SME Abstract Form - RRI - West Virginia University

Choosing the Discount Rate: A Fairy Tale
by
Thomas F. Torries
RESEARCH PAPER 2013
Session – Valuation I – Concepts
Paper presented to the 2000 SME Annual Meeting and Exhibit
Salt Lake City, Utah, February 29, 2000
Thomas F. Torries
Professor, Agricultural and Resource Economics, Division of Resource Management
and
Faculty Research Associate, Regional Research Institute
West Virginia University
Abstract: A tale involving a Prince, his wizard, a bargaining magical frog, the wise old
rabbit, and the Prince’s neighboring rulers illustrates a number of known and less well
known hazards in choosing the discount rate for evaluating investments. The tale reveals
a new twist to the determination of the actual value of the opportunity cost of capital that
involves deviations in estimated project value caused by the profession inability of
forecasters, evaluators and decision makers.
Choosing the Discount Rate: A Fairy Tale
Thomas F. Torries1
The Proposition
It was another beautiful day in the Enchanted Kingdom and the young Prince was
taking his daily walk past the pond. A large green frog sitting on a lily pad rolled his
eyes at the Prince and said “Hi, Prince.”
“Hi, frog.” the Prince replied without hesitation or surprise, for talking animals
were the norm in the Enchanted Kingdom. “What’s happening?” the Prince inquired.
“You are doing just a great job of making this Kingdom work, Prince.” the frog
replied. “I want to reward you. So, I am going to give you a choice of two alternatives
and you can take your pick of whichever one you want.” continued the frog. “Here’s the
deal.”
“I will give you the dam on the river that you wanted to build." continued the
frog. "The dam will supply energy to process your crops and make goods to sell in other
neighboring kingdoms continuously for the next fifty or more years. It will also provide
flood control and recreational activities."
“Wow!” exclaimed the Prince. “What is the other alternative?"
"Alternatively, I will give you a gold mine." explained the frog. "Unknown to
you, there is a rich deposit of gold and pyrite buried deep in the cliffs next to the river.
Mining this rich deposit will provide much wealth to you over the next ten to twenty
years. You can then use this wealth to help your subjects in any way you desire."
"However," continued the frog, "there are certain drawbacks to each offer. The
dam will also flood valuable land and disrupt aquatic life. The gold mine will require
reclamation when the mine closes. It will also produce acid water that will need
treatment on a declining cost basis for a thirty-year period. "
"The money it takes to build the dam or construct the gold mine is about equal.
While it appears the benefits strongly outnumber the costs in both alternatives, you may
conclude otherwise. Both projects should provide a means to a higher standard of living
for your subjects. I know you have wanted to improve living conditions for some time,
but whenever there was spare income, there were always other things that seem to be
more important. Although this is an Enchanted Kingdom, money still does not grow on
trees, you know.” joked the frog.
1
Dr. Torries is Professor of Resource Management at West Virginia University, P.O. Box 6106,
Morgantown, WV 26506-6108.
2
The frog shifted on the pad a bit as he eyed a nearby buzzing fly and replied, "So,
the choice is yours.”
“Oh, oh.” said the Prince to himself as much to the frog. “This is not going to be
easy. Frog, can I give you my answer at this time tomorrow? I need to consult my
Wizard about this.”
“Sure.” replied the frog as his attention clearly shifted toward the nearing fly.
“See you tomorrow.”
The Wizard's Valuation
The Prince ran back to the castle to summon the Wizard and his advisors. Once
they were gathered in the Great Room, the Prince explained the frog’s proposition. The
Wizard gathered his staff and assigned tasks required for the completion of a risk
adjusted discounted cash flow analyses complete with sensitivity and scenario analysis.
The Royal Computer hummed all night as the Wizard considered population projections,
future energy and agricultural product prices, forecast GKP (gross kingdom product),
employment levels, and other necessary data. Finally at mid-morning the next day amid
piles of computer output the Wizard declared that his analysis was complete.
“Prince, I have an answer for you.” the Wizard stated wearily, but with apparent
pride. “The Net Present Value of the gold mine is definitely higher than that of the dam.
You should choose the gold mine. The mine will last only twenty years, but will provide
significant revenue and jobs immediately. Although the benefits of the dam last for a full
fifty-year period, the annual benefits are smaller than those obtained from the gold mine.
In addition, the environmental costs of the mine are not paid until far into the future. Any
reasonable discount rate makes the present value of the dam small compared to the
present value of the gold mine.”
