Title:
Optimal investment in local networks of airports
Author:
Carles Vergara-Alert
Affiliation:
University of California Berkeley
Address:
Haas School of Business, University of California Berkeley
F605-1900 Berkeley, CA 94720-1900, USA
Phone:
(+1) 510-6045271
E-mail:
[email protected]
Submission: December 13, 2006
Revised:
April 30, 2007
ID:
1251
Track:
E5
Optimal investment in local networks of airports
Carles Vergara-Alert§
The strategy of an airport (as an independent agent that competes with other
independent airports to capture demand) depends on regional and global
macroeconomic variables and on the strategies and plans of the other airports that
conform the local network of airports in its urban area. Using a real option pricing
model, this paper establishes a methodology for the valuation of a local network
of airports from the two kinds of assets that every individual airport owns: assets
that are currently generating cash flows, and options on future investments with
positive net present value. Traditional project finance approaches have only taken
into account the first kind of assets. In our approach, we take into account that
changes in airport’s assets and their growth options are predictable if we assume
that they invest in enlargement projects using optimal strategies. Finally, this
paper introduces and quantifies the options on future airport enlargements and
improvements, which are important because of economic, ecologic and urban
planning constraints. As an example, the Oakland International Airport’s $1.4
billion expansion project has been valued in terms of the proposed model and its
location in the San Francisco Bay Area’s network of airports.
1 Introduction
Airports are very important assets for the economy in big cities and their metropolitan areas.
There is usually more than one airport in each metropolitan area and they all together stand as
a local network of airports. This is because, generally, any of these airports of the system is
good enough for passengers and hence the airlines to render connections to the relevant main
city/cities of the area. Nevertheless, these networks must perform optimally in order to offer
minimum standards of service and to obtain maximum returns at the minimum financial cost.
This paper values a system of airports, or a local network of airports located in the same
urban area, considering optimal investments and growth options. The approach is based on
the recent real option pricing literature and, in particular, on Berk, Green and Naik (1999) and
their valuation approach to industries and firms. Therefore, the network of airports, thought
of as an industry in financial terminology, is valued from the assets and the growth options of
the individual airports, which can be thought of as firms. The value of a single airport has
been modeled by combining the value of its real assets and the value of its growth options.
Hence, airports in the model own two kinds of assets:
i.
assets that are in place and currently generating cash flows
ii.
options to invest in future projects with positive net present value (NPV)
During each period, new investment projects can be presented to the airport. It is
important to take into account that investments with low systematic risk, other parameters
equal, are more attractive for the airports (see Dixit and Pindyck (1993) and McDonald and
Siegel (1986)) for more information on these topics). Hence, the average systematic risk of
their cash flows in subsequent periods is lowered, and this fact leads to lower returns on
average.
§
Haas School of Business, University of California Berkeley, [email protected]. The author would
like to acknowledge “La Caixa Foundation” and “Fundación Rafael del Pino” for their financial support.
1
Recent applications of the real option theory to the investment in airport projects are Smit
(2003) and Pereira et al. (2006). Smit (2003) solves a discrete-time model that combines real
option theory and game theory on the airport expansion with strategic interactions. He
focuses his analysis on the equilibrium obtained by a strategic game for enlarging airports
and studies different equilibrium scenarios. Pereira et al. (2006) focus on the construction of
new airports and derive the model for two stochastic variables and allow for positive and
negative jumps.
2.
The Model
2.1.
Value of the airport’s assets
The model is based on the partial equilibrium analysis of individual airports. Let’s assume
that the value of an airport’s assets at time t, A(t), is given by the sum of the value of all its
ongoing projects:
t
A( t )   Vj ( t )· j ( t )
[1]
j0
where V j ( t ) is the value of the assets at time t of the jth ongoing project (taken at time j) and
 j ( t ) is an indicator function describing whether the jth project is alive at time t or not (this
takes up the values of 1 for being alive and 0 otherwise). The function V j ( t ) can be written
as


