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					The Time Value of Money
Which would you prefer -- $10,000 today
        or $10,000 in 5 years?

               Obviously, $10,000 today.
 Time value of money -- Money to be paid out or received in the
    future is not equivalent to money paid out or received today

You already recognize that there is TIME VALUE TO
                     MONEY!!
             What is Time Value?
• We say that money has a time value because that money can
  be invested with the expectation of earning a positive rate of
  return
• In other words, “a dollar received today is worth more than a
  dollar to be received tomorrow”
• That is because today’s dollar can be invested so that we have
  more than one dollar tomorrow
      Intuition Behind Present Value
• There are three reasons why a dollar tomorrow is
  worth less than a dollar today
  –   Individuals prefer present consumption to future consumption. To induce people
    to give up present consumption you have to offer them more in the future.
  –   When there is monetary inflation, the value of currency decreases over time. The
    greater the inflation, the greater the difference in value between a dollar today and
    a dollar tomorrow.
  –   If there is any uncertainty (risk) associated with the cash flow in the future, the
    less that cash flow will be valued.

• Other things remaining equal, the value of cash
  flows in future time periods will decrease as
  – the preference for current consumption increases.
  – expected inflation increases.
  – the uncertainty in the cash flow increases.
    The Terminology of Time Value
• Present Value - An amount of money today, or the current
  value of a future cash flow
• Future Value - An amount of money at some future time
  period
• Period - A length of time (often a year, but can be a month,
  week, day, hour, etc.)
• Interest Rate - The compensation paid to a lender (or saver)
  for the use of funds expressed as a percentage for a period
  (normally expressed as an annual rate)
               Abbreviations
• PV - Present value
• FV - Future value
• Pmt - Per period payment amount
• N - Either the total number of cash flows or
             the number of a specific period
• i - The interest rate per period
                      Importance
• Shareholders’ wealth maximization requires to address timing
  of cash flow associated with any investment alternatives. So, it
  is the basic foundation for the goal.
• To compare the initial cash out flow (investment) and the
  future cash flow generated from it in long run. Such cash flow
  occurs at several point of time in future so, to make it
  comparable it must be discounted to present value.
• This concept also facilitates investors of securities to evaluate
  the price of securities before investing on them.
• Also used to evaluate the cost benefit of credit.
• Helps to compare cost of different sources of financing and
  help in financing decisions.
• To calculate effective rate on different cash flow compounded
  for different period.
          Cash flow time line
  Graphical presentation of cash flow at different
   point of time.
  Each tick represents one time period

  0 period means today

               i=10%

PV 0      1      2     3      4      5    FV 6

   -100   20     25    30     40     25    30
 Today
   Calculating the Future Value
• Suppose that you have an extra $100 today that you wish
  to invest for one year. If you can earn 10% per year on
  your investment, how much will you have in one year?



  -100      ?

   0        1      2       3       4      5

          FV1  1001  0.10  110
Calculating the Future Value (cont.)

 • Suppose that at the end of year 1 you decide to extend
   the investment for a second year. How much will you have
   accumulated at the end of year 2?

           -110     ?

    0        1      2       3      4       5

     FV2  1001  0.101  010  121
                              .
     or
     FV2  1001  0.10  121
                        2
  Generalizing the Future Value

• Recognizing the pattern that is developing, we can
  generalize the future value calculations as follows:
• FV=PV(FVIF,i%,n)

               FVN  PV1  i
                                    N



   If you extended the investment for a third
    year, you would have:

         FV3  1001  010  13310
                                3
                        .        .
            Compound Interest

• Note from the example that the future value is increasing
  at an increasing rate
• In other words, the amount of interest earned each year is
  increasing
   – Year 1: $10
   – Year 2: $11
   – Year 3: $12.10
• The reason for the increase is that each year you are
  earning interest on the interest that was earned in
  previous years in addition to the interest on the original
  principle amount
  Calculating the Present Value

• So far, we have seen how to calculate the
  future value of an investment
• But we can turn this around to find the
  amount that needs to be invested to achieve
  some desired future value:
PV=FV(PVIF,i%,n)
                        FVN
                PV 
                       1  i   N
    Present Value: An Example

• Suppose that your five-year old daughter has just
  announced her desire to attend college. After some
  research, you determine that you will need about
  $100,000 on her 18th birthday to pay for four years of
  college. If you can earn 8% per year on your investments,
  how much do you need to invest today to achieve your
  goal?

