1. CHAPTER 2 Time Value of Money
Future value
Present value
Annuities
Rates of return
Amortization

2. What is the future value (FV) of an initial $100 after 3 years, if I/YR = 10%?
Finding the FV of a cash flow or series of cash flows is called compounding.
FV = ?
(121x1.1)
=133.10 1 2 3
10%
100
(100x1.1)1
=110
(110x1.1)1
=121

3. Solving for FV: The step-by-step and formula methods
After 1 year:
FV1 = PV (1 + I) = $100 (1.10) = $110.00
After 2 years:
FV2 = PV (1 + I)2 = $100 (1.10) =$121.00
After 3 years:
FV3 = PV (1 + I)3 = $100 (1.10) =$133.10
After N years (general case):
FVN = PV (1 + I)N

4. FVn= PV (1+I)n FV3= PV (1+I)3 =100 (1+10%)3 =100 (1.10)3 =$133.10

5. or, using FVF table (A1)
FV3= PV (Fk% VFn) = 100 (F10% VF3) = 100 (1.331) = $

6. Time lines Show the timing of cash flows.1 2 3 I%
CF0
CF1
CF2
CF3
Show the timing of cash flows.
Tick marks occur at the end of periods, so Time 0 is today; Time 1 is the end of the first period (year, month, etc.) or the beginning of the second period.

7. Solving for FV: The calculator method
Solves the general FV equation.
Requires 4 inputs into calculator, and will solve for the fifth. (Set to P/YR = 1 and END mode.) 3 10
-100
INPUTS N
I/YR PV
PMT FV
OUTPUT
133.10

8. What is the present value (PV) of $100 due in 3 years, if I/YR = 10%?
Finding the PV of a cash flow or series of cash flows is called discounting (the reverse of compounding).
The PV shows the value of cash flows in terms of today’s purchasing power. 1 2 3
10%
PV = ?
82.64/(1.1)1
=75.13
100/(1.1)1
=90.91
90.91/(1.1)1
=82.64
100

9. Solving for PV: The formula method
Solve the general FV equation for PV:
PV = FVN / (1 + I)N
PV = FV3 / (1 + I)3
= $100 / (1.10)3
= $75.13

10. FV = PV (1+k)n 100 = PV (1+10%)3 100 = PV (1.1)3 75.13 = PV

11. or, using table (A-2)
PV= FV (Pk% VFn) = 100 (P10% VF3) = 100 (0.7513) = 75.13

12. Solving for PV: The calculator method
Solves the general FV equation for PV.
Exactly like solving for FV, except we have different input information and are solving for a different variable. 25 10
100
INPUTS N
I/YR PV
PMT FV
OUTPUT
-75.13

13. Solving for I: What interest rate would cause $100 to grow to $125
Solving for I: What interest rate would cause $100 to grow to $ in 3 years?
Solves the general FV equation for I.
Hard to solve without a financial calculator or spreadsheet. 3
-100
125.97
INPUTS N
I/YR PV
PMT FV
OUTPUT 8

14. FV = 1 2 3
PV=100
n = 3
i = ?
PV= 100
FV=

15. Solving for N: If sales grow at 20% per year, how long before sales double?
Solves the general FV equation for N.
Hard to solve without a financial calculator or spreadsheet. 20
-200
400
INPUTS N
I/YR PV
PMT FV
OUTPUT
3.8

16. FV = 400 ?
PV=200
i = 20%
PV= 200
FV= 400
n = ? ?

17. a) Based on the following arrangements, what interest rates are charged by the lending institution to the borrower. You borrow RM 500 and repay RM 551 in two years. FV = PV ( F V F ) n=2 k =? 551 = 500 ( F2 V Fk% ) 551 = ( F2 V Fk% ) 1.10 = ( F2 V Fk% ) k = 5 % from table A-1
PV = 500
FV = 551
n = 2
Pmt = 0
I/yr = ?

18. b) Borrow RM 100,000 and repay RM 146,932 in 5 years 146,932 = 100,000 ( F V F ) n = 5 k = ? 146,9325 = ( F5 V F k% ) 100, = ( F5V Fk% ) k = 8 % (see table A-1) c) Borrow RM 100,000 and repay RM 300,000 in 10 years 300,000 = 100,000 ( F V F ) n = 10 k = ? 300,000 = ( F10 V Fk% ) 3 = ( F10 V Fk% ) k = 11.6%
PV = -100,000
FV = 146,932
n = 5
Pmt = 0
I/yr = ?
PV = -100,000
FV = 300,000
n = 10
Pmt = 0
I/yr = ?

