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					  The Time Value of Money
Introduction to Time Value of Money, TVM
Future Value, FV
 Lump-sum amount
 Annuity
 Uneven cash flow
Present Value, PV
 Lump-sum amount
 Annuity
 Uneven cash flow
FV and PV Comparison
Solving for r and n
Intra-year Interest Compounding
Amortization


                                           1
        Time Value of Money
Why is it important to understand and apply time
value of money concepts?
What is the difference between a present value
amount and a future value amount?
What is an annuity?
What is the difference between the Annual
Percentage Rate and the Effective Annual Rate?
What is an amortized loan?
How is the return on an investment determined?


                                                   2
   The Time Value of Money
Time value of money is considered the
most important concept in finance
Mathematics of finance
“Nuts & Bolts” of financial analysis—
apply of TVM concepts to determine
value
Interest = Rate of return = r = i = k =
Y


                                          3
      The Time Value of Money

“$1 received today is more valuable than $1
received in one year.” Why?
Because if you have the opportunity to earn a
positive return, investing the $1 today will cause
it to grow to greater than $1 in one year.
For example, $1 invested at 5 percent will grow
to $1.05 in one year because 5¢ of interest will
be earned.



                                                     4
    Future Value and Present Value
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
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 (“deinterest” the FV)


                                                                 5
Lump-Sum Amounts, Annuities, and Uneven
          Cash Flow Streams
 Lump-sum amount—a single amount invested
 (received) today or in the future; growth in value is
 the result of interest only
 Annuity—equal payments made (received) at equal
 intervals; growth in value is the result of additional
 payments as well as interest
   ordinary annuity—end of period payments
   annuity due—beginning of period payments
 Uneven Cash Flows—payments that are not all equal
 that are generally made (received) at equal intervals;
 growth in value is the result of additional payments as
 well as interest


                                                           6
           Cash Flow Time Lines
   Helps you to visualize the timing of the
   cash flows associated with a particular
   situation
   Constructing a cash flow time line is easy:

    Time    0             1   2   3        4
                r = 10%

Cash Flows -500                       FVn = ?




                                                 7
    Approaches to TVM Solutions
Time line solution
 Solve using a cash flow time line
Equation (numerical) solution
 Use equations to solve the problem
Financial calculator solution
 Financial calculators are programmed to solve time value of
  money problems using the numerical solution
Spreadsheet solution
 Spreadsheets contain functions that can be used to solve
  time value of money problems using the numerical solution
Interest tables
 Obsolete



                                                                8
            Future Value
Determine to what amount an
investment will grow over a particular
time period if it is invested at a positive
rate of return.
Compounding
  Lump-sum amount
  Annuity
  Uneven cash flow stream




                                              9
Future Value, FV, of a Lump-Sum Amount

Example: If you invest $500 today at 10%, what
will the investment be worth in four years if interest
is paid annually?

      Time    0             1   2     3        4
                  r = 10%

  Cash Flows -500                         FVn = ?




                                                    10
                           Future Value
Graphically, these computations are:
               0            1            2            3            4
                    10%
                 ×
              -500 1.10         × 1.10       × 1.10       × 1.10
End of year            = 550.00      = 605.00    = 665.50      = 732.05
   amount

                   FV4 = 500(1.10 x 1.10 x 1.10 x 1.10)
                          = 500(1.10)4
                          = 732.05


                                                                       11
                       Future Value
The future value of an amount invested today for n
years, FVn, can be found using the following
equation:
      FVn = PV(1 + r)n = PV(interest multiple)

FVn   =   future value in period n
 PV   =   present, or current, value
  r   =   interest rate per period
  n   =   number of periods interest is earned




                                                     12
    Equation (Numerical) Solution

Determined by applying the appropriate
equation:
    FVn
      4    =      PV
                  500   x      (1 + r)n = 732.05
                               (1.10)4n



In our example: PV = $500, r = 10.0%, n = 4




                                                   13
        Financial Calculator Solution

In our example: PV = $500, r = 10.0%, n = 4

   4        10     -500       0         ?
    N        I/Y     PV      PMT        FV

                                   732.05




                                             14
      Future Value of an Annuity
Annuity—a series of equal payments that are
made at equal intervals
 Ordinary annuity—end of the period
 Annuity due—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



