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					                                                             Chapter 3

                                                              The Time
                                                                Value
                                                              of Money

Essentials of Managerial Finance by S. Besley & E. Brigham         Slide 1 of 48
 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?
             (a)An investment of €100,000 that would return
                         €200,000 after one year
             (b)An investment of €100,000 that would return
                         €220,000 after two years
Essentials of Managerial Finance by S. Besley & E. Brigham   Slide 2 of 48
 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
Essentials of Managerial Finance by S. Besley & E. Brigham            Slide 3 of 48
                           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            0                    1   2   3        4
                                            k = 10%

            Cash Flows -500                                              FVn = ?




Essentials of Managerial Finance by S. Besley & E. Brigham                        Slide 4 of 48
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= €5.00 + €100.00 = €105.00
• Year 2: 5% of €105.00= €5.25 + €105.00 = €110.25
• Year 3: 5% of €110.25= €5.51+ €110.25= €115.76
Essentials of Managerial Finance by S. Besley & E. Brigham                      Slide 5 of 48
                                               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)


Essentials of Managerial Finance by S. Besley & E. Brigham                 Slide 6 of 48
                        Effects of compounding




Essentials of Managerial Finance by S. Besley & E. Brigham   Slide 7 of 48
                        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 1.05. 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 = 110.25
     – By making the same assumptions, the FV in 3 years would
       be 115.76
Essentials of Managerial Finance by S. Besley & E. Brigham   Slide 8 of 48
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
   PV                                          100           (assumes compounded interest
   k                                        5,00%             as capital is being re-invested)
   n                                             3
                                                             =FV (interest, periods, pmt, PV)
   FV?                                      115,76
                                                             =FV (.05, 3, , 100)
Essentials of Managerial Finance by S. Besley & E. Brigham                             Slide 9 of 48
              Financial Calculator Solution
   In the previous example: PV = €100, k =
   5.0%, n = 3
       3         5     -100      0       ?
                N                             I              PV   PMT       FV

                                                                        115,76




Essentials of Managerial Finance by S. Besley & E. Brigham                   Slide 10 of 48
          A Graphic view of Future Value
              Relationship among Future Value, Growth or
                        Interest Rates and Time




Essentials of Managerial Finance by S. Besley & E. Brigham   Slide 11 of 48
                                             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)
Essentials of Managerial Finance by S. Besley & E. Brigham               Slide 12 of 48
                      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 = 8899.96




Essentials of Managerial Finance by S. Besley & E. Brigham   Slide 13 of 48
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?



   FV                          10.000                                 Excel Function
   k                                6,00%
                                                             =PV (interest, periods, pmt, FV)
   n                                     2
   PV?                            8.899,96                   =PV (.06, 2, , 10,000)


Essentials of Managerial Finance by S. Besley & E. Brigham                             Slide 14 of 48
              Financial Calculator Solution
   In the previous example: FV = €10000, k =
   6.0%, n = 2
       2         6        ?      0    10000
                N                             I              PV   PMT   FV

                                                    -8899,96




Essentials of Managerial Finance by S. Besley & E. Brigham               Slide 15 of 48
       A Graphic view of Present Value
            Relationship among Present Value, Growth or
                       Interest Rates and Time




Essentials of Managerial Finance by S. Besley & E. Brigham   Slide 16 of 48
 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%.




Essentials of Managerial Finance by S. Besley & E. Brigham   Slide 17 of 48
 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?
        1999                  1.000
        2000                 1.127
                                              PV              1.000      Excel Function

                                                                    =Rate(periods, pmt, PV,
        2001                 1.158            FV              5.525 FV)
        2002                 2.345
        2003                 3.985            n                   6 =Rate(6, ,1000, 5525)
        2004                 4.677
        2005                 5.525            k?             33,0%
Essentials of Managerial Finance by S. Besley & E. Brigham                           Slide 18 of 48
              Financial Calculator Solution
   In the previous example: PV = €1000, FV =
   €5525, n = 6
       6         ?    -1000      0     5525
                N                             I              PV   PMT   FV

                                         33




Essentials of Managerial Finance by S. Besley & E. Brigham               Slide 19 of 48
              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.




Essentials of Managerial Finance by S. Besley & E. Brigham   Slide 20 of 48
              Financial Calculator Solution
   In the previous example: PV = €712, FV =
   €848, k = 6%
        ?        6     -712      0     848
                N                             I              PV   PMT   FV

                3




Essentials of Managerial Finance by S. Besley & E. Brigham               Slide 21 of 48
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




Essentials of Managerial Finance by S. Besley & E. Brigham   Slide 22 of 48
                   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

Essentials of Managerial Finance by S. Besley & E. Brigham       Slide 23 of 48
   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)0
       + 100(1.03)1+ 100(1.03)2 + 100(1.03)3 + 100(1.03)4 = 530,91




Essentials of Managerial Finance by S. Besley & E. Brigham   Slide 24 of 48
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.




 PMT                                          100                    Excel Function
 k                                          3,0%             =FV (interest, periods, pmt, PV)
 n                                              5
 FV?                                       530,91            =FV (.03, 5,100, )

Essentials of Managerial Finance by S. Besley & E. Brigham                            Slide 25 of 48
              Financial Calculator Solution
   In the previous example: PMT = €100, k =
   3%, n=5
       5         3        0   -100      ?
                N                             I              PV   PMT       FV

                                                                        530,91




Essentials of Managerial Finance by S. Besley & E. Brigham                   Slide 26 of 48
      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
       + 100(1.03)2+ 100(1.03)3 + 100(1.03)4 + 100(1.03)5 = 546,84




Essentials of Managerial Finance by S. Besley & E. Brigham   Slide 27 of 48
    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.



