Estimating Cost of Capital in Firm Valuations with Arithmetic or Geometric
Mean – or Better Use the Cooper Estimator?
04/27/2009
Abstract: In order to calculate firm values a firm’s future expected costs of capital must be
estimated. A problem concerning the utilization of historical stock returns is that there does
not exist an unbiased estimator for expected cost of capital rates when they are employed as
discount rates. We test the bias effects of the most common statistics – the arithmetic and the
geometric mean of historical returns – on the basis of capital market data from Germany and
the US. Moreover, we examine the estimator suggested by Cooper (1996) who employs second order Taylor series to reduce the bias. Estimation problems are most severe, when – following practitioners’ usual approach – expected future cash flows are only separately considered for five to ten years and then terminal values for all following cash flows are computed
as a single net present value, because this one – as a hyperbolic function of the cost of capital
rate – is prone to severe estimation errors especially for high annual growth rates of cash
flows. A straightforward method to reduce estimation problems considerably is to discount
future cash flows up to, e.g., year 30 separately (even if estimated simply on the basis of a
constant annual growth rate) so that terminal value calculations are restricted to cash flows
beyond year 30 and then apply the Cooper estimator. Alternatively, one may completely neglect terminal value calculations and merely utilize the arithmetic mean estimator for the first
30 years. Both procedures prove to be superior compared to applying the arithmetic mean or
the geometric mean estimator for all expected future firm cash flows including terminal value
calculation. The goodness of the arithmetic mean estimator without taking terminal value calculation into account is quite robust against variations of assumptions in annual growth rates
of cash flows and thus may be preferable to an application even of the Cooper estimator with
explicit terminal value calculation for high annual cash flow growth rates. Regarding the
choice between estimating discount rates on the basis of historical total returns and on the basis of historical excess returns over corresponding riskless interest rates which then in turn are
combined with the current term structure of riskless interest rates, we recommend the latter
approach, as this allows us to employ the current term structure of riskless interest rates and
the Capital Asset Pricing Model. Moreover, it reduces the sensitivity to estimation risk with
respect to discount rates.
Keywords: arithmetic mean, cost of capital, estimation bias, firm valuation, geometric mean
JEL-Classification: G12, G34, G24
1 Introduction
In order to compute firm values it is necessary to estimate expected cash flows and corresponding costs of capital as discount rates with the latter typically being time and risk dependent. As firm values may amount to billions of euros, even small estimation errors may
lead to sums easily exceeding € 1 bn. or more. For example, when the German telecommunications company Deutsche Telekom merged with the internet provider T-Online, the equity
value of Deutsche Telekom was estimated to be € 118.77 bn. (see Breuer et al., 2007). Even a
small error of only 5 % would result in deviations from the “true” firm value which amount to
more than € 5 bn. This simple example highlights the practical importance of the search for
most adequate firm valuation techniques.
In the following, we want to focus on the special problem of adequate discount factor estimation (see, e.g. Fernández, 2008, for the paramount relevance of this issue). In principle,
there are two possible ways to quantify unknown variables. On the one hand, one may rely on
experts’ assessments. With respect to costs of (equity) capital, it is in particular possible to refer to the special knowledge of analysts, as they estimate firms’ future dividends and earnings.
These estimates may be utilized in order to compute corresponding costs of capital as internal
rates of return (see, e.g., Claus and Thomas, 2001, and Gebhardt et al., 2001). One major
weakness of this approach is that it leads to correct estimates only in the case of unbiased analysts’ forecasts that are in addition representative for overall market expectations. Both conditions are typically not satisfied (see, e.g., Breuer et al., 2008).
On the other hand, one may rely on historical return moments, as long as they are stationary, i.e. independently and identically distributed over time. However, even if these conditions are met, an unbiased estimation of discount factors for net present value computations is
impossible. This holds true irrespective of using the arithmetic or geometric mean of historical returns as an estimator. Although this problem has been documented in principle in the
theoretical literature, it has so far been widely neglected in practical applications of net pre1
sent value computations for firm value estimation, as its practical severity has not been sufficiently explored up to now. However, this problem is relevant for both, direct empirical observations of historical rates of stock return (“estimates on a total return basis”, henceforth)
and in the case that costs of capital are calculated with the help of the Capital Asset Pricing
Model (CAPM) with historical realizations of market risk premia as an adequate starting point
(“estimates on a partial return basis”, henceforth). The main goal of this paper is to close this
research gap by presenting an in-depth analysis of the three most important procedures for
discount rate determination with respect to their estimation goodness.
To this end, in Section 2, we formally introduce the underlying estimation problem when
referring to the arithmetic or geometric mean. Cooper (1996) suggested to solve the problem
with the help of a second-order Taylor approximation as a third estimation technique which
turns out to be a specific combination of the arithmetic and the geometric mean of historical
return realizations. This approach will also be presented in Section 2. Among other things,
one aim of the following sections is to find out whether the Cooper estimator is indeed superior to the arithmetic and the geometric mean. To this end, we determine numerically the resulting bias effects depending on the utilized estimation method in Section 3. Thereby, we assume stationary one-period returns which are either binomially, normally or lognormally distributed. Return distribution parameters are calibrated according to expected value and standard deviation of historical return realizations of the German stock indexes CDAX and DAX
and the US index S&P 500. CDAX and DAX data is extracted from Stehle (2004) ranging
from 1955 to 2003. Data for the S&P 500 is taken from http://pages.stern.nyu.edu/~adamodar/
(comprising the years 1928 to 2007). The Deutsche Aktienindex DAX represents the 30 largest (in terms of order book volume and market capitalization) listed German companies (blue
chips) whereas the Composite Deutsche Aktienindex CDAX consists of all German companies listed in the General Standard or the Prime Standard at the Frankfurt stock exchange (664
companies as per 7th January, 2009).
2
The numerical analysis of Section 3 confirms the general finding of Section 2 that the
arithmetic mean leads to downward biased discount rates and that the geometric mean is even
more biased. The differences to the actual discount rate increase almost linearly with growing
forecast horizon. This finding cannot be verified for the Cooper estimator which is superior to
the arithmetic and the geometric mean for short-term forecasting horizons, but gets progressively worse for long-term horizons (in particular when the forecast horizon exceeds 20
years). However, the performance of the Cooper estimator relatively improves if we allow for
the exclusion of unreasonable discount factor estimates below 0 and above 1. Moreover, we
find that estimation problems are more severe for Germany than for the US because of the
smaller historical sample length in Germany and because of a higher historical return standard
deviation. Nevertheless, for a general assessment of the various estimation procedures, it is
necessary to examine corresponding net present value computations.
We therefore take a closer look at the US and the German capital market in Section 4. In
contrast to the numerical analyses of Section 3, we now utilize directly the 49 historical (yearly) return realizations of the DAX and the CDAX index from Stehle (2004) as a proxy of the
true return distribution. In the same way, we make use of the 80 historical (yearly) return realizations of the S&P 500 from http://pages.stern.nyu.edu/~adamodar/. In addition, we extract
from this data excess return realizations in comparison to the corresponding riskless interest
rates as a proxy of the probability distribution of future market risk premia. Based on a bootstrapping approach, the power of the three different estimators on a total return basis and on a
partial return basis is assessed when computing the net present value for typified expected
cash flows of a German CDAX or DAX firm (and of a US S&P 500 firm). While the application of the Cooper estimator leads to lower value bounds, arithmetic means give upper
bounds. However, resulting estimation errors when applying the Cooper estimator are generally lower in absolute terms. Thereby, estimation errors depend to a great extent on the terminal value of future cash flows for years not separately considered. This becomes particular3
ly true for terminal value computations in the case of small time horizons beyond only five to
ten years. Moreover, in such situations, estimation errors are severely influenced by high
growth rates of expected future cash flows. In order to circumvent such problems, future cash
flows, even if not explicitly estimated but only determined via a fixed growth factor, should
be separately discounted for time horizons up to, e.g., 30 years by using the Cooper estimator.
Alternatively, one may completely neglect future cash flows beyond a time horizon of 30
years and then simply apply the arithmetic mean estimator. As the latter one typically leads to
upward biased present value computations, omitting future cash flows from year 31 on adjusts
the estimated firm value in the right direction and gives a good approximation of true overall
firm value. This alternative to an application of the Cooper estimator with explicit terminal
value consideration becomes even more interesting for higher annual growth rates of expected
future cash flow, as this increases estimation risk for terminal values. Section 5 concludes.
2 The basic problem of discount factor estimation
We assume forecasting horizons of n = 1, …, N years. The current price of a share of the firm
under consideration is V0 . Future one-period returns rη for years η−1 to η (η = 1, ..., n) from
are random variables satisfying
the holding of the share and corresponding future prices V
n
the following condition (see, e.g., Fama, 1996, or Stehle, 2004):
= V ⋅ (1 + r ) ⋅ (1 + r ) ⋅ ... ⋅ (1 + r ) .
V
n
0
1
2
n
(1)
Corresponding expectations as seen from t = 0 are
⎤ = V ⋅ E ⎡(1 + r ) ⋅ (1 + r ) ⋅ ... ⋅ (1 + r )⎤ .