                                                15
        Future Value of an Annuity, FVA
     Time    0         1                 2                 3
                 7%

Cash Flows            -100          -100               -100
                                                           x (1.07)0
                                                                    100.00
                                             x   (1.07)1
                                                                    107.00
                             x (1.07)2
                                                                     114.49
                                                               FVA = 321.49

FVA = 100(1.07)2 + 100(1.07)1 + 100(1.07)0           = 321.49
    = 100[(1.07)2 + (1.07)1 + (1.07)2] = 100(3.2149) = 321.49



                                                                        16
                FVA—
     Equation (Numerical) Solution
                n1
                                 (1  r)n - 1 
   FVA n  PMT  (1  r)t  PMT               
               0                      r       
In our example: PMT = $100, r = 7%, n = 3
                (1.07)3 - 1 
    FVA n  100             
                0.07 
           100(3.2149  321.49
                       )




                                                   17
         FVA—Annuity Due
Annuity due is an annuity with cash flows
that occur at the beginning of the period.
When compared to an ordinary annuity,
which has end-of-period cash flows, the cash
flows of an annuity due earn one additional
period of interest.




                                               18
                     FVA—Annuity Due

     Time     0           1              2           3
                    7%
Cash Flows   -100        -100
                         -100        -100        -100
                                             x (1.07)          x
                                                                  107.00
                                                         (1.07)0 100.00
                                x (1.07) x (1.07)1
                                                                107.00
                                                                 114.49
                    x (1.07) x (1.07)2
                                                          114.49
                                                           122.50
                                                    FVA = 343.99
                                               FVA(DUE) = 321.49




                                                                      19
  FVA(DUE)—Equation (Numerical) Solution

                  (1  r)n - 1           
 FVA(DUE)n = PMT 
 FVA                               (1  r)
                       r                 
In our example: PMT = $100, r = 7%, n = 3
                 (1.07)3 - 1        
 FVA(DUE)n  100               1.07
                 0.07               
            100(3.4399  343.99
                        )




                                                20
   FVA—Financial Calculator Solution
In our example: n = 3, r = 7%, PMT = $100

    3       7.0      0       -100      ?
   N        I/Y      PV       PMT     FV

                              FVA = 321.49
                             BEGIN
   3        7.0      0       -100      ?
   N        I/Y      PV       PMT     FV

                          FVA(DUE) = 343.99

                                              21
Solutions: Future value computations

FV of $25,000 lump-sum amount: N = 5, I/Y = 5,
                            31,907
PV = -25,000, PMT = 0, FV = ?


FV of $5,499.40 annuity: N = 5, I/Y = 5, PV = 0,
                      31,907
PMT = -5,499.40, FV = ?
    Uneven Cash Flow Streams

Uneven cash flow stream—cash flows
that are not all the same (equal)
Simplifying techniques (that is, using a
single equation) used to compute FVA
cannot be used




                                           23
       FV—Uneven Cash Flow Streams

0               1              2                   3
        4%

              -600            -400                -200
                                                        (1.04) 0
                                                                    200.00
                                              1
                                    (1.04)
                                                                    416.00
                  (1.04) 2
                                                                648.96
                                                              _______
    FV  600(1.04)2  400(1.04)1  200(1.04)0                1,264.96




                                                                         24
  FV of Uneven Cash Flow Streams—
    Equation (Numerical) Solution
                 1           2                      n
FVn  CF (1  r)  CF2 (1  r)    CFn (1  r)
        1
       n
      CFt (1  r)   t

      t 1
                 2            1            0
FVn  600(1.04)  400(1.04)  200(1.04)
     600(1.0816  400(1.0400  200(1.0000
               )            )            )
     1,264.96


                                               25
FV of Uneven CF Streams—Calculator Solution

  Input the cash flows, find the present
  value, PV, and then compute the future
  value, FV, of PV.
  Discussed in the next section.




