Henrik Anderson*, Stina Skogsvik*
If we believe in the dynamics of a competitive market economy, cash flows stemming from
sales of standardised goods should be mean reverting. This is not congruent with the use of a
constant risk-adjusted discount rate in valuation as the risk of a mean reverting cash flow is
asymptotically constant. Nevertheless, option theory can handle mean revering price
processes and be applied to determine an appropriate, but time dependent, risk-adjusted
discount rate for calculating the present value of operations. The advantage is the possibility
to correctly value the mean reverting cash flow without having to resort to option calculations
with all the complexities such a valuation would entail. Our analysis shows that with as
the risk adjusted return and r as the risk-free rate, the risk premium changes from the standard
))(( tTr to tT er
1 when cash flows are mean reverting according to the
exponential Ornstein-Uhlenbeck process. In other words, the risk premium is a function of
time but not a multiplication with time, and asymptotically constant as the risk is
asymptotically constant. Mean reversion normally leads to an increase in value and
differences are accentuated in a low interest rate environment even with a constant market
risk premium simply because future payments are more worth. It is therefore even more
important than historically to get this assumption right.
Key words: Present value, Discounting, Mean reversion, Market efficiency
JEL classification: C61, G12, G13, G14.
___________
* Henrik.Andersson@hhs.se, Stina.Skogsvik@hhs.se, Stockholm School of Economics, Box 6501, S –
113 83 Stockholm, Sweden. For discussions and valuable comments the authors are much indebted to
Peter Jennergren, Kenth Skogsvik and Håkan Thorsell.
1
1. Introduction
Discounting as it is taught in standard corporate finance textbooks is a rather straightforward
task. Assign a single risk-adjusted discount rate and use this to value the expected cash flows.
The risk-adjusted discount rate is typically found through the CAPM. In detail it is not so
simple. CAPM is a one-period equilibrium model and the extension to a multi-period setting
is not straightforward. Fama (1977) derives the necessary conditions and although the
mathematics gets rather involved, the result is easy enough. Discounting expected future
payments using a risk-adjusted rate of return requires the covariance with the market to be
non-stochastic and know at all points in time. The use of a single risk-adjusted discount rate
then requires the covariance with the market to be constant over time so that together with a
constant risk-free rate also the discount rate becomes constant.
This fits very nicely with the properties of a random walk time-series, where the return
distribution is constant over time. Assuming that cash flows follow a random walk is therefore
congruent with the standard corporate finance textbook recommendation of a single risk-
adjusted discount rate. In algebra we write the present value as
P:
(1)
where X(T) is the cash flow at time T and μ is the single risk-adjusted discount rate.
Continuous discounting is used for congruence with the continuous time modelling that
follows later. P: denotes that the discounting is done under real world probabilities, whereas
Q: later will denote risk-neutral probabilities.
Leaving the safety of the random walk assumption, we are essentially on lose ground. For a
mean reverting process, the expected return for the next interval depends on if we are above
or below the mean level, something that changes stochastically over time. The covariance
between asset i and market M, ,)()(, MMiiMi rErrErECov is then stochastic as )( irE
is stochastic. A mean reverting process is therefore not in line with the use of a risk-adjusted
discount rate. This problem has probably mostly been regarded as a slight inconvenience but
it is more serious.
2
For a random walk, which is congruent with a single risk-adjusted discount rate, the variance
increases linearly with time and therefore also the systematic variance (the part of the
variance that is priced) and the risk premium in the end becomes infinite as time approaches
infinity. This is not the case for mean reverting cash flows. Then the variance is bounded.
Obviously the systematic variance is also bounded, and it is therefore not possible to have a
constant risk-adjusted discount rate when cash flows are mean reverting. Future cash flows
are simply not that risky.
Students of corporate finance are often worried about that payments far into the future are not
adequately risk compensated, for example in a steady state calculation using the Gordon
growth formula. They see the distant future as very uncertain and do not realise that the
discount rate they use consists of both the risk-free rate and the risk premium. As time goes to
infinity so does the risk compensation through the risk premium, and it compounds
exponentially. In reality the problem is therefore the other way around. An overcompensation
for risk is a greater worry as even stocks, which traditionally have been view as random
walks, contains a mean-reverting component, see Fama and French (1988).
Many cash flows are supposed to be mean reverting. At least if we believe in the dynamics of
a competitive market where the price of standardized goods, for example commodities, are
supposed to revert towards the marginal cost of production.
