Real options pricing for the timing of IT investments under uncertainty+
David S. Shyu *
Kuo-Jung Lee
Department of Finance, National Sun Yat-sen University, 70 Lien-hai Rd. Kaohsiung,
Taiwan, R.O.C.
Abstract
Some researches have used Black-Scholes options model to evaluate IT investments. This study
establishes more flexible and precise model under real options analysis to analyze the optimal timing
decision of information technology (IT) investments when the benefits of IT investments are arisen
from the increasing output price, increasing sales, and cost saving. We derive the closed form
expression of the timing of IT investments and moreover prove that IT investments rise at an increasing
rate in economic booms and fall in economic busts. This study finds that increasing (decreasing) price
volatility will delay (advance) the timing of IT investments. Increasing IT investments, however, may
not delay the timing of IT investments. In addition, decreasing (increasing) efficiency and increasing
(decreasing) depreciation of IT investments will delay (advance) the timing of IT investments.
Keywords: Investment timing, IT investments, Real options
1. Introduction
IT has been developing very fast recently. The use of IT improves the process
quality, which in turn improves the level of output [18]. This type of impact is mainly
+
We are grateful for support from MOE Program for Promoting Academic Excellent of Universities
under the grant number 91-H-FA08-1-4.
*
Corresponding author. Tel: +886-7-525-2000 ext.4814; Fax: +886-7-525-4899.
E-mail addresses: dshyu@cm.nsysu.edu.tw (D. S. Shyu), leekj@pchome.com.tw (K. J. Lee)
on the operational level and results in cost reduction, higher productivity and
improved quality [17]. For example, online logistics can substantially reduce the costs
associated with purchasing, storing, administrating and distributing inventory through
better supply-chain management [3]. It provides customers Internet order and
increases sale performance. IT has likely led to higher productivity growth, lower
costs and reduced pricing power, which should allow a country to grow faster without
inflationary pressures [21]. On the other hand, it needs to invest tremendous amount
of money in software and hardware. A company, hence, has to consider the cost and
benefit before it implements IT investments.
The impact of IT on a firm’s performance has long been a subject of intense
research, with issues studied ranging from measurement of the impact, to the
conditions that are necessary to realize these impacts. Researchers, however, have
pointed out these impacts are neither guaranteed upon implementation of the system
nor are they uniform across the organization [5,25]. Realization of the value of the
system is conditional upon internal and external factors, some of which are
controllable by the organization [25]. When millions of dollars of investments are
being poured into e-business system projects in most larger companies, it is natural
for IT managers to face the challenges of quantifying the value of e-business
investments [20]. However, IT projects are often difficult to estimate and manage and
some projects are canceled or reduced in scope because of overruns in cost and time
or failure to produce anticipated benefits [14].
The studies with regard to the subject of valuation and investments decision in IT
investments are likely more and more. The traditional discount cash flow (DCF)
model is easy to value a firm and to make a decision of IT investments. But the DCF
method ignores the value of timing and operational flexibility [4,6,23]. The research
2
[10,11] has shown that traditional financial evaluation techniques such as net present
value (NPV) tend to undervalue IT investments and has proposed the use of real
options techniques for evaluating IT investments [12]. Using the real options theory to
account for the value of future development is now a standard approach in finance
among both academics and practitioners. Real options analysis is more focused on
describing uncertainty and in particular the managerial flexibility inherited in many
investments.
The real options approach was used in the field of natural resources, venture, and
R&D investments. Recently, some researches have used options pricing model to
evaluate IT investments. These literatures employ real options to stress the value of
flexibility resulting from IT investments [13], involve the deployment of point-of-sale
debit services by the Yankee 24 shared electronic banking network [1,2], value an
Internet company [19], select a software platform (SAP R/2 or SAP R/3) [22], manage
risks in IT projects [12], and make a decision of investments in online logistics [3].
The literatures above demonstrate the importance of real options analysis in IT
investments. The real options analysis, in addition to value the flexibility in IT
investments, is a useful tool for a manager to make the optimal decision, namely the
timing to invest in IT [1,2,3]. They employ a version of the Black-Scholes option
model. They, however, use the example or case to illustrate the concepts of real
options and don’t get the general and certain solution on the timing of IT investments.
In addition, applying the Black-Scholes formula doesn’t describe precisely the actual
situation of each firm. This study values the IT investments from the aspect of a firm’s
operation and the benefits of IT investments are arisen from the increasing output
price, increasing sales, and cost saving. That is, considering a firm’s output price,
production output and input, we derive the closed form expression of timing of IT
3
investments. In addition, we also explain the investments in IT are usually happened
in upward economic trend. When confronted with the availability of the investments
in IT, the managers have to choose not only if but also when to invest IT. Our model
that investigates a firm’s timing of IT investments and get a closed form expression
can help investors to make the optimal decision when confronted with IT investments.
The study is the first study to build a theoretical derivation to detect the impact of IT
investments on a company’s value and timing based on the real options model.
The main purposes of this study are threefold: Firstly, it attempts to derive a firm’s
value of IT investments using real option analysis when a firm’s output price is
uncertain. Secondly, it attempts to find the optimal timing of IT investments. Finally,
it attempts to perform sensitivity analysis to exploit the impact of some major
parameter to on a firm’s value and timing of IT investments.
The structure of the paper is as follows: Section 2 establishes the assumptions and
model about a firm’s value of IT investments under the Cobb-Douglas production
function. In section 3, this study uses the model developed in section 2 to derive a
closed form solution for the timing of IT investments. Extensions of the model are
investigated in Section 4. Section 4 studies the impact of depreciation of IT on a
firm’s timing of IT investments. In section 5, this study explores the sensitivity
analysis of the optimal IT investments, analyzes the simulation results and draws
some inferences by using numerical analysis. Section 6 sums up this paper with
highlights of important findings.
2. Firm’s Value of IT investments
We assume a firm’s value, V(pt), is determined by a stochastic output price pt. The
4
IT investments are irreversible, which implies that the firm’s value prior to the
investments includes an option to invest in IT, whereas the firm’s value after IT
investments does not. The firm’s value net of the option of IT investments is called the
stand-alone value and is denoted by Vs(pt). Hence, V ( pt )  Vs ( pt )  O( pt ) , where
O(pt) denotes the endogenously determined value of the option to IT investments.
The output price pt is assumed to follow the geometric Brownian motion:
dpt  pt dt  pt dzt ,
(1)
for constants   r ,   0 .  is the expected growth rate of the output price,  2 is
the variance of the growth rate, dzt is the random increment to the standard Brownian
motion zt, and r denotes the risk free rate. A firm’s instantaneous profit function is
given by1:
pt La K b  wL L  F ,
(2)
where L denotes the variable production input (i.e. labor), whereas K is fixed (i.e.
capital). The cost includes variable and fixed cost. Both variable cost and fixed cost
are denoted by wL L and F respectively. The cost per unit of input in L is denoted
by wL . The output is determined by a Cobb-Douglas production function, La K b ,
where a and b are elasticity parameters of labor and capital respectively. The
production function displays decreasing returns to scale with respect to the input (i.e.
a, b < 1). Then, the profit flow when the variable input, L, is chosen optimally is given
by:
1
 1aa
 a 1 1ba
1 a 

