1. Chapter 4
Time
Value
of Money

2. The Role of Time Value in Finance
Most financial decisions involve costs & benefits that are spread out over time.
Time value of money allows comparison of cash flows from different periods.
Which investment would you choose?
An investment of €100,000 that would return €200,000 after one year
An investment of €100,000 that would return €220,000 after two years

3. The Role of Time Value in Finance
In general, all else being equal, the sooner a € is received, the more quickly is re-invested
However a short-term investment is not necessarily more valuable because this depends on the re-investment interest rate
Answer
The investment (a) is more valuable if it can re- invested at an annual interest rate >10%, otherwise the investment (b) is more valuable

5. Cash Flow Time Lines
The cash flow time lines help to visualize the timing of the cash flows associated with a particular situation
The construction of a cash flow time line is fairly easy:
Time 1 2 3 4
k = 10%
Cash Flows
-500
FVn = ?

6. Difference between simple interest and compounded interest
With simple interest, an investor doesn’t earn interest on interest
Year 1: 5% of €100 = €5 + €100 = €105
Year 2: 5% of €100 = €5 + €105 = €110
Year 3: 5% of €100 = €5 + €110 = €115
With compounded interest, an investor earns interest on interest
Year 1: 5% of €100.00= € € = €105.00
Year 2: 5% of €105.00= € € = €110.25
Year 3: 5% of €110.25= €5.51+ € = €115.76

7. Definition and Formula
Future Value
Definition and Formula
Future Value (FV)—determine to what amount an investment will grow over a particular time period
re-invested interest (earned in previous periods) earns interest
compounding—interest compounds or grows the investment
FVn = PV0(1+k)n = PV(FVIFk,n)

8. Effects of compounding

9. Future Value - Example
Suppose an investment of €100 for one year at 5% per year. What is the future value in one year?
The compounding rate is given as 5%. Hence the value of current Euros in terms of future Euros is 1.05 future Euros per current Euro. Hence future value is 100(1.05) = €105.
Suppose that money is left in for another year. What is the future value in two years from now?
Assume that the money in one year as present value and the money in two years as future value. Therefore the price of one-year-from-now money in terms of two-years-from-now money is Therefore 105 of one-year-from-now Euros in terms of two years-from-now Euros is 105(1.05) = 100 (1.05)(1.05) = 100(1.05)2 =
By making the same assumptions, the FV in 3 years would be

10. (assumes compounded interest as capital is being re-invested)
Future Value– Example using Excel
An investment of €100 is made today at 5% interest. How much money will this investment yield in 3 years?
Excel Function
(assumes compounded interest as capital is being re-invested)
=FV (interest, periods, pmt, PV)
=FV (.05, 3, , 100)

11. Financial Calculator Solution
In the previous example: PV = €100, k = 5.0%, n = 3 ?
115,76  N  I
 PV
PMT FV

12. A Graphic view of Future Value
Relationship among Future Value, Growth or Interest Rates and Time

13. Definition and Formula
Present Value
Definition and Formula
Present value (PV)—determine the current value of an amount that will be paid, or received, at some time in the future
PV is the future amount restated in current dollars; future interest has not been earned, thus it is not included in the PV
discounting—deflate, or discount, the future amount by future interest that can be earned
PV0 = FVn[1/(1+k)n] = FV(PVIFk,n)

14. Present Value - Example
Suppose an investor needs €10,000 in two years for the down payment on a new car. If the investor can earn 6% annually, how much does he need to invest today?
PV = 10,000 / (1.06)2 =

15. Present Value– Example using Excel
How much is needed for an investment today in order to have €10,000 in 2 years if the investor can earn 6% interest on his investment?
Excel Function
=PV (interest, periods, pmt, FV)
=PV (.06, 2, , 10,000)

16. Financial Calculator Solution
In the previous example: FV = €10000, k = 6.0%, n = 2
2 6 ?
-8899,96  N  I
 PV
PMT FV

17. A Graphic view of Present Value
Relationship among Present Value, Growth or Interest Rates and Time

18. Solving for interest rates - Example
If a mutual fund investment that was bought six years ago at a price of €1000 is now worth €5525, what rate of return (k) has the investor already earned today?
FV = PV (1+k)n
5525 = 1000 (1+k)6
and hence k = 33%.