E ⎡⎣ V
n⎦
0
1
2
n ⎦
⎣
(2)
In the case of stationarity, i.e. if all returns rη are identically and independently distributed with E ⎡⎣ rη ⎤⎦ = r for η = 1, …, n, equation (2) simplifies to
⎤ = V ⋅ (1 + r )n .
E ⎡⎣ V
n⎦
0
(3)
4
Equation (3) is valid for all n = 1, …, N, so that the future share price in year n can always be determined as the product of all rates of return (adding 1) over the preceding n years.
From (3), we immediately get:
V0 =
⎤
E ⎡⎣ V
n⎦
(1 + r )
n
.
(4)
According to (4), for any future period n, the share price at t = 0 can be computed by discounting the future market value for n periods with the corresponding expected one-period return r as the relevant cost of capital. Following the well-known dividend discount model,
firm values or share prices can also be calculated on the basis of (4) when there are expected
cash flows in the numerator (instead of future market values) and when all discounted expected cash flows of future periods are summed up. These relationships apply accordingly for
portfolios of stocks instead of only single stocks.
In practical valuation problems, however, it is necessary to estimate expected future cash
flows and prices as well as expected future one-period returns in order to apply equation (4).
In what follows, we will focus on the special problems connected with the determination of
the discount factor
d n :=
1
(1 + r ) n
(5)
on the basis of historical return realizations.
To this end, one will typically refer to an estimation of r which can be used in (5) in order to determine dn. In the literature, there are advocates for a computation of r as the arithmetic mean of historical return realizations (see, e.g., Ross et al., 2005, p. 245, and Brealey et
al., 2008, p. 176) as well as authors who favor the geometric mean (see, e.g., Damodaran
2006, p. 98). In this context, the particular bias problems emerging by switching from the estimation of r to an estimation for dn are typically not discussed in detail. The reason for this
shortcoming might be that for known value of r , the discount rate in the denominator of (5)
5
actually is identical to this expected one-period return. However, this identity is not valid any
longer when r in (5) has to be estimated und thus is stochastic in itself.
A necessary condition for the utilization of historical return realizations for estimation
purposes is their stationarity as defined above. Moreover, estimates should be unbiased, i.e.,
estimates should be correct at least on average. Against this background, one argument in favor of the estimation of r as the arithmetic mean of historical returns is that it is an unbiased
estimator with respect to expected future one-period returns in the case of their stationarity.1
In contrast, the geometric mean is systematically downward biased. Nevertheless, we are not
primarily interested in estimators of expected future one-period returns, but – on this basis –
in the derivation of dn via (5). Even in the simple one-period case, neither the arithmetic nor
the geometric mean imply an unbiased estimation of dn or of r in (5) (see Butler and Schachter, 1989, as well as Cooper, 1996). This can be proven by Jensen’s inequality.
In the following, the index t = 1, ..., T denotes T historical return realizations. Despite
these variables being random from an ex-ante perspective, we refrain from labeling them by a
tilde in order to keep notation simple. It is now possible to describe the arithmetic mean ( Â T )
and the geometric one ( Ĝ T ) of one-period returns (increased by 1) in the following way:
 T =
1 T
⋅ ∑ (1 + rt )
T t =1
(6)
and
1/ T
⎡ T
⎤
Ĝ T = ⎢∏ (1 + rt ) ⎥
⎣ t =1
⎦
.
(7)
As there are no comparable biases with respect to the estimation of expected future cash
flows, we simply consider an expected payment of € 1 at the future point in time n (neverthe-
1
However, note that, if we have a multi-period forecasting horizon, there usually is a similar bias problem concerning the estimation of future expected compounding rates with a mean statistic of historical rates of return. In
a multi-period scenario, an unbiased estimation of future expected compounding rates is possible if and only if
the forecasting horizon has exactly the same length as the historical sample. See Blume (1974), Indro and Lee
(1997), Jacquier et al. (2005), and Missiakoulis et. al. (2008).
6
less, see Fama, 1996, for possible relationships between distributional properties of costs of
capital and future cash flows). Assuming D̂ n being an unbiased estimator for the discount
factor dn in order to value expected future payments, the following relationship must hold:
ˆ ⎤=
E ⎡⎣ D
n⎦
( )
ˆ
Since 1/ A
T
n
1
.
(1 + r ) n
(8)
( )
ˆ
and 1/ G
T
n
are convex functions for all n = 1, …, N, Jensen’s inequality
directly implies that both the arithmetic and the geometric mean lead to violations of (8):
⎡
1
ˆ
E AT = 1 + r ⇒ E ⎢
⎢
⎢⎣ Â T
( )
( )
1
⇔ r̂ :=
n
( )
ˆ
According to (9), 1/ A
T
n
⎤
1
⎥>
n ⎥
(1 + r ) n
⎥⎦
⎛
1
E⎜
⎜
⎜ Â T
⎝
( )
n
⎞
⎟
⎟
⎟
⎠
(9)
− 1 < r.
is merely an upward biased estimator for dn. Consequently,
the correspondingly implied estimator r̂ for the discount rate r in (5) is downward biased,
though the arithmetic mean of (6) is, according to the left side of the first line of (9), an unbiased estimator of the expected future one-period return (plus 1). Since the geometric mean
( )
ˆ
lies always below its arithmetic counterpart, (9) applies also for 1/ G
T
n
( ).
ˆ
instead of 1/ A
T
n
The above problem marks the starting point of the analysis by Cooper (1996), who utilizes a second-order Taylor approximation to derive an estimator for dn which is asserted to be
significantly superior to the standard estimators  T and Ĝ T . Cooper (1996) combines  T
und Ĝ T in such a way that there is no bias for his second-order Taylor approximation. Cooper
(1996) arrives at the following estimator Ĉn, T for dn for a sample of T periods:
Ĉ n, T := c n,T ⋅
1
( )
ˆ
A
T
n
+ (1 − c n,T ) ⋅
1
( )
ˆ
G
T
n
(10)
7
with c n,T := (T + n) /(T − 1) . The extent to which the estimators  T and Ĝ T are biased depends on the sample size T and on the forecasting horizon n. The same is true for the weights
of these two standard estimators in (10) and, consequently, for the Cooper estimator itself.
( )
ˆ
The weight of the component 1/ A
T
n
based on the arithmetic mean increases in forecasting
horizon n and decreases in sample size T. Moreover, it should be noted that c n,T > 1 so that
( )
ˆ
the component 1/ G
T
n
based on the geometric mean enters (10) negatively. This kind of
combination of the arithmetic and the geometric mean is a consequence of the fact that both
these standard estimators lead to upward biased net present values. In order to compensate
this problem, the more biased estimator, i.e. the geometric mean, enters (10) with a negative
weight.
Note that (10) still is an approximation for discount factors dn, whereby in particular all
moments of third and higher order are neglected. Therefore, one may not expect the Cooper
estimator to fully resolve the estimation problem sketched in this section. In the following
Section 3, we analyze – based on idealized return probability distributions – whether the
Cooper estimator succeeds in producing more accurate results than the traditional estimators.
3 Bias in the context of idealized return distributions: discount rate analysis
Following a typical approach in the theoretical literature, we assume uncertain rates of return
to be either binomially, normally, or lognormally distributed. Thereby, due to its simplicity,
we will start with a binomial process and present its implications.
Assume that in each period there are two possible rates of return, 38.40 % and −13.60 %
with probability of 0.5 each. This gives us a corresponding expected rate of return of 12.40 %
and a return standard deviation of 26.00 %. Both return moments are in line with estimates
based on arithmetic return means for the CDAX over a time period of 49 years according to
the data presented in Stehle (2004). We will call this situation simply the CDAX case.
8
>>> Insert Table 1 about here <<<
Moreover, we assume a history of ten periods which is identical to the length of our sample for our estimation procedures. For a historical starting stock price of € 100, possible terminal wealth levels evolve according to the results presented in Table 1. Corresponding
arithmetic and geometric return means are displayed in Table 1 as well. The expected arithmetic mean amounts to 12.40 %, while the expectation value of the geometric mean is only
9.66 %. As already pointed out in Section 2, even an unbiased estimator of the expected oneperiod return does not guarantee an unbiased estimation of the corresponding discount rate
and thus of the resulting net present value of expected future cash flows. To highlight this
problem in Table 1, we display resulting net present values of an expected payoff of € 100 after n = 10 future periods when discounted with the arithmetic or the geometric return mean.
An unbiased estimator should lead to a net present value of 100/1.124010 = € 31.07. However,
according to Table 1, expected net present values for computations on the basis of the arithmetic mean or the geometric mean amount to € 42.33 or € 53.85, respectively, indicating the
bias problem described in Section 2. In turn, we receive average discount rates (i.e. expected
estimators for r in (5)) ranging from (100/42.33)0,1−1 = 8.98 % (instead of the true value of
12.40 %) to (100/53.85)0,1−1 = 6.39 % per period. The expected estimation of the net present
value using the Cooper estimator equals € 28.26 and is thus nearer to the true value of € 31.07
than the outcomes of the two other approaches. Therefore, the corresponding one-period discount rate, 13.47 %, is relatively close to the true value of 12.40 %.
