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Zion Aviation
Rates:
Discount rate
Risk-free rate
4.5%
2.5%
Start
Cash from Operations
minus: Capital Expenditures
= Net Cash Flow (NCF)
Terminal Value
PV of NCF
Cash from Operations
minus: Capital Expenditures
= Net Cash Flow (NCF)
Terminal Value
PV of NCF
PV of Cap. Ex. (Yrs. 1-2)
1.0
1
3.0
5.0
Scenario: No Real Options
2
3
4
5
5.0
6.9
8.0
10.0
5.5
2.0
2.3
2.5
6
11.9
2.0
7
15.0
2.3
11.0
Start
1.0
1
3.0
Scenario: Real Options
2
3
4
5
5.0
6.9
8.0
10.0
2.0
2.3
2.5
6
11.9
2.0
7
15.0
2.3
11.0
Option Pricing:
PV of Cap. Ex. (Yrs. 1-2)
Maturity
5.0
PV of NCF
Risk free rate
Volatility
20%
BS calculations:
d1
#DIV/0!
N(d1)
#DIV/0!
d2
#DIV/0!
N(d2)
#DIV/0!
Price of call
#DIV/0!
Difference:
– Value of Option over PV
– % of PV
Start
Phase I
Network Expansion with New Aircraft
PV of
Phase II
Phase III
Revenues Costs
Success
165
70%
10
Utah
35%
7
Failure
30%
2
Success
55%
21
134
Failure
45%
4

Success
55%
17
119
Failure
45%
3

Colorado
40%
8
Success
75% 11
Arizona
15%
12
Start
6
Failure
10%
2
Failure
25%
5


rcraft
Net
Total
Expected
Probability
Value
No content – Intentionally left blank
Zion Aviation: Real Options and Decision Trees
Alfonso F. Canella Higuera
October 19, 2021
Randy Zeiss, the CEO of Zion Aviation, a regional airline serving the Rocky Mountain
states, was considering expanding his airline by buying additional gates at various airports it
served.
He knew that this expansion would not be done in one shot but would, instead, be done
incrementally. If things went well, Zion would keep expanding. If things went badly, Zion would
pull back and not buy gates or perhaps even sell gates. If things went ok but not spectacularly,
Zion would continue holding the existing gates.
Randy was cautious by nature and this had paid off as Zion had a streak of 60 straight
profitable quarters even during recessions and the 2020 pandemic. This “let’s see how it goes”
philosophy had worked well. What Randy did not know was that it was grounded in mathematics
and finance.
When Marina, his daughter, joined the company fresh out of school, she opened up Randy’s
eyes to the myriad ways that economics and finance had changed over the years since he had been
at school.
One of the key items that she introduced to him was the use of real options in decision
making. Real options use option pricing formulas, for example the Black-Scholes Option Pricing
Model, to inform capital investment decisions.
With real options, capital investments were not black and white anymore – invest or don’t
invest. Instead, these decisions became a lot more granular, dynamic, and incremental. With real
options instead of just do or not do, Zion could expand, contract, wait, add flights, remove flights,
change aircraft, lease or buy, abandon altogether, enter a new market from scratch, or stay in place.
The permutations were almost endless but more importantly, they showed that investments were
also places in a spectrum and were not anchored in place.
Randy’s decision-making process for airline operations assume a priori that the investment
just had to be made at the start as a negative cash flow, which was then subtracted from the present
value of the operations’ positive cash flows in the future. Marina clarified that under real options,
each decision was a node in a series of steps that offered different paths, none of them obligatory,
to the next level.
In effect, based on Marina’s presentation, Zion had to ask itself a number of questions. The
answers to those questions would then drive the decision-making process. The important part of
this was the “s” in questions as there were multiple questions that needed to be answered. In the
old days, there was only one question to be asked: should we invest, yes or no?
In the new world of real options, Marina set out a number of questions to be asked:
• Is Zion at full capacity at the target airport?
• Can Zion serve that airport with larger aircraft?
• Will the larger aircraft do the job without using more gates?
The answers to each of these questions would then inform the timing and the amount of
capital invested that needed to be made. Just as importantly, each of these answers had a
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system, or transmitted in any form, electronic, mechanical, photocopying, recording or otherwise without the
prior written consent of the University.
probability associated with it. These probabilities were based on Zion’s experience in the past in
these airports. Marina laid out the questions, the probable paths out of each question and the
projected probabilities to each path, as shown below:
Is Zion at full capacity at this
airport?
No
80%
Do not buy gates
Yes
20%
Can Zion use larger aircraft
to serve it?