                                           26
            Present Value
Determine the current value of an amount
that will be paid, or received, at a particular
time in the future.
Finding the present value (PV), or
discounting, an amount to be received
(paid) in the future is the reverse of
compounding, or determining the future
value of an amount invested today.
We find the PV by “de-interesting” the FV.


                                                  27
  Present Value—Lump-Sum Amount
What is the PV of $800 to be received in four
years if your opportunity cost is 8 percent?

Stated differently: How much would you be
willing to pay today for an investment that
will pay $800 in four years if you have the
opportunity to invest at 8 percent per year?



                                               28
     Present Value—Lump-Sum Amount


     Time    0         1   2   3     4
                  8%

Cash Flows   PV = ?                800




                                         29
Present Value—Equation (Numerical) Solution

Remember that FV is computed as follows:
          FV = PV x (1 + r)n
          800          (1.08)n
                             4

          800 = PV x 1.36049
In our example, FV4 = 800, n = 4, r = 8.0%
          PV = 800/1.36049 = 588.02




                                           30
Present Value—Equation (Numerical) Solution
                                     
                         
                               1      

  PV Equation: PV  FV
                                     

                                  n
                                     
                                     
                         
                         
                         
                             (1r)    
                                      
                                      


                          1      
                    800         
                                 4
                          (1.08) 
                         
                    800(0.7350 3)  588.02
In our example: FV = $800, r = 8.0%, and N = 4



                                              31
      Present Value—Time Line Solution

Graphically, this computation is:

               0                    1                   2                3                 4
                       8%
              588.02 1            635.07       1
                                                      685.87    1
                                                                       740.74       1
                                                                                       800.00
End of year                                                                     
                       1.08                  1.08             1.08                1.08
   amount




                                                                                               32
    PV Lump-Sum Amount—Financial
          Calculator Solution

In our example: FV = $800, r = 8.0%, n = 4

    4      8.0       ?       0     800
    N       I/Y     PV      PMT      FV
                  -588.02




                                             33
   Present Value of an Annuity, PVA
   0                1                    2               3
          7%

                1    100             100             100
                   1
                     
             (1.07)             1
 93.46                       (1.07)2
                                     
                                                1
 87.34                                               
                                             (1.07)3
 81.63
262.43 = PVA        1             1             1 
               100       1
                               100       2
                                               100         
                    (1.07)        (1.07)        (1.07)3 
                    1            1          1 
       262.43  100       1
                                     2
                                               3
                                                    100(2.6243)
                    (1.07)    (1.07)     (1.07) 


                                                             34
 PVA—Equation (Numerical) Solution
             n
               1            1 - (1r)n 
                                   1

PVA  PMT          t
                       PMT            
          1 (1  r)          r 
                                       
 In our example: PMT = $100, r = 7%, n = 3
         1 - (1.07) 3 
                 1

PVA  100             
          0.7 
                      
     100(2.6243)  262.43


                                             35
           PVA—Annuity Due

Annuity due is an annuity with cash flows
that occur at the beginning of the period.




                                             36
                 PVA—Annuity Due

           0            7%          1     2      3

                       1 1
          100            1   1
                                
                           (1.07)   100   100    100
                                                100
                       (1.07)
                    (1.07)
 93.46
100.00
                           1
               (1.07)           
                        (1.07)2
  87.34
 93.46
                           1
               (1.07)         3
                                
                        (1.07)
 81.63
 87.34
262.43
280.80 = PVA (DUE)




                                                     37
PVA(DUE)—Equation (Numerical) Solution
                  1 - (1r)n
                         1
                                  
  PVA(DUE)n = PMT 
  PVA                  r
                                   x (1 + r)
                  
                                 
                                  
  In our example: PMT = $100, r = 7%, n = 3
                 1 - (1.07) 3 
                 
                          1
                                        
                                        
  PVA(DUE)3  100               1.07
                  0.7 
                                     
                                        
             100(2.8080  280.80
                            )


                                                38
   PVA—Financial Calculator Solution
In our example: n = 3, r = 7%, PMT = $100
     3       7.0         ?      100       0
    N        I/Y      PV       PMT       FV

                   -262.43 = PVA
                             BEGIN
    3        7.0       ?       100       0
    N        I/Y      PV       PMT       FV
                    -280.80 = PVA(DUE)


