Time Value of Money

TIME VALUE OF MONEY (CHAPTER 4)

1. Future value (FV), the value of a present amount at a future date, is calculated by applying compound interest over a specific time period. Present value (PV), represents the dollar value today of a future amount, or the amount you would invest today at a given interest rate for a specified time period to equal the future amount. Financial managers prefer present value to future value because they typically make decisions at time zero, before the start of a project.

2. A single amount cash flow refers to an individual, stand alone, value occurring at one point in time. An annuity consists of an unbroken series of cash flows of equal dollar amount occurring over more than one period. A mixed stream is a pattern of cash flows over more than one time period and the amount of cash associated with each period will vary.

3. Compounding of interest occurs when an amount is deposited into a savings account and the interest paid after the specified time period remains in the account, thereby becoming part of the principal for the following period. The general equation for future value in year n (FVn) can be expressed using the specified notation as follows:
FVn ’ PV × (1 + i)n 4. A decrease in the interest rate lowers the future amount of a deposit for a given holding period, since the deposit earns less at the lower rate. An increase in the holding period for a given interest rate would increase the future value. The increased holding period increases the FV since the deposit earns interest over a longer period of time.

5. The present value of a future amount indicates how much money today would be equivalent to the future amount if one could invest that amount at a specified rate of interest. Using the given notation, the present value of a future amount (FVn) can be defined as follows:
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6. An increasing required rate of return would reduce the present value of a future amount, since future dollars would be worth less today. Looking at the formula for present value in Question 5, it should be clear that by increasing the i value, which is the required return, the present value interest factor would decrease, thereby reducing the present value of the future sum.

7. Present value calculations are the exact inverse of compound interest calculations. Using compound interest, one attempts to find the future value of a present amount; using present value, one attempts to find the present value of an amount to be received in the future.

8. An ordinary annuity is one for which payments occur at the end of each period. An annuity due is one for which payments occur at the beginning of each period.

The ordinary annuity is the more common. For otherwise identical annuities and interest rates, the annuity due results in a higher FV because cash flows occur earlier and have more time to compound. 9. The PV of an ordinary annuity, PVAn, can be determined using the formula:
PVAn ’ PMT × (PVIFAi%,n) where: PMT ’ the end of period cash inflows PVIFAi%,n ’ the PV interest factor of an annuity for interest rate i and n periods.

The PVIFA is related to the PVIF in that the annuity factor is the sum of the PVIFs over the number of periods for the annuity. For example, the PVIFA for 5% and 3 periods is 2.723, and the sum of the 5% PVIF for periods one through three is 2.723 (0.952 + 0.907 + 0.864).

10. The FVIFA factors for an ordinary annuity can be converted for use in calculating an annuity due by multiplying the FVIFAi%,n by 1 + i.

11. The PVIFA factors for an ordinary annuity can be converted for use in calculating an annuity due by multiplying the PVIFAi%,n by 1 + i.

12. A perpetuity is an infinite-lived annuity. The factor for finding the present value of a perpetuity can be found by dividing the discount rate into 1.0. The resulting quotient represents the factor for finding the present value of an infinite-lived stream of equal annual cash flows.

13. The future value of a mixed stream of cash flows is calculated by multiplying each year’s cash flow by the appropriate future value interest factor. To find the present value of a mixed stream of cash flows multiply each year’s cash flow by the appropriate present value interest factor. There will be at least as many calculations as the number of cash flows.

14. As interest is compounded more frequently than once a year, both (a) the future value for a given holding period and (b) the effective annual rate of interest will increase. This is due to the fact that the more frequently interest is compounded, the greater the quantity of money accumulated and reinvested as the principal value. In situations of intra-year compounding, the actual rate of interest is greater than the stated rate of interest.

15. Continuous compounding assumes interest will be compounded an infinite number of times per year, at intervals of microseconds. Continuous compounding of a given deposit at a given rate of interest results in the largest value when compared to any other compounding period.

16. The nominal annual rate is the contractual rate that is quoted to the borrower by the lender. The effective annual rate, sometimes called the true rate, is the actual rate that is paid by the borrower to the lender. The difference between the two rates is due to the compounding of interest at a frequency greater than once per year.

APR is the annual percentage rate and is required by “truth in lending laws” to be disclosed to consumers. This rate is calculated by multiplying the periodic rate by the number of periods in one year. The periodic rate is the nominal rate over the shortest time period in which interest is compounded. The APY, or annual percentage yield, is the effective rate of interest that must be disclosed to consumers by banks on their savings products as a result of the “truth in savings laws.” These laws result in both favorable and unfavorable information to consumers. The good news is that rate quotes on both loans and savings are standardized among financial institutions. The negative is that the APR, or lending rate, is a nominal rate, while the APY, or saving rate, is an effective rate. These rates are the same when compounding occurs only once per year.

17. The size of the equal annual end-of-year deposits needed to accumulate a given amount over a certain time period at a specified rate can be found by dividing the interest factor for the future value of an annuity for the given interest rate and the number of years (FVIFAi%,n) into the desired future amount. The resulting quotient would be the amount of the equal annual end-of-year deposits required. The future value interest factor for an annuity is used in this calculation:

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18. Amortizing a loan into equal annual payments involves finding the future payments whose PV at the loan interest rate just equals the amount of the initial principal borrowed. The formula is:

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19. a. Either the present value interest factor or the future value interest factor can be used to find the growth rate associated with a stream of cash flows. The growth rate associated with a stream of cash flows may be found by using the following equation, where the growth rate, g, is substituted for k.

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To find the rate at which growth has occurred, the amount received in the earliest year is divided by the amount received in the latest year. This quotient is the PVIFi%;n. The growth rate associated with this factor may be found in the PVIF table. b. To find the interest rate associated with an equal payment loan, the present value interest factors for a one-dollar annuity table would be used. To determine the interest rate associated with an equal payment loan, the following equation may be used:

PVn ’ PMT × (PVIFAi%,n)

Solving the equation for PVIFAi%,n we get:

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Then substitute the values for PVn and PMT into the formula, using the PVIFA table to find the interest rate most closely associated with the resulting PVIFA, which is the interest rate on the loan.

20. To find the number of periods it would take to compound a known present amount into a known future amount you can solve either the present value or future value equation for the interest factor as shown below using the present value:

PV ’ FV × (PVIFi%,n)

Solving the equation for PVIFi%,n we get:
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Then substitute the values for PV and FV into the formula, using the PVIF table for the known interest rate find the number of periods most closely associated with the resulting PVIF. The same approach would be used for finding the number of periods for an annuity except that the annuity factor and the PVIFA (or FVIFA) table would be used. This process is shown below.
PVn ’ PMT × (PVIFAi%,n)

Solving the equation for PVIFAi%,n we get: [pic]