1. Chapter 5 The Time Value of Money
Pr. Zoubida SAMLAL

2. The Interest Rate
Which would you prefer -- $10,000 today or $10,000 in 5 years?
Obviously, $10,000 today.
You already recognize that there is TIME VALUE TO MONEY!!

3. Why is TIME such an important element in your decision?
Why TIME?
Why is TIME such an important element in your decision?
TIME allows you the opportunity to postpone consumption and earn INTEREST.

4. Types of Interest Simple Interest Compound Interest
Interest paid (earned) on only the original amount, or principal, borrowed (lent).
Compound Interest
Interest paid (earned) on any previous interest
earned, as well as on the principal borrowed
(lent).

5. Simple Interest Formula
Formula SI = P0(i)(n)
SI: Simple Interest
P0: Deposit today (t=0)
i: Interest Rate per Period
n: Number of Time Periods

6. Simple Interest (FV) What is the Future Value (FV) of the deposit?
FV = P0 + SI = $1,000 + $ = $1,140
Future Value is the value at some future time of a present amount of money, or a series of payments, evaluated at a given interest rate.

7. Simple Interest (PV)
What is the Present Value (PV) of the previous problem?
The Present Value is simply the $1,000 you originally deposited.
That is the value today!
Present Value is the current value of a future amount of money, or a series of payments, evaluated at a given interest rate.

8. Types of TVM Calculations
There are many types of TVM calculations
The basic types will be covered in this review module and include:
Present value of a lump sum
Future value of a lump sum
Present and future value of cash flow streams
Present and future value of annuities
Keep in mind that these forms can, should, and will be used in combination to solve more complex TVM problems

9. Future Value Single Deposit
FV1 = P0 (1+i)1 = $1,000 (1.07) = $1,070
Compound Interest
You earned $70 interest on your $1,000 deposit over the first year.
This is the same amount of interest you would earn under simple interest.

10. Future Value Single Deposit
FV1 = P0 (1+i)1 = $1,000 (1.07) = $1,070
FV2 = FV1 (1+i) = P0 (1+i)(1+i) = $1,000(1.07)(1.07) = P0 (1+i)2 = $1,000(1.07) = $1,144.90
You earned an EXTRA $4.90 in Year 2 with compound over simple interest.

11. General Future Value Formula
FV1 = P0(1+i)1
FV2 = P0(1+i)2
General Future Value Formula:
FVn = P0 (1+i)n
or FVn = P0 (FVIFi,n)
etc.

12. Single-Sum Future Value
What factor do we use?

13. Single-Sum Future Value
Table A-1- FV of 1$
$10, x = $12,597
Present Value
Factor
Future Value

14. Example of FV of a Lump Sum
How much money will you have in 5 years if you invest $100 today at a 10% rate of return?
Draw a timeline
Write out the formula using symbols:
FVt = CF0 * (1+r)t
i = 10%
$100 ? 1 2 3 4 5

15. Example of FV of a Lump Sum
Substitute the numbers into the formula:
FV = $100 * (1+.1)5
Solve for the future value:
FV = $161.05
Check answer using a financial calculator:
i = 10%
n = 5
PV = $100
PMT = $0
FV = ?

16. General Present Value Formula
PV0 = FV1 (1+i)-1
PV0 = FV2(1+i)-2
General Future Value Formula:
PV0 = FVn (1+i)-n
or PV0 = FVn (PVIFi,n)
etc.

17. Single-Sum Present Value
Table A-2- PV of 1$
What factor do we use?

18. Single-Sum Problems Table A-2 Table A-2- PV of 1$
$20, x = $12,710
Future Value
Factor
Present Value

19. Example of PV of a Lump Sum
How much would $100 received five years from now be worth today if the current interest rate is 10%?
Draw a timeline
The arrow represents the flow of money and the
numbers under the timeline represent the time period.
Note that time period zero is today.
i = 10%
$100 ? 1 2 3 4 5

20. Example of PV of a Lump Sum
Write out the formula using symbols:
PV = CFt / (1+r)t
Insert the appropriate numbers:
PV = 100 / (1 + .1)5
Solve the formula:
PV = $62.09
Check using a financial calculator:
FV = $100
n = 5
PMT = 0
i = 10%
PV = ?

21. Annuities
Annuity requires the following:
Periodic payments or receipts (called rents) of the same amount,
The same-length interval between such rents, and
Compounding of interest once each interval.
Two Types
Ordinary annuity - rents occur at the end of each period.
Annuity Due - rents occur at the beginning of each period.

22. Annuities Rents occur at the end of each period.
Future Value of an Ordinary Annuity
Rents occur at the end of each period.
No interest during 1st period.
Present Value
Future Value
$20,000
20,000
20,000
20,000
20,000
20,000
20,000
20,000 1 2 3 4 5 6 7 8

23. Future Value of an Ordinary Annuity
Present Value
Future Value
$20,000
20,000
20,000
20,000
20,000
20,000
20,000
20,000 1 2 3 4 5 6 7 8
Bayou Inc. will deposit $20,000 in a 12% fund at the end of each year for 8 years beginning December 31, Year 1. What amount will be in the fund immediately after the last deposit?
What table do we use?