Vj (t )  I·exp C   j ·Dr(t )
[2]
where:
- I is the investment required for undertaking the project (considered constant at any
time i.e. the investment required for undertaking a project is the same at different
periods of time). The airport operates with a finite horizon in discrete time, such that
at each date t a project becomes available. If the project is undertaken, then an
irreversible investment, that cannot be postponed, is required.
- C is the mean of the cash flows. The cash flows from a project that was undertaken
at date j<t are governed by the following stochastic process:
1


C j ( t )  I·exp C   2j   j  j ( t ) 
2


[3]
where the innovation parameter  j ( t ) with t>j are independent standard normal
distributions and  j controls the variance of the cash flows.
-
-
1
j
1
is
the
systematic
risk
of
a
project’s
(see [10] for further notation) given by the following expression:
cash
 j   j  z cov j ( t ),  ( t ) 
flow
[4]
Dr(t ) represents the value of a perpetual riskless bond where its payments
depreciate at a constant rate 1   (see also equation [12]), according to the following
formula:
Note that this formula is very similar than the market beta in the Capital Asset Pricing Model (CAPM).
1
Dr ( t ) 


(s t )
Bs  t , r ( t )
[5]
s  t 1
The airport’s group of projects determines how its risk and expected return change over
time. If the book value of the assets of the airport is defined as
t
b( t )  I·  i ( t )
[6]
i 1
then it is straightforward to demonstrate that the value of the airport’s assets is
A(t )  b(t )·e C ( t ) Dr(t )
[7]
where ( t ) represents the average systematic risk of the airport’s existing assets
 t  j ( t )  j 
( t )   ln 
e 
 j0 n( t )

[8]
where n ( t ) is the number of projects alive2 and  j the systematic risk of the cash flows.
2.2.
Option of enlarging the airport
On the other hand, we must also value the option of enlarging the airport that “belongs” to
the operator. These options strongly depend on time and not only on the economic results of
the considered airport, but the performance of the other ones within the local networks. Real
option theory must be applied for the valuation of these kinds of options, known as growth
options.
The value of the airport’s growth options on the investment date, that is the value of the
options to invest in such projects in the future, is the maximum of the NPV of the project or
zero, since the investment decision cannot be delayed:
k
  z(s)


V( t )  E t  
max Vs (s)  I   G i (s),0 
i 1


s t 1z( t )
[9]
k
The variable
 G (s )
i 1
i
introduces the values of the synergies among the airports in the
local network3 and is one of the main contributions of this paper. Hence, the values of Gi(S)
are based on the amounts of money that the airport will not make due to the existence of other
airports close to it, that is, due to the competition among the airports of the network. These
values must be assessed for every particular case and carefully calibrated. Competition
among airports should be studied under an equilibrium framework for strategic option
exercise games. For example, Grenadier (2002) studies the symmetric Nash equilibrium real
2
3
Note that each project of enlargement of an airport has a start and end dates.
For example, whenever an airport is evaluating each of its enlargement projects, it must take into account
whether there is another airport within the local network that may potentially grow and may be planning its
expansion as well. In this case, competition among airports decreases the value of the growth option because
and earlier enlargement of any of the other airports of the network will decrease the cash flows of the
considered airport. On the other hand, a closure of one of the other airports of the system will potentially
increase its cash inflows. Hence, this variable depends on the options and performance of the other airports of
the local network and it can hold positive or negative value.
3
estate development strategy for the exercise of options and concludes that competitions
eliminates option values and pushes firms back to the zero NPV rule. Similar results are
obtained by Leahy (1993) and Kogan (2001). They choose the type of industry that they
model has linear and incremental technologies. If we considered that competition pushes the
results back to the NPV zero rule, the investor will wait just until NPV=0. However, the zero
NPV case will happen just in the hypothetical case that the industry has a continuum of equal
firms which creates a scenario of perfect competition. In real cases, competition will
decrease the value of the growth option but not to the lowest boundary of NPV=0.
Equation [9] comes from the assumption that the price at time t of a random cash flow
C(T) delivered at T>t is given by E t (z(T) / z(t )) , where
1