"Great!" exclaimed the Prince. "But, tell me more about your assumptions and
the choice of the discount rates. What rates were used and how were the cash flows
calculated?"
To that, the Wizard replied with the following explanation:
"Figure 1 compares the constant dollar cash flows of the mine and dam. Both
projects require the same amount of capital and last for the same number of years. The
mine has large positive cash flows in early years, a large reclamation cost in year 20, and
declining negative cash flows in years 21 to 50. The dam has lower cash flows than the
mine in the early years, but continues to have positive constant cash flows for the entire
50-year period. Because of the different patterns of benefits and costs, the choice of
discount rate will be important in choosing between the two projects."
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300
MK Dollars
100
-100
0
10
20
30
40
50
-300
-500
-700
-900
-1100
Years
Dam
Mine
Figure 1. Cash Flow Profiles of the Dam and Mine (constant dollars)
"While we all agree that the opportunity cost of capital (OCC) is the proper
discount rate, determining the value of OCC is difficult. There are numerous rates we
can consider, as are shown in Table 1.”
Table 1. Candidate Rates of Return (Current After Tax)
WACC (calculated)
13.97%
Risk free alternative
5.50%
Cost of debt
11.00%
Historical rate of return
14.00%
Risk adjusted rate of return
14.00% + risk
Hurdle rate
15.00%
Social rate
14.00% - social
Temporal rate (Intergenerational)
14% - temporal adjustment
Tax rate
0.00% (does not tax itself)
Inflation rate
2.50%
“Note that any rate in current dollars, such as the historical rate and the cost of
debt, must be adjusted for inflation before using to discount the constant dollar cash
flows of the two projects. Also, these rates must all treat taxes consistently. Of these
rates, the weighted average cost of capital (WACC) is the one we believe most accurately
portrays the OCC.”
"We used a discount rate of 12.0%, which is close to the actual calculated value of
11.47% of the after tax, constant dollar WACC. Although, both projects have different
risks, we believe the overall amount of risk for each project is the same."
“At any reasonable discount rate (above 9.0%) that approaches the WACC, the
mine will have the higher NPV. This is shown in Figure 2. Therefore, we recommend
the mine."
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3000
2500
NPV
2000
1500
1000
500
0
0.0%
-500
5.0%
10.0%
15.0%
20.0%
Discount Rate
Dam
M ine
Figure 2. Sensitivity Analysis of NPV and Discount Rates of Mine and Dam
Pollution Abatement Sinking Fund at Safe Rate
The Prince considered this explanation for several moments, and then replied “I
still have a number of questions. First, I am concerned that the process of discounting at
a higher discount rate disguises the problems that may be encountered with the pollution
abatement in years 20 through 50 for the mine. I believe the cost of the future pollution
abatement should be paid as a current expense to produce a sinking fund that will finance
the pollution abatement that occurs at the end of the mine’s useful life. Since we must
invest these funds in a safe manner, the discount rate for these funds should be around
3.0% rather than the WACC. The WACC should be used to determine the present value
of the cash flows after deducting for the sinking fund contributions. What does this do to
NPV of the mine?”
“Oh!” replied the Wizard. “What you suggest is based on the same principle as
the Hoskold formula, but instead of creating a sinking fund at a safe rate to finance new
operations after depletion of the current mine, you want to finance a pollution abatement
program. This is easily done.” the Wizard continued as he pressed a few keys on his
laptop.
“The new cash flow profile of the mine is now shown in Figure 3. The mine now
has lower cash flows in years 1 though 20 but no negative cash flows due to pollution
abatement from years 20 through 50. Figure 4 shows the NPV of the dam and the two
versions of the mine. The modified mine cash flows are discounted at 12% and the
sinking fund is accumulated at 3%. This means the NPV of the mine will decrease. The
5
MK Dollars
mine is still the preferred project at any discount rate over 10%, although by a smaller
margin.”
300
200
100
0
-100 0
-200
-300
-400
-500
10
20
30
40
50
Years
Mine
Mine Rec Pmt
Figure 3. Cash Flow Profile of Mine with Reclamation Payment
3000
2500
NPV
2000
1500
1000
500
0
-5000.0%
5.0%
10.0%
15.0%
20.0%
Discount Rate
Dam
Mine
Mine"
Figure 4. Sensitivity Analysis of NPV and Discount Rates of Mine Options and Dam
Intergenerational Equity
“Now,” continued the Prince, “I am also concerned with the choice of discount
rate since both projects span so many years. Should we use a lower discount rate to
account for intergenerational equity, such as described in Portney and Weyant (1999)?”