                     100,000
              PV                  $36,769.79
                     108
                       .
                             13
• Find the future value of an initial Rs. 500 compounded for
  1 year at 10%.
• Find the future value of an initial Rs. 500 compounded for
  5 year at 10%.
• Find the present value of Rs. 500 due in 1 year at 10%.
• Find the present value of Rs. 500 due in 5 years at 10%.
• Repeat all above if compounding is for each six months.
• If you deposit Rs. 20000 at XYZ bank’s saving account
  which provide 10%, how long will it take to be the deposit
  double?
• If you borrow Rs. 700 today and promise to pay back Rs.
  749 at the end of year what is the interest rate you are
  paying?
                          Annuities
• An annuity is a series of nominally equal payments occurred at equal
  interval of time for a given number of periods.
• There are two types of annuities:
  a) ordinary annuity : Series of equal payment at the end of each period.
  b) Annuity due: Series of equal payment at the beginning of each period.
• Annuities are very common:
   – Rent
   – Mortgage payments
   – Car payment
   – Pension income
• The timeline shows an example of a 5-year, $100 annuity


                   100    100     100     100     100

            0       1       2      3       4      5
  The Principle of Value Additivity
• How do we find the value (PV or FV) of an
  annuity?
• First, you must understand the principle of
  value additivity:
  – The value of any stream of cash flows is equal to
    the sum of the values of the components
• In other words, if we can move the cash flows
  to the same time period we can simply add
  them all together to get the total value
Present Value of an Ordinary Annuity

• We can use the principle of value additivity to find the
  present value of an annuity, by simply summing the
  present values of each of the components:
            N

            1  i
                  Pmt t            Pmt 1            Pmt 2                   Pmt N
   PVA                                                          
           t 1
                          t
                                  1  i  1
                                                   1  i   2
                                                                           1  i   N
 Present Value of an Ordinary Annuity (cont.)


 • Using the example, and assuming a discount rate of 10%
   per year, we find that the present value is:
                  100            100             100             100             100
         PVA                                                                            379.08
                 110
                   .
                        1
                                110
                                  .
                                        2
                                                110
                                                  .
                                                        3
                                                                110
                                                                  .
                                                                        4
                                                                                110
                                                                                  .
                                                                                        5




62.09
68.30
75.13
82.64
90.91
379.08             100            100            100             100            100

          0         1              2              3               4               5
Present Value of an Ordinary Annuity (cont.)

• Actually, there is no need to take the present
  value of each cash flow separately
• We can use a closed-form of the PVA equation
  instead:
• PVA=Pmt(PVIFA,i%,n)
                                       1  1          
               N
                                             1  i 
                                                     N


               1  i
                     Pmt t
      PVA                   t
                                  Pmt                
              t 1                           i        
                                                      
Present Value of an Ordinary Annuity (cont.)


• We can use this equation to find the present
  value of our example annuity as follows:
                     1  1         
                     
           PVA  Pmt 
                             110   379.08
                               .
                                  5

                                    
                          .
                          010       
                                   

   This equation works for all regular annuities,
    regardless of the number of payments
          Future Value of an Ordinary Annuity


  • We can also use the principle of value additivity to find the
    future value of an annuity, by simply summing the future
    values of each of the components:

        N

              Pmt t 1  i          Pmt 1 1  i           Pmt 2 1  i
                               Nt                     N 1                     N 2
FVA                                                                                        Pmt N
        t 1
  The Future Value of an Ordinary Annuity (cont.)


    • Using the example, and assuming a discount rate of 10%
      per year, we find that the future value is:
FVA  100110  100110  100110  100110  100  610.51
                4         3          2          1
           .          .          .          .