No
60%
Yes
40%
Will the larger aircraft do the
job without using more gates?
No
Yes
80%
Use larger aircraft.
Do not buy gates
20%
Buy one more gate. If not
enough, buy another
Note: Probabilities are based on Zion’s route development group’s analysis
Each of these nodes, then, has a probability. Depending on the problem, the boxes are
going to have a cost or a payoff to them. In the example above, the box “Do not buy gates” has a
cost of zero to it.
The box “Buy one more gate. If not enough, buy another” has a cost, in so far as Zion is
concerned, of approximately $500,000. The box “Use larger aircraft. Do not buy gates” doesn’t
really have an explicit dollar cost to it but it does have an opportunity cost to it. That opportunity
cost is the cost of those larger aircraft being used in other routes with a potentially higher payoff
to Zion.
As you can see, in economics and finance, everything has a cost and a benefit and the costs
can either be explicit (in dollars) or implicit (in forsaking larger gains elsewhere). The benefits,
for their part, are explicit (in dollars) and perhaps in improved synergies within the firm (that is,
they allow the firm to be more efficient and that efficiency will translate into profits). In general,
synergies are estimated but not explicitly quantified as they can be rather amorphous. (Still, many
firms acquire other firms expecting to gain value from synergies – in many cases, they will pay as
much as 20 or 30 per cent above the target firm’s share price because of that perceived synergy.)
These decisions showed a nodal network that resembles the binomial tree that is used to
model options and is the basis for the Black-Scholes Option Pricing Model:
As you can see, these nodes (that is, the forks in the graph) exhibit what is called a binomial
tree distribution. The shape of this tree is a function of just five variables, each of which is fairly
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easy to calculate in and of itself. These variables are:
•
•
•
•
•
Stock Price (S) – present value (PV) of future cash flows derived from the new gates
Strike Price (X) – PV of the incremental capital expense for the new gates
Time to Maturity (t) – time, in years, that the project spending may be deferred
Risk-free Rate (Rf) – yield on a Treasury with maturity t (i.e. the opportunity cost)
Volatility (?2) – the standard deviation of the returns associated with airport gates
The Black-Scholes Model uses these variables to estimate the price of an option as follows:
Option price = SN(d1) – Xe-rtN(d2)
???????? =
????
????????
???????? ?????? + (???????? + ???? )????
?????????
???????? = ???????? ? ???? ?????
Marina remarked that the purpose of her introduction to the Black-Scholes was not to
overwhelm the folks at Zion. Instead, she noted that the Black-Scholes was a lot less intimidating
if one followed her proposed methodology for calculating the value of the real options available
to Zion.
Embedded in this methodology was the use of spreadsheets that made quick work of this
model and, just as importantly, allowed for changing the input variables depending on the various
projects being evaluated and coming up with a quick answer.
In particular, the N(d1) and N(d2) were rapidly handled with the spreadsheet function that
yields the area under the distribution curve – NORMSDIST. Implicit in these terms is the fact that
the Black-Scholes takes the binomial tree and recreates it using the five variables. This new tree
has a probability distribution all of its own. The NORMSDIST function gives this probability
distribution with minimal pain.
So, Marina summarized her suggestions as follows:
•
•
•
1st – create the d1 and d2 formulas in separate cells in a spreadsheet; if desired, break up
d1 into its main components and then join these up in another cell to calculate d1
2nd – use Excel’s NORMSDIST function in two cells: one with d1 and another with d2
3rd – in a fifth cell, calculate the option price
[Note to the student: for the purposes of this exercise, the template you will use in the assignment
already has the Black-Scholes formula set out for you. All you need to do is provide the five
variables and the call option value will be calculated automatically.]
Marina then went on to show how different airports might be combined into a single,
probability-adjusted Net Present Value (NPV) depending on each airport’s characteristics. For
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system, or transmitted in any form, electronic, mechanical, photocopying, recording or otherwise without the
prior written consent of the University.
each airport, she took Zion’s network development department’s estimates of the costs of obtaining
new gates. From the engineering department, she got the projected capital expenditures that would
be required to outfit those gates for Zion’s use. Finally, she talked to Zion’s revenue management
department and got their estimates of what would be the probabilities of being successful if the
airline expanded in those airports.
The beauty of this is that her methodology could be replicated for multiple airports within
Zion’s service area. By making the methodology almost automated, it allowed different staff at
Zion to run different scenarios in different markets under different conditions. By offering this,
Marina recognized that Zion’s size was large enough that the methodology had to allow for use in
different offices, each with different staff but all targeting the same outcome, namely the profitable
expansion of the airline in those airports where it made the most sense.