z( t  1)  z( t ) exp  r ( t )   2z   z  ( t  1)
2


[10]
with z(0)=1, and  ( t ) are series of independent standard random normal distributions. The
process describing the pricing methodology is taken to be exogenous and it is assumed that
the distribution of the pricing is consistent with aggregated cash flows consequent to
investment decisions made at the individual airport level. Nevertheless, it will allow us to
focus on the relative risks of individual airports.
Imposing that the project will be taken if the value at time t of the option to invest at date
s>t (represented by Vs (s) ) minus the investment required to undertake the project I, is
positive:
Vs (s)  I  Iexp( C  s )Dr(s)  1  0
[11]
it is easy to obtain the value of V( t ) as follows.
The value of new investment opportunities for the airport has been calculated according to
the procedure outlined below. The following equations represent notations and quantities that
will be used to derive the quantities in other equations later in this paper:
1 (n  1)  1 (n)   2 (n)
 2 (n  1)   2 (n)
[12]
[13]
1
1 (n  1)  1 (n )   2 (n )   2z
2
[14]
2 (n  1)   2 (n)  (1  )r
[15]
12 (n  1)  12 (n)   22 (n)  212 (n)   2z
 (n  1)   (n)  
2
2
2
2
2
r
12 (n  1)  12 (n)   22 (n)   zr
1
(n )  1 (n )  12 (n )
2
[16]
[17]
[18]
[19]
The equations [12]-[19] must accomplish the following initial conditions:
1 0  1 0  2 0  12 0   22 0  12 0   r t,0   z t,0  0
4
[20]
 2 0  1
[21]
The price at time t of a discount bond that matures at date s is given by:
Bs  t, r(t )  exp 1 (s  t)r(t )  (s  t )
[22]
which can be substituted in equation [5].
The risk of a project is denoted as s and it is assumed to follow an exponential
distribution with the following probability distribution function:
   * 

exp  *
   

f () 
*  
[23]
To find the parameters  * and  of this distribution, two quantities have been used:
(a) the probability that a project is alive given that the interest rate is r ,
Prx t ( t )  1 | r(t )  r   0.05 , and
(b) the probability that a project is alive given that the interest rate is 0,
Prx t (t )  1 | r(t )  0  0.10 .
We know that a project will be alive only if its risk   T , where  T is a threshold
value of beta that solves the following equation:
Vs (s)  I  Iexp C   s Dr(s)  1  0
[24]
Substituting the two levels of interest rates ( r and 0) into this equation gives us two
values for T. Integrating the exponential probability distribution function from negative
infinity to T and setting the value of this cumulative density value to either 0.05 or 0.10
gives us two equations that can be solved simultaneously to obtain the values of the
parameters * and 
For a given  s , a r*s is defined as the solution of the equation:



exp C   s   k B k , r*s  1
[25]
k 1
The values of bond options are determined by the equations:




J r(t ), s  t, k, r*s  Bs  t  k, r(t )N(d1 )  B k, r*s Bs  t, r(t )N(d 2 )
where,
5
[26]


1
ln Bs  t  k, r ( t )  ln Bs  t , r ( t )  ln B k, r* s  12 (k ) 22 (s  t )
2
d1 
1 (k ) 2 (s  t )
[27]
d 2  d1  1 (k) 2 (s  t )
[28]
Finally, putting all the equations of this subsection together, the present value of all future
growth opportunities can be computed as follows:
V * ( t )  Ie C
   Jr(t ), s  t, k, r e