However, Figure 1 reveals that with increasing forecasting horizon the average error of
the Cooper estimator increases progressively, while the bias of the two mean estimators only
grows (approximately) “linearly”. Following the break-even point, the Cooper estimator ends
up being inferior to the two mean estimators. Not surprisingly, with growing sample length all
9
three estimation approaches lead to better results with the Cooper estimator being most favored by longer sample periods.
>>> Insert Figure 1 about here <<<
The problem of the Cooper estimator apparently is caused by too low a weight of the
geometric mean in the case of long-term forecasting horizons (compared to the sample period
length). Linearly in n growing absolute weights of the geometric mean discount rate estimator
imply progressively increasing underestimations of the true discount factor.
In order to identify critical influential factors for the goodness of estimation outcomes,
Table 2 presents the results of ceteris paribus variations of several parameters. In this context,
(for equally probable two return outcomes) we choose an expectation value of one-period return of 12.50 % and a corresponding return standard deviation of 25.00 % with a sample
length of 50 periods as the base case (which thus is approximately identical to the CDAX
case) and then adjust the two possible return outcomes in order to vary in a certain way
(1) the return standard deviation for given expected return of 12.50 % and given 50:50 chance
of both possible return outcomes,
(2) the expected return for given return standard deviation of 25.00 % and given 50:50 chance
of both possible return outcomes, and
(3) the probability of the “good” return outcome for given return expectation of 12.50 % and
given return standard deviation of 25.00 %.
>>> Insert Table 2 about here <<<
Apparently, variations of the return standard deviation have a crucial impact on the goodness of estimation results. While for n = 30 and a return standard deviation of 25 %, i.e. near
to the CDAX case, the arithmetic mean estimator of the true expected rate of return of 12.5 %
is 1.7581 percentage points too low and the geometric one is even 4.4757 percentage points
10
too small, the Cooper estimator is 2.7828 percentage points too high. For a return standard
deviation of 27.50 %, however, corresponding errors are −2.1370 percentage points and
−5.4237 percentage points for the arithmetic and the geometric mean estimator, respectively,
while the Cooper estimator is even 8.3515 percentage points too high. With expected future
one-period returns of 12.96 % and a return standard deviation of 27.98 % based on historical
DAX data (the “DAX case”, henceforth), we may immediately conclude that estimation problems for the DAX case are more pronounced than for the CDAX case. For the S&P 500, we
have a historically estimated expected one-period return of 11.69 % and a return standard deviation of only 19.80 % (the “S&P 500 case”, henceforth). Following Table 2, a return standard
deviation of 20 % for given expected return of 12.5 % implies arithmetic and geometric mean
return estimators for n = 30 to be only 1.1167 or 2.8572 percentage points too small, while the
Cooper return estimator is merely 0.6851 percentage points too high. Moreover, estimation
problems for the S&P 500 case are even more reduced (though to a limited degree) due to the
accessibility of a sample length of about 80 periods. To be more precise, for the S&P 500 case
with T = 75 and n = 30, we get respective estimation errors of −1.1023, −2.8205, and 0.6701
percentage points (for the CDAX case with T = 75 instead of T = 50 see Table Ad1, available
from the authors upon request). Apparently, estimation problems are not a critical issue for
the S&P 500 case, while the problems may be more severe for the CDAX case and the DAX
case. In what follows, we therefore focus primarily on the German capital market data.
According to the second part of Table 2, a ceteris paribus increase of expected returns reduces estimation problems. However, the relevance of such parameter variations seems considerable lower than that of variations of the return standard deviation. Only for long forecasting horizons (see n = 30) in connection with the Cooper estimator a ceteris paribus increase of
one-period expected returns reduces the estimation problem significantly.
In contrast to these first two influential factors, variations of the probability of the high
return realization do not exhibit a monotone impact on estimation results. At any rate, the re11
levance of probability variations is – at most – of a comparable size like that of variations of
expected one-period return. In general, estimation results based on the arithmetic mean are
best for rather high probabilities of the good return outcome, while low probabilities of the
good return outcome favor the Cooper estimator.
In order to analyze whether the above results are a consequence of the underlying return
distribution or if similar results may be obtained for alternative distributional assumptions, we
test normal and lognormal one-period return distributions for the CDAX case with a sample
length of 50 periods, simulating once again the length of the available German time series
(see Table 3). Computations are based on Monte Carlo simulations with 10,000 runs, as – because of the assumption of continuous return distributions – it is no longer possible to offer a
complete reproduction of all conceivable future price developments. According to Table 3,
qualitative results are the same for all three probability distributions: Estimation results become worse for longer forecasting horizons with the geometric mean being by far the least
precise estimator. Moreover, the Cooper estimator is superior to both “conventional” estimation approaches unless very long forecasting horizons (of about 20 to 25 periods and more)
are considered.
>>> Insert Table 3 about here <<<
Table 3 indicates that estimation problems for the arithmetic mean estimator are almost
independent of the specific return distribution while, with respect to the geometric mean and
the Cooper estimator, they are least severe for lognormal return distributions and are worst for
normally distributed one-period returns. Taking the numerical results of Tables 2 and 3 together, the goodness of these estimation results seems to be influenced in the first place by the
relevance of extreme return realizations. Moreover, extremely small return realizations seem
to affect estimation performance stronger than extremely high return realizations, as is implied by the positive ceteris paribus influence of a higher expected one-period return moment.
12
Against this background, it is obvious that binomially distributed one-period returns lead to
better estimation results than in a situation with normal return distribution, as in the former
case there are only two possible one-period return realizations. Although lognormally distributed one-period returns imply extreme return realizations as well, they offer the advantage of
effectively avoiding the more relevant outliers at the lower end of the probability distribution.
This circumstance may explain the quite good performance of estimation procedures in situations with lognormal one-period return distributions.
The rather poor performance of the Cooper estimator for long forecasting horizons roots
in the quite strong impact of some extreme outliers on the corresponding average estimation
power. This is highlighted in more detail in Figures 2 and 3 which present the distribution of
realized net present values of expected future cash flows of € 1 at time n, i.e. the estimates D̂ n
of discount factors, on the basis of the different estimators depending on the various paths of
return realizations and corresponding estimation results for the arithmetic and the geometric
mean. Thereby, we refer to a situation with normally distributed one-period returns for a sample length of 50 periods in the CDAX case with a forecasting horizon of 10 or 30 periods, respectively.
>>> Insert Figures 2 and 3 about here <<<
As a consequence of the hyperbolic character of discounting, even in the case of almost
symmetrically distributed results for the arithmetic and the geometric mean, corresponding
net present value estimations exhibit a strong positive skewness emphasizing the importance
of outliers. Moreover, these figures highlight the underlying logic of the Cooper estimator.
This estimator utilizes the more upward biased realizations of the geometric mean discount
factor estimator as a correction factor by subtracting it from the (less but also upward biased)
arithmetic mean estimator. This adjustment reduces the skewness of the resulting estimates.
While effectively working for a forecasting horizon of 10 periods (see Table 4), in the case of
13
30 periods forecasting horizon, the Cooper estimator is inferior to the simple arithmetic mean
and leads to an economically meaningless negative average discount factor of –0.0009 (for
sample length T = 50). The reason for this shortcoming of the Cooper estimator lies in an
overcorrection of the simple arithmetic mean. In particular, the adjustment by the geometric
mean produces net present value distributions which are negatively skewed implying outliers
at the lower tail instead of the upper tail (see Figure 3 in connection with Table 4). In addition
to this unfavorable skewness effect, the kurtosis of the net present value estimators may also
become extremely high when using the Cooper estimator. Certainly, the quality of secondorder Taylor approximations is adversely affected by these effects.
>>> Insert Table 4 about here <<<
A closer analysis of the distributions of the resulting net present values on the basis of the
arithmetic mean and of the geometric mean reveals that the reason for this failure of the
Cooper estimator is in particular rooted in situations with relatively high arithmetic means accompanied by simultaneously low geometric means. For example, for a situation with 50 periods sample length and normally distributed one-period returns, a maximum biased Cooper
discount factor estimator of −2.2994 results for a forecasting horizon of 30 periods with the
geometric mean being minimal with a value of −4.49 % and the corresponding arithmetic
mean amounting to 7.05 %. In contrast, situations with low values of the arithmetic return
mean, i.e. high values of the corresponding discount factor, do not lead to such extremely
poor estimations based on the Cooper approach. In such a situation, the subtraction of a correction factor based on the geometric mean effectively reduces the bias from the arithmetic
mean. Summarizing, the high sensitivity of the quality of the Cooper estimator with respect to
variations of return standard deviations seems to be a consequence of greater possible differences between arithmetic mean estimators and their geometric counterparts.
14
In fact, these findings hint at a simple method to improve the power especially of the
Cooper estimator. In practical applications, it is hard to conceive that decision-makers will
make use of economically meaningless estimations D̂ n of discount factors dn being smaller
than zero or greater than one. An easy way to improve the three estimators under consideration is thus to allow for the additional restriction of discount factors being acceptable only for
values between 0 and 1 so that estimators beyond these boundaries are replaced by the otherwise violated upper or lower bound. In Figure 4, we reconsider the CDAX (binomial) case
with the additional restriction of discount factors lying in the interval [0, 1] (corresponding results are valid for the CDAX case with normally or lognormally distributed one-period returns; see Figures Ad1 to Ad4, available from the authors upon request). Apparently, this
simple modification effectively avoids extreme average results for the Cooper estimator, in
particular for small sample lengths, while the arithmetic or geometric mean estimators on average do not benefit very much from this adjustment.