She summarized the results in the table below using five airports where Zion already had
a presence but suspected that there was room for profitable expansion:
PV of future cash flows
PV of incremental capital expense
Maturity
Risk free rate
Volatility
Price of option
Probability of option
Value of option
Average value for all airports
Symbol
S
X
t
Rf
?2
12.1
COS
24.0
12.0
2.0
2.5%
Airport
SLC
IWA
48.0
27.0
10.7
6.2
1.5
1.5
2.5%
2.5%
GCN
21.0
2.0
1.5
2.5%
JAC
38.0
8.1
2.0
2.5%
15%
12.6
33%
4.2
22%
37.7
50%
18.9
15%
19.1
90%
17.2
10%
30.3
20%
6.1
10%
21.0
67%
14.1
After showing the table above, Marina made the following points:
• Zion did not necessarily have to go with the highest option value in SLC. This was
•
•
•
because to get that highest option value, it would have to invest $10.7 million in capital
expenditures. She suspected that her father was not quite ready to invest that large a
sum at the start of this project
Zion could conceivably (and this was a gesture to her father’s risk-averse nature) invest
in GCN, the airport that required the lowest capital expenditure, and still get a high
option value. In fact, GCN showed the most promise of all the options she had laid out
for Zion as it was a busy tourist area and one area that would always be popular with
tourists no matter the state of the economy
The option values did not consider unquantifiable, yet still valid, strategic factors. For
example, investing in JAC would allow Zion to fend off competitors in that high
visibility airport
Increased competition in certain airports (SLC, for example) might reduce the projected
PV of future cash flows, thus making that airport less desirable. While SLC had the
highest volatility (in other words, risk), the PV of its future cash flows were still almost
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University, Daytona Beach, Florida, 32114. No part of this material may be reproduced, stored in a retrieval
system, or transmitted in any form, electronic, mechanical, photocopying, recording or otherwise without the
prior written consent of the University.
•
•
twice of those in other airports. This was a direct result of SLC’s higher passenger
traffic, which fed off the population growth around that city as well as higher tourist
traffic there
Finally, some airports really wanted Zion to expand there. This was recognized in the
probability of option per cent. In the case of GCN, the airport had the highest
probability but one of the lowest option prices. Marina remarked that this was mostly
due to the airport’s strong seasonality, with months of limited traffic and months of
heavy traffic. She noted that it was hard to get a solid option value when the monthly
cash flows showed such strong variability. In the end, she admitted that while that
variability was a risk, the low cost of entry (that is, the incremental capital expense)
was a strong mitigating factor in favor of GCN
She wrapped up her comments by saying that the average value for all five airports
($12.1 million) was a benchmark that the company should keep in mind when making
investment decisions on expanding its operations. These five airports were part of
Zion’s core operations and thus represented the company’s strongest position vis a vis
its competitors. If Zion were to continue its streak of profitable quarters, it would
behoove it to stick to its core to handle the cross winds of this competitive industry!
After presenting these points, Marina went a step further and argued that Zion could look
at its expansion plans in a different, less quantitative view. This view combined the traditional
way of looking at investments with the option view of accepting volatility as a friend.
In the traditional method, the decision was based on the NPV of the cash flows. If the NPV
was positive, it was a go. If negative, it was a no go. With the options pricing, the volatility meant
that some projects might be less doomed than originally thought. The argument was that higher
volatility meant higher option values and the same concept could be applied to a traditional
discounted cash flow calculation.
In a low volatility environment, the cash flows would most likely not change appreciably
if at all. In a high volatility environment, the cash flows would most likely change and, if they
changed toward the better, then the investment might conceivably become viable. Marina
presented the mathematics behind the conceptual construct using a new term, NPV’:
????????????? =
???????? ???????? ???????????????????????? ???????????????? ????????????????????
????
=
???????? ???????? ???????????????????????????????????????????? ???????????????????????????? ????????????????????????????
????
Under this construct, if NPV’ > 1.0 and the traditional NPV was > 0, then this was a no
brainer – go forward now! Under both constructs, the investment was worth making. But her
presentation wasn’t about no brainers, it was about all the exceptions to these no brainer than
warranted being considered.
She showed these visually:
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system, or transmitted in any form, electronic, mechanical, photocopying, recording or otherwise without the
prior written consent of the University.