       PMT                              100                          Excel Function
       k                             3,00%                   =FV (interest, periods, pmt, PV)
       n                                  5
                                                             =FV (.03, 5,100, )
       FV                            530,91
       FVA?                          546,84                  =530.91*(1.03)
Essentials of Managerial Finance by S. Besley & E. Brigham                            Slide 28 of 48
              Financial Calculator Solution
   In the previous example: PMT = €100, k =
   3%, n=5 (switch calculator to BEGIN)
       5         3        0    -100     ?
                N                             I              PV   PMT       FV

                                                                        546,84




Essentials of Managerial Finance by S. Besley & E. Brigham                   Slide 29 of 48
                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

Essentials of Managerial Finance by S. Besley & E. Brigham                 Slide 30 of 48
                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 + 1000/(1.1)2+ 1000/(1.1)3 + 1000/(1.1)4
       + 1000/(1.1)5 = 3790,79




Essentials of Managerial Finance by S. Besley & E. Brigham   Slide 31 of 48
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?



 PMT                                     1000                        Excel Function
 I                                     10,0%                 =PV (interest, periods, pmt, FV)
 n                                          5
 PV?                                 3.790,79                =PV (.10, 5, 1000, )
Essentials of Managerial Finance by S. Besley & E. Brigham                            Slide 32 of 48
              Financial Calculator Solution
   In the previous example: PMT = €1000, k =
   10%, n=5
       5        10      ?     -1000     0
                N                             I              PV   PMT   FV

                                                       3790,79




Essentials of Managerial Finance by S. Besley & E. Brigham               Slide 33 of 48
       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)0 + 1000/(1.1)1+ 1000/(1.1)2 + 1000/(1.1)3
       + 1000/(1.1)4 = 4169,87




Essentials of Managerial Finance by S. Besley & E. Brigham   Slide 34 of 48
      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?


              PMT                      1000                          Excel Function
              k                     10,00%
                                                             =PV (interest, periods, pmt, FV)
              n                           5
              PV                    3790,79                  =PV (.10, 5, 1000, )
              PVA?                  4169,87
                                                             = 3790,79*(1,1)
Essentials of Managerial Finance by S. Besley & E. Brigham                            Slide 35 of 48
              Financial Calculator Solution
   In the previous example: PMT = €1000, k =
   10%, n=5 (switch calculator to begin)
       5        10      ?      -1000     0
                N                             I              PV   PMT   FV

                                                      4169,87




Essentials of Managerial Finance by S. Besley & E. Brigham               Slide 36 of 48
              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.


Essentials of Managerial Finance by S. Besley & E. Brigham    Slide 37 of 48
              Financial Calculator Solution
   In the previous example: PV = €846,80, PMT
   = €250, n = 4
       4         ? -846,80     250      0
                N                             I              PV   PMT   FV

                                             7




Essentials of Managerial Finance by S. Besley & E. Brigham               Slide 38 of 48
 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.




Essentials of Managerial Finance by S. Besley & E. Brigham    Slide 39 of 48
              Financial Calculator Solution
   In the previous example: PV = €1685, PMT =
   €400, k = 6
        ?        6    -1685     400      0
                N                             I              PV   PMT   FV

                5




Essentials of Managerial Finance by S. Besley & E. Brigham               Slide 40 of 48
            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
Essentials of Managerial Finance by S. Besley & E. Brigham         Slide 41 of 48
               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




Essentials of Managerial Finance by S. Besley & E. Brigham   Slide 42 of 48
               Present Value of an uneven
               Cashflow Stream - Example
• Calculate the present value of the following uneven
   cashflow stream assuming a required return of 9%.


                                  Year Cash Flow PVIF9% ,N             PV
                                      1                400   0,917      366,80
                                      2                800   0,842      673,60
                                      3                500   0,772      386,00
                                      4                400   0,708      283,20
                                      5                300   0,650      195,00
                                                              PV     1.904,60


Essentials of Managerial Finance by S. Besley & E. Brigham                       Slide 43 of 48
                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%.

       Ye ar Cas h Flow
            1                     400
                                                                    Excel Function
            2                     800
            3                     500                    =NPV (interest, cells containing CFs)
            4                     400
            5                     300
                                                         =NPV (.09,B3:B7)
        NPV                    1.904,76
Essentials of Managerial Finance by S. Besley & E. Brigham                            Slide 44 of 48
        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.




Essentials of Managerial Finance by S. Besley & E. Brigham   Slide 45 of 48
                      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:                                     100 x (1 + .03)5 =    €115.92
     Semiannually:                                   100 x (1 + .015)10=   €116.05



Essentials of Managerial Finance by S. Besley & E. Brigham                     Slide 46 of 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
Essentials of Managerial Finance by S. Besley & E. Brigham        Slide 47 of 48
                   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 = (1 + .18/12) 12 -1
                                        EAR = 19.56%




Essentials of Managerial Finance by S. Besley & E. Brigham         Slide 48 of 48

				
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