>>> Insert Figure 4 about here <<<
4 Bias in the context of empirical return distributions: net present value analysis
After the analysis of some idealized return distributions in the preceding section, we now turn
to a closer examination for the case of empirically observable return distributions especially
in Germany, as here estimation biases seem to be more relevant due to a smaller sample
length. To be more precise, we primarily rely on yearly return realizations for the DAX and
the CDAX over the time period from 1955 to 2003 as presented in Stehle (2004). The utilization of this return data seems adequate if one assumes stationary return distributions. Alternatively, one may assume market risk premia to be stationary and thus try to estimate expected
future market risk premia on the basis of historical excess returns. After having added the relevant term structure of riskless interest rates as of the date of the evaluation, we again would
receive estimators for discount rates. The latter procedure may be interpreted as a simplified
15
application of the CAPM for a beta factor of just one. Both approaches lead to different valuation results unless the term structure of riskless interest rates is flat. Our estimations of market
risk premia also rely on the data of Stehle (2004) in order to calculate differences between
CDAX or DAX returns on the one side and REXP returns on the other. The REXP is a performance index reflecting the return of a synthetic portfolio of 30 idealized German government bonds with maturities ranging from 1 up to 10 years. Moreover, the current term structure for a time horizon up to N = 30 years is computed with the help of the Svensson method
on the basis of data from the Deutsche Bundesbank (see Svensson, 1994).
We make use of an ADF-test as well as a KPSS-test to check whether the time series given in our index samples are stationary. With respect to all time series of returns and market
risk premia, both tests confirmed that the time series are stationary on a significance level of 1
% or even better. Thus, we examine both estimation approaches, although from an economic
point of view the estimation of market risk premia should generally be preferable, as this approach offers the opportunity to allow for non-flat term structures of interest rates in Germany
and it can be used as a starting point for the application of the CAPM for firm valuation (after
determining the beta factor of a certain firm or peer group).
We perform a bootstrapping approach, by making, once again, use of Monte Carlo simulations under the assumption that historical return realizations correctly describe the probability distribution of future returns. To do so, we randomly determine 10,000 paths of return realizations each with a length of 49 periods. For each selected return path, we compute all three
estimators under consideration in this paper: the arithmetic mean, the geometric mean and the
Cooper estimator. According to our findings of the previous section, we do so with the additional restriction of discount factors being in the interval [0, 1]. Resulting estimators for discount rates as well as market risk premia in comparison to the “true” values are presented in
Table 5. For simulations that refer to historical excess returns, we simply assume current riskless interest rates to be 0 %. As we only want to assess the power of estimates, the current
16
term structure of riskless interest rates would only be of secondary importance for the analysis
of computations based on partial return considerations.
At first glance, it might be surprising that the “true” discount rates in Table 5 are not exactly the same for all forecasting horizons. This is simply a consequence of the approximative
character of the Monte Carlo simulation. Even 10,000 runs are not sufficient for a perfect replication of the assumed return distribution. However, this lack of precision is of negligible relevance and, at any rate, does not affect the analysis of the consequences of the three possible
estimation methods, as they are assessed in relation to the results of the Monte Carlo simulation.
>>> Insert Table 5 about here <<<
In principle, there are two possible ways to derive discount rates from estimations of
market risk premia. On the one hand, one may utilize arithmetic and geometric means of historical market risk premia in order to employ formula (10) to derive the corresponding Cooper
estimator and eventually add the current term structure of interest rates to this Cooper estimator of market risk premium. On the other hand, one may first add the relevant current term
structure of riskless interest rates to the estimators for the market risk premia as derived from
arithmetic and geometric mean estimators and then apply the Cooper formula according to
(10). However, with respect to Table 5, we simply assume all current riskless interest rates to
be zero so that there is no difference between both approaches. Nevertheless, for the more
general case with current riskless interest rates being not all identical to zero, we recommend
only the first approach, as this offers the advantage of a better distinction between the unknown market risk premium component and the known term structure component. Moreover,
this is, for example, helpful when applying the CAPM (for situations with a beta factor not
being equal to one). Interestingly, the restriction to discount factor estimates in the range of 0
to 1 is of much higher importance in the partial return situation than in the total return situa17
tion. The reason simply is that market risk premia are systematically smaller than total returns
so that a binding upper boundary of 1 for the corresponding discount factors is ceteris paribus
more frequently implied. According to Table 5, on a total return basis, the Cooper estimator is
advantageous only for forecasting horizons up to about 25 years as in Section 3, while, on a
partial return basis, the Cooper estimator seems to be superior for almost all forecasting horizons.
In any case, we now want to compute corresponding firm values for our different estimation approaches. Generally speaking, we assume that expected cash flows cf n at future times
n up to a forecast horizon N1 are explicitly estimated by decision-makers. From N1 on, expected cash flows are assumed to grow at a constant growth rate g per period. Moreover, up to
a forecast horizon of N2 ≥ N1, one period discount rates r̂n from time n−1 to n may be explicitly estimated. From N2 on, one-period discount rates are assumed to be identical to the last
one which has been explicitly estimated, i.e. r̂N2 . Against this background, firm value is computed as:
N1
V=∑
n =1
cf n
n
∏ (1 + rˆ )
η=1
η
+
N2
∑
n = N1 +1
cf N1 ⋅ (1 + g) n − N1
n
∏ (1 + rˆ )
η=1
η
+
cf N1 ⋅ (1 + g) N 2 − N1 +1
N2
(rˆN2 − g) ⋅ ∏ (1 + rˆη )
⋅
(11)
η=1
We start with the simple case of a firm with a constant expected cash flow of € 1 bn. per period from time n = 1 on until N1 = 30. Beyond time 30, all expected cash flows are identical to
zero, i.e., we have g = −1, so that only the first summand on the right side of equation (11) is
relevant. The market value of this firm is to be determined for both the DAX case and the
CDAX case and on both a total return basis and a partial return basis (with the term structure
of riskless interest rates as of 31st December, 2003). This gives us four different scenarios for
which we compute five different average net present values, each. Each average net present
value is based on 10,000 runs of our Monte Carlo simulation. For each run, we use the simulated history of (excess) returns to compute the corresponding arithmetic mean, the corres18
ponding geometric mean, and the Cooper estimator. Based on these estimators, for each
(excess) return path, five net present values are calculated. The first one follows from a direct
application of the arithmetic mean estimator, the second one is a consequence of applying the
geometric mean estimator. The third net present value is computed based on the Cooper estimator. The fourth one uses the Cooper estimator only for the first twenty years and then
switches to the arithmetic mean estimator. The fifth calculation differs from the fourth by
postponing the respective switch to the end of year 25. Both combined estimation procedures
are motivated by our findings of the previous section and of Table 5. Once again, we restrict
admissible discount factors to the interval [0, 1].
Overall, this gives us five different calculations for each return path for which we calculate average net present values across all 10,000 simulation runs (see Table 6). Moreover, we
calculate the “true” net present value (under the assumption that the historical return distribution is identical to the true probability distribution as simulated by the bootstrapping approach). This enables us to compute average estimation errors for all estimation procedures
under consideration. We simply rely on average estimation errors, as we believe that practitioners are most interested in valuation procedures that produce on average highly precise results (for a general discussion of measuring estimation errors see, e.g., Ditmann and Maug,
2008). While the left columns of Table 6 only refer to firm cash flows over the first 30 years,
in the right columns, we also have incorporated as a second case the average net present values of expected cash flows beyond year 30. Thereby, for each of the 10,000 simulation runs,
expected payments after year 30 are evaluated under the assumption of constant costs of capital from year 30 on and on the basis of the estimation result calculated for year 30. This
means, for the second case we apply equation (11) with g = 0 and N1 = N 2 so that the second
summand on the right side of (11) vanishes.
>>> Insert Table 6 about here <<<
19
Let us first take a look at the left columns of Table 6. The results of the upper lines for estimations in the CDAX and the DAX case on a total return basis confirm our previous findings. First of all, the geometric mean obviously is not suited for the estimation of firm values,
as errors amount to 36.02 % (CDAX) or 41.93 % (DAX), respectively. The arithmetic mean
overestimates the true net present value by 6.70 % (CDAX) or 7.41 % (DAX), while the
Cooper estimator falls short of the true value by −3.30 % (CDAX) or −4.24 % (DAX), respectively. In the case of an unconditional decision among the three different estimation procedures therefore, the application of the Cooper estimator is preferable. Depending on the issue
whether under- or overvaluation is more important to be avoided, one may also decide to employ the arithmetic mean. However, a combination of the Cooper estimator (for the first twenty or twenty-five years) and the arithmetic mean estimator (for the following years) leads to
better results. In the first case, the average estimation error is reduced to 1.37 % (CDAX) or
1.14 % (DAX). In the latter case, estimation errors decrease to −0.97 % (CDAX) or −1.59 %
(DAX). Up to now, we would recommend to employ the Cooper estimator at first, and to
switch to the arithmetic mean for forecasting periods beyond 20 to 25 years.