19. Finding Implied Interest And Number Of Factoring Years
How long does it take for the followings to happen a) RM 856 to grow into RM1,122 at 7 % FV = PV ( Fk% V Fn ) RM 1,122 = RM 856 ( F 7%V Fn=? ) RM 1,122 = ( F7% V Fn=? ) RM = ( F7% V Fn=? ) 4 = n (use table A1) b) RM10,000 to grow into RM100,000 at 8 % 100,000 = 10,000 ( F V F ) 100,000 = ( F8% V Fn=? ) 10, = ( F8% V Fn=? ) n = 30 (use table A1)
PV = -856
FV = 1,122
k = 7%
Pmt = 0
n = ?
PV = -10,000
FV = 100,000
k = 8%
Pmt = 0
n = ?

20. c) RM10,000 to grow into RM200,000 at 8 % 200,000 = 10,000 ( F V F ) 200,000 = ( F8% V Fn=? ) 10, = ( F8% V Fn=? ) n = 38 (use table A1)
PV = -10,000
FV = 200,000
k = 7%
Pmt = 0
n = ?

21. Example of present value problem
1. En. Kamal is planning ahead for his son’s education. The boy is eight now and will start college in 12 years time. How much must he set aside now to have RM 100,000 when his son starts schooling. The interest rate is 8 % k = 8% n= 12 years FV = RM 100,000 PV = ? FV = PV ( 1 + k )n PV = RM 100,000 ( )12 = RM 100, = RM 39,711
FV = 100,000
k = 8%
Pmt = 0
n = 12
PV = ?

22. or, using present value factor table
PV = FV ( P V F ) K% n PV = RM 100,000 ( P V F ) 8 % 12 PV = RM 100,000 ( ) from Table A-2 PV = RM 39,710 ~ RM 39,71

23. Drawing time lines 100 (100x1.1)1 =110 (110x1.1)1 =121 1 2 31 2 3 I%
FV 3 year $100 ordinary 10%
Σ = 331

24. Solving for FV: 3-year ordinary annuity of $100 at 10%
$100 payments occur at the end of each period, but there is no PV. 3 10
-100
INPUTS N
I/YR PV
PMT FV
OUTPUT
331

25. Problems of future value annuity (for all future value annuity problems, please refer to table A-3 )
1. Assume that you have deposited RM100 each year in a bank account that pays 15% interest rate. How much will you have in your account at the end of the fifth year? F VF = P m t ( F V F A ) k% n k = 15% n = 5 F VF A = 100 ( F 15%V F A 5 ) = 100 ( ) (use table A-3) F V A =
PV = 0
k = 15%
Pmt = -100
n = 5
FV = ?

26. 2. Assume you have deposited RM200 each year in a bank account that pays 20% interest rate. How much will you have in your account after 25 years. F VF = P m t ( F V F A ) k% n k = 20% n =2 5 F VF A = 200 ( F 20% V F A25 ) = 200 ( ) (use table A-3) F V A = 94,396
PV = 0
k = 20%
Pmt = -200
n = 25
FV = ?

27. 3. Assume you have deposited RM1000 each year in a bank account that pays 20% interest rate. How much will you have in your account at the end of the 25 year F VF = P m t ( F V F A ) k % n k = 20% n =2 5 F VF A = 1000( F 20%V F A 25 ) = 1000 ( ) F V A = 471,980
PV = 0
k = 20%
Pmt = -1,000
n = 25
FV = ?

