ALAN ANDERSON, Ph.D.
     ECI RISK TRAINING
www.ecirisktraining.com
The time value of money formulas can be
used to solve for the appropriate rate of
interest or time horizon given the prese...
The present and future value
formulas can be used to solve
for the rate of interest.




                   (c) ECI RISK T...
Suppose that an investor deposits $10,000
in a bank account.

The investor plans to keep these funds in
the bank for ten y...
This can be determined
algebraically as follows:

   FVN = PV(1 + I)N


    FVN
        = (1 + I ) N

    PV



          ...
FVN
N       = (1 + I )
    PV

        FVN
    N       −1= I
        PV



               (c) ECI RISK TRAINING 2009
     ...
In this example,



     20, 000
  10         − 1 = 0.07177 = 7.177%
     10, 000




                    (c) ECI RISK TRA...
The present and future value formulas can
also be used to solve for the time horizon.




                         (c) ECI...
Suppose that an investor deposits
$5,000 in a bank account that pays 6%
interest per year. The investor wants
to know how ...
This can be determined
algebraically as follows:

   FVN = PV(1 + I)N

    FVN
        = (1 + I ) N

    PV



           ...
⎛ FVN ⎞
ln ⎜
   ⎝ PV ⎠⎟ = N ln(1 + I )


          ⎛ FVN ⎞
       ln ⎜      ⎟
          ⎝ PV ⎠
    N=
        ln(1 + I )

...
In this example,



           ⎛ 10, 000 ⎞
        ln ⎜         ⎟
           ⎝ 5, 000 ⎠
     N=                = 11.896
  ...
The Rule of 72 is a quick method for estimating
the time horizon or the interest rate needed to
double the value of an inv...
Dividing the interest rate into 72 gives the
approximate number of years that it would
take to double the value of an inve...
Dividing the number of years into 72 gives the
approximate interest rate that would be required
to double the value of an ...
In the case of a stream of cash flows that
are not equal, computing the future and
present value of the cash flows is a mo...
The two basic types of uneven cash
flows of interest in finance are:

1) an annuity with an additional
payment during the ...
The cash flows from most bonds take
the form of an annuity with an additional
payment during the final period.

Investment...
Suppose that a bond offers investors cash
flows of $100 each year for the next three
years, with an additional payment of ...
In this case,

     N=3
     I=5
     PMT = $100
     FV3 = $1,000




                (c) ECI RISK TRAINING 2009
        ...
⎡      1                         ⎤
             1−
           ⎢ (1 + I )N                      ⎥
PVAN = PMT ⎢             ...
⎡       1    ⎤
             1−
           ⎢ (1 + .05)3 ⎥
PVA3 = 100 ⎢            ⎥ = $272.32
           ⎢    .05     ⎥
   ...
FVN         1, 000
PV =            =
     (1 + I ) N
                  (1.05) 3




    1, 000
  =        = $863.84
    1....
Combining these results gives the present
value of the cash flow stream:

PVA3 + PV = 272.32 + 863.84 = $1,136.16




    ...
Suppose that an investment project
produces cash flows of $200 at the end
of the next two years, and $300 at the
end of th...
If the periodic rate of interest is 4%, what is
the present value of these cash flows?

In this case, the present value of...
In this case, the present value is:


  200     200     300     300     300
      1
        +     2
                +     ...
Each of the examples considered so far has
been based on the assumption that interest
is paid annually.

When interest is ...
Two adjustments must be made:

    1) the periodic interest rate
    2) the number of periods




                      (c...
 The   periodic interest rate equals:

 annual rate / number of periods per year



 The   number of periods equals:

 (...
Suppose that a sum of $1,000 is invested for
two years at an annual rate of interest of 4%.
Compute the future value of th...
With annual compounding,

         N=2
         I=4
         PV = $1,000




                (c) ECI RISK TRAINING 2009
  ...
Using the future value formula,


    FVN = PV(1+I)N
    FV2 = 1,000(1+.04)2
    FV2 = 1,000(1.081600)
    FV2 = $1081.60
...
With semi-annual compounding,

         N=4
         I=2
         PV = $1,000




                  (c) ECI RISK TRAINING ...
Using the future value formula,


    FVN = PV(1+I)N
    FV4 = 1,000(1+.02)4
    FV4 = 1,000(1.082432)
    FV4 = $1082.43
...
The more frequently interest is paid
each year, the greater will be the
future value of a sum or an annuity.