 ( pt )   a  a  wL 1a pt 1a K  F ,


  ( wL , a ) pt K b  F ,
1
The analogy settings can be seen in [6] and [16].
5
(3)

a
1

a
where  ( wL , a )  a 1a  a 1a wL1a and   11a denotes elasticity of the price. Because
a < 1, it follows that  ( wL , a)  0 and   1 . Therefore, the profits are a convex
increasing function of the output price, pt.
The stand-alone value of the firm, VS, can be seen as the discounted value of all
future profits and is given by2:
Vs ( pt ) 
 ( wL , a ) pt K b F
F
  pt  ,
r  g ( )
r
r
where g ( )   ( r   )  12  2 (  1) and  
 ( wL ,a ) K b
r  g ( )
(4)
is a constant. For all this to
make economic sense we need to assume r  g ( ) , or specifically    , where  is
the positive root to the quadratic equation3:
1
2
 2 x ( x  1)  ( r   ) x  r  0 . Moreover,
one will note the equation (4) is similar with the dividend growth model (also known
as the Gordon growth model). The main difference is the somewhat more complicated
nature of the profit growth rate, g ( ) , which is a function of the stochastic output
price process driving the profits, the risk free interest rate and the elasticity of scale, a
and b.
The simplest and most common way to describe the technology of a firm is the
production function (see [24] in page 1). By taking account of production function of
IT investments, the Cobb-Douglas production function can be expanded by the early
research [8] as:
h( L, I )  L( I ) a K b I c ,
(5)
where I denotes IT investments, c denotes elasticity parameter of IT investments on
output, and I > 0 under the existence of IT investments. Since the IT will also affect
2
3
Please refer to appendix 1 or [6] in pp. 196-198.
For further details of  , please refer to [6] in pp. 142-143. In addition, the restriction   
imposes an upper bound on the elasticity of production, i.e. a 
 1