k
s  t 1 k 1
2.3.
*
s
B
 s
dF ( s )  Ie C J * r ( t )
[29]
Total value of the airport
This paper develops and analyzes a model of a multi-stage investment project for airport
operators. In section 2.2, the growth projects have been simplified as future cash flows
coming from enlargements of the airport, for example the construction of a new runway for
the taking off and landing of airplanes and the necessary growing up of terminals and
services. A new runway will provide the possibility of obtaining more cash inflows not only
from the fees paid by the airlines that will be able to schedule more flights to this airport
because of the increase in the availability of spots but also from the increasing number of
passengers who will use the airport.
An important common feature to these kinds of investments is that the firm learns about
the potential profitability of the project throughout its life, but that technical uncertainty about
the development effort is only resolved through additional investment by the airport operator.
Hence, the risks associated with the cash flows of the airport, at maturity of the project, have
a systematic component even while the purely technical and management risks are
idiosyncratic. The model captures these different sources of risk, and allows the study of
their interaction in determining the risk premium earned during the project.
So far, we have developed a very simple methodology to estimate both the value of the
airport’s assets and the value of the options of enlarging the airport (or growth options). By
adding the results of subsections 2.2 and 2.3, the total value of the single airport P(t) can be
estimated as the sum of the following two variables:
P( t )  A(t )  V * (t )
[30]
Note that this model is based on changes in individual airport’s risks through time and
may include risks as diverse as traffic forecast risk (idiosyncratic and systematic), terrorism
risk, catastrophic risks and market risks. The model relates these changes in risk to airportspecific variables that empirically explain the cross-sectional variation in expected returns. It
also considers the fact that airports that economically perform well tend to be those that have
developed valuable investment opportunities or, equivalently, those that have optimally
exercise their growth options.
3.
Application of the Model to the S.F. Bay Area Network of Airports
This model was applied to the San Francisco Bay Area Network of Airports and, in
6
particular, to the analysis of the Oakland International Airport and its $1.4 billion expansion
project option has been valued4.. This project is currently underway and it is expected to be
completed by 2009.
The airport of Oakland is located in the San Francisco Bay Area. It covers 2,900 acres
and has four runways spread across two fields. In recent years it has been one of the fastest
growing airports in the US. About 7.04 million people lived in 2000 in the San Francisco
Consolidated Metropolitan Statistical Area comprised almost 2.6 percent of the U.S.
population, up from 6.25 million and 2.5 percent in 1990 as shown in U.S. Department of
Commerce, Census Bureau (2001). The areas surrounding the three main airports of the area
– the San Francisco area, the San Jose Area, and the Oakland area – contained the vast
majority of that population: 5.81 million in 2000 and 5.18 million in 1990. There are two
other important airports that can infer in the demand associated to the Oakland International
Airport: the San Francisco International Airport and the San Jose International Airport.
The airport enlargement project forecasts an increase of 5.8 million of passengers per year
(from 8 to 13 million/year). The phase 1 of the project (around $500 million) is the terminal
improvement project, which includes the expansion and improvement of terminal facilities,
including the addition of five passenger gates, a 6,000-space multi-level automobile parking
garage, installation of new baggage and passenger security screening systems and a modern
centralized food, beverage, and retail shopping area. Additionally, the Bay Area Rapid
Transit Authority (BART) is planning to build a light-rail system to connect the airport to the
BART network.
The program is being financed through Port-issued municipal bonds, passenger facility
charges, airline fees and federal airport improvement fund5 as pointed in County of Oakland
(2002). It is expected to generate more than 4,000 new construction and aviation-related jobs
and generate nearly $2 billion in business revenue, $600 million in personal income and $76
million in additional annual state and local tax.
Tables 1a and 1b list the input data that has been used to price the value of the Oakland’s