The determination of typified firm values in Table 6 is based on net present values which
have been computed by discounting expected future cash flows with discount rates estimated
by the various methods under consideration. Thereby, a Monte Carlo simulation is utilized to
model the random character of the sample. In practical applications, however, there is only
one time series of historical return distribution given. In the DAX case and the CDAX case,
this is the data of Stehle (2004). If we now apply the arithmetic mean estimator to this time
series of return or the Cooper estimator using the 31st of December 2003 as the relevant valuation date, we get estimations for firm values that are between € 7,215,880,077 (Cooper) and
€ 8,064,516,129 (arithmetic mean) for the CDAX case and € 6,812,401,234 (Cooper) and
€ 7,716,049,383 (arithmetic mean) for the DAX case. Apparently, the range of possible firm
values is almost € 1 bn. which emphasizes the relevance of adequate estimation procedures
20
even if expected future cash flows are unambiguously given. When applying a combination of
the Cooper estimation and the arithmetic mean with a switch after 25 years, we get firm values of € 7,542,646,351 (CDAX case) or € 7,139,026,392 (DAX case).
In general, our qualitative findings for a total return basis are confirmed for estimates on
a partial return basis. To be more precise, we get average estimation errors of 28.97 %
(CDAX case) or 33.46 % (DAX case) for the geometric mean. The corresponding results for
the arithmetic mean are 5.30 % (CDAX case) and 5.89 % (DAX case) and for the Cooper estimator −2.84 % (CDAX case) and −3.47 % (DAX case). By applying the Cooper estimator
only to the first 20 or 25 years and then switching to the arithmetic mean, the average estimation error amounts to 0.85 % or −1.03 % (CDAX case) and 0.69 % or −1.48 % (DAX case),
respectively. The combination of an upward biased return estimator with a downward biased
one thus proves advantageous even on a partial return basis.
Without simulation, net present value estimates as of 12/31/2003 now range from
€ 8,480,449,074 (Cooper estimator) to € 9,851,908,938 (arithmetic mean) for the CDAX case
and from € 7,953,585,795 to € 9,357,560,423 for the DAX case. Actually, we now receive
differences of more than € 1.3 bn. As a point estimator when utilizing the Cooper approach
for the first 25 years and ever after applying the arithmetic mean we get € 9,097,573,942
(CDAX case) and € 8,532,839,806 respectively (DAX case).
Interestingly, when, for example, comparing the results for applying the Cooper estimator
for the first 25 years and ever after utilizing the arithmetic mean, the net present value point
estimators on a partial return basis are considerably greater than the corresponding ones on a
total return basis (+20.62 % for the CDAX case, +19.52 % for the DAX case). Although both
approaches may be justified in the light of statistical significance tests, economic as well as
practical reasons suggest an application of estimations on a partial return basis (because of the
possibility to take the current term structure of interest rates into account and in order to apply
the CAPM).
21
Up to now, we have been silent on tax considerations, although they are of great practical
importance. Certainly, in the above example, cash flows could also be understood as determined after corporate and personal income taxes. Therefore, the tax issue focuses on the consequences of tax considerations on resulting discount rates. The lower part of Table 6, thus,
presents additionally all corresponding results when applying a personal income tax rate of
35 % on interest payments and 0 % on capital gains, as is assumed to be typical for private
German investors as of 12/31/2003 (see Breuer et al., 2007, for more details on German tax
rules). Qualitative results are identical to pre-tax considerations. Moreover, for German tax
laws at the end of 2003, firm values could only be computed on a partial return basis and not
on a total return basis which emphasizes the practical relevance of the former approach.
Finally, the last two lines of Table 6 present pre-tax outcomes for the S&P 500 case as a
further benchmark. Thereby, we approximate riskless interest rates in the US by the interest
rates of the State and Local Government Series (SLGS) securities as of 12/31/2007 (see
https://www.treasurydirect.gov/SZ/SPESDailyRate?requestType=R). Obviously, now the
“simple” Cooper estimator is superior to all other estimation alternatives which confirms our
finding of the previous section that for the US there is no severe estimation problem. In contrast, up to now, for the German case, we would recommend a combination of the arithmetic
mean and the Cooper estimator.
However, due to the hyperbolic character of discounting, things change when explicitly
taking expected payments after year 30 into account, as is depicted by the right columns of
Table 6. In particular, biases when applying the arithmetic or the geometric mean estimator
become considerably larger. At first glance, somewhat surprisingly, as the Cooper estimator
tends to underestimate the true net present value, this additional bias in the traditional estimators may lead to even better estimation results for the Cooper estimator. In fact, not only for
the US, but also for Germany, now the Cooper estimator becomes the most preferable estimation approach. As after-tax considerations with infinite time horizon are practically most im22
portant, we arrive at the recommendation of applying the Cooper estimator for both the DAX
and the CDAX case. Estimation errors are then as low as −1.23 % (CDAX case) or −2.35 %
(DAX case) and not only clearly superior to an application of the geometric mean estimator,
but also to a utilization of the simple arithmetic one (or combinations with this estimator).
As indicated by our numerical analysis so far, the influence of the terminal value of future cash flows beyond year 30 on the severity of the estimation problem seems to be rather
great. In fact, as this issue leads to too high arithmetic mean and geometric mean estimators
with the latter one being relatively more increased and thus biased than the former one, this
generally improves the goodness of the Cooper estimator. However, there would be another
possibility to tackle this issue by simply neglecting the terminal value for future cash flows
from year 31 on, i.e. the last summand in equation (11). Apparently, this renders firm value
estimators for the situation with infinite time horizon ceteris paribus too low, but as in particular the arithmetic mean is upward biased, the overall effect of this simplification may be positive. In fact, when doing so, in the right columns, estimation errors for the arithmetic mean for
the cases 1 to 8 of Table 6 are reduced to 3.36 %, 4.51 % (CDAX and DAX on a total return
basis, before taxes), 0.69 %, 1.64 % (CDAX and DAX on a partial return basis, before taxes),
1.01 %, 1.91 % (CDAX and DAX on a partial return basis, after taxes), −0.19 %, and 0.10 %
(S&P 500, total or partial return basis, before taxes). Apparently, the arithmetic mean estimator may be justified for determining total firm value when arbitrarily neglecting future cash
flows beyond year 30 in order to compensate for the upward bias of this estimator.
As can be seen by a comparison of the last two columns of Figure 5 (please be aware of
the differences in scale in the various graphs!) for the CDAX case on a partial return basis
with and without taxes and for the S&P 500 case on a partial return basis before taxes (the
same holds true for the other cases), in particular, the arithmetic mean estimator for future
cash flow discounting without terminal value consideration does not react very sensitively to
variations of the assumed annual growth rate of future cash flows and thus may prove supe23
rior even to an application of the Cooper estimator with explicit terminal value calculation for
sufficiently high annual growth rates of expected future cash flows. Thereby, geometric mean
estimates are depicted by the upper (solid) lines, while arithmetic mean estimates are described by the middle (dotted) lines and Cooper estimates by the lower (dashed) lines.
>>> Insert Figure 5 about here <<<
However, in practical applications, explicit future cash flow estimations seldom reach
beyond 10 years. In fact, in most cases, they are restricted to three to five future years.
Beyond these time horizons, a constant annual growth rate of future cash flows is typically assumed and there is just one terminal value computed for the last year of explicit recognition
which then is discounted to get its present value. Against the background of our results so far,
this specific situation should lead to ceteris paribus more pronounced estimation problems.
This conjecture is confirmed by Figure 5 as well. In the first two columns of Figure 5, on a
partial return basis, we assume detailed cash flow planning for five or ten years for the CDAX
case before and after taxes and for the S&P 500 case before taxes. After this detailed planning
period, terminal values are calculated for a specific growth rate g between 0 % and 2 % from
time 5 or 10 on (with cost of capital of time 5 or 10, respectively, assumed to be constant for
all future periods) and are then discounted to time 0. For the detailed planning period, we assume the same growth rate as for the computation of the terminal value. Referring to equation
(11), we thus have N1 = 1 and N2 = 5 or N2 = 10, respectively. In this regard, in the third column, we also have N1 = 1, but N2 = 30. The same is true for the fourth column, but in addition
the last summand of equation (11) is arbitrarily neglected, although we assume that expected
cash flows for n greater than 30 are the same as for the computations underlying the third column of Figure 5. Computations for the first two columns and the last two columns thus differ
with respect to the length of the explicit consideration of future cash flows (5, 10, or 30 years,
in the latter case with a computation of the terminal value of later cash flows or not). Moreo24
ver, in order to assure a “fair” comparison, for the first two columns, we assume actual cost of
capital per period to be constant from year 5 or year 10 on, respectively, while for the last two
columns this is assumed to hold from year 30 on.
