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					Business Funding & Financial
        Awareness
Time Value of Money – The Role of Interest
        Rates in Decision Taking


                J R Davies
                 May 2011
Time Value of Money – The Role of Interest Rates
               in Decision Taking
                     May 2011


                    Dick Davies
Investment/Financing Decisions
 The time dimension –

 Many financial and investment decisions involve costs and
 benefits spread out over time

 An investment decision involves the commitment of resources on
 the expectation of future benefits
     Costs    Benefits       Benefits         Benefits……
                                                             Time
       0           1             2               3
  This implies it is necessary to allow for

  •The time value of money
  •The impact of risk and uncertainty
                                                                    3
Time Value of Money
• A pound today is worth more than a pound to-
  morrow………even in the absence of
   – Risk and uncertainty
   – Inflation
• The time value of money stems from the interest rate –
  effectively the price that balances the supply and demand for
  loans- and this will positive in a world of constant prices and no
  uncertainty




                                                                       4
Determining Interest Rates – A Simple Model
         Interest rate
                          Saving = Deposits = Lending




     r

                          Investment = Borrowing

                                Saving and Investment



                                                        5
Interest rates adjust to changes in savings
and investment – eg an increase in savings
          Interest rate
                           Saving = Deposits = Lending




     r
     r1
                           Investment = Borrowing

                                 Saving and Investment



                                                         6
  Interest Rates
• The riskless real rate of interest (r0): the rate of interest that can
  be expected in the absence of
   – Risk and uncertainty
   – Inflation

• A premium is added to the “real” rate of interest for
   – Risk and uncertainty – this will vary across borrowers
   – Inflation.


         rM  r0  f  u
                            eal                                  ation
         for example if the r rate is 4 per cent, the rate of infl
                            he
         is 6 per cent, and t risk premium is 5 per cent
         rM  r0  f  u  0.04  0.06  0.05  0.15
                                                                           7
Time Dimension – Investment/Financing Decisions
  Capital Budgeting (Real Investments)
    -C0     +C1    +C2    +C3    +C4      +C5   Time
     0       1      2       3      4       5

  Share Purchase (Financial Investment)
     -C0    +C1     +C2    +C3    +C4     +C5   Time
     0       1       2      3      4       5

   Loan (Financing Decision)
     +C0 -C1        -C2 -C3       -C4     -C5   Time
     0      1       2      3       4       5           8
Evaluating Cash Flows Arising at Different Points
in Time
  •   Cash flows that occur at different points in time
      cannot be summed to determine the net benefit
      position
  •   If a cash payment is made now on the expectation of
      receiving cash inflows in the future it is necessary to
      •   borrow the funds to make the payment now, and this implies
          an interest cost will be incurred, or
      •    use your own funds - and this implies foregoing the interest
          income that could have been earned on these funds
      •   in either case there is an interest cost to consider.




                                                                          9
 Adjusting Values to Allow for Interest (1)

 Assume the interest rate is 10 % ie 0.10

 What are the equivalent values at the end of one year, year two,
 etc to a sum of £100 available today ?

100        ?            ?              ?
 0         1            2              3




                                             time



                                                                    10
Adjusting Values to Allow for Interest (2)

  Given an interest rate of 10 % what is the equivalent value at the
  end of one year of £100 that is available today ?

100     (10)    110              ?                ?
0                1               2                3




                                                                   time



The original sum (£100) plus interest that can be earned
over one year (£10).
                                                                          11
Adjusting Values to Allow for Interest (3)


    Given an interest rate of 10 % what is the equivalent
    value at the end of two years of £100 that is available
    today ?

  100    (10)   110    (11)   121           ?
   0             1             2            3
                                                           Time



   The original sum (£100) plus interest (£10) for year one,
   and interest of £10 for year two on the initial £100, plus £1
   of interest on the interest of £10 earned in period one.        12
Adjusting Values to Allow for Interest (4)

   Given an interest rate of 10 %
              £100.00       today
              £110.00       next year
              £121.00       two years from now
              £133.10       three years from now


    all have the same real value (in principle) and are
    equally acceptable
    (assume no risk and no inflation for simplicity).



