1. 9 Chapter The Time Value of Money McGraw-Hill Ryerson
©2003 McGraw-Hill Ryerson Limited

2. Chapter 9 - Outline Time Value of Money Future Value and Present Value
PPT 9-2
Chapter 9 - Outline
Time Value of Money
Future Value and Present Value
Annuities
Time-Value-of-Money Formulas
Adjusting for Non-Annual Compounding
Compound Interest Tables
Summary and Conclusions

3. Time Value of Money Money Has Time Value
In 1624 Manhattan Island was sold for $24.
$24 invested at 6% annual interest rate could be $130 billion in 2009 : Buy New York City partly
$ 1 received 2009 years ago invested at 6%
Source:

4. PPT 9-3
Time Value of Money
The basic idea behind the concept of time value of money is:
$1 received today is worth more than $1 in the future OR
$1 received in the future is worth less than $1 today
Why?
because interest can be earned on the money
The connecting piece or link between present (today) and future is the interest or discount rate

5. Future Value and Present Value
PPT 9-4
Future Value and Present Value
Future Value (FV) is what money today will be worth at some point in the future
Present Value (PV) is what money at some point in the future is worth today

6. Figure 9-1 Relationship of present value and future value
PPT 9-5
Figure 9-1 Relationship of present value and future value $
$1,464.10
future
value
$1,000 present value
10% interest 1 2 3 4
Number of periods

7. C. 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.

8. Future Value of a Lump Sum
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.
B. 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

9. C. Interest Factor Tables
Using the interest factor tables, Appendix A and the following formula:
FV = PV X FVIF
Where: FV = Future Value
PV - Present Value
IF = factor from interest factor table
Sample Problem: You invest Tk for four years at 10 percent interest. What is the value at the end of the fourth year?

10. 2. Present Value of a Lump Sum 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.

11. B. Present Value Formula
B. Present Value Formula. You can calculate the present value of a lump sum using this formula:
PV = FV X 1/(1 + i)n
Where: FV = Future Value
PV = Present Value
I = interest rate and n = time
C. Interest Factor Tables. Using interest factor tables, Appendix B, use the following formula:
PV = FV X PVIF
IF = factor from interest factor table
Sample problem: You will receive Tk.1,000 after four years at a discount rate of 10 percent. How much is this worth today?

12. 3. Future Value of An Annuity
An annuity is a series of equal, consecutive payments.
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.

13. B. Future Value of An Annuity Formula Using Appendix C:
FVA = A X FVIFA
Where FVA = Future Value of An Annuity
A = Annuity Payment
FVIFA = Interest factor given an interest rate and time period
Sample Problem: You will receive Tk. 1,000 at the end of each period for four periods. What is the accumulated value (future worth) at the end of the fourth period if money grows at 10 percent?

14. 4. Present Value of An Annuity
The present value of an annuity is the reverse of the future value of an annuity.
B. Present Value of An Annuity Formula:
Using the Appendix D:
PVA = A X PVIVA
Where PVA = Present Value of An Annuity
A = Annuity Payment
PVIFA = interest factor given an interest rate and time period
Sample Problem: You will receive Tk. 1,000 at the end of each period for four years. At a discount rate of 10 percent, what is the cash flow currently worth?

15. 5. Solving for the value of the Annuity
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.
B. 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/FVIFA
Where A = Annuity Value
FVA = Future Value of the Annuity
FVIFA = Interest factor
Sample problem: You need Tk. 1,000 after four periods. With an interest rate of 10 percent, how much must be set aside at the end of each period to accumulate this amount

16. C. 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/PVIFA
Where A = Annuity Value
PVA = Present Value of the Annuity
PVIFA = interest factor
Sample problem: You deposit Tk. 1,000 today and wish to withdraw funds equally over four years. How much can you withdraw at the end of each year if funds earn 10 percent?

17. 6. Solving for the yield or interest rate
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.
Sample problem: You invest Tk. 1,000 now, and the funds are expected to increase to Tk. 1,360 after four periods. What is the yield on the investment?

18. 6. Solving for the yield or interest rate
B. 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.

19. 7. Special types of time value of money problems
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 2 (the number of times interest is compounded) and multiply the number of years to maturity by 2.
Sample problem: You invest Tk. 1,000 compounded semiannually at 8 percent per annum over four years. Determine the future value.

20. 7. Special types of time value of money problems
B. 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.
Sample problem: you will receive Tk per period, starting at the end of the fourth period and running through the end of the eighth period. With a discount rate of 8 percent, determine the present value.

21. Summary and Conclusions
PPT 9-20
Summary and Conclusions
The financial manager uses the time value of money approach to value cash flows that occur at different points in time
A dollar invested today at compound interest will grow a larger value in future. That future value, discounted at compound interest, is equated to a present value today
Cash values may be single amounts, or a series of equal amounts (annuity)