Despite these rather small differences in the four approaches, estimation errors for the
first two calculations are considerably greater and, moreover, react quite sensitively to increased annual growth rates. The reason is simply the relatively higher importance of the terminal value computation for short detailed planning periods and its great dependence on the
discount rate estimator due to the hyperbolic influence of the latter. Estimation errors for the
first two columns certainly would be even greater, if we assumed that corresponding actual
costs of capital per period would not be constant until year 30. Apparently, a quite
straightforward way to improve the ex ante performance of estimation procedures is to extend
the time period of separately computed cash flow discounts: Even if practitioners only plan in
detail cash flows for up to five to ten years, one should explicitly consider cash flows for
years 6 to 30 with constant growth rates and then compute the terminal value only as from
year 31 on. With respect to equation (11), this means that even for small values of N1, e.g., N1
= 5, one should not choose N2 = N1, but better compute (11) for N2 = 30 and thereby rely on
the Cooper estimator. Moreover, as has already been mentioned, in addition to choose N2 = 30
even for small values of N1, it may pay (in particular for high annual growth rates of expected
future cash flows) to neglect the last summand in (11) completely and then to rely on the
arithmetic mean estimator.
5 Conclusion
The aim of this paper was to address the issue of estimating adequate discount rates for expected future cash flows in firm valuations. Even in situations with independently and identically distributed historical rates of stock return or risk premia, an unbiased estimation of future discount rates as cost of capital is not possible. Arithmetic and geometric mean estima-
25
tors lead on average to an underestimation of the correct discount factor, whereby the bias of
the geometric mean is considerably larger. In 1996, Cooper suggested a combination of these
two mean estimators as a new estimation procedure.
Rather interestingly, due to the hyperbolic character of discounting, the goodness of estimation procedures is particularly depending on whether or not one restricts himself to time
horizons up to, e.g., 30 years. In the latter (more practically relevant) situation, estimation
quality deteriorates considerably for the arithmetic mean and the geometric mean estimator
and makes the Cooper estimator comparatively more attractive. In fact, the Cooper estimator
produces quite satisfactory results for German and US capital market data for annual growth
rates of expected future cash flows being not too high. Since arithmetic mean estimators of
firm values are upward biased, a possible alternative to referring to the Cooper estimator is an
application of the arithmetic mean estimator with – at the same time – completely neglecting
the terminal value of future cash flows beyond 30 years. This alternative becomes more interesting for higher annual growth rates of expected future cash flow, as this increases estimation
risk for terminal values. In any case, one should not follow typical practical approaches which
compute terminal values after future time periods as short as five to ten future years, because
such terminal value estimations are extremely prone to estimation errors. Even if only future
cash flows for five years are explicitly estimated, future cash flows for years 6 to 30 should be
separately discounted.
Regarding the choice between estimating discount rates on the basis of historical total returns and on the basis of historical excess returns which then in turn are combined with the
current term structure of riskless interest rates in order to derive adequate discount rates, we
recommend the latter approach, as this allows us to employ the current term structure and the
CAPM to derive adequate costs of capital. Moreover, it reduces the sensitivity to estimation
risk with respect to discount rates, as very low values near zero are effectively avoided by the
utilization of the current term structure of riskless interest rates.
26
References
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27
CDAX case: Binom ial distribution
(T = 10 periods )
70%
60%
50%
40%
30%
20%
10%
0%
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
(T = 25 periods)
30%
25%
20%
15%
10%
5%
0%
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
(T = 50 periods)
20%
15%
10%
5%
0%
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
(T = 75 periods)
20%
15%
10%
5%
0%
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Forecasting horizon
Expected value
Figure 1:
Arithmetic mean
Geometric mean
Cooper estimator
Average estimated discount rates for binomially distributed one-period returns in the
CDAX case (expected return: 12.40 %, return standard deviation: 26.00 %) for sample
lengths of T = 10, 25, 50, and 75 periods
Average estimated discount rates according to the arithmetic mean estimator, the geometric mean estimator, and the corresponding Cooper estimator for binomially distributed one-period returns in the CDAX case (i.e. for a one-period expected rate
of return of 12.40 % and corresponding return standard deviation of 26.00 %) for sample lengths of T = 10, 25, 50, and 75
periods are presented as a function of forecasting horizon n = 1, …, 30. “Expected value” stands for the “true” discount rate
which would result in the case of knowing the true return distribution.
28
Discount factor estim ations: arithm etic m ean
(T = 50)
0.05
0.04
0.03
0.02
0.01
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.0
1.1
1.2
1.3
1.0
1.1
1.2
1.3
n = 10
Discount factor estim ations: geom etric m ean
(T = 50)
0.05
0.04
0.03
0.02
0.01
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
n = 10
Discount factor estim ations: Cooper
(T = 50)
0.05
0.04
0.03
0.02
0.01
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
n = 10
Figure 2:
Distribution of discount factor estimations as derived by a Monte Carlo simulation for the
three estimation methods in the CDAX case with normally distributed one-period returns
(sample length T = 50 periods and forecasting horizon n = 10 periods)
Based on a Monte Carlo simulation with 10,000 runs assuming normally distributed one-period returns according to the CDAX case (i.e. expected one-period return of 12.40 % and corresponding return standard deviation of
26.00 %), for a sample length T = 50 and a forecasting horizon of n = 10, the distribution of discount factor estimates according to the arithmetic mean estimator, the geometric mean estimator, and the corresponding Cooper
estimator are computed.
29
Discount factor estim ations: arithm etic m ean
(T = 50)
0.30
0.25
0.20
0.15
0.10
0.05
0.00
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.4
0.5
0.4
0.5
n = 30
Discount factor estim ations: geom etric m ean
(T = 50)
0.30
0.25
0.20
0.15
0.10
0.05
0.00
-0.2
-0.1
0.0
0.1
0.2
0.3
n = 30
Discount factor estim ations: Cooper
(T = 50)
0.30
0.25
0.20
0.15
0.10
0.05
0.00
-0.2
-0.1
0.0
0.1
0.2
0.3
n = 30
Figure 3:
Distribution of discount factor estimations as derived by a Monte Carlo simulation for the
three estimation methods in the CDAX case with normally distributed one-period returns
(sample length T = 50 periods and forecasting horizon n = 30 periods)
Based on a Monte Carlo simulation with 10,000 runs assuming normally distributed one-period returns according to the CDAX case (i.e. expected one-period return of 12.40 % and corresponding return standard deviation of
26.00 %), for a sample length T = 50 and a forecasting horizon of n = 30, the distribution of discount factor estimates according to the arithmetic mean estimator, the geometric mean estimator, and the corresponding Cooper
estimator are computed.
30
CDAX case without extreme results: Binomial distribution
(T = 10 periods)
20%
15%
10%
5%
0%
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
(T = 25 periods)
20%
15%
10%
5%
0%
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
(T = 50 periods)
20%
15%
10%
5%
0%
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
(T = 75 periods)
20%
15%
10%
5%
0%
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Forecasting horizon
Expected value
Figure 4:
Arithmetic mean
Geometric mean
Cooper estimator
Average estimated discount rates for binomially distributed one-period returns in the CDAX case
(expected return: 12.40 %; return standard deviation: 26.00 %) when discount factor proxies
smaller than 0 and greater than 1 are substituted by 0 and 1, respectively (sample lengths of T = 10,
25, 50, and 75 periods)
Average estimated discount rates according to the arithmetic mean estimator, the geometric mean estimator, and the corresponding Cooper estimator for binomially distributed one-period returns in the CDAX case (i.e. for a one-period expected rate
of return of 12.40 % and corresponding return standard deviation of 26.00 %) for sample lengths of T = 10, 25, 50, and 75
periods are presented as a function of forecasting horizon n = 1, …, 30. Thereby, discount factor proxies smaller than 0 and
greater than 1 are substituted by 0 and 1, respectively. “Expected value” stands for the “true” discount rate which would result in the case of knowing the true return distribution.
31
CDAX pre-tax: 5 years planning period
CDAX pre-tax: 10 years planning period
140%
120%
CDAX pre-tax: 30 years plan. per., not truncated
70%
25%
80%
60%
20%
50%
15%
40%
10%
30%
5%
70%
100%
60%
80%
50%
40%
60%
20%
0%
10%
-5%
10%
0%
-10%
0%
-10%
30%
40%
20%
20%
0%
0.000
0.005
0.010
0.015
0.020
CDAX after-tax: 5 years planning period
CDAX pre-tax: 30 years plan. per., truncated
90%
0.000
0.005
0.010
0.015
CDAX after-tax: 10 years planning period
-15%
0.000
0.020
0.005
0.010
0.015
0.020
CDAX after-tax: 30 years plan. per., not truncated
0.000
400%
180%
140%
30%
350%
160%
120%
25%
140%
300%
250%
200%
150%
100%
20%
0%
0%
0%
-20%
0.005
0.010
0.015
0.020
S&P 500 pre-tax: 5 years planning period
0%
20%
40%
50%
0.000
5%
40%
60%
0.000
0.005
0.010
0.015
S&P 500 pre-tax: 10 years planning period
-5%
-10%
-15%
0.000
0.020
0.005
0.010
0.015
0.020
S&P 500 pre-tax: 30 years plan. per., not truncated
0.000
0.005
0.010
0.015
0.020
S&P 500 pre-tax: 30 years plan. per., truncated
70%
60%
60%
20%
60%
50%
50%
15%
40%
40%
10%
30%
30%
5%
20%
20%
0%
10%
10%
-5%
50%
0.020
10%
60%
80%
0.015
15%
80%
100%
0.010
20%
100%
120%
0.005
CDAX after-tax: 30 years plan. per., truncated
40%
30%
20%
10%
0%
0%
0.000
0.005
0.010
0.015
0.020
0%
0.000
0.005
0.010
0.015
0.020
-10%
0.000
0.005
0.010
0.015
0.020
0.000
0.005
0.010
0.015
0.020
Figure 5: Valuation error [%] in dependence of annual growth rate of expected cash flows
Relative errors in firm value estimation on a partial return basis (vertical axis) are depicted as a function of annual growth rates of expected future cash flows between 0 % and 2 % (horizontal axis)
for different lengths of detailed cash flow planning periods (5, 10, and 30 years). “Truncation” denotes a situation where cash flows beyond year 30 are arbitrarily neglected for estimation purposes.
