# 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,

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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