19. Solving for interest rates with Excel
What rate of return has an investor earned from a €1000 investment bought 6 years ago that is worth today €5525?
Excel Function
=Rate(periods, pmt, PV, FV)
=Rate(6, ,1000, 5525)

20. Financial Calculator Solution
In the previous example: PV = €1000, FV = €5525, n = 6
6 ? 33  N  I
 PV
PMT FV

21. Solving for period - Example
If a security worth €712 is invested at 6 percent, how long will it take to grow to €848?
FV = PV (1+k)n
848 = 712 (1+0.06)n
and hence n = 3.

22. Financial Calculator Solution
In the previous example: PV = €712, FV = €848, k = 6% ? 3  N  I
 PV
PMT FV

23. Relationship between interest rates and present value
For a given interest rate – the longer the time period, the lower the present value
For a given time period – the higher the interest rate, the smaller the present value

24. Future Value of an annuity
Definition and Formula
Annuity—a series of equal payments that are made at equal intervals
Ordinary annuity—has cash flows that occur at the end of each period
Annuity due—has cash flows that occur at the beginning of the period
The future value of an annuity, FVA, can be computed by solving for the future value of a lump-sum amount
FVAn = A (1+k)n - 1 = A(FVIFAk,n) k

25. FV of Ordinary Annuity - Example
Suppose an equal cash flow of deposits of €100 at the end of each year for five years at 3% per year. How much will these deposits grow?
The growth rate is given as 3%. Therefore FVA = 100(1.03) (1.03)1+ 100(1.03) (1.03) (1.03)4 = 530,91

26. Future Value of an Ordinary Annuity – using Excel
How much will the deposits grow if the initial deposit is €100 at the end of each year at 3% interest for five years.
Excel Function
=FV (interest, periods, pmt, PV)
=FV (.03, 5,100, )

27. Financial Calculator Solution
In the previous example: PMT = €100, k = 3%, n=5 ?
530,91  N  I
 PV
PMT FV

28. FV of an Annuity Due - Example
Suppose an equal cash flow of deposits of €100 at the beginning of each year for five years at 3% per year. How much will these deposits grow?
The growth rate is given as 3%. Therefore FVA = 100(1.03) (1.03)2+ 100(1.03) (1.03) (1.03)5 = 546,84

29. Future Value of an Annuity Due – using Excel
How much will the deposits grow if the initial deposit is €100 at the beginning of each year at 3% interest for five years.
Excel Function
=FV (interest, periods, pmt, PV)
=FV (.03, 5,100, )
=530.91*(1.03)

30. Financial Calculator Solution
In the previous example: PMT = €100, k = 3%, n=5 (switch calculator to BEGIN) ?
546,84  N  I
 PV
PMT FV

31. Present Value of an annuity
Definition and Formula
The present value of an annuity, FVA, can be computed by solving for the future value of a lump-sum amount
Annuity due is an annuity with cash flows that occur at the beginning of the period.
PVA0 = A 1 - [1/(1+k)n] = A(PVIFAk,n) k

32. PV of an Ordinary Annuity - Example
Suppose an equal cash flow of payments of €1000 at the end of each year. How much could an investor borrow if he could afford annual payments of $1,000 (which includes both principal and interest) at the end of each year for five years at 10% interest?
The present value of the annuity is calculated as follows : PVA = 1000/(1.1) /(1.1) /(1.1) /(1.1) /(1.1)5 = 3790,79

33. Present Value of an Ordinary Annuity – using Excel
How much could an investor borrow if he could afford annual payments of €1,000 (which includes both principal and interest) at the end of each year for five years at 10% interest?
Excel Function
=PV (interest, periods, pmt, FV)
=PV (.10, 5, 1000, )

34. Financial Calculator Solution
In the previous example: PMT = €1000, k = 10%, n=5 ?
3790,79  N  I
 PV
PMT FV

35. PV of an Annuity Due- Example
Suppose an equal cash flow of payments of €1000 at the beginning of each year. How much could an investor borrow if he could afford annual payments of $1,000 (which includes both principal and interest) at the end of each year for five years at 10% interest?
The present value of the annuity is calculated as follows : PVA = 1000/(1.1) /(1.1) /(1.1) /(1.1) /(1.1)4 = 4169,87