The upper (solid) line stands for geometric mean estimates, the middle (dotted) line for arithmetic mean estimates and the lower (dashed) line for Cooper estimates.
32
Table 1: Net present value estimates for a sample length and forecasting horizon of 10 periods each
(equally probable rates of return 38.40 % and –13.60 %, “CDAX case”)
Observable data
per return path
Sample T =10
Present value estimation
n = 10
Accumulated value
Arithmetic
mean
Geometric
mean
Arithmetic
mean
Geometric
mean
Cooper
estimator
10 × 38.40%
(0.000977)
2,578.46
38.40%
38.40%
3.88
3.88
3.88
9 × 38.40%, 1 × −13.60%
(0.009766)
1,609.68
33.20%
32.03%
5.69
6.21
5.05
8 × 38.40%, 2 × −13.60%
(0.043945)
1,004.88
28.00%
25.95%
8.47
9.95
6.66
7 × 38.40%, 3 × −13.60%
(0.117188)
627.33
22.80%
20.16%
12.82
15.94
9.01
6 × 38.40%,4 × −13.60%
(0.205078)
391.63
17.60%
14.63%
19.77
25.53
12.72
5 × 38.40%, 5 × −13.60%
(0.246094)
244.48
12.40%
9.35%
31.07
40.90
19.05
4 × 38.40%, 6 × −13.60%
(0.205078)
152.63
7.20%
4.32%
49.89
65.52
30.80
3 × 38.40%,7 × −13.60%
(0.117188)
95.28
2.00%
-0.48%
82.03
104.95
54.02
2 × 38.40%, 8 × −13.60%
(0.043945)
59.48
-3.20%
-5.06%
138.44
168.12
102.15
1 × 38.40%, 9 × −13.60%
(0.009766)
37.13
-8.40%
-9.43%
240.46
269.30
205.21
10 × −13.60%
(0.000977)
23.18
−13.60%
−13.60%
431.38
431.38
431.38
321.86
12.40%
9.66%
42.33
53.85
28.26
8.98%
6.39%
13.47%
Return path
(probability of return path)
Expected value
Expected estimation:
Discount rate
Net present value estimates for a sample length and forecasting horizon of 10 periods each are presented for a binomial model with (equally probable) rates of return per period of 38.40 % and −13.60 %. Rates of return per period have been chosen in
such a way as to guarantee an expected one-period return and a corresponding return standard deviation that are equal to
those estimated for the German stock index CDAX based on historical return realizations over the last 49 years from 1955 to
2003 (“CDAX case”). The probability p of a return path with h realizations of the “good” return 38.40 % and l realizations of
the “bad” return –13.60 % with n = h+l total future periods (i.e. forecasting horizon) is thus computed as
p = 0.5h ⋅ 0.5l ⋅ (h!/(l!⋅ (n − l)!)) . For each return path, the resulting terminal value for an initial investment of € 100 after ten
periods sample length is displayed as well as the corresponding resulting arithmetic mean and geometric mean estimators.
Based on these two estimators and the corresponding Cooper estimator, the last three columns give net present value estimators for an expected future cash flow of € 100 at time n = 10 that is discounted for ten periods (forecasting horizon) based on
the three return estimators in question. The last but one line of Table 1 presents the true expectation values of the one-period
arithmetic mean and the corresponding geometric mean, but also the expected net present value of € 100 expected future cash
flow according to the three return estimators. The corresponding expected discount rates are displayed in the last line of Table 1. It can be seen that the expected discount rates as implied by the arithmetic mean and the geometric mean are smaller
than the correct discount rate of 12.40 % and the discount rate as implied by the Cooper estimator is too high. However, the
Cooper estimation of € 28.26 is nearest to the true value of 100/1.12410 = € 31.07.
33
Table 2: Sensitivity analysis of expected return estimation errors in the binomial model for varying return standard deviation, varying expected value of one-period
returns and varying probabilities of “good” and “bad” returns
n=1
Arit.
mean
Geo.
mean
n=2
Cooper
Arit.
mean
Geo.
mean
n=5
Cooper
Arit.
mean
Geo.
mean
n = 10
Cooper
Arit.
mean
n = 20
n = 30
Geo.
mean
Cooper
Arit.
mean
Geo.
mean
Cooper
Arit.
mean
Geo.
mean
Cooper
(1) Return standard deviation ranging from 15 to 35 percent; expected return 12.5 percent, probability of high return 0.5
-0.0400
-1.0245
0.0005
-0.0601
-1.0446
0.0011
-0.1202
-1.1047
0.0040
-0.2206
15.0
-0.0545
-1.3968
0.0010
-0.0818
-1.4241
0.0020
-0.1637
-1.5060
0.0075
-0.3006
17.5
-0.0712
-1.8278
0.0017
-0.1068
-1.8635
0.0034
-0.2140
-1.9706
0.0130
-0.3932
20.0
-0.0901
-2.3183
0.0027
-0.1353
-2.3635
0.0056
-0.2710
-2.4992
0.0211
-0.4983
22.5
-0.1113
-2.8690
0.0042
-0.1671
-2.9249
0.0086
-0.3349
-3.0927
0.0327
-0.6162
25.0
-0.1348
-3.4808
0.0062
-0.2023
-3.5486
0.0127
-0.4057
-3.7518
0.0489
-0.7469
27.5
-0.1604
-4.1547
0.0088
-0.2409
-4.2356
0.0182
-0.4833
-4.4778
0.0709
-0.8906
30.0
-0.1884
-4.8919
0.0123
-0.2829
-4.9870
0.0255
-0.5679
-5.2717
0.1002
-1.0474
32.5
-0.2186
-5.6936
0.0168
-0.3284
-5.8041
0.0349
-0.6596
-6.1350
0.1386
-1.2176
35.0
-1.2049
-1.6424
-2.1488
-2.7249
-3.3715
-4.0895
-4.8800
-5.7443
-6.6835
0.0140
0.0265
0.0465
0.0768
0.1212
0.1846
0.2736
0.3968
0.5657
-0.4222
-0.5758
-0.7537
-0.9563
-1.1840
-1.4372
-1.7164
-2.0222
-2.3553
-1.4048
-1.9144
-2.5040
-3.1743
-3.9261
-4.7603
-5.6780
-6.6802
-7.7683
0.0615
0.1202
0.2189
0.3796
0.6376
1.0536
1.7417
2.9549
5.4471
-0.6245
-0.8523
-1.1167
-1.4184
-1.7581
-2.1370
-2.5559
-3.0164
-3.5197
-1.6040
-2.1852
-2.8572
-3.6204
-4.4757
-5.4237
-6.4653
-7.6014
-8.8332
0.1686
0.3479
0.6851
1.3401
2.7828
8.3515
-*
-*
-*
(2) Expected return ranging from 0 to 20 percent; return standard deviation 25 percent, probability of high return 0.5
-0.1253
-3.2386
0.0060
-0.1881
-3.3017
0.0124
-0.3773
-3.4907
0.0479
-0.6947
0.0
-0.1222
-3.1571
0.0056
-0.1835
-3.2186
0.0115
-0.3680
-3.4030
0.0442
-0.6774
2.5
-0.1193
-3.0797
0.0052
-0.1791
-3.1397
0.0106
-0.3591
-3.3196
0.0409
-0.6610
5.0
-0.1165
-3.0060
0.0048
-0.1749
-3.0647
0.0099
-0.3507
-3.2403
0.0379
-0.6453
7.5
-0.1139
-2.9359
0.0045
-0.1709
-2.9931
0.0092
-0.3426
-3.1647
0.0352
-0.6304
10.0
-0.1113
-2.8690
0.0042
-0.1671
-2.9249
0.0086
-0.3349
-3.0927
0.0327
-0.6162
12.5
-0.1089
-2.8050
0.0039
-0.1635
-2.8598
0.0080
-0.3276
-3.0238
0.0305
-0.6026
15.0
-0.1066
-2.7440
0.0037
-0.1600
-2.7975
0.0075
-0.3206
-2.9580
0.0285
-0.5895
17.5
-0.1043
-2.6855
0.0034
-0.1566
-2.7379
0.0070
-0.3138
-2.8951
0.0267