                                                    }
                                          146.41
                                          133.10
                                          121.00        = 610.51
                                          110.00
                                                        at year 5
           100      100       100   100   100

    0       1       2         3     4      5
The Future Value of an Annuity (cont.)

• Just as we did for the PVA equation, we could
  instead use a closed-form of the FVA equation:
• FVA=Pmt(FVIFA,i%,n)
             N
                                         1  i N  1 
             Pmt 1  i
                            Nt
    FVA           t               Pmt                
            t 1                        
                                               i       
                                                        
   This equation works for all regular annuities,
    regardless of the number of payments
The Future Value of an Annuity (cont.)

• We can use this equation to find the future
  value of the example annuity:
                  1105  1 
                     .
        FVA  100              610.51
                  010 
                      .      
                      Annuities Due

     • Thus far, the annuities that we have looked at begin their
       payments at the end of period 1; these are referred to as
       regular annuities
     • A annuity due is the same as a regular annuity, except that
       its cash flows occur at the beginning of the period rather
       than at the end

5-period Annuity Due     100   100    100   100    100
5-period Regular Annuity        100   100    100   100   100

                         0      1      2     3      4    5
Present Value of an Annuity Due
• We can find the present value of an annuity due in the
  same way as we did for a regular annuity, with one
  exception
• Note from the timeline that, if we ignore the first cash
  flow, the annuity due looks just like a four-period regular
  annuity
• Therefore, we can value an annuity due with:
• PVAD=Pmt(PVIFA,i%,n-1)+Pmt or
• PVAD=Pmt(PVIFA,i%,n)(1+i)
                        1  1         N 1
                                              
                        
                   Pmt 
                               1  i   Pmt
           PVAD                               
                                i            
                        
                                             
                                              
Present Value of an Annuity Due (cont.)


• Therefore, the present value of our
  example annuity due is:
                    1  1       51
                                       
                    
                100
                           110   100  416.98
                              .
        PVAD                           
                        010.          
                    
                                      
                                       

   Note that this is higher than the PV of the,
    otherwise equivalent, regular annuity
Future Value of an Annuity Due
• To calculate the FV of an annuity due, we
  can treat it as regular annuity, and then
  take it one more period forward:
• FVAD=Pmt(FVIFA,i%,n)(1+i)
                          1  i N  1 
            FVAD    Pmt                1  i
                         
                                i       
                                         

      Pmt    Pmt     Pmt   Pmt    Pmt

      0       1       2     3      4     5
Future Value of an Annuity Due (cont.)


• The future value of our example annuity is:

                  1105  1 
                     .
     FVAD    100            110  67156
                                 .        .
                  010 
                      .      

   Note that this is higher than the future value
    of the, otherwise equivalent, regular annuity
        Deferred Annuities

• A deferred annuity is the same as any other
  annuity, except that its payments do not
  begin until some later period
• The timeline shows a five-period deferred
  annuity
                 100   100   100   100   100

    0    1   2    3     4     5     6     7
      PV of a Deferred Annuity

• We can find the present value of a deferred annuity in the
  same way as any other annuity, with an extra step
  required
• Before we can do this however, there is an important rule
  to understand:
     When using the PVA equation, the resulting PV
     is always one period before the first payment
     occurs
PV of a Deferred Annuity (cont.)

• To find the PV of a deferred annuity, we
  first find use the PVA equation, and then
  discount that result back to period 0
• Here we are using a 10% discount rate
               PV2 = 379.08
PV0 = 313.29
           0     0    100     100   100   100   100

     0     1      2    3       4     5     6     7
PV of a Deferred Annuity (cont.)


                     1  1         
                     
            PV2  100
                             110   379.08
                               .
                                  5

  Step 1:                           
                         010
                           .        
                     