Figure 1. A mean reverting price process reverts towards the equilibrium price, the
marginal cost of production.
Intuitively, exceptionally high prices will attract new producers and therefore reduce the
price. Unusually low prices will drive some producers out of business, something that tends to
increase the price. This intuition not only motivates mean reverting prices, it is also the
fundamental process for how a market economy solves the production problem of what and
how much the society should produce of each good. Of course, technological innovations and
marginal cost changes may destroy this structure but it is possible to argue that the economy
seems to function quite nicely and in such a refined way that one may put some trust in the
assumptions underlying a market economy, something also implying that prices of
standardised goods should be mean reverting.
From a valuation point of view, instead of having a single risk-adjusted discount rate
multiplied by time as the discount factor, it is more general to assume that the risk adjustment
in the discounting factor is function of time – and not necessarily a multiplication with time.
We could then write
(2)
where r is the risk-free rate and ),,( tTMRPf is the risk premium as a function of the
systematic risk, the market risk premium, and the time to maturity. Of course, under the
random walk assumption of equation (1), the risk premium is just
),)((),,( tTrtTMRPf the difference between the risk-adjusted required return
and the risk-free rate multiplied by time. The question is what this risk premium will look like
for other cash flow processes?
The idea of this paper is to utilize option theory to arrive at the arbitrage free value of a future
mean reverting cash flow and rewrite it in the style of equation (2). The technical complexity
of an option approach is in most cases overly excessive and it is in practice skipped excepted
in the most advanced situations. Equation (2) provides more intuition from a present value
perspective than solving the Black-Scholes differential equation. Equation (2) would provide
usability and a shortcut that cannot be obtained by an option valuation.
The essence of the Feynman-Kac solution to the Black-Scholes differential equation is that
the value of the contract payment can be seen as the expected payment in a risk-neutral world
4
)(TXE Q
discounted at the risk-free rate r. We can then express the value of the cash flow

, )()(
(3)

. )(
TXE tTMRPf
Q (4)
We show that for the exponential Ornstein-Uhlenbeck process the risk premium becomes
tT ertTMRPf
1),,( (5)
where μ is the risk-adjusted required return, r the risk-free rate and η is the speed of reversion
for the mean reverting process. The compensation for risk approaches asymptotically the
constant value r when time tends towards infinity because in a mean reverting process the
variance is asymptotically constant.
A natural question is of course why option pricing theory can be used to solve the discounting
problem of the mean reverting process? The reason is the construction of the instantaneous
risk-free portfolio from where the Black-Scholes differential equation is derived. Even though
the risk changes over time in a stochastic way, the portfolio can be maintained as risk-free by
revising its composition, thus allowing valuation of the future payment. The cost of this
ability is the requirement of an underlying tradable asset, following a specified stochastic
process, and the continuous updating of the portfolio. 1 The advantage is, however, the ability
to value any arbitrary contract as long as the above conditions are met. When these conditions
are not met, as is the case for most non-financial contracts, the result should not be seen as
arbitrage free but as a much weaker equilibrium – a value that can be expected to hold, but
there are no explicit forces driving it towards this value if there are deviations.
1 Rubinstein (1976) showed that the Black-Scholes analysis also holds also when trading can take place only at
discrete points in time.
5
The difficulty of valuing options lies in the fact that the payment is a non-linear function of
the underlying asset. Such a payment is often referred to as an asymmetric payment, a
terminology used by, for example, Trigeorgis and Mason (1987). In order for the CAPM to
handle asymmetric payments, it would have to be updated instantaneously, which, in fact, was
one of the ideas leading to the Black and Scholes differential equation, see Black (1989). In
their original paper from 1973, Black and Scholes provide a derivation of their differential
equation using the CAPM.
Mean reversion and therefore lower risk translates into higher present values and lower option
values as the outcome is more certain. It should be stressed that equation (3) provides no
simplification for valuing options. In principle, we could exchange )(TXE for )(TXgE
with g as the payment function of a call option, but in the case of a random walk, the result
would just be the Black-Scholes formula. The point is that the Black-Scholes option pricing
formula is a quick way to value an option when the price process is a random walk. Equation
(3) is a quick way to value symmetric payments when the price process is mean reverting.