                                              39
Calculator Solution:
N = 5, I/Y = 5, PMT = -5,499.40, FV = 0, PV = ?
25,000


Numerical Solution:
                     1               
                 1                  
                 (1.05 )5    x(1.05 )  5,499 .40( 4.54595 )
PVA  5,499 .40                        
                 0.05                
                
                
                              
                                       
                                        
     25,000
    Uneven Cash Flow Streams
Uneven cash flow stream—cash flows
that are not all the same (equal)
Simplifying techniques (that is, using a
single equation) used to compute PVA
cannot be used




                                           41
            Present Value of an
          Uneven Cash Flow Stream
      0              1                  2               3
            4%

                 1    600           400             200
                      
              (1.04)1          1
 576.92                     (1.04)2
                                    
                                               1
 369.82                                           3
                                                    
                                            (1.04)
 177.80
               1             1             1 
1,124.54  600       1
                          400       2
                                          200       3
               (1.04)        (1.04)        (1.04) 



                                                            42
  PV of Uneven Cash Flows— Equation
          (Numerical) Solution
          1                 1                      1 
PV  CF 
       1          1
                      CF2    (1  r)2     CFn    (1  r)n 
          (1  r)                                           
      n
            1 
     CFt 
      1     (1  r)t 
                      
         1               1             1 
    600        1
                      400       2
                                      200
         (1.04)          (1.04)        (1.04)3 
                                                    
    1,124.54



                                                                     43
    PV of Uneven Cash Flows—Financial
            Calculator Solution
   Use the cash flow (CF) register (see calculator
    instructions)
   Input CFs in the order they occur—that is,
    first input CF1, then input CF2, and so on
   CF0—most calculators require you to input a
    value for before entering any other cash flows
   Enter the value for I
   NPV = PV of uneven cash flows


                                                      44
PV of Uneven Cash Flows—Financial
        Calculator Solution
     CF0 =   0
     CF1 = 600
     CF2 = 400
     CF3 = 200
      r =    4%
     Compute NPV = 1,124.54



                                    45
FV of Uneven CF Streams—Calculator Solution

    Input the cash flows, find the present
    value, PV, and then compute the future
    value, FV, of PV
    In our example, PV = $1,124.54, so the
    future value is:
        FV = $1,124.54(1.04)3 = $1,264.95




                                             46
Numerical solution:
PV = ($2 million)/(1.06)1 + ($4 million)/(1.06)2
      + ($5 million)/(1.06)3
    = ($2 million)(0.943396) + ($4 million)(0.889996)
      +($5 million)(0.839619)
    = 1.8868 + 3.5600 + 4.1981 = 9.6449
Calculator solution:
CF0 = 0                   I=6
CF1 = 2 million           NPV = 9.6448
CF2 = 4 million
CF3 = 5 million
           Comparison of PVA, FVA,
            and Lump-Sum Amount
     PMT = $100; r = 7%; n = 3
     FVA = $321.49
     PVA = $262.43
        0             1     C           2             3
              7%
A                     100              100            100 B
PVA = 262.43                                    FVA = 321.49
               FV = 262.43 x (1.07)3 = 321.49

               PV = 321.49/(1.07)3 = 262.43




                                                               48
     PVA, FVA, and Lump-Sum Amount
PMT = $100; r = 7%; n = 3; PVA = $262.43

       Beginning Interest    Ending   Payment/
Year    Balance   @ 7%      Balance   Withdrawal
 1      $262.43   $18.37    $280.80    $100.00
 2       180.80    12.66     193.46     100.00
 3        93.46     6.54     100.00     100.00

FVA = 321.49




                                                   49
Solving for Time (n) and Interest Rates
            (r)—Lump Sums
 The computations for lump-sum amounts
 included four variables: n, r, PV, and FV.
 If three of the four variables are known, then
 we can solve for the unknown variable—e.g., if
 n, PV, and FV are known, we can solve for r.




                                                  50
Solving for Interest Rates, r—Lump-Sum
                 Amount
If $200 that was invested three years ago is now
worth $245, what rate of return (r) did the
investment earn?