                                                          13
 Future Value Factors

To obtain the equivalent value at a point in time in the
future of a sum available today we must multiply this sum
by a future value factor – also referred to as a compound
interest factor, or more simply as the interest factor – to
allow for interest that can be earned on the sum available
to-day:
                   FVFn/r = (1 + r)n
where       r    is the rate of interest
            n   is the number of time periods in the
                future
                                                              14
Developing Future Value Factors

  V1  V0  V0 r  V0 (1  r )
  V2  V1  V1r  V1 (1  r )
      V0 (1  r )(1  r )
      V0 (1  r ) 2
  Vn  V0 (1  r )   n




                                  15
Developing Future Value Factors - An
Example
   V1  100  100 x 0.10  100(1  0.10)  110
   V2  110  110 x 0.10  110(1  0.10)
      100(1  0.10)(1  0.10)
      110(1  0.10)  121
                     2




   V3  121  121 x 0.10
      121(1 x 0.10)
      [100(1  0.10)(1  0.10)] (1  0.10)
      100(1  0.10) 3  100 x 1.331  133.10
                                                 16
   Developing Future Value Factors
                    V1  V0 (1  r )
                    V2  V0 (1  r ) 2
                    V3  V0 (1  r ) 3
This implies one more interest factor is introduced for each added
time period and the value at the end of period n is given by

          Vn  V0 (1  r ) n  (1  r )(1  r )(1  r )....(1  r )

                                  Multiply together n interest factors
                                                                         17
     Using Future Values (1)
    What will £800 deposited in a bank account at an interest rate of 12 per cent
    grow to by the end of year 5 if all interest income is reinvested?




.
    Determining the Future Value of a Sum (2)
   What will £800 invested at interest rate is
   12 per cent grow to by the end of year 5 ?
                      Vn  V0 (1  r ) n
                      V 6  800(1  0.12)   5


                            800(1.7623)
                            1409.87
The interest rate for these calculations must be written in decimal form.
In principle this implies that £800 today is of equivalent value to £1410
to be received after five years.
                                                                            19
 Use factors taken from table 1
    Determining the Future Value of a Sum (3)
   You expect to receive £1000 at the end two years
   and you expect to be able to invest this to earn an
   interest rate of 8 per cent. What can the sum be
   expected to grow to by the end of year 5 ?
                              V5  V2 (1  r ) 5 2
                              V5  1000 (1  0.08 ) 3
                                   1000 (1.2597 )
                                   1259 .70
The interest rate for these calculations must be written in decimal form.
In principle this implies that £1000 after two years is of equivalent value
to £1259.70 to be received after five years.
                                                                          20
 Use factors taken from table 1
Annuities
An annuity is a constant payment at the end in
each time period for a specified number of periods.

A constant periodic NCF      Constant net cash flows



       A     A    A     A     A      A      A     A

 0     1     2     3     4     5     6      7      8 …
                                                       21
 Annuities
An annuity is a constant payment at the end (or the start) of
each time period for a specified number of time periods.
A constant periodic NCF



 An Example of an Annuity

      0               1              2               3
                    500            500             500


    An annuity of £500 for three years

                                                                22
 Investment Example and the use of Annuity Factors


FV (3)  V3  500 x (1.1) 2  500 x (1.1)1  500 x (1.0)
V3  500 x [(1.1000) 2  (1.1000)1  (1.0000)]
V3  500 x [(1.2100)  (1.100)  (1.000)]
V3  500 x [3.3100]  1655



                                    Future value annuity factor
                                    for three years at 10 per cent

                                                                     23
Future Value Annuity Factors




                               24
     Using Future Value Annuity Factors
What will be the accumulated value of annual savings of £1200
deposited in a savings account at the end of each of the next 8
years if the interest rate is 7 percent ?


       1200 1200       1200    1200     1200     1200     1200    1200
 0       1       2       3       4        5       6       7       8


                                                      Accumulated Value ?



                                                                            25
    Using Future Value Annuity Factors (2)
What will be the value of annual savings of £1200 for the next 8 years if the interest
rate is 7 percent ? (Interest being reinvested at 7 per cent.)
                   Opening                    Closing
          Year      Value        Interest      Value       Savings
            1         0.00         0.00         £0.00        1200
            2       1200.00       84.00       1,284.00       1200
            3       2484.00       173.88      2,657.88       1200
            4       3857.88       270.05      4,127.93       1200
            5       5327.93       372.96      5,700.89       1200
            6       6900.89       483.06      7,383.95       1200
            7       8583.95       600.88      9,184.83       1200
            8      10384.83       726.94     11,111.76       1200
            9      12311.76
       FV (8) = 1200 times FVAF8/0.07
                                                                                         26
              = 1200 times 10.2598 = 12311.76
Example: Using Time Value Concepts (3)


 Determining Pension Income
 An individual pays £3,000 per annum into a pension
 fund (a defined contribution scheme) for thirty years.
 The scheme guarantees a minimum return of 5 per cent.


 How much will have been accumulated in the fund by
 the end of the 30 year period.