. Note that a positive labor
elasticity, a, satisfying this condition will always exist as one can show that   1 .
6
labor input, we assume L( I )  L / I d , where d denotes elasticity parameter of IT
investments on labor input. One would expect IT investments to lead to a decrease in
unit costs which implied d > 0. To make our discussion more general, this research
doesn’t impose any restrictions on the value of d. This is because the labor input will
increase to support IT operation in the early stage. As time goes on, the labor input
could be reduced through the efficiency of IT investments. In addition, the influence
of IT on production function depends upon parameter c. We assume 0  c  1 .
Equation (5) converges to original Cobb-Douglas production function when c and d
are zero. The variable cost function of IT investments can be expressed
as c( L, I )  wL L / I d . Similarly, the profit flow,  I ( pt ) , of IT investments when the
variable input is chosen optimally is then given by:
 I ( pt )   ( wL , a ) pt K b I   F ,
where  
a (1 a ) d  c
1 a
(6)
.
By using the same method above, we can derive the firm’s value of IT investments
below:
Proposition 1: Assuming investors are risk neutral and there exists a risk free asset
yielding a constant interest rate, r, at which investors may borrow or lend freely, then
the value of a firm to invest in IT is given by:
VI ( pt )  pt I  
F
.
r
(7)
The benefit of IT investments is the difference of value before and after investing in IT:
VI ( pt )  VS ( pt ) 
 pt K b ( I   1)
  pt ( I   1) .
r  g ( )
(8)
Equation (7) indicates that the firm’s value is affected by the IT investments
7
through the parameter  . The following differential result makes it easier to
understand how IT investments affect the firm’s value:
VI ( p )
 pt I  1 .
I
(9)
The result is positive if   0 . This implies that the IT investments can enhance a
firm’s value only when  is positive. This result also indicates that the firm’s value
may decrease after investing in IT due to labor inputs keep on increasing and
efficiency of IT has not developed.
One can realize the character of benefit from IT investments more easily from
Equation (8). Equation (8) shows that the benefit from IT investments is positive if
  0 or consequently if a (1  a )d  c  0 . Since the benefit from IT investments is
a convex increasing function of the output price, pt, it is procyclical. That is, IT
investments rise at an increasing rate in economic booms and fall in economic busts.
The main implication of the proposition 1 is that IT investments should optimally
happen in a rising product market, creating IT activity at high output prices and IT
inactivity at low output prices. In considering IT investments, a firm makes a tradeoff
between the stochastic benefit and the cost of IT investments. As a firm has the option
but not the obligation to invest in IT, one will do so only when it is in one’s interest.
This implies that the benefit from IT investments has call option characteristics, which
is discussed in the next section.
3. The Timing of IT Investments
When the value of the option which depends on the output price is large enough,
the option will be exercised. That is, the firm’s decision to invest in IT is contingent
8
upon the output price. The firm’s value without IT investments, V(pt), equals the sum
of the stand-alone value plus the value of the option to invest in IT is:
V ( pt )  pt 
F
 O ( pt ) .
r
(10)
The option value of IT investments can be denoted as follows:
O( pt )  max[ VI ( pt )  VS ( pt )  I , 0] ,
(11)
The option value is worth to invest in the IT only if the firm’s value with IT
investments is greater than that without IT investments. We follow the steps of
contingent claims valuation [6,15], then the option, O(p), to invest in IT must satisfy
the following differential equation:
1
2
 2 p 2O '' ( p )   pO ' ( p )  rO ( p )  0 ,
(12)
subject to the boundary conditions:
O( p )  Ve ( p )  Vs ( p)  I ,
O ' ( p )  Ve' ( p )  Vs' ( p ) .
(13)
The first equation is the value-matching condition and it means that the value of the
option must equal the net value obtained by exercising it. The second equation is the
smooth-pasting condition and that is, the graphs of O(p) and Ve(p) - V(p) – I should
meet tangentially at p . Essentially, this condition ensures that p is the trigger that
maximizes the value of the option. The option value of IT investments is gained in a
closed form expression below:
Proposition 2: IT investments take place the first when the state variable pt hits the
threshold4 p from below. The value of firm’s option to invest in IT is given by:
4
Since the factor

