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BUDGETING TECHNIQUES

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					 Mathematics for Asset Valuation
                 Budi Frensidy
  Faculty of Economics, University of Indonesia




 4th International Conference on Research
        and Education in Mathematics
Renaissance Hotel Kuala Lumpur, Malaysia
             21-23 October 2009
                  Budi Frensidy - FEUI            1
              Introduction
Present value concept is needed for:
 Project evaluation
 Asset valuation (real and financial)
 Recording and reporting an asset on the
  balance sheet

 We usually do the discounting one by
  one by using the basic equation
 This method is impractical as the
  number of periods can be infinite (~)
                Budi Frensidy - FEUI        2
       The Purpose of the Paper
 We actually can use the short-cut
  equations if certain conditions are met
  namely equal cash flows or equal
  growth (in percentages)
 There are at least 13 other equations
  besides the basic and continuous
  equation that can be used for this
 This paper tries to enumerate and give
  examples of all the equations using a
  simple and interesting case on valuation

                Budi Frensidy - FEUI         3
  Definition of Annuity and Perpetuity
 Annuity is a series of cash payments or cash
  receipts, generally in equal amounts, in
  equal periodic intervals. Installments on
  home loan or car loan and bond coupon are
  some examples
 Whereas perpetuity or perpetual annuity is
  infinite annuity which is a special type of
  annuity when the periods are countless like
  stock dividends, pension allowances,
  royalties, and copyrights

                 Budi Frensidy - FEUI            4
              Valuation Case (1)
  A young investor is contemplating investment in
  the following financial assets which are offered at
  the same price i.e. Rp 100 million. With his budget
  constraint, he can choose one and only one of the
  alternatives given.
a. Zero-coupon bond with the nominal value of Rp
  250 million, due in 8 years
b. No-par bond that pays cash Rp 18 million every
  year for 10 times starting next year
c. No-par bond that pays cash Rp 16 million every
  year for 10 times starting right now
d. No-par bond with the payoff Rp 50 million every
  year for five times, but starting in five years
                    Budi Frensidy - FEUI                5
                 Valuation Case (2)
e. Discretionary fund that gives cash Rp 12.5 million every
    year for the whole life starting next year
f. Discretionary fund that gives cash Rp 11.5 million every
    year for the whole life starting next week
g. Hedge fund with the payoff Rp 15 million every year for
    the whole life starting in three years
h. Hedge fund that pays cash Rp 15 million next year, then
    becomes Rp 15.9 million the following year and steadily
    rises 6% every year, and the payments are for ten times
    only
i. Banking product with the payoff Rp 13 million next week
    then Rp 14.04 million next year and rises consistently 8%
    every year, and the payments are for ten times only
j. Banking product that promises cash Rp 40 million in five
    years then becomes Rp 43.2 million a year after and
    steadily rises 8% every year, and the payments are for five
    times only
                        Budi Frensidy - FEUI                      6
               Valuation Case (3)
k. Investment trust that promises cash Rp 4 million
   next year, and then rises 8% every year for the
   whole life
l. Investment trust that gives cash Rp 7 million next
   week, then Rp 7.35 million next year and rises on
   at 5% every year for the whole life
m. Private equity with the payoff Rp 12 million
   starting in three years and rises to Rp 12.36 million
   the following year and at 3% every year for the
   whole life
n. Private equity that distributes cash Rp 12 million
   next year and rises Rp 60,000 every year for 50
   years


                     Budi Frensidy - FEUI                  7
               Valuation Case (4)
 The security, risk, and certainty of the above financial
  assets are assumed the same. The investor is expected
  to act rationally and will base his decision only on the
  mathematical calculation. Using a different formula
  for each alternative, which asset should be chosen if
  the relevant discount rate is 12% p.a.?
 To decide which asset should be chosen, the investor
  must calculate the present value of all the assets. As
  long as the present value of the cash flows generated
  from an asset is more than the cost which is Rp 100
  million, the asset is worth buying
 The problem is when there is a budget constraint, as
  we usually encounter, we have to rank all the choices
  and choose the highest value. This is what is meant by
  mutually exclusive projects in finance

                      Budi Frensidy - FEUI                   8
                Equations and Results
         Present
Asset                                    Equation                       Use
        value (Rp)
                               FV                         Basic equation (To calculate
 A      100,970,807   PV =
                             (1  i) n                    PV from a single FV)

                            1  (1  i)  n   
 B      101,704,014   PV  
                                              A
                                                         Ordinary annuity
                                  i           
                            1  1  i  n 1    
 C      101,251,997   PV  
                                                1A
                                                   
                                                          Annuity due
                                   i              
                             1  (1  i)  n
                            
                                               A
                                                
 D      114,545,022
                      PV           i                   Deferred annuity
                                 (1  i)  m -1


                           A
 E      104,166,667   PV                                 Ordinary perpetuity
                            i
                           A
 F      107,333,333   PV     A                          Perpetuity due
                            i
                                A
                      PV 
                           i1  i 
 G      99,649,235                   m 1                 Deferred perpetuity

                               1 g 
                                         n
                                               
                           1               
                      PV       1 i        A
 H      105,848,743                                       Ordinary growing annuity
                                ig            1
                                              
                           
                                              
                                               

                                   Budi Frensidy - FEUI                                  9
           Equations and Results
                           1 g  
                                       n 1

                       1            
I   110,977,751   PV       1 i   A  A        Growing annuity due
                             ig            1 0

                                           
                       
                                           
                                            
                           1 g  
                                        n

                       1            
                            1 i   A
J   105,663,644             ig           1       Deferred growing annuity
                                         
                       
                                         
                                          
                  PV 
                             1  i m 1


                        A1
K   100,000,000   PV                               Ordinary growing perpetuity
                       ig
                        A1
L   112,000,000   PV          A0                  Growing perpetuity due
                       ig
                                 A
                  PV 
                         (i  g )1  i 
M   106,292,517                          m 1       Deferred growing perpetuity

                                                    Variable annuity where A1 =
                            1  
                                     n
                                                    (a1 + d/i + nd)
                       1          
N   103,719,727   PV       1  i   A  nd
                                                    d is the nominal difference
                              i        1   i      between periods and a1 is
                                      
                                                  the first cash flow



                           Budi Frensidy - FEUI                                   10
                    Summary
 Based on the above results, the investor must
  choose asset d which is a no-par bond with the
  payoff Rp 50 million every year for five times,
  but starting in five years. This asset gives the
  highest present value namely Rp 114,545,022
 Present value is the basics to understand financial
  management, investment management, and
  accounting
 Certainly, there are some assumptions to be met
  in order for us to use the 13 equations. The
  amount of periodic cash flow must be equal, or
  they grow at an equal percentage or in equal
  difference from one period to another period. If
  one of these assumptions is fulfilled, the use of
  mathematical equations will give the same result
  as the present value derived from one by one
  calculation        Budi Frensidy - FEUI             11
THANK YOU




  Budi Frensidy - FEUI   12

				
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