                                                          27
Assessing Pension Payments
            Period for contributions


     3000    3000…….                       3000

 0    1         2 ………………...…………………… 30


            V30 = £3000 times FVAF30/.05
                = £3000 x 66.4388
                = £199,317




                                                  28
    Using Future Value Annuity Factors (5)
Hendy Hotels Ltd

Hendy Hotels is a family owned concern that avoids the use of external
funding. The owners recognise that they will have to undertake a major
investment five years from now to meet EU safety regulations. This
investment will cost £600,000 and the company’s management intend
putting aside funds at the end of each of the next five years so as to be
able to cover the expenditure. The funds can be invested at 7 per cent
until needed. If the same amount is saved each year how much has to
saved on an annual basis to cover the expenditure?




                                                                            29
   Using Future Value Annuity Factors (5)
Hendy Hotels Ltd

Hendy Hotels is a family owned concern that avoids the use of external
funding. The owners recognise that they will have to undertake a major
investment five years from now to meet EU regulations. This investment
will cost £600,000 and the company’s management intend putting aside
funds at the end of each of the next five years so as to be able to cover
the expenditure. The funds can be invested at 7 per cent until needed. If
the same amount is saved each year how much has to saved on an annual
basis to cover the expenditure?

      FV (5) = X times FVAF5/0.07 = £600,000
            = X times 5.7507      = £600,000
          X = 600,000 / 5.7507 = £104,334                               30
Present Value or Discount Factors (1)

  To derive the value today, the present value, of a sum
  expected in the future this future sum must be
  multiplied by a present value or discount factor.

                            1
                        (1  r ) n

  This has a value of less than one as the denominator (1+r) is greater
  than one when r is positive, and applying this to a future NCF will
  allow for the loss of interest as a result of the delay in the receipt of
  the cash..
                                                                          31
Discount Rates - Terminology
• The discount rate
• The opportunity cost of funds – interest
  foregone by waiting.
• The required rate of return.
• The cost of capital.




                                             32
Present Value Factors


                          1
         PVFn / r   
                      (1  r ) n




                                   Time
0    1     2              n


                                          33
   Present Value Factors
All financial arithmetic is based on the future value equation.


                           Vn  V0 (1  r ) n

If a future value is known the equivalent value today is derived
by multiplying the future value by the discount factor, one over the
interest factor
                                     1
                            Vn x             V0
                                 (1  r ) n



  i.e.

                                       1
                           V0  Vn
                                   (1  r ) n                          34
      Present Value Factors at 10 %

1.2




  1




0.8



                                                                           Interest lost in the
0.6
                                                                           delay in receiving
                                                                           cash.
0.4




0.2




 0
      0   1   2   3   4   5   6   7   8   9     10    11   12   13   14   15   16   17   18   19   20

                                              YEARS


                                                                                                        35
    Determining Present Values
What is the equivalent value today of £650 to be received
three years from now if the interest rate (discount rate) is
14 percent ?



                               1
                   V0  Vn
                           (1  r ) n
                                   1
                   V0  650
                             (1  0.14) 3
                       650  0.6750
                       438.73
                                                          36
Present Value Factors




                        37
Net Present Value of an Investment

• The surplus expected from a project, measured in
  today’s values ….after appropriate allowances have
  been made for the
   – recovery the capital outlay
   – the interest charges

• It can also be defined as the increment of wealth
  generated created by an investment



                                                       38
Net Present Value Equation



                      1              1
  NPV   I 0  C1           C2            ....
                   (1  r )      (1  r ) 2




                                                   39
Assessing Investment Proposals Using NPV

    An investment of 1200 is expected to produce cash flows of 500 at the end
    of years 1, 2 and 3. The required rate of return is 10 per cent.
    Determine the investment’s NPV

         Time      NCF       PVF(10%)          PV

            0       -1,200      1.000       -1,200.0
            1          500      0.909          454.5
            2          500      0.826          413.0
            3          500      0.751          375.5

                                  NPV =          43.0


                                                                                40
              Present Value Annuity Factors at 10%
12.0000



                                                              Annuity Factors
10.0000




8.0000




6.0000
          Discount Factors

4.0000




2.0000




0.0000
          1   2   3   4   5   6   7   8    9   10   11   12     13   14   15   16   17   18   19   20

                                          Years                                                    41
Present Value Annuity Factors As The Sum
of Discount Factors


                      Present Value        Sum of Present
            Year
                       Factor 10%           Value Factors

             1            0.9091               0.9091
             2            0.8264               1.7355
             3            0.7513               2.4869
             4            0.6830               3.1699
             5            0.6209               3.7908
             6            0.5645               4.3553
             7            0.5132               4.8684
             8            0.4665               5.3349
             9            0.4241               5.7590
             10           0.3855               6.1446
                                                            42
Using Present Value Annuity Factors

What is the equivalent value today of £840 to be received at the
end of each year for the next seven years if the interest rate is 6 percent ?