airport with a growth option. We used annual reports of airports and data from the Bureau of
Transportation Statistics to estimate the values of the following parameters: Investment (I),
number of passengers before expansion and after expansion, C and  j . Tables 1a and
Figure 1 provide more details about it. As for the rest of the parameters, we use standard
values in the finance literature.
The probability that a project terminate, 1   , has a natural interpretation as the
depreciation rate for the firm. Business cycle models are typically calibrated with
depreciation rates of 8 to 10 percent, as pointed, for instance, in Kydland and Prescott (1982).
The value of 0.01 for 1   was chosen. Hence, as the model is calibrated to monthly
intervals, the depreciation annual rate is slightly above 10 percent.
Using this data and the model previously exposed, we obtain that the value of all ongoing
projects (hypothetical) is equal to $62,650,900. The value of new investment opportunities
obtained from the simulations is $3,270,000. The total value of the airport is then just the
sum of these two values, which is $65,920,900. We notice immediately that the value of the
4
5
The Port of Oakland Board of Commissioners approved the concept and $1.4 billion budget for the Terminal
Expansion Program in 1999.
Apart from the budgeted funds that were needed for the expansion project, the County of Oakland established
a fund for Airport Facilities in order to account for operations of the County’s Oakland International,
Oakland/Troy, and Oakland/Southwest airports. Revenues are primarily derived from leases, hangar rentals,
landing fees, and other rental or service charges.
7
new growth opportunities is 5.22% of the value of the real assets of the airport. In this case,
this fraction may seem low. This could be due to several factors, probably the most
important of them being the high risk of the projected cash flows from expansion. Another
factor that increases this risk and reduces the real option of new growth opportunities is the
competition that this airport faces from the big airports of the region, namely San Francisco
International Airport and San Jose International Airport. The competition among airports in
the San Francisco Bay Area decreases the value of the growth options of its three main
airports. This makes the growth options to be exercised earlier than in non-competition
scenarios and, therefore, airports would presumably be expanded earlier in competitive
scenarios because delays in capacity increase would decrease their possibilities to capture
demand from the other airports.
4.
Conclusions
This paper introduces the need of taking into account both the assets that are in place and
currently generating cash flows and the options to invest in future projects with positive net
present value, when valuing the assets of an airport. Besides, it shows the dependency to
other airports in the same area when enlargement or improvement projects come into picture.
The model developed in this paper describes investment decision making by individual
airports being part of a local network of airports. In a similar way than the real option pricing
literature and, in particular, Berk, Green and Naik (1999), we obtain that changes in airports'
assets and growth options (enlargement projects) can be priced and predicted because every
single airport operator6 chooses its investments in an optimal way. The valuation of cash
flows that result from the investment decisions, together with the airport’s growth options in
the future, leads to dynamics for conditional expected returns.
The assets of the individual airport turn into as new investment opportunities with
different risk levels, existing projects expirations, interest rate change and the assumption that
firms choose optimally their investments.
Moreover the simulations reproduce
simultaneously several important cross-sectional and time-series behaviors for returns of the
projects, such that interest rates, market value and momentum strategies at different horizons.
As an application of the model, the Oakland International Airport’s $1.4 billion expansion
project has been valued taking into account its location in the San Francisco Bay Area’s
network of airports that includes San Francisco International and San Jose International
airports. Using the model presented in the paper, the estimated value of all ongoing projects
for the Oakland International airport is equal to $62,650,900 and the value of new investment
opportunities obtained from the simulations is $3,270,000. Therefore total value of the
airport is then just the sum of these two values, which is $65,920,900. The value of the new
growth opportunities is just 5.22% of the value of the real assets of the airport, which may