28. Drawing time lines 100 100/(1.1)1 =90.91 1 2 31 2 3 I%
PV 3 year $100 ordinary 10%
100/(1.1)1
=90.91
90.91/(1.1)1
=82.64
82.64/(1.1)1
=75.13
100
100/(1.1)1
=90.91
90.91/(1.1)1
=82.64
Σ =

29. Solving for PV: 3-year ordinary annuity of $100 at 10%
$100 payments still occur at the end of each period, but now there is no FV. 3 10
100
INPUTS N
I/YR PV
PMT FV
OUTPUT

30. Example:
We would like to know what is the present value of four payments in the amount of RM1,000 each to be received in the next four years. The current market interest rate is 10% P V F = P m t ( P V F A ) Pmt = RM1,000, k%= 10, n = 4 k % n PVF = 1000 (P10%VFA4) (please refer to table A-4 for this value) PVF = 1000 (3.1699) PVF = 3,169.9

31. What is the difference between an ordinary annuity and an annuity due?
PMT 1 2 3 i%
PMT
100
100(1.1)
=110
100(1.1)1
110 (1.1)1
=121 1 2 3 i%
Annuity Due
100(1.1)1
110 (1.1)1
121 (1.1)1

32. Solving for FV: 3-year annuity due of $100 at 10%
Now, $100 payments occur at the beginning of each period.
FVAdue= FVAord(1+I) = $331(1.10) = $
Alternatively, set calculator to “BEGIN” mode and solve for the FV of the annuity:
BEGIN 3 10
-100
INPUTS N
I/YR PV
PMT FV
OUTPUT
364.10

33. FV Annuity due
3 year annuity due of $100 at 10%, n= 3, Pmt=100, k=10%, FV=? FVA = Pmt (F10% VFA3) = 100 (3.31) table A-3 FVA = 331 (1 + 10%) FVAdue = 331 (1.10) =

34. Present value annuity due
The process of discounting the future payment back to the present time in the case of Present value annuity due is different than the normal present value annuity. The figure below illustrate the difference Present value annuity due P m t 1 P m t 2 P m t 3 PVA 1/(1 + k)1 1/ (1 + k)2 Present value annuity 0 P m t 1 P m t 2 P m t 3 PVA 1/(1 + k)1 1/(1 + k)2 1/(1 + k)3

35. Because the first payment took place at the beginning of year one and not the end of year one, that payment will be discounted one less period. Hence that factor has to be taken into consideration when we compute the present value annuity due amount. Illustrated below, we have added (1+k) factor to the original formula use to compute an ordinary present value annuity amount P V A due = P m t ( P V F A ) ( 1 + k ) Annuity due factor N K%

36. PV Annuity due
3 year annuity due of $100 at 10% n= 3, Pmt= 100, k=10%, PV= ? PVA = Pmt ( PV10%FA3) = 100 ( ) table A-4 = PVAdue = ( 1.10 ) =

37. Solving for PV: 3-year annuity due of $100 at 10%
Again, $100 payments occur at the beginning of each period.
PVAdue= PVAord(1+I) = $248.69(1.10) = $
Alternatively, set calculator to “BEGIN” mode and solve for the PV of the annuity:
BEGIN 3 10
100
INPUTS N
I/YR PV
PMT FV
OUTPUT

38. What is the present value of a 5-year $100 ordinary annuity at 10%?
Be sure your financial calculator is set back to END mode and solve for PV:
N = 5, I/YR = 10, PMT = 100, FV = 0.
PV = $379.08

39. Present value annuity
Pmt =100, n =5, i =10%, FV =0, PV =? PVA = 100 ( PVk% FAn ) = 100 ( PV10% FA5 ) = 100 ( ) Table A-4 PVA =

40. What if it were a 10-year annuity? A 25-year annuity? A perpetuity?
N = 10, I/YR = 10, PMT = 100, FV = 0; solve for PV = $
25-year annuity
N = 25, I/YR = 10, PMT = 100, FV = 0; solve for PV = $
Perpetuity
PV = PMT / I = $100/0.1 = $1,000.

41. A perpetuity ∞ 1 100 2 ∞ PV =Σ ? Pmt = 100 i = 10 PV = ? Pmt = 100
= 1,000

42. Will the FV of a lump sum be larger or smaller if compounded more often, holding the stated I% constant?
LARGER, as the more frequently compounding occurs, interest is earned on interest more often.
Yr= 1
Yr= 2
Yr= 3
10%
100
133.10
Annually: N= 3, PV= -100, I/YR= 10, PMT=0, FV =133.10 1 2 3 5% 4 5 6
134.01
Yr= 1
Yr= 2
Yr= 3
100
Semiannually: N= 2x3, PV= -100, I/YR= 10/2, PMT=0, FV =134.01

43. Quaterly: N= 4x3, PV= 100, I/yr= 10/4, FV=. = 12 = 2. 5 134
Quaterly: N= 4x3, PV= 100, I/yr= 10/4, FV= ? = 12 = Monthly: N= 12x3, PV= 100, I/yr= 10/12, FV= ? = 36 =