         ...
Compute the present value of $1,000 to
be received in four years using an annual
interest rate of 6% with:

   a) annual c...
With annual compounding,

         N=4
         I=6
         FV4 = $1,000




                (c) ECI RISK TRAINING 2009
 ...
Using the present value formula,


       FVN          1000
PV =            =             = $792.09
     (1 + I ) N
      ...
With semi-annual compounding,

         N=8
         I=3
         FV8 = $1,000




                  (c) ECI RISK TRAINING...
Using the present value formula,



       FVN          1000
PV =            =             = $789.41
     (1 + I ) N
     ...
The more frequently interest is paid
each year, the smaller will be the
present value of a sum or an annuity.




        ...
As the frequency of compounding
increases, the present value of a sum or
annuity decreases, while the future value
of a su...
The limiting compounding frequency is known as
continuous compounding. In this case, interest is
compounded at every insta...
FVN = eIN

    FVN         − IN
PV = IN = FVN e
     e

e = 2.7182818......




                 (c) ECI RISK TRAINING 200...
The present value of $1,000 to be
received in four years with an annual
rate of interest of 5% compounded
continuously is ...
PV = 1,000e-(0.05)(4) =

1,000e-(0.20) = $818.73




                 (c) ECI RISK TRAINING 2009
                     www....
The future value of $1,000 invested for
three years at an annual rate of interest of
4% compounded continuously is compute...
FV3 = 1,000e(0.04)(3) =

1,000e(0.12) = $1,127.50




                  (c) ECI RISK TRAINING 2009
                      w...
In order to compare interest rates with different
compounding frequencies, they can be converted
into the effective annual...
This is done with the following formula:

                                       M
          ⎛     APR ⎞
    EAR = ⎜ 1 +  ...
where:

    APR = the annual percentage rate




                     (c) ECI RISK TRAINING 2009
                         ...
If a bank charges an APR of 6% per year,
compounded quarterly for a loan, what is
the effective annual rate?




         ...
This can be determined with the
formula, as follows:

                                M
      ⎛     APR ⎞
EAR = ⎜ 1 +     ...
4
      ⎛     .06 ⎞
EAR = ⎜ 1 +     ⎟ − 1 = 0.06136
      ⎝      4 ⎠




                    (c) ECI RISK TRAINING 2009
  ...
This indicates that the borrower is actually
paying 6.136% per year for this loan.




                        (c) ECI RIS...
With continuous compounding,
the EAR formula becomes:



  EAR = eAPR - 1




                (c) ECI RISK TRAINING 2009
 ...
If a bank charges an APR of 5% per year,
continuously compounded, what is the
effective annual rate?

        EAR = eAPR –...
For free problem sets based on this material
along with worked-out solutions, write to
info@ecirisktraining.com. To learn ...
Upcoming SlideShare
Loading in...5
×

Time Value Of Money Part 2

3,486

Published on

The Time Value of Money

Published in: Economy & Finance, Business
1 Comment
2 Likes
Statistics
Notes
No Downloads
Views
Total Views
3,486
On Slideshare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
Downloads
1,189
Comments
1
Likes
2
Embeds 0
No embeds

