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Chapter - 2


 Concepts of Value
    and Return
Chapter Objectives
 Understand what gives money its time value.
 Explain the methods of calculating present
  and future values.
 Highlight the use of present value technique
  (discounting) in financial decisions.
 Introduce the concept of internal rate of
  return.




            Financial Management, Ninth   2
Time Preference for Money
 Time preference for money is an
  individual’s preference for possession of a
  given amount of money now, rather than the
  same amount at some future time.
 Three reasons may be attributed to the
  individual’s time preference for money:
     risk
     preference for consumption
     investment opportunities



             Financial Management, Ninth   3
Required Rate of Return
 The time preference for money is generally
  expressed by an interest rate. This rate will be
  positive even in the absence of any risk. It may
  be therefore called the risk-free rate.
 An investor requires compensation for assuming
  risk, which is called risk premium.
 The investor’s required rate of return is:
       Risk-free rate + Risk premium.



            Financial Management, Ninth   4
Time Value Adjustment
 Two most common methods of adjusting
 cash flows for time value of money:
     Compounding—the process of calculating future
      values of cash flows and
     Discounting—the process of calculating present
      values of cash flows.




              Financial Management, Ninth   5
Future Value
 Compounding is the process of finding the future
  values of cash flows by applying the concept of
  compound interest.
 Compound interest is the interest that is received on
  the original amount (principal) as well as on any
  interest earned but not withdrawn during earlier
  periods.
 Simple interest is the interest that is calculated only
  on the original amount (principal), and thus, no
  compounding of interest takes place.



              Financial Management, Ninth   6
Future Value
 The general form of equation for calculating
  the future value of a lump sum after n periods
  may, therefore, be written as follows:
         Fn = P (1 + i ) n
 The term (1 + i)n is the compound value
  factor (CVF) of a lump sum of Re 1, and it
  always has a value greater than 1 for positive
  i, indicating that CVF increases as i and n
  increase.
          Fn =P × CVFn,i
            Financial Management, Ninth   7
Example
 If you deposited Rs 55,650 in a bank, which
  was paying a 15 per cent rate of interest on a
  ten-year time deposit, how much would the
  deposit grow at the end of ten years?

 We will first find out the compound value
  factor at 15 per cent for 10 years which is
  4.046. Multiplying 4.046 by Rs 55,650, we get
  Rs 225,159.90 as the compound value:
  FV = 55,650 × CVF10, 0.12 = 55,650 × 4.046 = Rs 225,159.90




                 Financial Management, Ninth           8
Future Value of an Annuity
 Annuity is a fixed payment (or receipt) each
  year for a specified number of years. If you rent
  a flat and promise to make a series of
  payments over an agreed period, you have
  created an annuity.
                       (1 + i ) n − 1 
               Fn = A                 
                             i        
 The term within brackets is the compound
  value factor for an annuity of Re 1, which we
  shall refer as CVFA.
               Fn =A× CVFA n, i
            Financial Management, Ninth    9
Example
 Suppose that a firm deposits Rs 5,000 at the
  end of each year for four years at 6 per cent
  rate of interest. How much would this annuity
  accumulate at the end of the fourth year? We
  first find CVFA which is 4.3746. If we multiply
  4.375 by Rs 5,000, we obtain a compound
  value of Rs 21,875:
  F4 = 5,000(CVFA 4, 0.06 ) = 5,000 × 4.3746 = Rs 21,873




              Financial Management, Ninth   10
Sinking Fund
 Sinking fund is a fund, which is created out of
  fixed payments each period to accumulate to a
  future sum after a specified period. For
  example, companies generally create sinking
  funds to retire bonds (debentures) on maturity.
 The factor used to calculate the annuity for a
  given future sum is called the sinking fund
  factor (SFF).
                          i        
            A = Fn 
                    (1 + i ) n − 1 
                                    
            Financial Management, Ninth   11
Present Value
 Present value of a future cash flow (inflow or
  outflow) is the amount of current cash that is
  of equivalent value to the decision-maker.
 Discounting is the process of determining
  present value of a series of future cash flows.
 The interest rate used for discounting cash
  flows is also called the discount rate.