Unlikely to go forward
NPV < 0 + NPV’ < 1;
lower volatility will not
work in favor
High
Volatility
Low
NPV’
1.0
Go forward now
NPV > 0 + NPV’ > 1;
odds are it’s a go
NPV < 0 + NPV’ < 1;
NPV < 0 + NPV’ > 1;
high volatility may work high volatility may work
in favor
in favor
NPV = 0
The key areas for Zion to focus on were the spaces at the bottom of the graphic. On the
right side were the projects with a clear NPV below zero that should not be totally written off as
the higher volatility environment might allow them to cross the NPV = 0 line, thus becoming
viable.
On the other side of the NPV’ = 1.0 line, projects that had an NPV < 0 and an NPV’ < 1.0
should not be totally written off either. The concept here is that these investments might migrate
and become NPV’ > 0 thanks to the higher volatility environments they were in. Conceivably,
then, these investments might migrate up and to the right of the graphic and cross the NPV = 0
line, thus becoming investable for Zion.
Everyone looked at each other in a mixture of understanding and disbelief. Marina’s
arguments were persuasive but where to go from here?
At this juncture, Randy piped up and suggested that Marina should form and head the
airline’s data analytics unit. This unit would work with operating departments to get the latest and
most extensive data with which to inform strategic and tactical decisions going forward. The
department would also work closely with the finance unit to get the most accurate data on revenues
and costs. Lastly, the department would work with network development so as to integrate these
new methodologies into a robust decision-making system for airplane assignments, slot utilization,
and route expansion.
Six months later…
Marina was settling in nicely but was finding out that some of the calculations that she was
making for Zion were not to her liking. They just didn’t seem to be descriptive enough of the
airline’s options in various airports.
Her analysis using real options was working well and some projects that seemed doomed
were still alive but hanging by a thread. Those that were a go from the start had done well so at
the very least, she knew that the fundamental data behind the decision-making process were
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system, or transmitted in any form, electronic, mechanical, photocopying, recording or otherwise without the
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accurate. Still, she was looking for an additional methodology for analyzing Zion’s options going
forward.
Looking back at the binomial distribution underlying the Black-Scholes model, she
realized that perhaps she could use that model to forecast the airline’s network development
directly. To do so, she got all the data on Zion’s operations in various markets and tried to come
up with a general forecast of the outcome for the introduction of a new airplane that promised to
revolutionize travel to small airports.
The airplane allowed Zion to fly turboprops to certain markets that were not particularly
competitive. If Zion could enter these markets and get a foothold, those footholds would likely
continue for decades before the competition would start eroding Zion’s margins.
Marina’s department researched the new plane and the new markets and came up with three
phases that would inform whether or not the long-term viability of those airports would be a payoff.
Also, she would be able to blend in the probability assumptions that she had originally used in the
evaluation of the five airports in her presentation some six months earlier.
She got to work on putting together the underlying market data and got the following:
•
•
•
•
Start – research the best routes, airports, and seating configurations for the new plane.
Train the following departments, maintenance, cabin crews, pilots, and runway staff in
the optimal operation of the new plane
Phase I – lease three new planes with the following destinations: Utah, Colorado, and
Arizona
Phase II – operate the new planes in these three markets and actively promote those
routes with passengers. Provide price incentives to passengers to use those planes and
destinations
Phase III – refine the airplane’s operations after all the kinks have been worked out.
Expand the schedule so as to keep the planes in the air as often as possible
After many meetings and long nights sorting out the details with the various department
heads, Marina put together her department’s best estimate for the new aircraft’s operations in the
Zion network.
The plan was comprehensive but it made sense:
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University, Daytona Beach, Florida, 32114. No part of this material may be reproduced, stored in a retrieval
system, or transmitted in any form, electronic, mechanical, photocopying, recording or otherwise without the
prior written consent of the University.
Phase I
Start
Network Expansion with New Aircraft
PV of
Phase II
Phase III
Revenues Costs
26
128
Success
70%
8
Utah
40%
6
20
Failure
30%
2
Colorado
20%
7
Success
65%
8
Start
Arizona
25%
10
4
Failure
15%
3
Failure
35%
4
Net
102
Expected
Probability Value
18.2%
18.6
(20)
7.8%
(1.6)
Success
55%
15
145
34
111
7.2%
7.9
Failure
45%
3

22
(22)
5.9%
(1.3)
35
60
8.9%
5.4

25
(25)
7.3%
(1.8)

15
(15)
9.8%
(1.5)

8
(8)
35.0%
(2.8)
100.0%
22.9
Success
55%
13
Failure
45%
3
95
The plan hinged on a slow roll-out in the starting phases but long-term solid results in
Colorado and Utah in particular because of the more limited competition in those routes. Arizona
was profitable but nowhere near as profitable as these two states.
Marina deliberately made the probabilities of success sobering because overstating them
could be costly to Zion. After all, launching a new airplane in three new routes could be very
problematic for the airline if the projections did not pan out.
