuity due. Comparing an ordinary annuity and an annuity due, the PV of each payment would be larger, because each payment would be discounted back one year less, so the PV of the annuity due would be larger. Similarly, the FV of the annuity due would also be larger because each payment would be compounded for an extra year.
A perpetuity is a special breed of annuities. In a perpetuity, the annuity payment is streamed out forever. The value of a perpetuity in the present can be obtained by dividing the perpetuity payment by the relevant interest rate.
If a cash flow stream is unequal (all of the payment are not the same), we could not use the annuity formulas. To find the PV or FV of an uneven series, find the PV or FV of each individual cash flow and then sum them. Note, though, that if some of the cash flows constitute an annuity, then the annuity formula can be used to calculate the present value of that part of the cash flow stream.
TVM calculations generally involve equations which have four variables, and if you know three of the values, you (or you calculator) can solve the fourth.
Thus far, we have assumed that payments are made, and interest is earned, annually. However, many contracts call for more frequent payments; for example, mortgage and auto loans call for monthly payments and most bonds pay interest semiannually. Similarly, most banks compute interest daily. When compounding occurs more frequently than once a year this fact must be recognized. Often, you will have to determine the periodic interest rate to make calculations. The periodic interest rate is arrived upon by dividing the nominal rate by the number of periods in a year. Furthermore, to carry out calculations, you will also have to change the number of years to the appropriate number of periods, corresponding with the periodic rate.
If we are to compare the costs of loans that require different numbers of payments thorough the year (a.k.a. one investment makes more payments in a year), we must conduct comparisons based upon the equivalent (or effective) rates of return.
An amortized loan is one that is paid off in equal payments over a specified period. An amortization schedule shows how much of each payment constitutes interest, how much is used to reduce the principal, and the unpaid balance at each point in time.
The concepts covered in this chapter will be used throughout the remainder of the book. For example, in Chapters 8 and 9, we will apply present value concepts to the process of valuing bonds and stocks, and we will see that the market prices of securities are established by determining the present values of the cash flows they are expected to provide. In later chapters, the same basic concepts are applied to corporate decisions involving expenditures on capital assets, to the types of capital that should be used to pay for assets, to leasing decisions, and so forth.
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