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ALAN ANDERSON, Ph.D.
     ECI RISK TRAINING
www.ecirisktraining.com
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   2
The time value of money is one of the most
fundamental concepts in finance; it is based
on the notion that receiving a sum of money
in the future is less valuable than receiving
that sum today.

This is because a sum received today can be
invested and earn interest.



                        (c) ECI RISK TRAINING 2009
                            www.ecirisktraining.com   3
The four basic time value of
 money concepts are:

 future value of a sum
 present value of a sum
 future value of an annuity
 present value of an annuity




                 (c) ECI RISK TRAINING 2009
                     www.ecirisktraining.com   4
If a sum is invested today, it will earn interest
and increase in value over time. The value that
the sum grows to is known as its future value.

Computing the future value of a sum is known
as compounding.




                          (c) ECI RISK TRAINING 2009
                              www.ecirisktraining.com   5
The future value of a sum depends on
the interest rate earned and the time
horizon over which the sum is invested.

This is shown with the following formula:


        FVN = PV(1+I)N




                       (c) ECI RISK TRAINING 2009
                           www.ecirisktraining.com   6
where:

 FVN     = future value of a sum
         invested for N periods
 I       = periodic rate of interest
 PV      = the present or current
         value of the sum invested




                       (c) ECI RISK TRAINING 2009
                           www.ecirisktraining.com   7
Suppose that a sum of $1,000 is invested for
four years at an annual rate of interest of 3%.
What is the future value of this sum?




                          (c) ECI RISK TRAINING 2009
                              www.ecirisktraining.com   8
In this case,

     N=4
     I=3
     PV = $1,000




                (c) ECI RISK TRAINING 2009
                    www.ecirisktraining.com   9
Using the future value formula,


    FVN = PV(1+I)N
    FV4 = 1,000(1+.03)4
    FV4 = 1,000(1.125509)
    FV4 = $1,125.51



                      (c) ECI RISK TRAINING 2009
                          www.ecirisktraining.com   10
The present value of a sum is the amount
that would need to be invested today in
order to be worth that sum in the future.

Computing the present value of a sum is
known as discounting.




                      (c) ECI RISK TRAINING 2009
                          www.ecirisktraining.com   11
The formula for computing the
present value of a sum is:


           FVN
    PV =
         (1 + I ) N




                  (c) ECI RISK TRAINING 2009
                      www.ecirisktraining.com   12
How much must be deposited in a bank
account that pays 5% interest per year in
order to be worth $1,000 in three years?





                        (c) ECI RISK TRAINING 2009
                            www.ecirisktraining.com   13
In this case,

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




                (c) ECI RISK TRAINING 2009
                    www.ecirisktraining.com   14
FVN         1, 000
PV =            =
     (1 + I ) N
                  (1.05) 3



     1, 000
   =        = $863.84
     1.1576


                (c) ECI RISK TRAINING 2009
                    www.ecirisktraining.com   15
An annuity is a periodic stream of
equally-sized payments.

The two basic types of annuities are:

       ordinary annuity
       annuity due



                       (c) ECI RISK TRAINING 2009
                           www.ecirisktraining.com   16
With an ordinary annuity, the first
payment takes place one period in
the future.

With an annuity due, the first
payment takes place immediately.




                    (c) ECI RISK TRAINING 2009
                        www.ecirisktraining.com   17
The formulas used to compute the
future value and present value of a
sum can be easily extended to the
case of an annuity.




                    (c) ECI RISK TRAINING 2009
                        www.ecirisktraining.com   18
The formula for computing the future
value of an ordinary annuity is:


           ⎡ (1 + I ) − 1 ⎤           N
FVAN = PMT ⎢              ⎥
           ⎣       I      ⎦


                     (c) ECI RISK TRAINING 2009
                         www.ecirisktraining.com   19
where:



 FVAN    = future value of an
         N-period ordinary annuity

 PMT     = the value of the
         periodic payment




                      (c) ECI RISK TRAINING 2009
                          www.ecirisktraining.com   20
Suppose that a sum of $1,000 is invested at
the end of each of the next four years at an
annual rate of interest of 3%. What is the
future value of this ordinary annuity?