            Financial Management, Ninth   12
Present Value of a Single Cash Flow
 The following general formula can be employed to
  calculate the present value of a lump sum to be
  received after some future periods:
                     Fn
              P=             = Fn (1 + i ) −n 
                                              
                  (1 + i ) n
 The term in parentheses is the discount factor or
  present value factor (PVF), and it is always less
  than 1.0 for positive i, indicating that a future amount
  has a smaller present value.

              PV = Fn × PVFn ,i


              Financial Management, Ninth    13
Example
 Suppose that an investor wants to find out
  the present value of Rs 50,000 to be
  received after 15 years. Her interest rate is
  9 per cent. First, we will find out the present
  value factor, which is 0.275. Multiplying
  0.275 by Rs 50,000, we obtain Rs 13,750 as
  the present value:
    PV = 50,000 × PVF15, 0.09 = 50,000 × 0.275 = Rs 13,750




               Financial Management, Ninth        14
Present Value of an Annuity
 The computation of the present value of an
  annuity can be written in the following general
  form:
                 1     1 
            P = A −          
                 i i (1+ i) 
                            n
                             
 The term within parentheses is the present
  value factor of an annuity of Re 1, which we
  would call PVFA, and it is a sum of single-
  payment present value factors.
           P = A × PVAFn, i


            Financial Management, Ninth   15
Capital Recovery and Loan
Amortisation
 Capital recovery is the annuity of an investment
  made today for a specified period of time at a
  given rate of interest. Capital recovery factor
  helps in the preparation of a loan amortisation
  (loan repayment) schedule.
                1 
           A= P          
                PVAFn ,i 

           A = P × CRFn,i
The reciprocal of the present value annuity factor
  is called the capital recovery factor (CRF).

            Financial Management, Ninth   16
Present Value of an Uneven
Periodic Sum
 Investments made by of a firm do not
  frequently yield constant periodic cash flows
  (annuity). In most instances the firm receives
  a stream of uneven cash flows. Thus the
  present value factors for an annuity cannot be
  used. The procedure is to calculate the
  present value of each cash flow and
  aggregate all present values.



            Financial Management, Ninth   17
Present Value of Perpetuity
 Perpetuity is an annuity that occurs
  indefinitely. Perpetuities are not very common
  in financial decision-making:
                                    Perpetuity
 Present value of a perpetuity =
                                   Interest rate




               Financial Management, Ninth         18
Present Value of Growing Annuities
 The present value of a constantly growing
  annuity is given below:
              A  1+ g          
                              n

         P=       1 −    ÷      
            i − g   1+ i 
                                 
                                  
 Present value of a constantly growing
  perpetuity is given by a simple formula as
  follows:

                A
          P=
               i–g

            Financial Management, Ninth   19
Value of an Annuity Due
 Annuity due is a series of fixed receipts or
  payments starting at the beginning of each
  period for a specified number of periods.
 Future Value of an Annuity Due
        Fn = A × CVFA n , i × (1 + i )
 Present Value of an Annuity Due
        P = A × PVFA n, i × (1 + i )




              Financial Management, Ninth   20
Multi-Period Compounding
 If compounding is done more than once a
  year, the actual annualised rate of interest
  would be higher than the nominal interest rate
  and it is called the effective interest rate.
                          n× m
                    i
           EIR = 1 +           –1
                  m




            Financial Management, Ninth   21
Continuous Compounding
 The continuous compounding function takes
  the form of the following formula:
         Fn = P × ei× n = P × e x
 Present value under continuous compounding:

             Fn
         P = i n = Fn × e − i ×n
            e




              Financial Management, Ninth   22
Net Present Value
 Net present value (NPV) of a financial
  decision is the difference between the present
  value of cash inflows and the present value of
  cash outflows.
                  n
                       Ct
         NPV = ∑             t
                               − C0
               t =1 (1 + k )




            Financial Management, Ninth   23
Present Value and Rate of Return
 A bond that pays some specified amount in
  future (without periodic interest) in exchange for
  the current price today is called a zero-interest
  bond or zero-coupon bond. In such situations,
  you would be interested to know what rate of
  interest the advertiser is offering. You can use
  the concept of present value to find out the rate
  of return or yield of these offers.
 The rate of return of an investment is called
  internal rate of return since it depends
  exclusively on the cash flows of the investment.