                                   
                                    


                    379.08
  Step 2:   PV0                 313.29
                    110
                      .
                            2
       FV of a Deferred Annuity
• The future value of a deferred annuity is
  calculated in exactly the same way as any
  other annuity
• There are no extra steps at all
          Uneven Cash Flows
• Very often an investment offers a stream of
  cash flows which are not either a lump sum or
  an annuity
• We can find the present or future value of
  such a stream by using the principle of value
  additivity
Uneven Cash Flows: An Example (1)

• Assume that an investment offers the following cash
  flows. If your required return is 7%, what is the maximum
  price that you would pay for this investment?

                  100       200         300

           0       1            2           3       4           5


                  100            200             300
          PV                                              513.04
                 107
                   .
                        1
                                107
                                  .
                                        2
                                                107
                                                  .
                                                        3
Uneven Cash Flows: An Example (2)

• Suppose that you were to deposit the following amounts
  in an account paying 5% per year. What would the
  balance of the account be at the end of the third year?

                300   500     700

           0     1        2    3        4   5


         FV 300105  500105  700  1,555.75
                      2             1
                  .          .
                 Class Discussion
• Here are the three contracts offered by different
  companies for an Engineer who has to select one. The
  cash flow (Rs. In million) from each contract is as follows:

         Year    CF Contract 1   CF Contract 2   CF Contract 3
          1                  3               2               7
          2                  3               3               1
          3                  3               4               1
          4                  3               5               1

• If discounting interest rate is 10%, which contract should
  the engineer accept?
                 Class Discussion
• Assume that it is now Jan-1, 2010 and you need Rs. 1000
  on Jan-1, 2014. Your bank account interest rate is 8%
  annual. If you want to make equal payment on each Jan-1,
  2011 through 2014 to accumulate the Rs. 1000, how large
  must each deposit be? If your father offer either to make
  the payments calculated above or to give you a lump sum
  of Rs. 750 on Jan-1, 2011, which would you choose?
• If you won a lottery which will pay you Rs. 1.75 million per
  year over next 20 years and the first installment is
  received immediately, find the following:
   – If interest rate is 8% what is the present value of the lottery?
   – If interest rate is 8% what is the future value of the lottery?
   – How would your answer change if the payments were received at
     the end of each year?
        Non-annual Compounding
• So far we have assumed that the time period is equal to a
  year
• However, there is no reason that a time period can’t be any
  other length of time
• We could assume that interest is earned semi-annually,
  quarterly, monthly, daily, or any other length of time
• The only change that must be made is to make sure that the
  rate of interest is adjusted to the period length
Non-annual Compounding (cont.)

• Suppose that you have $1,000 available for investment.
  After investigating the local banks, you have compiled the
  following table for comparison. In which bank should you
  deposit your funds?

   Bank                Interest Rate   Compounding
   First National          10%           Annual
   Second National         10%           Monthly
   Third National          10%            Daily
Non-annual Compounding (cont.)

• To solve this problem, you need to determine which bank
  will pay you the most interest
• In other words, at which bank will you have the highest
  future value?
• To find out, let’s change our basic FV equation slightly:

                                    Nm
                               i
                 FV  PV 1  
                               m
  In this version of the equation ‘m’ is the number of
  compounding periods per year
Non-annual Compounding (cont.)

• We can find the FV for each bank as follows:


                           FV  1,000110  1100
                                           1
First National Bank:                   .      ,
                                                   12
                                          010 
                                            .
Second National Bank:      FV  1,000 1                1104.71
                                                            ,
                                          12 
                                                   365
                                          010 
                                            .
Third National Bank:       FV  1,000 1                110516
                                                            , .
                                          365 
 Obviously, you should choose the Third National Bank
            Effective Annual Rate
• If a rate is quoted at 16%, compounded semiannually, then the actual
  rate is 8% per six months. Is 8% per six months the same as 16% per
  year?
• Normally there are two kinds of interest rate: simple interest rate and
  effective annual rate.
• Simple interest rate also called annual percentage rate (APR) is the
  quoted interest rate which is used to compute the interest paid per
  period.
• Effective annual rate (EAR) is the annual interest rate actually being
  earned, as opposed to the quoted rate, considering the compounding
  of interest.
• EAR is calculated to compare between two simple interest rate when
  the compounding period is different.
                                     m
                            i 
                   EAR  1    1
                          m
                Class Discussion
• Suppose Bank of Kathmandu pays 7 percent interest,
  compounded annually on saving deposit. Everest Bank Ltd
  pays 6.5 percent interest compounded quarterly. Based on
  the effective annual rate in which bank would you prefer
  to deposit your money?
• You have an option to invest at a development bond
  issued by Nepal Rastra Bank paying interest 8% but the
  interest is paid semiannually. Similarly, a bank is offering
  you a fixed deposit scheme with 7.5% interest rate
  compounded monthly. If you have Rs. 100000 investable
  fund where will you invest and what will be the total
  amount with you after one year?
• A bank charges 1% per month on car loans. What is the
  APR? What is the EAR?
               Loan Amortization
• One of the most important applications of compound
  interest involves loans that are paid off in installments
  over time.
• Example: automobile loans, student loans, home
  mortgage loan etc.
• If a loan is to be repaid in equal periodic amounts
  (monthly, quarterly or annually) then it is said to be
  amortized loan.
        Loan Amortization (Cont.)
• Suppose a loan of Rs. 10000 is to be repaid in 4 equal
  installments including principal and 10% interest per
  annum.
• The time line for the loan will be:

                 Inst1 Inst 2 Inst 3 Inst 4
   10000
           0      1      2      3      4
          Loan Amortization (Cont.)
  • All four installment will be equal and they contain
    principal repayment as well as interest amount.
  • Installment amount is calculated by using the following
    formula:

              Amount of Loan 10000
Installment                           3154 .67
               (PVIFA, i%, n)   3.1699
  • The installment of Rs. 3154 includes both principal and
    interest.
  • But the interest and principal amount differs each year.
  • Loan amortization schedule provides the amount of
    interest and principal repayment for each period.
           Loan Amortization (Cont.)
Year (A)    Loan (B)   Installment Interest (D)   Principal Loan
                       (C)         =Bx10%         Repayment Balance
                                                  (E)=C-D
1           10000      3154.67      1000          2154.67   7845.30


2           7845.30    3154.67      784.53        2370.14   5475.16

3           5475.16    3154.67      547.52        2607.15   2868.01

4           2868.01    3154.67      286.66        2868.01   0
                  Class Discussion
• ABC inc. just borrowed Rs. 25000. Loan is to be repaid in
  equal installment at the end of each of the next 5 years
  and the interest rate is 10%.
    Set up an amortization schedule for the loan.
    How large must each annual payment be if the loan is for Rs.
     50000?
    How large be the installment be if the loan is Rs. 50000 and equal
     installment would be paid at the end of next 10 years?
    Although the loan is for the same amount but the payment is
     spread over twice as many periods. Why are these payments not
     half as large as the payment of the loan in the earlier case?
                 Class Discussion
• You want to buy a Mazda. It costs $25,000. With a 10%
  down payment, the bank will loan you the rest at 12% per
  year (1% per month) for 60 months. What will your monthly
  payment be?
• Consider Bill’s retirement plan
  Assume he just turned 40, but, recognizing that he has a lot
  of time to make up for, he decides to invest in some high-risk
  ventures that may yield 20% annually. (Or he may lose his
  money completely!) Anyway, assuming that Bill still wishes
  to accumulate $1 million by age 65, and will begin making
  equal annual deposits in one year and make the last one at
  age 65, now how much must each deposit be?
                     Class Discussion
• To complete your last year in Business school and then go
  through Law school, you need Rs. 10000 per year for next
  4 years starting from next year i.e. you need to withdraw
  first 10000 one year from now. Your rich uncle offers to
  put you through the school and he will deposit in a bank
  paying 7% interest a sum of money that will be sufficient
  to provide the four payments of Rs. 10000 each. His
  deposit will be made today.
    How large must the deposit be?
    How much will be in the account immediately after you make the first withdrawal?
     After the last withdrawal?

				
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