No notes for slide

Time Value Of Money Part 2

  1. 1. ALAN ANDERSON, Ph.D. ECI RISK TRAINING www.ecirisktraining.com
  2. 2. The time value of money formulas can be used to solve for the appropriate rate of interest or time horizon given the present and future value of a sum. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 40
  3. 3. The present and future value formulas can be used to solve for the rate of interest. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 41
  4. 4. Suppose that an investor deposits $10,000 in a bank account. The investor plans to keep these funds in the bank for ten years, with a goal of having $20,000 at the end of that time. What rate of interest would he have to earn to double his money in ten years? (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 42
  5. 5. This can be determined algebraically as follows: FVN = PV(1 + I)N FVN = (1 + I ) N PV (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 43
  6. 6. FVN N = (1 + I ) PV FVN N −1= I PV (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 44
  7. 7. In this example, 20, 000 10 − 1 = 0.07177 = 7.177% 10, 000 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 45
  8. 8. The present and future value formulas can also be used to solve for the time horizon. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 46
  9. 9. Suppose that an investor deposits $5,000 in a bank account that pays 6% interest per year. The investor wants to know how long it will take for these funds to be worth $10,000. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 47
  10. 10. This can be determined algebraically as follows: FVN = PV(1 + I)N FVN = (1 + I ) N PV (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 48
  11. 11. ⎛ FVN ⎞ ln ⎜ ⎝ PV ⎠⎟ = N ln(1 + I ) ⎛ FVN ⎞ ln ⎜ ⎟ ⎝ PV ⎠ N= ln(1 + I ) (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 49
  12. 12. In this example, ⎛ 10, 000 ⎞ ln ⎜ ⎟ ⎝ 5, 000 ⎠ N= = 11.896 ln(1 + .06) (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 50
  13. 13. The Rule of 72 is a quick method for estimating the time horizon or the interest rate needed to double the value of an investment. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 51
  14. 14. Dividing the interest rate into 72 gives the approximate number of years that it would take to double the value of an investment. For the example of the investor who needs to know how many years it would take to double his money at an interest rate of 6%, dividing 72 by 6 gives a result of 12, which is very close to the actual value of 11.896 years. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 52
  15. 15. Dividing the number of years into 72 gives the approximate interest rate that would be required to double the value of an investment. For the example of the investor who needs to know what rate of interest is required to double his money in ten years, dividing 72 by 10 gives a result of 7.2%, which is very close to the actual value of 7.177%. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 53
  16. 16. In the case of a stream of cash flows that are not equal, computing the future and present value of the cash flows is a more complex process. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 54
  17. 17. The two basic types of uneven cash flows of interest in finance are: 1) an annuity with an additional payment during the final period 2) a cash flow stream with no pattern, known as an irregular stream of cash flows (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 55
  18. 18. The cash flows from most bonds take the form of an annuity with an additional payment during the final period. Investment projects often generate irregular streams of cash flows to firms. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 56
  19. 19. Suppose that a bond offers investors cash flows of $100 each year for the next three years, with an additional payment of $1,000 at the end of the third year. If the periodic rate of interest is 5%, what is the present value of this stream of cash flows? (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 57
  20. 20. In this case, N=3 I=5 PMT = $100 FV3 = $1,000 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 58
  21. 21. ⎡ 1 ⎤ 1− ⎢ (1 + I )N ⎥ PVAN = PMT ⎢ ⎥ ⎢ I ⎥ ⎢ ⎣ ⎥ ⎦ (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 59
  22. 22. ⎡ 1 ⎤ 1− ⎢ (1 + .05)3 ⎥ PVA3 = 100 ⎢ ⎥ = $272.32 ⎢ .05 ⎥ ⎢ ⎣ ⎥ ⎦ (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 60
  23. 23. FVN 1, 000 PV = = (1 + I ) N (1.05) 3 1, 000 = = $863.84 1.1576 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 61
  24. 24. Combining these results gives the present value of the cash flow stream: PVA3 + PV = 272.32 + 863.84 = $1,136.16 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 62
  25. 25. Suppose that an investment project produces cash flows of $200 at the end of the next two years, and $300 at the end of the following three years. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 63
  26. 26. If the periodic rate of interest is 4%, what is the present value of these cash flows? In this case, the present value of each cash flow is computed using the PV formula; these results are combined to give the present value of the stream of irregular cash flows. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 64
  27. 27. In this case, the present value is: 200 200 300 300 300 1 + 2 + 3 + 4 + 5 (1.04) (1.04) (1.04) (1.04) (1.04) = 192.31 + 184.91 + 266.67 + 256.44 + 246.58 = $1,146.91 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 65