24. Future Value of an Ordinary Annuity
Table A-3- FV of an annuity payment of 1$ per year
What factor do we use?

25. Future Value of an Ordinary Annuity
Table A-3- FV of an annuity payment of 1$ per year
Table A-3
$20, x = $245,994
Deposit
Factor
Future Value

26. Example of FV of an Annuity
Write out the formula using symbols:
FVAt = PMT * {[(1+r)t –1]/r}
Substitute the appropriate numbers:
FVA20 = $100 * {[(1+.15)20 –1]/.15
Solve for the FV:
FVA20 = $100 *
FVA20 = $10,244.36

27. Example of FV of an Annuity
Check using calculator:
Make sure that the calculator is set to one period per year
PMT = $100
n = 20
i = 15%
FV = ?

28. Present Value of an Ordinary Annuity
$100,000
100,000
100,000
100,000
100,000
100,000 1 2 3 4 19 20
Jaime Yuen wins $2,000,000 in the state lottery. She will be paid $100,000 at the end of each year for the next 20 years. How much has she actually won? Assume an appropriate interest rate of 8%.
What table do we use?

29. Present Value of an Ordinary Annuity
Table A-4
Table A-4- PV of an annuity payment of 1$ per year
What factor do we use?

30. Present Value of an Ordinary Annuity
Table 6-4
Table A-4- PV of an annuity payment of 1$ per year
$100, x = $981,815
Receipt
Factor
Present Value

31. Formulas of Annuities Present value of an annuity:
PVA = PMT * {[1-(1+r)-t]/r}
Future value of an annuity:
FVA t = PMT * {[(1+r)t –1]/r}

32. How about a stream of payments that are NOT equal??

33. Formulas of Cash Flow stream
Future value of a cash flow stream: n
FV = S [CFt * (1+r)n-t]
t=0
Present value of a cash flow stream:
PV = S [CFt / (1+r)n-t]

34. Example of PV of a Cash Flow Stream
Joe made an investment that will pay $100 the first year, $300 the second year, $500 the third year and $1000 the fourth year. If the interest rate is ten percent, what is the present value of this cash flow stream?
Draw a timeline:
$100
$300
$500
$1000 1 2 3 4
A-2 r=10% , n= 1
A-2 r=10% , n= 2
i = 10%
A-2 r=10% , n= 3
A-2 r=10% , n= 4

35. Example of PV of a Cash Flow Stream
Table A-2- PV of 1$
CF1 * Factor 1 +
CF2 * Factor 2 +
CF3 * Factor 3 +
CF4* Factor 4 =
Present value of a cash flow stream: n
PV = S [CFt / (1+r)n-t]
t=0

36. Example of PV of a Cash Flow Stream
Write out the formula using symbols: n
PV = S [CFt / (1+r)t]
t=0 OR
PV = [CF1/(1+r)1]+[CF2/(1+r)2]+[CF3/(1+r)3]+[CF4/(1+r)4]
Substitute the appropriate numbers:
PV = [100/(1+.1)1]+[$300/(1+.1)2]+[500/(1+.1)3]+[1000/(1.1)4]

37. Example of PV of a Cash Flow Stream
Solve for the present value:
PV = $ $ $ $683.01
PV = $
Check using a calculator:
Make sure to use the appropriate rate of return, number of periods, and future value for each of the calculations. To illustrate, for the first cash flow, you should enter FV=100, n=1, i=10, PMT=0, PV=?. Note that you will have to do four separate calculations.

38. Example of FV of a Cash Flow Stream
Joe made a decision to start saving money. He will pay $100 now year, $300 the first year, $500 the second year and $1000 the third year. If the interest rate is ten percent, what is the future value of this cash flow stream?
Draw a timeline:
i = 10%
$100
$300
$500
$1000
A-1 r=10% , n= 4 + 1 2 3 4
A-1 r=10% , n= 3 +
A-1 r=10% , n= 2 +
A-1 r=10% , n= 1

39. Example of FV of a Cash Flow Stream
Table A-1- FV of 1$
CF3* Factor 1 +
CF2 * Factor 2 +
CF1 * Factor 3 +
CF0* Factor 4 =
Future value of a cash flow stream: n
FV = S [CFt * (1+r)n-t]

40. Rule of Thumb
The following are simple rules that you should always use no matter what type of TVM problem you are trying to solve:
Stop and think: Make sure you understand what the problem is asking. You will get the wrong answer if you are answering the wrong question.
Draw a representative timeline and label the cash flows and time periods appropriately.
Write out the complete formula using symbols first and then substitute the actual numbers to solve.
Check your answers using a calculator.
While these may seem like trivial and time consuming tasks, they will significantly increase your understanding of the material and your accuracy rate.

41. We will use the “Rule-of-72”.
Double Your Money!!!
Quick! How long does it take to double $5,000 at a compound rate of 12% per year (approx.)?
We will use the “Rule-of-72”.

42. Approx. Years to Double = 72 / i%
The “Rule-of-72”
Quick! How long does it take to double $5,000 at a compound rate of 12% per year (approx.)?
Approx. Years to Double = 72 / i%
72 / 12% = 6 Years
[Actual Time is 6.12 Years]

43. Steps to Solve Time Value of Money Problems
1. Read problem thoroughly
2. Create a time line
3. Put cash flows and arrows on time line
4. Determine if it is a PV or FV problem
Determine if solution involves a single
CF, annuity stream(s), or mixed flow
6. Solve the problem
7. Check with financial calculator (optional)