So, for example, the probability of success in Colorado was only 7.2%. Marina thought
this was low but she knew that when she presented the numbers to the company’s top managers
there would be push back. These were a cautious lot but they had a track record of good results
so one had to recognize and defer to their opinions.
Still, what Marina wanted to remove was the opinion from the system. She had read
“Moneyball” and wanted to bring that sort of data-driven analysis to Zion. If a small market team
like Oakland could make a go of it, Zion could too, but in its industry. So, with the conference
room booked and the presentation ready to go, Marina got rolling.
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University, Daytona Beach, Florida, 32114. No part of this material may be reproduced, stored in a retrieval
system, or transmitted in any form, electronic, mechanical, photocopying, recording or otherwise without the
prior written consent of the University.
During the presentation, Marina presented her assumptions and explained the
methodology. This group had already bought into real options so the overall philosophy was not
the issue. The issue was the level of expenditures, the three markets being targeted in one swoop,
and the fact that the table itself showed the word “Failure” in five separate rows!
Marina started by explaining the methodology using Utah as the proxy:
•
•
•
•
•
The Start period’s costs were explained first. These costs applied to all three states but
it was the first of four phases so there were many other costs on the way!
She explained the Phase I costs. This phase could blow up – there was a one in three
chance of that – so this was not a gimme. Still the most the company would lose if
things blew up in Phase I was $8 million – a high enough loss to make everyone shuffle
in their seats
If Phase I worked out, the company would then enter a three-prong Phase II. Despite
the spread over time and space, the projected probabilities are only 15% as the Phase I
and II combined efforts reduced the probabilities of failure as a result of proper
planning and execution of the airline’s operating plans
Phase III represented a more long-term operation in each state with the new airplanes.
This phase would deliver the revenues that Zion would collect as a result of its entry
into these destinations with the new planes. The margins, if the operations went
according to plan, would be better than Zion’s other operations. This is because the
new airplane will allow for higher yields, lower CASM, higher load factors, and less
head-to-head competition
She explained the way in which the decision tree worked by explaining the intricacies
of the methodology:
? For Utah, for example, the total costs were the sum of the Start ($4M), Phase I
$8M), Phase II ($6M), and Phase III ($8M) costs associated with the Utah path
alone. This tied to the $26M total costs that would be applied against $128M in
revenues for Utah
? The Utah probabilities were the multiplication of the probabilities of success for
Phase I (65%), Phase II (40%) and Phase III (70%). When these three probabilities
are multiplied, they yield the overall success probability of 18.2% for the Utah path
alone
? Continuing with the Utah example, the probability of failure in Utah was the
multiplication of Phase I (65%), Phase II (40%) and Phase III (30%) to give a failure
probability of 7.8%
? The Utah Phase III failure costs were $2M so to calculate the costs of failure in
Utah, these $2M were added to the Start, Phase I, and Phase II costs associated with
the Utah-specific path
? To make sure that the calculations were all correct, the sum of all probabilities –
failure and success- for all three states as well as the Phase I failure had to tally up
to 100%
Despite the depth of the analysis, everyone in the conference room thought that the threeprong effort would be a big step for Zion. Marina replied that the advantage of the new airplane
and the way in which Zion would use it and finance it required that there be “scale” – in other
words, for the project to be optimized, it really had to be launched in all three states. The results
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system, or transmitted in any form, electronic, mechanical, photocopying, recording or otherwise without the
prior written consent of the University.
would be approximately $23 million, which would represent a significant increase to Zion’s equity
position in the balance sheet.
She added, further, that in the worst case, the airline would lose $8M if the project did not
work out. This number was the sum of the Start costs ($4M) and Phase I Failure costs ($4M). So,
in a way, if the company were going to fail, it would be best if it failed early rather than later!
This is because failure in Phase II would cost $15M, almost twice that of Phase I. She
noted that it was ironic to see this sort of cost progression but the methodology was sound and the
data underlying it was vetted by the various departments that were present at the meeting.
With that said, the management team gave the go ahead for the new project with the new
airplane. Marina left the room and return to her department and called out: “Wheels up, everyone!”
Case Questions
Use the template provided to calculate:
Part 1
The value of Real Options vs. No Real Options scenarios for Zion using the data contained
in the template. Note that the Black-Scholes is already contained in the template so there is no
need to recreate it. You will calculate the value of the Real Options scenario and compare that
value to the No Real Options scenario.
Part 2
The overall expected value of a new set of data inputs for the decision tree calculations.
All the information you need to solve this case is contained in the text provided in the template.
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