                        (c) ECI RISK TRAINING 2009
                            www.ecirisktraining.com   21
In this case,

     N=4
     I=3
     PMT = $1,000




                (c) ECI RISK TRAINING 2009
                    www.ecirisktraining.com   22
Using the formula,



           ⎡ (1 + I ) − 1 ⎤              N
FVAN = PMT ⎢              ⎥
           ⎣       I      ⎦



                     (c) ECI RISK TRAINING 2009
                         www.ecirisktraining.com   23
⎡ (1 + .03) − 1 ⎤
                         4
FVA4 = 1,000 ⎢               ⎥ = $4,183.63
             ⎣       .03     ⎦




                     (c) ECI RISK TRAINING 2009
                         www.ecirisktraining.com   24
The future value of the annuity can
also be obtained by computing the
future value of each term and then
combining the results:




                     (c) ECI RISK TRAINING 2009
                         www.ecirisktraining.com   25
1,000(1.03)3 + 1,000(1.03)2 +
1,000(1.03)1 + 1,000(1.03)0

= 1,092.73 + 1,060.90 +
  1,030.00 + 1,000.00

= $4,183.63



                 (c) ECI RISK TRAINING 2009
                     www.ecirisktraining.com   26
The future value of an annuity due
is computed as follows:

   FVAdue = FVAordinary(1+I)




                      (c) ECI RISK TRAINING 2009
                          www.ecirisktraining.com   27
Referring to the previous example, the
future value of an annuity due would be:

   4,183.63(1+.03) = $4,309.14




                     (c) ECI RISK TRAINING 2009
                         www.ecirisktraining.com   28
The formula for computing the present
value of an ordinary annuity is:


              ⎡      1                            ⎤
                1−
              ⎢ (1 + I )N                         ⎥
   PVAN = PMT ⎢                                   ⎥
              ⎢    I                              ⎥
              ⎢
              ⎣                                   ⎥
                                                  ⎦
                    (c) ECI RISK TRAINING 2009
                        www.ecirisktraining.com       29
where:



 PVAN    = future value of an
         N-period ordinary annuity

 PMT     = the value of the
         periodic payment




                    (c) ECI RISK TRAINING 2009
                        www.ecirisktraining.com   30
How much must be invested today in a bank
account that pays 5% interest per year in order
to generate a stream of payments of $1,000 at
the end of each of the next three years?





                          (c) ECI RISK TRAINING 2009
                              www.ecirisktraining.com   31
In this case,

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




                (c) ECI RISK TRAINING 2009
                    www.ecirisktraining.com   32
Using the formula,


              ⎡      1                             ⎤
                1−
              ⎢ (1 + I )N                          ⎥
   PVAN = PMT ⎢                                    ⎥
              ⎢    I                               ⎥
              ⎢
              ⎣                                    ⎥
                                                   ⎦


                     (c) ECI RISK TRAINING 2009
                         www.ecirisktraining.com       33
⎡       1    ⎤
                1−
              ⎢ (1 + .05)3 ⎥
PVA3 = 1, 000 ⎢            ⎥ = $2, 723.25
              ⎢    .05     ⎥
              ⎢
              ⎣            ⎥
                           ⎦




                      (c) ECI RISK TRAINING 2009
                          www.ecirisktraining.com   34
The present value of the annuity can
also be obtained by computing the
present value of each term and then
combining the results:




                    (c) ECI RISK TRAINING 2009
                        www.ecirisktraining.com   35
1,000(1.05)-3 + 1,000(1.05)-2 + 1,000(1.05)-1
= 863.84 + 907.03 + 952.38
= $2723.25




                       (c) ECI RISK TRAINING 2009
                           www.ecirisktraining.com   36
The present value of an annuity
due is computed as follows:

   PVAdue = PVAordinary(1+I)




                    (c) ECI RISK TRAINING 2009
                        www.ecirisktraining.com   37
Referring to the previous example, the
present value of an annuity due would be:

   2,723.25(1+.05) = $2,859.41




                       (c) ECI RISK TRAINING 2009
                           www.ecirisktraining.com   38