We proceed in Section 2 by showing that equation (1) is indeed the present value of a future
cash flow from a random walk process. The exponential Ornstein-Uhlenbeck process, where
the log cash flow follows an Ornstein-Uhlenbeck process is then introduced. As cash flows
tends to increase over time, if for no other reason by the rate of inflation, a drift in the
equilibrium is introduced. After that, equation (5), i.e. the risk premium that must be used for
valuing mean reverting cash flows in a real world (as opposed to a risk-neutral world) is
derived.
Section 3 is used to assess the magnitude of error in the valuation that can result if a cash flow
is mistakenly identified as a random walk when in fact it is mean reverting. Section 4
summarises, and the expectation of the exponential Ornstein-Uhlenbeck process is deferred to
the appendix.
2. Deriving the risk premium in continuous time
Assume that a cash flow process tX follows the Ito process
P: ., XdwXdtXtdX (6)
We allow for an arbitrary drift rate Xt, in order to accommodate both a random walk and
mean reversion, but specify the diffusion rate σ as constant since there is no need for
generality here. The value of an asset V(t, X) dependent on the cash flow X, is the solution to
the B-S differential equation,
1 rVVXXVXtrV XXXt (7)
where subscript denotes partial derivatives, The term ),( Xt requires some extra
attention. is the instantaneous risk-adjusted return associated with X. If an investor requires
and gets Xt, as an expected drift, the difference between these two must be the
dividend yield the investor is expected to receive or else we are not in equilibrium. 2 As r is the
instantaneous risk-free rate, the expression within the curly brackets must be the drift rate of X
in a risk neutral world – the difference between the required rate of return r and the dividend
yield. Defining a new Ito process
Q: ,),( dvXXdtXtdX (8)
),,(),( XtrXt (9)
),(),( )(
(10)
2 See Dixit and Pindyck (1994, page 161-162) for a mean reverting example. McDonald and Siegel (1984)
provides a more detailed discussion on option pricing when assets earns a below equilibrium rate of return.
7
The Feynman-Kac formula states that the value of an asset today, equals the discounted value
of the expected value at maturity. However, note that the expectation should be computed for
a cash flow X(t) following the diffusion process (8). Hence the notation )].([ TXE Q
As argued
this is the process that X would follow in a risk-neutral world and (10) then becomes the risk-
neutral solution to derivative pricing.
A random walk is congruent with the standard present value calculation.
A random walk with drift, or its continuous time equivalent the Geometric Brownian motion,
is congruent with the standard present value application and we here sketch the argument. The
diffusion process is,
P: ,XdwdtXdX (11)
where now is a constant. The value of receiving X(T) is then )(),( )(
TXEeXtV QtTr
Q: .)( XdvdtXrdX (12)
. )(
)()(),( )(
)( )()()()(
The numerator )( )(
tT etX
is nothing else then the expectation of X and we could therefore
write
(14)
8
In other words, the assumption that the cash flow process X follows a random walk in discrete
time and the geometric Brownian motion in continuous time is congruent with the use of a
single risk-adjusted discount rate.
Introducing a mean reverting process
As an alternative to the geometric Brownian motion we chose the exponential Ornstein-
Uhlenbeck process for example used by Schwartz (1997) and Al-Harthy (2007) for
commodities, and Ekvall, Jennergren and Näslund (1995) and Tvedt (2012) for exchange
rates:
P: .lnln dwdtXXd (15)
When Xln is smaller than the equilibrium level the expected change is positive and the
other way around. Thus, there is always a drift towards the equilibrium level. In order to cope
with inflation and other reasons for a trend, we follow Andersson (2007) and allow for a drift
t in the mean reversion level. The processes then becomes
P: .lnln dwdtXtXd (16)
Writing the exponential Ornstein-Uhlenbeck process in the cash flow X rather than in Xln it
becomes
(17)
something that is easily verified by applying Ito’s lemma to a function Xf ln to equation
(17). Just as for the geometric Brownian motion this process is lognormal and from equations
(7) and (8), we then have the risk-neutral process as
Q: .ln 2
(18)
9
It is shown in the appendix A that the expectations of (17) and (18) are,
P: )(2
(20)
The risk premium appropriate for discounting the mean reverting cash flow was in (4) given
as

10
3. Valuation differences
As an economic necessity, the risk premium can approach zero for two reasons: Either that no
risk compensation is required ,r or that time approaches zero .0)( tT Naturally, this
holds true for both the risk premium of the random walk ),)(( tTr and for the mean
reversion process .1 tT
When the time horizon is short, the term ).(1 tTe
tT
The mean reversion process
can therefore actually imply a larger risk premium when the mean reversion parameter is
greater than one and the time horizon short. Generally however, the risk premium will be
smaller for the mean reverting process and is asymptotically constant and equal to ),( r as
the variance of a mean reverting process also is asymptotically constant.