            Financial Management, Ninth   24
Internal Rate of Return
 The formula for Internal Rate of Return is
  given below. Here, all parameters are given
  except ‘r’ which can be found by trial and
  error.
                 n
                      Ct
         NPV =   ∑ (1 + r )t − C0 = 0
                 t =1




            Financial Management, Ninth   25

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Time value of money

  • 1. Chapter - 2 Concepts of Value and Return
  • 2. Chapter Objectives  Understand what gives money its time value.  Explain the methods of calculating present and future values.  Highlight the use of present value technique (discounting) in financial decisions.  Introduce the concept of internal rate of return. Financial Management, Ninth 2
  • 3. Time Preference for Money  Time preference for money is an individual’s preference for possession of a given amount of money now, rather than the same amount at some future time.  Three reasons may be attributed to the individual’s time preference for money:  risk  preference for consumption  investment opportunities Financial Management, Ninth 3
  • 4. Required Rate of Return  The time preference for money is generally expressed by an interest rate. This rate will be positive even in the absence of any risk. It may be therefore called the risk-free rate.  An investor requires compensation for assuming risk, which is called risk premium.  The investor’s required rate of return is: Risk-free rate + Risk premium. Financial Management, Ninth 4
  • 5. Time Value Adjustment  Two most common methods of adjusting cash flows for time value of money:  Compounding—the process of calculating future values of cash flows and  Discounting—the process of calculating present values of cash flows. Financial Management, Ninth 5
  • 6. Future Value  Compounding is the process of finding the future values of cash flows by applying the concept of compound interest.  Compound interest is the interest that is received on the original amount (principal) as well as on any interest earned but not withdrawn during earlier periods.  Simple interest is the interest that is calculated only on the original amount (principal), and thus, no compounding of interest takes place. Financial Management, Ninth 6
  • 7. Future Value  The general form of equation for calculating the future value of a lump sum after n periods may, therefore, be written as follows: Fn = P (1 + i ) n  The term (1 + i)n is the compound value factor (CVF) of a lump sum of Re 1, and it always has a value greater than 1 for positive i, indicating that CVF increases as i and n increase. Fn =P × CVFn,i Financial Management, Ninth 7
  • 8. Example  If you deposited Rs 55,650 in a bank, which was paying a 15 per cent rate of interest on a ten-year time deposit, how much would the deposit grow at the end of ten years?  We will first find out the compound value factor at 15 per cent for 10 years which is 4.046. Multiplying 4.046 by Rs 55,650, we get Rs 225,159.90 as the compound value: FV = 55,650 × CVF10, 0.12 = 55,650 × 4.046 = Rs 225,159.90 Financial Management, Ninth 8
  • 9. Future Value of an Annuity  Annuity is a fixed payment (or receipt) each year for a specified number of years. If you rent a flat and promise to make a series of payments over an agreed period, you have created an annuity.  (1 + i ) n − 1  Fn = A    i   The term within brackets is the compound value factor for an annuity of Re 1, which we shall refer as CVFA. Fn =A× CVFA n, i Financial Management, Ninth 9
  • 10. Example  Suppose that a firm deposits Rs 5,000 at the end of each year for four years at 6 per cent rate of interest. How much would this annuity accumulate at the end of the fourth year? We first find CVFA which is 4.3746. If we multiply 4.375 by Rs 5,000, we obtain a compound value of Rs 21,875: F4 = 5,000(CVFA 4, 0.06 ) = 5,000 × 4.3746 = Rs 21,873 Financial Management, Ninth 10
  • 11. Sinking Fund  Sinking fund is a fund, which is created out of fixed payments each period to accumulate to a future sum after a specified period. For example, companies generally create sinking funds to retire bonds (debentures) on maturity.  The factor used to calculate the annuity for a given future sum is called the sinking fund factor (SFF).  i  A = Fn   (1 + i ) n − 1   Financial Management, Ninth 11
  • 12. Present Value  Present value of a future cash flow (inflow or outflow) is the amount of current cash that is of equivalent value to the decision-maker.  Discounting is the process of determining present value of a series of future cash flows.  The interest rate used for discounting cash flows is also called the discount rate. Financial Management, Ninth 12