36. Present Value of an Annuity Due– using Excel
How much could an investor borrow if he could afford annual payments of €1,000 (which includes both principal and interest) at the beginning of each year for five years at 10% interest?
Excel Function
=PV (interest, periods, pmt, FV)
=PV (.10, 5, 1000, )
= 3790,79*(1,1)

37. Financial Calculator Solution
In the previous example: PMT = €1000, k = 10%, n=5 (switch calculator to begin) ?
4169,87  N  I
 PV
PMT FV

38. Solving for interest rates with annuities- Example
If an investor pays €846,80 for an investment that promises to pay €250 per year for the next four years, what rate of return (k) will the investor earn on the investment?
Assuming that payments are made at the end of each year, this is an ordinary annuity. The solution from the annuity equation provides k=7%. Beware that trial and error process should be used.

39. Financial Calculator Solution
In the previous example: PV = €846,80, PMT = €250, n = 4
4 ? -846, 7  N  I
 PV
PMT FV

40. Solving for interest rates - Example
If an investor pays €1685 for an investment that promises to pay him back €400 per year, how many payments must he receive to earn a 6% return?
Assuming that payments are made at the end of each year, this is an ordinary annuity. The solution from the annuity equation provides n=5%. Beware that trial and error process should be used.

41. Financial Calculator Solution
In the previous example: PV = €1685, PMT = €400, k = 6 ? 5  N  I
 PV
PMT FV

42. Present Value of a Perpetuity
A perpetuity is a special kind of annuity.
With a perpetuity, the periodic annuity or cash flow stream continues forever.
PV = Annuity/k
For example, how much would an investor have to deposit today in order to withdraw €1,000 each year forever if the investor can earn 8% return?
PV = €1,000/.08 = $12,500

43. Uneven Cash Flow Streams
In an uneven cash flow stream, the cash flows are not the same (equal).
Simplifying techniques, i.e. the use of a single equation to compute PV cannot be used

44. Present Value of an uneven Cashflow Stream - Example
Calculate the present value of the following uneven cashflow stream assuming a required return of 9%.

45. Present Value of an uneven Cashflow Stream - Example using Excel
Find the present value of the following uneven cashflow stream assuming a required return of 9%.
Excel Function
=NPV (interest, cells containing CFs)
=NPV (.09,B3:B7)

46. Compounding More Frequently than Annually
Interest is compounded more than once per year—quarterly, monthly, or daily. The more frequent the compounding, the more the investor earns because he is earning on interest more frequently
Therefore, the effective interest rate (the rate of return per year considering interest compounding) is greater than the nominal (annual) interest rate.

47. Annual and Semi-annual compounding
For example, what would be the difference in future value if the depositors puts €100 for 5 years in the bank and earns 3% annual interest compounded (a) annually, (b) semiannually?
Annually: x ( )5 = €115.92
Semiannually: 100 x ( )10= €116.05

48. Nominal & Effective Rates
The nominal interest rate is the stated or contractual rate of interest charged by a lender or promised by a borrower.
The effective interest rate is the rate actually paid or earned.
In general, the effective rate > nominal rate whenever compounding occurs more than once per year
EAR = (1 + k/m) m -1

49. Nominal & Effective Rates
Example: Paul bought a vacation package for winter holidays and charged it to his credit card. What is the effective rate of interest on Paul’s credit card if the nominal rate is 18% per year, compounded monthly?
EAR = ( /12) 12 -1
EAR = %

50. Real Rate Rr = [(1 + INOM)/(1 + inflation)] – 1
Example: If the nominal annual interest rate is 10% and the expected inflation rate is 5% per annum, what is the expected real rate of return?
Rr = [1.10/1.05] – 1 = = 4.76%

51. Canadian Mortgage Loans
The previous example illustrates an annual amortized loans.
However, in Canada, mortgage loans are a little bit complicated.
While a borrower is required to make monthly payments, the annual rate is semiannual compounding.

52. Canadian Mortgage Loans (cont'd)
A two-step procedure is applied to calculate the monthly rate in order to calculate the monthly payment:
First, convert the APR to EAR:
EAR = (1 + 6%/2)2 – 1 = 6.09%
Second, calculate the monthly rate (r) that has the same EAR as 6.09%:
6.09% = (1 + r)12 – 1
Solve the above equation, r =