How large is then the effect on valuation? From a practical viewpoint, assume that we
somehow have estimated the future expected cash flow and are now pondering which
stochastic process that is related to this expectation. This is actually very realistic, not only
from a practical point of view but also statistically. Volatility estimations are more or less
identical independent of process assumptions and if the equilibrium drift parameter in the
exponential Ornstein-Uhlenbeck process is set equal to the drift rate of the random walk,
expectations for the two processes becomes very similar.
The only question then is our belief about the risk, bounded or unbounded as time goes to
infinity? Mean reversion or random walk? The difference is in the risk premium of the
denominator used in discounting as we already have settled on equal expected payments.
Algebraically it becomes:
11
The difference will be quite large for distant cash flows but this is mitigated by the fact that in
most cases there are intermediate cash flows until the project ends.
Figure 2 on the next page tries to illustrate the difference between the two processes. Shown
are the prices of the two commodities oil and pulp, as well as the NYSE composite index
between 1980 and 2016 where data is provided by Datastream. Depicted in solid lines is the
95 % confidence interval for the random walk / geometric Brownian motion process of (11)
where parameters are estimated in the standard way. See for example Björk (2009, page 109).
Dashed lines are used to demark the 95 % confidence interval for the mean reverting process
(15) where parameters are estimated in accordance with Andersson (2007, equations 23 and
24). Differences are striking. The point of a 95 % confidence interval is of course that the
process should be within this interval 95 % of the time and outside 5 % of the time. Yet, not a
single time it the 95 % border violated in the case of the geometric Brownian motion.
12
Figure 2 95 % confidence intervals for the geometric Brownian motion (solid line) and
the mean reverting process (dashed line).
13
In fact, the mean reversion process seems to more in line with the data also for the New York
Stock Exchange in the third diagram. It should perhaps be mentioned that statistical tests that
try to distinguish between a random walk and mean reversion are notorious for their very low
power. So called unit root tests, where the Dickey-Fuller and the Phillips-Perron tests are the
most well know suffers from the problem that they can only reject the null hypothesis of a
random walk for many years of data. 3 We may therefore be tempted to overuse the standard
assumption of a random walk. However, confidence intervals like in figure 2 clearly shows
the difference.
For some actual numbers on valuation differences, we use the Ibbotson data of the U.S.
market from 1926 until 2016. The continuously compounded average return on treasury bills
have been 3.4 %, the S&P 500 have returned 11.4% and the market risk premium therefore 8
% historically. The data used in figure 2 with prices of crude oil and paper pulp between 1980
and 2016 gives ,25.0,1.0,3.0,2.0 PulpPulpOilOil with continuous compounding.
The systematic risk as measured by β is rather low and this is quite typical for commodities.
Stock markets react in expectation of change in economic climate whereas commodity prices
react when the change has been realised and altered the demand and supply situation for the
specific commodity. When it comes to the speed of reversion for the mean reverting process,
the interpretation of 3.0Oil is that without any new information arriving,
%26)1( 3.0
e of the difference between the actual price and the equilibrium price has
disappeared after 1 year. Mean reversion is therefore rather slow which of course makes it
even harder to detect except for very long horizons.
Figure 3 provides two diagrams for crude oil. The first is the value of one unit of oil received
after n years and the other is the value of one unit of oil received every year up to year n. As
can be expected the mean reversion assumption provides the higher values. We assume the
expected cash flow equal to unity in both cases so the only difference is due to the risk
assumption. Concentrating on the first diagram with only one payment, differences are the
largest when the payments occur after approximately 30 years for this specific parameter
setting. When time approaches infinity, neither process will add any value due to the
discounting effect of the risk-free rate. However, taking about differences in percent has little
3 Hamilton (1994) provides a very clear and comprehensive description of the different unit root tests applicable
to different situations. Kwiatkowski, Phillips, Schmidt and Shin (1992) has a test where stationarity, mean
reversion, is the null hypothesis but the logic is somewhat unclear.
