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1. Introduction to Time Value of Money
1. What does time value of money means? The concept of time value of money simply means that a dollar received today is worth more than a dollar to be received in the future. This is because you can invest that dollar today and earn interest on it. Gradually, interest is earned on that dollar plus the interest it has already generated. This is the concept of ____(Critical Concept). See Slide Time Value of Money (52.0K) . See Future and Present Value (53.0K)
2. Why is time value of money important in finance? Financial managers must make a decision whether or not to purchase new plant and equipment for capital projects. Future benefits must be large enough to justify those capital outlays. Time value of money is a tool that allows the financial manager to calculate the current value of future benefits generated by capital projects.
3. Solving time value of money problems. You can solve time value of money problems in three different ways. First, you can use a financial calculator. Second, you can use the actual formulas given in Chapter 9. Third, you can use the interest factor tables in the back of your textbook. For a good tutorial on time value of money, check out this virtual lecture here for high-bandwidth users, as well as here for low-bandwidth users.
2. Future Value of a Lump Sum
1. Future value of a single amount. First, you must understand that if a single lump sum of money is deposited in a bank, or other financial institution, and allowed to grow at a given interest rate over a period of time, the end result will be its future value. See Slide 3 (42.0K)
2. Future value formula. You can calculate the future value of a single amount given this formula:

FV = PV X (1 + i)ⁿ

Where: FV = Future Value

PV = Present Value

i = interest rate and n = time

3. Interest Factor Tables. Using the interest factor tables, Appendix A, use the following formula:

FV = PV X FV_IF Slide 4 (56.0K)

Where: FV = Future Value

PV - Present Value

IF = factor from interest factor table

3. Present Value of a Lump Sum
1. Present value of a single amount. Present value is the mathematical reciprocal of future value. If you expect to receive a sum of money at some future time, what is the value of that sum of money today? In other words, what would you pay for the opportunity of receiving that money today? A good example is the lottery. If you win the lottery, you may be given the option of receiving your proceeds over 20 years or as a lump sum. Money that you receive in the lump sum now is worth more to you than money you receive over 20 years due to the time value of money. See Slide for relationship between present value and future value Slide 2 (29.0K)
2. Present Value Formula. You can calculate the present value of a lump sum using this formula:

PV = FV X 1/(1 + i)ⁿ (See Slide 6 (56.0K) )

Where: FV = Future Value

PV = Present Value

I = interest rate and n = time

3. Interest Factor Tables. Using interest factor tables, Appendix B, use the following formula:

PV = FV X PV_IF (See Slide 7 (56.0K) )

Where: FV = Future Value

PV = Present Value

IF = factor from interest factor table

D. Example of Present Value of a Lump Sum. See Slide 8 (56.0K) .

4. Future Value of An Annuity
1. Future Value of An Annuity. An annuity is a ____(Key Term). In calculating the future value of an annuity, you must consider the value of compounding for each time period in which the annuity is deposited. The best methods are to use either a financial calculator or the interest factor tables at the end of your textbook. See Annuity (44.0K) .
2. Future Value of An Annuity Formula. Using Appendix C:

FVA = A X FV_IFA

Where FVA = Future Value of An Annuity

A = Annuity Payment

FVIFA = interest factor given an interest rate and time period

See Slide 5 (36.0K) to look at a graphical representation of the compounding process for an annuity.

3. Example. If you decide to deposit $100 per month every month for 5 years in to your savings account, you would use the Future Value of An Annuity method to calculate the value of your savings at the end of five years.
5. Present Value of An Annuity
1. Present Value of An Annuity. The present value of an annuity is the reverse of the future value of an annuity.
2. Present Value of An Annuity Formula: Using the Appendix D:

PVA = A X PV_IVA

Where PVA = Present Value of An Annuity

A = Annuity Payment

PVIFA = interest factor given an interest rate and time period

3. Example. Again, using the lottery example, if you win the lottery and choose to have your proceeds paid to you in equal payments over 20 years, the payment in year 20 is not worth as much to you today as the payment in year 1.
6. Solving for the value of the Annuity
1. Value of the annuity. So far, in time value of money problems, we've been solving for future and present values. You can also solve for the value of an annuity.
2. Annuity equaling a future value. If you want to calculate how much you should save to have a given sum of money at some date in the future, use this formula:

A = FVA/FV_IFA

Where A = Annuity Value

FVA = Future Value of the Annuity

FVIFA = interest factor

3. Annuity equaling a present value. If you already know the present value of an amount, this formula will help you determine how many withdrawals you can make from that amount over a period of time.

A = PVA/PV_IFA

Where A = Annuity Value

PVA = Present Value of the Annuity

PVIFA = interest factor

7. Solving for the yield or interest rate
1. Solving for the yield or return on a lump sum. We can also manipulate the formulas used so far to determine the yield or interest rate we will earn if we know the present value and future vale of the investment:

PVIF = PV/FV

Go to the table, present value of a lump sum, and find the PVIF. The column under which the PVIF lies is the approximate interest rate.

2. Solving for the yield or return on an annuity. We can also solve for yield or return if the cash flows from the investment are annuity payments.

PVIFA = PVA/A

Go to the table, present value of an annuity, and find the PVIFA. The column under which the PVIFA lies is the approximate interest rate.

8. Special types of time value of money problems
1. Compounding more often than annually. If interest is compounded, for example, semi-annually, solve for the future or present value in the same way. However, when determining the correct interest factor to use, divide the interest rate by ____ (the number of times interest is compounded) and multiply the number of years to maturity by ____(Critical Concept).
2. Deferred Annuity. A deferred annuity is an annuity paid sometime in the future. First, calculate the present value of the annuity using Appendix D. Then, calculate the present value using Appendix B.