 
exceeds unity the price threshold exceeds the Marshallian break-even threshold.
9


p 
O ( pt , p )  p  ( I   1)  I  t 
 p

for pt  p .
(14)
The optimal threshold, p , of IT investments is given by:
1


I
p  
 ,

 (    ) ( I  1) 
(15)
This proposition gives a closed form expression for a firm’s manager. This allows
us to get a clear insight into the determinants of the IT investments. The optimal
strategy is to begin investing in IT the first moment that p(t) equals or exceeds the
trigger value p , as defined in Equation (15). That is, the optimal timing of IT
investments, TI, can be written as:
TI  inf[ t  0 : p(t )  p] .
(16)
Under the dynamic situation that output price is stochastic, the higher the price the
higher a firm profit is. The equation (16) indicates that when the output price reaches
the threshold p from lower price, it’s the optimal timing to invest in IT. Besides, we
further express the price threshold equivalently in terms of a value threshold,
I  1
.
VI ( p ) 
(    )( I   1)
(17)
It means that it’s not the timing to invest in IT unless the firm’s value has reached the
threshold from the lower firm value.
The comparative static with respect to the investments threshold in IT is very much
in line with the predictions of real options theory. Namely, increasing uncertainty
( ) will lower  and therefore increase the (hysteresis) factor  /(    ) and hence
the threshold, p . Consequently, higher volatility delays investments, a well-known
result from the real options literature [6,16,23]. In addition, increasing volatility also
The firm’s net present value is therefore strictly positive. One can see the reference [6] for further
details.
10
raises the growth rate, g ( ) , which in its turn speeds up investments. The other effect
results from the convexity of profit in the output price and has been termed the
Jensen’s Inequality effect (see [6] in page 199) on investments. For economically
meaningful parameters, the former effect usually dominates. An increase in volatility
therefore delays the timing of IT investments.
The larger IT costs usually delay the timing to invest in IT. Differentiating
threshold p with respect to I, we can more understand the effect of IT:
1
   I  (1   )  1
p 1 1 1 

 I  
.
1
 
1
I 
 (    )   ( I   1)  
Let f ( I ,  ) 
. Since  
a (1 a ) d  c
1 a
,    ,   1 , and  0 , then
p
I
 () 0
f ( I ,  )  () 0 . When 0    1 , then f ( I , ) is greater than zero. It
only if
implies
I  (1 ) 1
( I  1)( 1 /  ) 1
p
I
is positive. This indicates that the larger the IT investment, the higher the
threshold is. This case is consistent with real option theory. When   1 ,
then f ( I , ) is smaller than zero. It implies
p
I
is negative. Equation (6) shows that the
larger the value of  , the higher the company’s profit. Therefore, we can conclude
that a large amount of IT investment may advance or delay the timing of IT
investment depending upon the value of  in which has to do with the relative
magnitude of the elasticity parameter of IT investment on output, c, and the elasticity
parameter of IT investment on labor input, d.
4. Depreciation Effect of Timing of IT Investments
We have assumed thus far that the IT investments last forever. In reality, physical
decay or technological obsolescence limits the development and life of IT investments.
11
Capital thereby depreciates through age, use, or advance of competing technologies.
One would expect that opportunity investments in a depreciable project of IT would
be less valuable.
This paper assumes the form of depreciation to follow the previous research [6].
Exponential decay at any time T, if the IT investments have lasted that long, there is
probability dT that it will die during the next short interval of time dT. Now, from the
initial perspective, the probability that the project dies before T, or the cumulative
probability distribution function of the random lifetime T, is 1  e  T . The
corresponding probability density function of T is  e  T .
The benefits of IT investments on cash flow can be denoted by  I ( pt )   ( pt ) and
the risk-adjusted discount rate appropriate to p is  ' , which is defined in appendix 1.
If the IT investments last T years, the expected present value of its profit flows is
T
E  e  t [ I ( pt )   ( pt )]dt
'
0