      Time     0     r=?   1      2         3

 Cash Flows   -200                         245




                                                   51
Solving for r for a Lump-Sum—Equation
          (Numerical) Solution
      FV    = PV (1+r)n
       245 = 200 (1+r)3

     (1 r)3  245
               200
                245 
                       1/3
      (1 r)  
               
                     
                     
                200 
                    

         r  2.45
                 0.333 - 1.0 = 7.0%
                    
                   
                   




                                         52
Solving for r for a Lump-Sum—Financial
           Calculator Solution

In our example: PV = $200, FV = $245, n = 3

   3         ?    -200       0     245
    N       I/Y     PV      PMT      FV

           7.00




                                          53
Solving for Number of Years, n—Lump-
             Sum Amount
If $712 is invested at 6 percent, how long will it
take to grow to $848?

      Time    0 r = 6% 1            2        n=?
                                        …
Cash Flows   -712                             848




                                                     54
Solving for n for a Lump-Sum—Equation
          (Numerical) Solution

             FV  PV(1  r)n

            848  712(1.06)n

        (1.06)n  848
                  712



                                    55
Solving for n for a Lump-Sum—Financial
           Calculator Solution

In our example: PV = $712, FV = $848, r = 6%

    ?      6.0      -712      0       848
    N       I/Y      PV      PMT      FV

  3.00




                                               56
Solving for Time (n) and Interest Rates
             (r)—Annuities
   The computations for annuities included
   four variables: n, r, PMT, and PVA or FVA.
   If three of the four variables are known,
   then we can solve for the unknown
   variable—e.g., if n, PVA (or FVA), and PMT
   are known, we can solve for r.




                                                57
Solving for Interest Rates, r, for Annuities

 The current value of an investment that will pay
 $300 each year for three years is $817. What rate
 of return (r) will the investment earn?
             0           1        2        3
                   r=?
      PVA = -817         300     300       300




                                                 58
Solving for r for Annuities—Equation
        (Numerical) Solution
              1 - (1r)n 
                      1

    PVA  PMT            
               r 
                         
             1 - (1r)3 
                     1

    817  300           
              r 
                        
   To solve, use a trial-and-error process
   Solution = 5.0%



                                              59
  Solving for r for Annuities—Financial
           Calculator Solution

In our example: PMT = $300, PVA = $817, n = 3


    3         ?      -817      300       0
    N        I/Y      PV       PMT       FV
            5.00




                                                60
  Solving for Number of Years, n, for
               Annuities
 If $480 is invested each year at 8 percent,
 how long will take to grow to $2,816?

     Time    0 r = 8% 1           2         n=?
                                        …
Cash Flows            -480       -480        -480
                                      FVA = 2,816




                                                    61
Solving for n for Annuities—Equation
        (Numerical) Solution
                (1  r)n - 1
     FVA  PMT              
                     r      
               (1.08)n - 1
   2,816  480               
               0.08 

    To solve, use a trial-and-error process
    Solution = 5 years



                                               62
  Solving for n for Annuities—Financial
           Calculator Solution

In our example: PMT = $480, FVA = $2,816, r = 8%

    ?       8.0         0     -480   2,816
    N        I/Y      PV       PMT       FV

  5.00




                                              63
Solving for r for Uneven Cash Flows

 Internal Rate of Return (IRR)—average rate
 of return an investment earns
 Capital budgeting decisions—decisions
 concerning what investments a firm should
 purchase




                                              64
Intra-Year Interest Compounding
Interest is compounded more than once per
year—quarterly, monthly, or daily
Adjustments to computations:
 Use the interest rate per compounding period and
  the number of interest compounding periods
  during the life of the investment, or
 Use the effective annual rate, EAR, and the
  number of years to maturity




                                                     65
    Intra-Year Interest Compounding—
                  Example
How much will an amount invested today grow to in two years if
interest is paid quarterly? PV = $200 and r = 8%
        Quarterly interest = 8%/4 = 2% = r/m
        Number of interest payments = 2 years x 4 = 8 = n
   Year   0                                                            1         2
Quarter   0               1              2              3              4         8
              r = 2%
                                                                           …
               x 1.02         x 1.02         x 1.02         x 1.02
      200.00            204.00         208.08         212.24         216.49    234.33