  28. 28. Each of the examples considered so far has been based on the assumption that interest is paid annually. When interest is paid more often than once per year, the present value and future value formulas must be adjusted. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 66
  29. 29. Two adjustments must be made: 1) the periodic interest rate 2) the number of periods (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 67
  30. 30.  The periodic interest rate equals: annual rate / number of periods per year  The number of periods equals: (number of years)(number of periods per year) (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 68
  31. 31. Suppose that a sum of $1,000 is invested for two years at an annual rate of interest of 4%. Compute the future value of this sum based on the assumption of: a) annual compounding b) semi-annual compounding (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 69
  32. 32. With annual compounding, N=2 I=4 PV = $1,000 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 70
  33. 33. Using the future value formula, FVN = PV(1+I)N FV2 = 1,000(1+.04)2 FV2 = 1,000(1.081600) FV2 = $1081.60 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 71
  34. 34. With semi-annual compounding, N=4 I=2 PV = $1,000 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 72
  35. 35. Using the future value formula, FVN = PV(1+I)N FV4 = 1,000(1+.02)4 FV4 = 1,000(1.082432) FV4 = $1082.43 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 73
  36. 36. The more frequently interest is paid each year, the greater will be the future value of a sum or an annuity. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 74
  37. 37. Compute the present value of $1,000 to be received in four years using an annual interest rate of 6% with: a) annual compounding b) semi-annual compounding (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 75
  38. 38. With annual compounding, N=4 I=6 FV4 = $1,000 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 76
  39. 39. Using the present value formula, FVN 1000 PV = = = $792.09 (1 + I ) N (1 + .06) 4 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 77
  40. 40. With semi-annual compounding, N=8 I=3 FV8 = $1,000 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 78
  41. 41. Using the present value formula, FVN 1000 PV = = = $789.41 (1 + I ) N (1 + .03) 8 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 79
  42. 42. The more frequently interest is paid each year, the smaller will be the present value of a sum or an annuity. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 80
  43. 43. As the frequency of compounding increases, the present value of a sum or annuity decreases, while the future value of a sum or annuity increases. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 81
  44. 44. The limiting compounding frequency is known as continuous compounding. In this case, interest is compounded at every instant in time. As a result, the number of compounding periods is infinite. The present and future value formulas with continuous compounding are: (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 82
  45. 45. FVN = eIN FVN − IN PV = IN = FVN e e e = 2.7182818...... (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 83
  46. 46. The present value of $1,000 to be received in four years with an annual rate of interest of 5% compounded continuously is computed as follows: (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 84
  47. 47. PV = 1,000e-(0.05)(4) = 1,000e-(0.20) = $818.73 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 85
  48. 48. The future value of $1,000 invested for three years at an annual rate of interest of 4% compounded continuously is computed as follows: (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 86
  49. 49. FV3 = 1,000e(0.04)(3) = 1,000e(0.12) = $1,127.50 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 87
  50. 50. In order to compare interest rates with different compounding frequencies, they can be converted into the effective annual rate (EAR). (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 88
  51. 51. This is done with the following formula: M ⎛ APR ⎞ EAR = ⎜ 1 + ⎟ −1 ⎝ M ⎠ (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 89
  52. 52. where: APR = the annual percentage rate (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 90
  53. 53. If a bank charges an APR of 6% per year, compounded quarterly for a loan, what is the effective annual rate? (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 91
  54. 54. This can be determined with the formula, as follows: M ⎛ APR ⎞ EAR = ⎜ 1 + ⎟ −1 ⎝ M ⎠ (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 92
  55. 55. 4 ⎛ .06 ⎞ EAR = ⎜ 1 + ⎟ − 1 = 0.06136 ⎝ 4 ⎠ (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 93
  56. 56. This indicates that the borrower is actually paying 6.136% per year for this loan. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 94
  57. 57. With continuous compounding, the EAR formula becomes: EAR = eAPR - 1 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 95
  58. 58. If a bank charges an APR of 5% per year, continuously compounded, what is the effective annual rate? EAR = eAPR – 1 = e.05 – 1 = 0.051271 = 5.1271% (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 96
  59. 59. For free problem sets based on this material along with worked-out solutions, write to info@ecirisktraining.com. To learn about training opportunities in finance and risk management, visit www.ecirisktraining.com (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 97
  1. A particular slide catching your eye?

    Clipping is a handy way to collect important slides you want to go back to later.

×