 ( wL , a ) pt K b ( I   1)(1  e [ r  g ( )]T )
.
r  g ( )
(18)
By assuming the probability density of the lifetime to follow a Poisson process, we
can obtain the expected value of a firm with depreciable IT, V ( p ) :

V ( p)    e  T
0

 ( wL , a ) pt K b ( I   1)(1  e [ r  g ( )]T )
dT  Vs ( p)
r  g ( )
 ( wL , a ) pt K b ( I   1)
 Vs ( P)
  r  g ( )
   pt ( I   1)   pt 
F
,
r
b
(19)
where    ( w rL,ag)(K ) . The firm’s value with depreciable IT compared with that in
12
Equation (7), the discount rate of firm value is increased by  . That is, the result of
Equation (7) is the case of   0 .
By using the same method, we can get the optimal timing of IT investments under
the case of depreciation:
Proposition 3: IT investments take place the first when the state variable pt hits the
threshold p from below. The value of firm’s option to invest in IT under the case of
depreciation is given by:
 p 
O ( pt , p )    p ( I  1)  I  t 
 p 






for pt  p .
(20)
The optimal threshold, p , of IT investments under the case of depreciation is given
by:
1


I
 .
p  

 (    )  ( I  1) 
(21)
The threshold with depreciable case above is similar with that in Equation (15). The
proposition gives the managers the explicit timing of IT investments when
considering the depreciation of IT. It means that it’s not the optimal timing to invest in
IT unless the output price has reached the threshold from the lower price.
5. Numerical Results
Although we derive the closed form solutions of a firm’s value and threshold of IT
investments, sensitivity analysis makes it easier to understand the inference of
parameter to a firm’s value and threshold under complicate calculations. In this
13
section we perform some sensitivity analysis to the impact of parameter on a firm’s
value and threshold of IT investments. We set parameters as following: r = 0.05,
return shortfall  = 0.045,  = 0.1, a = 0.4, b = 0.55, c = 0.003, d = 0.001, wL = 300,
F = 5000, K = 1000000, and current output price p = 10.
Among settings of parameters, the setting value of c and d are much small than
value of a and b. This is because a and b are elasticity parameter of labor and capital
respectively and labor and capital are essential factors of a firm. On the contrary, c
and d are related elasticity parameters of IT investments, and IT investments are not
essential inputs of a firm although they will become more and more important for a
company. We thus set the value of a and b are much larger than value of c and d. One
of course may set the value of c and d larger especially when efficiency of IT has
developed.
That the effects of IT investments on thresholds are presented in figure 1. We can
find that increasing IT investments (I) increase the threshold and delay the timing of
IT investments under both considering depreciation (   0 ) and not considering
depreciation (   0 ). This is consistent with the analysis of the model because of  =
0.054, 0 <  <1. For general case (   0 ), if IT investment is 50 thousand dollars, the
price threshold to invest IT is 7.32. Since the current price, 10, is larger than the price
threshold, 7.32, it thus means that now is the optimal timing to invest IT. If the
efficiency of IT investments is not affected by the size of IT investments, it isn’t
suitable to invest IT immediately unless the IT investments are lower than 68
thousand dollars (threshold is 10.02) for depreciable case with   0.01 and the IT
investments are lower than about 45 thousand dollars (threshold is 9.94) for
depreciable case with   0.03 respectively.
14
Price threshold
λ =0
λ = 0.01
λ = 0.03
30
25
20
15
10
5
0
5
10
15
20
25
30
IT investment (10 thousand)
Figure1 The effect of IT investments on threshold
The effects of IT investments on firm’s value are presented in figure 2. We find that
increasing IT investments increase a firm’s value slowly under both general (   0 )
and depreciation situations. Besides, depreciation lowers a firm’s value seriously and
Firm value (10
thousand)
decreases a firm’s value as  increasing.
λ =0
λ = 0.01
λ = 0.03
310
305
300
295
290
5
10
15
20
25
30
IT investment (10 thousand)
Figure 2 The effect of IT investments on a firm's value
The effects of price uncertainty ( ) on threshold are shown in figure 3. The plot
illustrates that increasing (decreasing) price uncertainty will increase (decrease) the
threshold and delay (advance) the timing of IT investments especially for depreciable
case with   0.03 . The price threshold will increase rapidly when  becomes bigger.
In the case of   0 , the adopting threshold, for instance, is 9.28 when   0.05 and
the threshold is 13.77 when   0.2 . This is consistent with the results of real option
theorem [6,16,23].
15
Price threshold
40
λ =0
30
λ = 0.01
20
λ = 0.03
10
0
0.01
0.05
0.09
0.13
0.17
0.21
0.25
σ
Figure 3 The effect of  on thresholds of IT investments
Figure 4 plots the joint effect of IT elasticity c and d on the thresholds. The
elasticity c denotes the efficiency of IT in the production output and d denotes the
efficiency of IT in labor input. The larger the elasticity c, the efficient the IT
investments is, and therefore the threshold is getting lower when elasticity d is given.
But, when c is large enough (e.g. c = 0.005), a firm can adopt IT immediately in the
range of –0.003 < d < 0.003. This is because adopting threshold is smaller than
current price, 10 in the range of –0.003 < d < 0.003. Besides, the small the elasticity d,
the higher the threshold of IT investments is especially when d is negative and c is
Price threshold
small (i.e. c = 0.001).
50
c = 0.001
40
c = 0.003
c = 0.005
30
20
10
0
-0.003
-0.002
-0.001
0
0.001
0.002
0.003
d
Figure 4 The effect of IT elasticity c and d on thresholds
6. Conclusions
16
Undoubtedly, real options analysis is important to value and manage in IT
investments. This study gets the general and explicit solution for decision-maker in IT
investments by using more complicated real options model. This paper applies a real
options approach to analyze the impact of IT investments on a firm’s value and timing
when the output price is uncertain. Under the condition of Cobb-Douglas production
function, this research derives a closed form solution for a firm’s timing of IT
investments. By using numerical analysis to explore the sensitive analysis of optimal
IT investments, the results of simulation are as follows: (1) the larger (smaller) the IT
investments, the higher (lower) the threshold of IT investments, and the greater
(smaller) a firm value is; (2) we find that increasing (decreasing) price volatility will
increase (decrease) the threshold, delay (advance) the timing of investing IT. This is
consistent with the results of real option theorem; (3) depreciation of IT investments
will make a firm’s value less valuable, increase the threshold, and delay the timing of
IT investments.
17
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20
Appendix
1. Proof of Proposition 1
By the method of literature [6]:


VI ( p )      I ( p )e  ' t dt     I ( p )e 't e  't dt
 0
 0

 ( wL , a ) pt K b E 
,
 ' '
(a1)
where  ' is a risk-adjusted discount rate,  ' is a growth rate of output price, and they
are to be determined as follow.:
d I ( p ) dp 
  .
 I ( p)
p
(a2)
By using Ito’s Lemma,
dp     p  dt    p  dz  12  (  1) 2 p  dt ,
dp 
 [   12  (  1) 2 ]dt    dz .

p
(a3)
 '     12  (  1) 2 .
(a4)
We get
In the other hands, we get  ' by Dixit and Pindyck (1994) as
 '  r   (  r ) .
(a5)
Then the return shortfall or convenience yield on p  must be
 '   ' '
 r  (r   )  12  (  1) 2 .
(a6)
Finally, we get the firm value of IT investments
21
 ( wL , a ) pt K b I 
 ( wL , a ) pt K b I 
.
VI 

r  ( r   )  12  (  1) 2
r  g ( )
(a7)
2. Proof of Proposition 2
The solution for the value of the option is assumed as:

O( pt )  Ap ,
(a8)
where A is a constant to be determined and  is the positive root of quadratic equation:
1
2
 2 x ( x  1)  ( r   ) x  r  0 .
The solution of option must satisfy the value-matching and smooth-pasting conditions
defined in Equation (13). Using the specific functional forms of O(p), VI(p), and V(p),
we can write the value-matching and smooth-pasting conditions as
O ( p )  Ap

  p  ( I   1)  I
(a9)
and
O ' ( p )  Ap
 1
   p  1 ( I   1) .
By dividing the equation above each other, A will be eliminated and get p :
1


I
p  
 .

 (    ) ( I  1) 
Substituting p backs to the equation above and arrange, then we can get A as


I
I
A



 (    )  (    ) ( I  1) 
22


.
(a10)