seem a low fraction. This could be mainly due to the high risk of the projected cash flows
from expansion and the competition that this airport faces from the other big airports of the
Bay Area. Hence, the competition among airports in the San Francisco Bay Area decreases
the value of the growth options of its three main airports, which makes the growth options to
be exercised earlier than in non-competition scenarios.
8
REFERENCES
BERK J., R. GREEN and V. NAIK (1999), Optimal Investment, Growth Options and
Security Returns. Journal of Finance, 54 (1999), 1153-1607
COUNTY OF OAKLAND (2002) Comprehensive Annual Financial Report. Fiscal Year
Ended September 30, 2002. With Independent Auditors’ Report Thereon. Prepared by
the Department of Management and Budget Fiscal Services Division.
DIXIT, A., and R. PINDYCK (1993) Investment under Uncertainty. Princeton University
Press, Princeton, N.J. ISBN 0-691-03410-9
GRENADIER, S.R., 2002, Option Exercise Games: An Application to the Equilibrium
Investment Strategy of the Firms, Review of Financial Studies 15, pp. 691-721.
KOGAN, L., 2001, An Equilibrium Model of Irreversible Investment, Journal of Financial
Economics 62, pp. 201-145.
LEAHY, J.V., 1993, Investment in Competitive Equilibrium: The Optimality of Myopic
Behavior, The Quarterly Journal of Economics 108, pp. 1105-1133.
McDONALD, R. and D. SIEGEL, 1986, The Value of Waiting to Invest, Quarterly Journal of
Economics 101, 707-727.
KYDLAND, FINN E., AND EDWARD C. PRESCOTT (1982), Time to Build and
Aggregate Fluctuations. Econometrica 50, 1345-1370.
PEREIRA P., A. RODRIGUES and M.J. ROCHA ARMADA (2006), The Optimal Timing
for the Construction o fan Internacional Airport: a Real Options Approach with
Multiple Stochastic Factors and Shocks. Working Paper
SMIT, H. (2003), Infrastructure Investment as a Real Options Game: The Case of European
Airport Expansion. Financial Management 32(4), 5-35
U.S. DEPARTMENT OF COMMERCE, CENSUS BUREAU (2001) Census 2000
Redistricting Data (P.S. 94-171) Summary File and 1990 Census. Publication PHC-T3, Ranking Tables for Metropolitan Areas: 1990 and 2000, Table 1, Metropolitan Areas
and their Geographic Components in Alphabetic Sort, 1990 and 2000 Population, and
Numeric and Percent Population Change: 1990 to 2000. Washington, D.C.: The
Department (April 2). Available at http://www.census.gov/population/cen2000/phct3/tab01.xls.
6
Airport operators are either public entities depending on the government and/or municipalities or private
concessionary companies that coordinate the management –including design, built, operation, maintenance
and finance– of the airport infrastructure (e.g. runways and terminals).
9
Table 1. Data and parameter values used in the model
Table 1a: Airport’s data used to price the Oakland Airport growth option
Parameter
Investment required for the project
Number of passengers (before expansion)
Number of passengers (after expansion)
Log of the mean cash flow yield from the project
Cash flow volatility
Notation
I
Probability that a project terminate
1 
C
j
Value
$1,400 million
8 million/year
20 million/year
0.2055
0.223
0.01
Table 1b: Parameters for the interest rate process
Parameter
Long-term mean of the short rate
Mean reversion
Covariance between the pricing and the short rate
Volatility of the short rate
Notation
r
K
r
 jr
Value
0.006236
0.95
0.002
Volatility of the pricing
z
0.4
-0.00014
Note: The investment required I and the number of passengers before the expansion are
particular to every project and airport. In 2004, when part of the expansion of the terminals
was on service, 14.3 million passengers used the airport. We will use data from 1959 to 2006
on the overall available Ton-miles provided by the Bureau of Transportation Statistics in
order to estimate some of the parameters. We will use the volatility of the return on overall
available ton-miles (which includes passengers and freight) as a proxy for the volatility of the
returns of cash flows. Figure 1 shows the process of these returns. We also used
performance data of the San Francisco, Oakland and San Jose airports in as a proxy for the
cash flows in order to estimate the parameter C .
10
Figure 1: Returns on revenue passenger and overall available ton-miles
35%
30%
Yearly returns on revenue passenger ton-miles (%)
25%
Yearly returns on overall available ton-miles (%)
20%
15%
10%
5%
-5%
-10%
Returns on revenue passenger Ton-miles
Returns on overall available Ton-miles
11
Mean
7.0%
6.6%
Volatility
8.8%
22.3%
1998
1997
1996
1995
1994
1993
1992
1991
1990
1989
1988
1987
1986
1985
1984
1983
1982
1981
1980
1979
1978
1977
1976
1975
1974
1973
1972
1971
1970
1969
1968
1967
1966
1965
1964
1963
1962
1961
1960
1959
0%