  • 13. Present Value of a Single Cash Flow  The following general formula can be employed to calculate the present value of a lump sum to be received after some future periods: Fn P= = Fn (1 + i ) −n    (1 + i ) n  The term in parentheses is the discount factor or present value factor (PVF), and it is always less than 1.0 for positive i, indicating that a future amount has a smaller present value. PV = Fn × PVFn ,i Financial Management, Ninth 13
  • 14. Example  Suppose that an investor wants to find out the present value of Rs 50,000 to be received after 15 years. Her interest rate is 9 per cent. First, we will find out the present value factor, which is 0.275. Multiplying 0.275 by Rs 50,000, we obtain Rs 13,750 as the present value: PV = 50,000 × PVF15, 0.09 = 50,000 × 0.275 = Rs 13,750 Financial Management, Ninth 14
  • 15. Present Value of an Annuity  The computation of the present value of an annuity can be written in the following general form: 1 1  P = A −  i i (1+ i)  n    The term within parentheses is the present value factor of an annuity of Re 1, which we would call PVFA, and it is a sum of single- payment present value factors. P = A × PVAFn, i Financial Management, Ninth 15
  • 16. Capital Recovery and Loan Amortisation  Capital recovery is the annuity of an investment made today for a specified period of time at a given rate of interest. Capital recovery factor helps in the preparation of a loan amortisation (loan repayment) schedule.  1  A= P   PVAFn ,i  A = P × CRFn,i The reciprocal of the present value annuity factor is called the capital recovery factor (CRF). Financial Management, Ninth 16
  • 17. Present Value of an Uneven Periodic Sum  Investments made by of a firm do not frequently yield constant periodic cash flows (annuity). In most instances the firm receives a stream of uneven cash flows. Thus the present value factors for an annuity cannot be used. The procedure is to calculate the present value of each cash flow and aggregate all present values. Financial Management, Ninth 17
  • 18. Present Value of Perpetuity  Perpetuity is an annuity that occurs indefinitely. Perpetuities are not very common in financial decision-making: Perpetuity Present value of a perpetuity = Interest rate Financial Management, Ninth 18
  • 19. Present Value of Growing Annuities  The present value of a constantly growing annuity is given below: A  1+ g   n P= 1 −  ÷  i − g   1+ i      Present value of a constantly growing perpetuity is given by a simple formula as follows: A P= i–g Financial Management, Ninth 19
  • 20. Value of an Annuity Due  Annuity due is a series of fixed receipts or payments starting at the beginning of each period for a specified number of periods.  Future Value of an Annuity Due Fn = A × CVFA n , i × (1 + i )  Present Value of an Annuity Due P = A × PVFA n, i × (1 + i ) Financial Management, Ninth 20
  • 21. Multi-Period Compounding  If compounding is done more than once a year, the actual annualised rate of interest would be higher than the nominal interest rate and it is called the effective interest rate. n× m  i EIR = 1 +  –1  m Financial Management, Ninth 21
  • 22. Continuous Compounding  The continuous compounding function takes the form of the following formula: Fn = P × ei× n = P × e x  Present value under continuous compounding: Fn P = i n = Fn × e − i ×n e Financial Management, Ninth 22
  • 23. Net Present Value  Net present value (NPV) of a financial decision is the difference between the present value of cash inflows and the present value of cash outflows. n Ct NPV = ∑ t − C0 t =1 (1 + k ) Financial Management, Ninth 23
  • 24. Present Value and Rate of Return  A bond that pays some specified amount in future (without periodic interest) in exchange for the current price today is called a zero-interest bond or zero-coupon bond. In such situations, you would be interested to know what rate of interest the advertiser is offering. You can use the concept of present value to find out the rate of return or yield of these offers.  The rate of return of an investment is called internal rate of return since it depends exclusively on the cash flows of the investment. Financial Management, Ninth 24
  • 25. Internal Rate of Return  The formula for Internal Rate of Return is given below. Here, all parameters are given except ‘r’ which can be found by trial and error. n Ct NPV = ∑ (1 + r )t − C0 = 0 t =1 Financial Management, Ninth 25