                  Cash     Present Value
       Year       Flow       Factor 6%         Present Value
        1          840         0.9434              792.45
        2          840         0.8900              747.60
        3          840         0.8396              705.28
        4          840         0.7921              665.36
        5          840         0.7473              627.70
        6          840         0.7050              592.17
        7          840         0.6651              558.65
                          Present Value =         4689 20
     Using PVAF
       1 to 7   840            5.5824              4689.20                  43
Example: Using Time Value Concepts (1)


 Arrangements for repaying a bank loan
 A bank makes a loan at £10,000 at a fixed interest rate of 12 per
 cent and this is to be repaid in five equal instalments. (Each
 instalment covers repayment of the loan as well as the interest on
 the outstanding balance of the loan.



 Determine the annual instalment. (Convert a capital sum
 into a constant cash flow.)
 The instalment is the equivalent constant annual cash
                                                                      44
 flow to a capital sum.
Bank Loan – the Required Annual Payments

 Loan      = Present Value of Repayments at 12 per cent

 10,000    = X . PVAF5|.12
 10,000    = X times 3.6048
  X        = 10,000/3.6048
           = 2,774


                                                          45
  Internal Rate of Return
The rate of discount at which the NPV is equal to
zero.
This may be interpreted as the highest rate of
interest that can be paid on a loan used to finance
a project without making a loss.

                         1            1
 NPV  0   I 0  C1          C2            ....
                      (1  i)      (1  i ) 2



                                                      46
Investment Appraisal (IRR)


    Time   NCF    PVF12%)    PV
      0    -1,200   1.000    -1,200.0
      1       500   0.893       446.4
      2       500   0.797       398.6
      3       500   0.712       355.9

                    NPV =    0



                                        47
Loan Analysis (2)


Period Loan at Interest Loan at End   Repayment
       Outset (12%)       of Year

  1      1200      144        1,344         500
  2       844      101          945         500
  3       445      55*          500         500

                         Surplus              0



                                                  48
NPV and IRR


              PRODUCTIVITY OF
              CAPITAL (IRR)


NPV



              SIZE OF THE
              INVESTMENT
                                49
    Investment Appraisal (IRR)
 Consider the simple investment considered earlier - an outlay of 1200 that is
 expected to produce three annual NCFs of 500 and a discount rate of 10 per
 cent. The NPV was 43 and the IRR was 12 per cent. Now double the size of
 all the NCFs – the NPV doubles but the IRR remains at 12 per cent.

Time NCF     PVF(12%)     PV                Time NCF      PVF(10%)    PV
  0 -2,400    1.0000    -2,400.0              0  -2,400    1.000      -2,400
  1   1000    0.8928       892.8              1    1000    0.909         909
                                              2    1000    0.826         826
  2   1000    0.7972       797.2
                                              3    1000    0.751         751
  3   1000    0.7118       711.8

               NPV =       0                              NPV =          86




                                                                                 50
 Determining the IRR (1)
An investment of £400,000 is expected to a constant annual NCF
of £102,865 for the next 6 years. Determine the approximate value
of the investment’s IRR.


       NPV  0  400,000  102,863 x PVAF |i
                                          6

       PVAF  400,000/102,863  3.8887
           6/i

       PVAF6/0.14  3.8887          t
                           (Look up able, row 6)
       i  0.14
    Determining the IRR (2)
An investment of £750,000 is expected to produce NCFs at the end of years 1
to 5 of £150,000, £200,000, £300,000, £300,000 and £100,000 respectively.
Determine the approximate value of the investment’s IRR.

NPV  0  750,000  150,000 x PVF|i  250,000 x PVF|i  300,000 x PVF|i  300,000 x PVF|i  100,000 x PVF|i
                                  1                 2                 3                 4                 5



                                    1                     1                      1                      1                      1
NPV  0  750,000  150,000 x              250,000 x             300,000 x             300,000 x             100,000 x
                                 (1  i )1
                                                       (1  i ) 2
                                                                              (1  i ) 3
                                                                                                     (1  i ) 4
                                                                                                                            (1  i )5

                                          Time         NCF
                                            0        -750,000
                                            1        100,000
                                            2        250,000
                                            3        300,000
                                            4        300,000                                         Use Excel!!
                                            5        100,000
                                          IRR =        12%

				
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