 FVn = PV(1 + r)n = 200(1.02)8 = 200(1.17166) = 234.33




                                                                                     66
Intra-Year Interest Compounding

Financial calculator solution:
     8       2.0     -200        0    ?
     N       I/Y      PV     PMT       FV

                                     234.33




                                              67
Intra-Year Interest Compounding—Effective
                Annual Rate
   Interest = 8%, compounded quarterly
    r = 8% per year is the simple, or non-compounded rate
    r/m = 2% per quarter is the effective rate per
     compounding period (each quarter; m=4)
   Effective Annual Rate (EAR), rEAR
           rEAR = (1 + r/m)m – 1.0
   m = number of interest payment periods per year
   rEAR = (1.02)4 – 1.0 = 0.08243 = 8.243%




                                                             68
Intra-Year Interest Compounding—Using
         Effective Annual Rate
  In our example:
      PV = $200
      n    = 2 years
      rEAR = 8.243%

     FVn =   PV(1 + rEAR)n
         =   200(1.08243)2
         =   234.33
         =   200(1.02)8



                                        69
Annual Percentage Rate, APR, Versus
    Effective Annual Rate, EAR
EAR—the rate of return per year considering
interest compounding
    rEAR = (1 + r/m)m – 1.0
APR—simple rate of return; does not consider
compounding
    APR = r/m x m = r = simple interest
rEAR = APR only if interest is paid once per year—
that is, annual compounding
rEAR > APR if interest is paid more than once per
year


                                                     70
    FVA—Financial Calculator Solution

In our example: n = 365, PV = -$1, PMT = 0, FV
= $1.06168

 365        ?        -1        0      1.06168
    N        I       PV       PMT       FV
         0.016399
 APR = 0.016399 x 365 = 5.986
 Advertise: 5.986% compounded daily
          Amortized Loans
Loan agreement requires equal periodic
payments
A portion of the payment represents interest
on the debt and the remainder is applied to
the repayment of the debt
Amortization schedule—used to determine
what portion of the total payment is interest
and what portion is repayment of principal
Mortgage payment—only the interest portion
is considered an expense for tax purposes


                                                72
  Amortization Schedule—Example
Example: $6,655 home-equity loan; r = 6%; n = 2
years; payments are made quarterly
The constant payments per quarter represent an annuity
and the amount of the loan ($6,655) represents the
present value of the loan payments
PVA = $6,655; (n x m) = 2 x 4 = 8 payments; r/m =
6%/4 = 1.5%
Financial calculator solution:
       8       1.5     6,655      ?       0
        N       I/Y      PV      PMT      FV

                                -889



                                                     73
      Amortization Schedule—Example
Payment = $889
              Begin.   Payment    Interest (I)    Loan Repay.
Year Pmt # Balance      (Pmt)    [1.5% x Beg Bal]  = Pmt – I
 1    1     $6,655.00   $889         $99.83        $789.17
      2      5,865.83    889          87.99          801.01
      3      5,064.82    889          75.97          813.03
      4      4,251.79    889          63.78          825.22
 2    5      3,426.57    889          51.40          837.60
      6      2,588.97    889          38.83          850.17
      7      1,738.80    889          26.08          862.92
      8        875.88*   889          13.14          875.86*
 * Rounding difference




                                                           74
    Answers to TVM Questions

Why is it important to understand and apply
time value of money concepts?
 To be able to compare various investments.
What is the difference between a present
value amount and a future value amount?
 Future value adds interest (compounds); present
  value subtracts interest (“de-interests”).
What is an annuity?
 A series of equal payments that occur at equal
  time intervals.


                                                    75
      Answers to TVM Questions
What is the difference between the Annual
Percentage Rate and the Effective Annual Rate?
 APR is a simple interest rate quoted on
  loans/investments, whereas EAR is the actual interest or
  rate of return.
What is an amortized loan?
 A loan paid off in equal payments over a specified period,
  which include payments of interest and principal
How is the return on an investment determined?
 Compute the annual rate based on the amount to which
  an investment will grow in the future.



                                                               76

				
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