14
meaning as the contribution to the present value becomes smaller and smaller the further into
the future we get. The second diagram in figure 3 gives the present value of one unit received
every year. If this stream of payments continues for 50 years, the mean reverting process will
provide a value estimation that is some 25 % higher, although it is perhaps difficult to project
payment streams that long for most business ventures. Asymptotically the difference becomes
constant as payments very far into the future contribute less and less to the present value.
Figure 3. Value difference for the random walk and mean reversion processes when
receiving one unit of oil after n years in the first diagram, and one unit each year
in the second diagram.
15
Figure 4 beneath provides the same diagrams as in figure 3 but for paper pulp instead.
Figure 4. Value difference for the random walk and mean reversion processes when
receiving one unit of paper pulp after n years in the first diagram, and one unit
each year in the second diagram.
Here differences are smaller as the systematic risk in pulp is smaller, Pulp is only 0.1,
compared to Oil which was equal to 0.2. Naturally the differences also changes with the
assumption of the market risk premium being higher or lower than 8 %. The speed of mean
16
reversion , however, has no practical influence on the valuation at all. Setting 0.1Oil
instead of 0.3 would produce more or less identical results. The speed of reversion measures
how fast the process returns to the mean after a deviation has occurred. If a future deviation,
be it up or down, from the mean occurs is does not really matter how long time it takes to
reverse. The best guess before any deviation has occurred is still the mean.
One thing that matters though is the interest rate environment. The earlier diagrams were
based on the historic average return on treasury bills, 3.4 %. Currently we experience very
low risk-free rates. In Figure 5 the risk-free rate is set as 0.4 % while keeping the market risk
premium constant at 8 %.
Figure 5. Value differences increases in a low interest rate environment
0
0,2
0,4
0,6
0,8
1
1,2
0 5 10 15 20 25 30 35 40 45 50
The value of one unit of oil received after n years
Random walk Mean reversion
0
10
20
30
40
50
0 5 10 15 20 25 30 35 40 45 50
The value of one unit of oil received yearly for n years
Random walk Mean reversion
17
Naturally the present value is higher in a low interest rate environment as future payments are
more worth, but the interesting observation is that this also contribute to larger valuation
differences. If cash flows behave like a random walk or mean reversion process is even more
important today than historically.
4. Summary
Prices of standardised goods should be mean reverting according to the logic of a market
economy. Then also cash flows stemming from the sales or purchase of these goods should be
mean reverting. This is not congruent with the use of a constant risk-adjusted discount rate in
valuation as the risk in a mean reverting cash flow is asymptotically constant.
Option valuation techniques can handle a vast number of stochastic processes and has been
used for a long time. In this paper, we use option theory to indirectly determine an
appropriate, but time dependent, risk-adjusted discount rate for calculating the present value
when cash flows follow an exponential Ornstein-Uhlenbeck process, possibly with drift. For a
standard present value calculation, and when cash flows follow a random walk, the risk
premium is multiplied by time. More general is to let the risk premium be a function of time
and the purpose of this paper is to derive the risk premium for a mean reverting process.
Lower risk will translate into a higher value, all other things being equal.
The present value of future mean reverting cash flows is therefore higher than under a random
walk assumption. Differences are substantial but values are still of the same magnitude as
under the normal present value calculations but accentuated in a low interest rate
environment.
18
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19
Appendix - Expectation of the mean reverting process
Assume the cash flow to follow the exponential Ornstein-Uhlenbeck process with an
additional drift term such that:
.ln 2
dwXFdtFXFXXtFdF XXXXt
,dwdtFtdF
(A.2)
which is an ordinary Ornstein-Uhlenbeck process if the equilibrium drift rate is zero. In
)(),( )(
20




(A.3)
In order to derive the distribution of F(T), a useful lemma from stochastic calculus will be
used:
,)()()(
0
T
t
tdwthTY
.)()(
0
2
T
t
dtthTYVar
This result can, for example, be found in Björk (2009; p. 57). Applied to the process F, we
immediately get









XdwXdtXtdX
ln
2
2
can by defining F(t) = ln X(t) be reduced to solving the process
.lnln dwdtXtXd
)( )(ln1)(ln
Getting the expectations:
We thereafter apply the result for a lognormal distribution, see for example Aitchinson and
Brown (1957), that if ),,(~ln baNY then . 2
2
P: ,ln 2
Q: ,ln 2
(A.11)