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Time Value Of Money Part 1

  • 1. ALAN ANDERSON, Ph.D. ECI RISK TRAINING www.ecirisktraining.com
  • 2. 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 2
  • 3. The time value of money is one of the most fundamental concepts in finance; it is based on the notion that receiving a sum of money in the future is less valuable than receiving that sum today. This is because a sum received today can be invested and earn interest. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 3
  • 4. The four basic time value of money concepts are:  future value of a sum  present value of a sum  future value of an annuity  present value of an annuity (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 4
  • 5. If a sum is invested today, it will earn interest and increase in value over time. The value that the sum grows to is known as its future value. Computing the future value of a sum is known as compounding. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 5
  • 6. The future value of a sum depends on the interest rate earned and the time horizon over which the sum is invested. This is shown with the following formula: FVN = PV(1+I)N (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 6
  • 7. where: FVN = future value of a sum invested for N periods I = periodic rate of interest PV = the present or current value of the sum invested (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 7
  • 8. Suppose that a sum of $1,000 is invested for four years at an annual rate of interest of 3%. What is the future value of this sum? (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 8
  • 9. In this case, N=4 I=3 PV = $1,000 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 9
  • 10. Using the future value formula, FVN = PV(1+I)N FV4 = 1,000(1+.03)4 FV4 = 1,000(1.125509) FV4 = $1,125.51 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 10
  • 11. The present value of a sum is the amount that would need to be invested today in order to be worth that sum in the future. Computing the present value of a sum is known as discounting. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 11
  • 12. The formula for computing the present value of a sum is: FVN PV = (1 + I ) N (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 12
  • 13. How much must be deposited in a bank account that pays 5% interest per year in order to be worth $1,000 in three years?
 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 13
  • 14. In this case, N=3 I=5 FV3 = $1,000 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 14
  • 15. FVN 1, 000 PV = = (1 + I ) N (1.05) 3 1, 000 = = $863.84 1.1576 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 15
  • 16. An annuity is a periodic stream of equally-sized payments. The two basic types of annuities are:  ordinary annuity  annuity due (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 16
  • 17. With an ordinary annuity, the first payment takes place one period in the future. With an annuity due, the first payment takes place immediately. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 17
  • 18. The formulas used to compute the future value and present value of a sum can be easily extended to the case of an annuity. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 18
  • 19. The formula for computing the future value of an ordinary annuity is: ⎡ (1 + I ) − 1 ⎤ N FVAN = PMT ⎢ ⎥ ⎣ I ⎦ (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 19
  • 20. where: FVAN = future value of an N-period ordinary annuity PMT = the value of the periodic payment (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 20
  • 21. Suppose that a sum of $1,000 is invested at the end of each of the next four years at an annual rate of interest of 3%. What is the future value of this ordinary annuity? (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 21
  • 22. In this case, N=4 I=3 PMT = $1,000 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 22
  • 23. Using the formula, ⎡ (1 + I ) − 1 ⎤ N FVAN = PMT ⎢ ⎥ ⎣ I ⎦ (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 23
  • 24. ⎡ (1 + .03) − 1 ⎤ 4 FVA4 = 1,000 ⎢ ⎥ = $4,183.63 ⎣ .03 ⎦ (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 24
  • 25. The future value of the annuity can also be obtained by computing the future value of each term and then combining the results: (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 25
  • 26. 1,000(1.03)3 + 1,000(1.03)2 + 1,000(1.03)1 + 1,000(1.03)0 = 1,092.73 + 1,060.90 + 1,030.00 + 1,000.00 = $4,183.63 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 26
  • 27. The future value of an annuity due is computed as follows: FVAdue = FVAordinary(1+I) (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 27
  • 28. Referring to the previous example, the future value of an annuity due would be: 4,183.63(1+.03) = $4,309.14 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 28
  • 29. The formula for computing the present value of an ordinary annuity is: ⎡ 1 ⎤ 1− ⎢ (1 + I )N ⎥ PVAN = PMT ⎢ ⎥ ⎢ I ⎥ ⎢ ⎣ ⎥ ⎦ (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 29
  • 30. where: PVAN = future value of an N-period ordinary annuity PMT = the value of the periodic payment (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 30
  • 31. How much must be invested today in a bank account that pays 5% interest per year in order to generate a stream of payments of $1,000 at the end of each of the next three years?
 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 31
  • 32. In this case, N=3 I=5 PMT = $1,000 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 32
  • 33. Using the formula, ⎡ 1 ⎤ 1− ⎢ (1 + I )N ⎥ PVAN = PMT ⎢ ⎥ ⎢ I ⎥ ⎢ ⎣ ⎥ ⎦ (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 33
  • 34. 1 ⎤ 1− ⎢ (1 + .05)3 ⎥ PVA3 = 1, 000 ⎢ ⎥ = $2, 723.25 ⎢ .05 ⎥ ⎢ ⎣ ⎥ ⎦ (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 34
  • 35. The present value of the annuity can also be obtained by computing the present value of each term and then combining the results: (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 35
  • 36. 1,000(1.05)-3 + 1,000(1.05)-2 + 1,000(1.05)-1 = 863.84 + 907.03 + 952.38 = $2723.25 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 36
  • 37. The present value of an annuity due is computed as follows: PVAdue = PVAordinary(1+I) (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 37
  • 38. Referring to the previous example, the present value of an annuity due would be: 2,723.25(1+.05) = $2,859.41 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 38