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“Prince,” replied the Wizard, “I have read this book. It contains valuable
contributions from Solow, Arrow, Dasgupta, Smith, Lind, and many other very notable
economists. One of the major conclusions was that intergenerational discounting at some
lower positive discount rate should be used with projects that have lives of 50 or more
years. While a lower rate could be used in the present case, the time period is marginally
long, and there is considerable disagreement about just how and how much to reduce the
discount rate. I recommend that we do not use any discount rate lower than 10% in this
case. This means we choose the mine.”
Adjusting OCC to Account for Imperfect Analysis
The Prince considered this explanation and then replied “The discount rate still
bothers me. What does King Richard in the adjoining kingdom use as his discount rate?
His OCC should be much the same as ours since we both have similar investment
opportunities.”
“King Richard,” answered the Wizard uses 20%, which is higher than ours is. He
would choose the mine as well.”
“Why is his discount rate higher than ours?” asked the Prince?
There was silence as the Wizard and his workers looked at each other and
shrugged unknowingly.
From the corner of the room stepped forward an old floppy eared rabbit who had
been around the castle as long as anyone could remember. “Prince, I can answer that.
The truth is, you have a better Wizard than King Richard, so there is a greater risk of his
projects unexpectedly failing or succeeding. King Richard knows this and has added an
additional risk factor to his discount rate in the hope of avoiding unexpected losses and
thereby increasing his overall rate of return. While this adjustment process has wide
appeal, the truth is that it is invalid in this case. Such an adjustment would be valid only
if the King’s Wizard consistently under or over estimated the NPV. However, his
Wizard’s errors are random.”
“As a result,” continued the rabbit, “increasing the King’s discount rate means
more projects are rejected and more capital is left for the default safe investment. If the
King raises the discount rate sufficiently high, all projects will be rejected except the
default safe investment. This means his resulting OCC becomes the safe rate. You and
your Wizard do much better. You should not consider using a higher rate just because
King Richard does to value similar projects.”
“Thank you, Rabbit.” exclaimed the Prince. “I now have a considerable amount of
information I need to make a choice. I must leave now to meet the magical frog at the
pond.”
7
The Decision
The Prince hurriedly left the castle to seek the frog, who was easily found sitting
on the same lily pad as the day before.”
“Hi, Prince!” greeted the frog. “Have you made a decision?”
“I have!” replied the Prince. “I am going to choose the dam.”
“That is a good project, Prince. But, why did you choose the alternative with the
lower NPV?” asked the frog.
The Prince replied “The NPV of the mine is higher even with a sinking fund to
pay for future pollution control. If all works as planned, the mine would be the correct
choice. However, if something happens to me, there may be a temptation for the next
ruler to use the accumulated sinking fund for other purposes before the pollution is
abated. Alternatively, pollution abatement may be more difficult than expected in spite
of what my Royal Engineers have to say.”
“I simply do not want to take the chance of saddling the next generation with a
non-productive payment that this current generation has caused, especially since an
alternative investment is available. Therefore, I choose the dam.” explained the Prince.
“Congratulations!” exclaimed the frog. “You have considered both the normative
economics, which favor the mine, and your own subjective judgement. The application
of both normative and subjective economics is important in almost all important
investment decisions. Your Wizard supplied the numbers that describe the costs,
benefits, and risks of both projects. You then used your subjective reasoning to help
interpret what the numbers mean to you and your subjects. This is the proper role of a
decision maker. You did well. The dam will now be constructed.”
“Thank you for the project as well as the compliments, frog.” replied the Prince.
“But, I worry that if I had additional information, I might have made a better choice.”
“Well, you might have considered option pricing.” replied the frog. “Gotta go
now. The Missus is waiting.” And with that, the frog plunged into the pond leaving the
Prince scratching his head.
“Option pricing. What’s that?” the Prince asked himself as he turned and started
walking back to the castle. “Hmmm,” mused the Prince. “I wonder where the Wizard
is?”
Thus ends this fairy tale, as all fairy tails must. And they lived happily ever after.
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References
Portney, P. and J. Weyant, ed. (1999). Discounting and Intergenerational Equity,
Resources for the Future, Washington, DC.
Torries, T. (1998). Evaluating Mineral Projects: Applications and Misconceptions,
SME, Littleton, CO.
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