44. Will the PV of a lump sum be larger or smaller if discounted more often, holding the stated I% constant
Annually FV= 100, I/yr= 10%, n= 3, PV= ? Semiannually FV= 100, I/yr= 10%/2, n= 3x2, PV= ? = 5 =

45. Quarterly FV= 100, I/yr= 10%/4, n= 3x4, PV=. = 2. 5 = 12 74
Quarterly FV= 100, I/yr= 10%/4, n= 3x4, PV= ? = 2.5 = Monthly FV= 100, I/yr= 10%/12, n= 3x12, PV= ? = =

46. Yr 1
Yr 1
Yr 1 3
100 1 2 “ “
100
(1+i )n
Yr 1
Yr2
Yr 3 1 2 3 4 5 6
100 “ “ “ “ “
100
(1+i )n

47. The Power of Compound Interest
A 20-year-old student wants to save $3 a day for her retirement. Every day she places $3 in a drawer. At the end of the year, she invests the accumulated savings ($1,095) in a brokerage account with an expected annual return of 12%.
How much money will she have when she is 65 years old?

48. 20 21 22 23 64 65
1,095
1,095
1,095
1,095
1,095
1,095
n= 65-20
= 45
i / yr = 12%
PV = 0
Pmt = 1,095
FV = ?

49. Solving for FV: If she begins saving today, how much will she have when she is 65?
If she sticks to her plan, she will have $1,487, when she is 65.
INPUTS
OUTPUT N
I/YR
PMT PV FV 45 12
-1095
1,487,262

50. Solving for FV: If you don’t start saving until you are 40 years old, how much will you have at 65?
If a 40-year-old investor begins saving today, and sticks to the plan, he or she will have $146, at age 65. This is $1.3 million less than if starting at age 20.
Lesson: It pays to start saving early. 25 12
-1095
INPUTS N
I/YR PV
PMT FV
OUTPUT
146,001

51. 40 41 42 43 44 64 65
1,095
1,095
1,095
n= 65-40
= 25
i / yr = 12%
PV = 0
Pmt = ?
FV = 146,000.59

52. Solving for PMT: How much must the 40-year old deposit annually to catch the 20-year old?
To find the required annual contribution, enter the number of years until retirement and the final goal of $1,487,261.89, and solve for PMT. 25 12
1,487,262
INPUTS N
I/YR PV
PMT FV
OUTPUT
-11,154.42

53. 40 41 42 43 44 64 65 ? ? ?
n= 65-40
= 25
i / yr = 12%
PV = 0
Pmt = ?
FV = 1,487,261.89

54. The Effective Annual rate(EAR)
Example Borrow: RM 100, Payback time: 12% Compounding Final balance Annual Semiannual Quarterly Monthly

55. When is each rate used?
INOM written into contracts, quoted by banks and brokers. Not used in calculations or shown on timelines.
EAR Used to compare returns on investments with different compounding periods.
i.e. A credit card holder must pay 18% monthly compounding interest on the amount outstanding. What is the EAR of the credit card?

56. Effective Annual Rate (EAR)
EAR = ( 1 + I nom )m – 1.0 m = ( /12 )12 – 1.0 = ≈ 19.56% ie: A credit card holder must pay 18% monthly compounding interest on the amount outstanding. What is the EAR of the credit card? I nom = 18% annually m = Compounding period

57. Loan amortization
Amortization tables are widely used for home mortgages.
Ie. A home owner borrows, RM 200,000 on a mortgage loan. The loan is to be repaid in 180 ( 12months x 15 years) equal payments at the end of each month. The bank charges 12% on the loan per year ( 1% monthly). What is the monthly repayment?

58. ? ? ? ? ? ?
200,000
Pmt 1
Pmt 2
Pmt 3
Pmt 178
Pmt 179
Pmt 180
n= 12 months x 15
= 180
i / yr = 12% / 12
= 1%
PV = 200,000
FV = 0
Pmt = ?

59. Step 1: Find the required annual payment
All input information is already given, just remember that the FV = 0 because the reason for amortizing the loan and making payments is to retire the loan.
180 1
-200,000
INPUTS N
I/YR PV
PMT FV
OUTPUT
2,400