-0.5771
20.0
-3.8047
-3.7093
-3.6186
-3.5322
-3.4500
-3.3715
-3.2966
-3.2249
-3.1564
0.1815
0.1665
0.1533
0.1414
0.1307
0.1212
0.1125
0.1047
0.0976
-1.3374
-1.3035
-1.2714
-1.2408
-1.2117
-1.1840
-1.1575
-1.1322
-1.1080
-4.4283
-4.3178
-4.2126
-4.1125
-4.0172
-3.9261
-3.8392
-3.7560
-3.6764
1.0611
0.9483
0.8526
0.7705
0.6995
0.6376
0.5834
0.5356
0.4932
-1.9892
-1.9382
-1.8897
-1.8437
-1.7999
-1.7581
-1.7183
-1.6803
-1.6440
-5.0447
-4.9195
-4.8004
-4.6870
-4.5789
-4.4757
-4.3771
-4.2827
-4.1924
-*
7.2734
5.1207
4.0074
3.2916
2.7828
2.3993
2.0986
1.8561
(3) Probability of high return ranging from 30 to 70 percent; expected return 12.5 percent, return standard deviation 25 percent
-0.1022
-2.3363
-0.0089
-0.1533
-2.3791
-0.0121
-0.3060
-2.5068
-0.0157
-0.5592
-2.7171
30
-0.1082
-2.5506
-0.0061
-0.1623
-2.5981
-0.0075
-0.3244
-2.7398
-0.0041
-0.5940
-2.9739
35
-0.1116
-2.7116
-0.0028
-0.1675
-2.7629
-0.0023
-0.3351
-2.9162
0.0084
-0.6146
-3.1698
40
-0.1126
-2.8182
0.0007
-0.1690
-2.8723
0.0032
-0.3385
-3.0343
0.0210
-0.6217
-3.3030
45
-0.1113
-2.8690
0.0042
-0.1671
-2.9249
0.0086
-0.3349
-3.0927
0.0327
-0.6162
-3.3715
50
-0.1079
-2.8626
0.0074
-0.1620
-2.9193
0.0135
-0.3250
-3.0896
0.0430
-0.5986
-3.3732
55
-0.1025
-2.7978
0.0102
-0.1539
-2.8541
0.0177
-0.3089
-3.0233
0.0512
-0.5697
-3.3060
60
-0.0952
-2.6731
0.0125
-0.1430
-2.7278
0.0209
-0.2872
-2.8922
0.0567
-0.5302
-3.1676
65
-0.0861
-2.4871
0.0139
-0.1294
-2.5388
0.0229
-0.2600
-2.6944
0.0592
-0.4805
-2.9557
70
0.0035
0.0341
0.0654
0.0951
0.1212
0.1418
0.1557
0.1622
0.1606
-1.0601
-1.1304
-1.1737
-1.1912
-1.1840
-1.1534
-1.1004
-1.0264
-0.9322
-3.1284
-3.4334
-3.6698
-3.8351
-3.9261
-3.9400
-3.8734
-3.7230
-3.4852
0.1981
0.3220
0.4440
0.5526
0.6376
0.6915
0.7099
0.6918
0.6400
-1.5526
-1.6620
-1.7318
-1.7634
-1.7581
-1.7175
-1.6430
-1.5361
-1.3981
-3.5270
-3.8810
-4.1597
-4.3592
-4.4757
-4.5049
-4.4425
-4.2837
-4.0234
0.8877
1.3759
1.9032
2.4018
2.7828
2.9635
2.9072
2.6418
2.2381
For binomial one-period returns, average errors (in percentage points deviation from the correct discount rate) in estimating expected one-period returns for discounting purposes by the arithmetic mean estimator, the geometric mean estimator, and the Cooper estimator are presented. Thereby, ceteris paribus variations of return standard deviations, expectation values of one-period returns, and probabilities of “good”
and “bad” one-period returns are considered for forecasting horizons n = 1, 2, 5, 10, 20, and 30. The sample length is assumed to be T = 50 in all cases.
*
Resulting average discount factors would be negative, i.e. corresponding average discount rates would be smaller than −100 %.
34
Table 3: Return estimation errors when assuming a binomial, normal or lognormal distribution in the
CDAX case (sample length: T = 50)
Binomial distribution
n
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Arith.
mean
-0.12
-0.18
-0.24
-0.30
-0.36
-0.42
-0.48
-0.55
-0.61
-0.67
-0.73
-0.79
-0.85
-0.91
-0.97
-1.04
-1.10
-1.16
-1.22
-1.28
-1.35
-1.41
-1.47
-1.53
-1.59
-1.66
-1.72
-1.78
-1.84
-1.91
Geo.
mean
-3.11
-3.17
-3.23
-3.29
-3.35
-3.41
-3.47
-3.53
-3.59
-3.65
-3.71
-3.77
-3.83
-3.89
-3.95
-4.01
-4.07
-4.13
-4.19
-4.25
-4.31
-4.37
-4.43
-4.49
-4.55
-4.61
-4.67
-4.73
-4.79
-4.85
Cooper
0.00
0.01
0.02
0.03
0.04
0.05
0.07
0.09
0.12
0.14
0.18
0.21
0.26
0.31
0.36
0.43
0.50
0.58
0.68
0.78
0.91
1.05
1.22
1.41
1.64
1.92
2.25
2.68
3.22
3.97
Normal distribution
Arith.
mean
-0.32
-0.07
-0.02
-0.02
-0.20
-0.19
-0.29
-0.41
-0.53
-0.65
-0.73
-0.74
-0.81
-0.86
-0.94
-1.00
-1.06
-1.11
-1.17
-1.26
-1.31
-1.36
-1.45
-1,51
-1.59
-1.64
-1.71
-1.77
-1.85
-1.87
Geo.
mean
-2.88
-3.14
-3.24
-3.25
-3.44
-3.44
-3.54
-3.68
-3.80
-3.93
-4.02
-4.05
-4.12
-4.18
-4.27
-4.34
-4.41
-4.46
-4.53
-4.64
-4.70
-4.76
-4.86
-4.93
-5.01
-5.07
-5.16
-5.23
-5.32
-5.36
Cooper
0.46
0.28
0.26
0.34
0.24
0.33
0.32
0.29
0.28
0.26
0.29
0.40
0.46
0.55
0.62
0.72
0.84
0.99
1.15
1.30
1.53
1.79
2.08
2.47
2.95
3.61
4.51
5.94
8.82
-*
Lognormal distribution
Arith.
mean
-0.06
-0.10
-0.03
-0.09
-0.28
-0.45
-0.44
-0.48
-0.55
-0.59
-0.68
-0.78
-0.86
-0.94
-0.96
-1.02
-1.09
-1.17
-1.22
-1.28
-1.35
-1.41
-1.50
-1.59
-1.65
-1.74
-1.79
-1.88
-1.94
-1.98
Geo.
mean
-2.87
-2.71
-2.78
-2.90
-3.08
-3.25
-3.24
-3.28
-3.34
-3.37
-3.47
-3.56
-3.63
-3.72
-3.73
-3.79
-3.85
-3.93
-3.98
-4.04
-4.10
-4.16
-4.24
-4.33
-4.39
-4.47
-4.52
-4.61
-4.66
-4.70
Cooper
0.06
0.28
0.27
0.21
0.10
0.00
0.07
0.10
0.12
0.16
0.15
0.14
0.16
0.17
0.25
0.30
0.35
0.39
0.47
0.56
0.64
0.75
0.85
0.96
1.13
1.30
1.54
1.80
2,15
2.62
For binomially, normally, and lognormally distributed one-period returns, average errors (in percentage points deviation from
the correct discount rate) in estimating expected one-period returns for discounting purposes by the arithmetic mean estimator, the geometric mean estimator, and the Cooper estimator are presented. Thereby, forecasting horizons from n = 1 to n =
30 are considered. The sample length is assumed to be T = 50 in all cases. Expectation value of one-period returns (12.40 %)
and corresponding return standard deviation (26.00 %) are assumed to be identical to those parameter values as estimated for
the German stock index CDAX based on historical return data (CDAX case). For computations regarding normal and lognormal one-period return distributions, a Monte Carlo simulation with 10,000 runs has to be applied.
*
Resulting discount factor would be negative.
35
Table 4: Distributional parameters for discount factor estimates in the CDAX case with normally distributed return rates
Arith.
mean
Minimum
Maximum
Arith. mean
Standard dev.
Skewness
Kurtosis
Minimum
Maximum
Arith. mean
Standard dev.
Skewness
Kurtosis
Minimum
Maximum
Arith. mean
Standard dev.
Skewness
Kurtosis
Minimum
Maximum
Arith. mean
Standard dev.
Skewness
Kurtosis
Expected
value
T = 10
T = 25
T = 50
T = 75
Geo.
mean
Discount factor estimation
n=1
Discount factor estimation
n = 10
Discount factor estimation
n = 20
Discount factor estimation
n = 30
Arith.
mean
Geo.
mean
Cooper
Arith.
mean
Arith.
mean
Arith.
mean
Geo.
mean
Cooper
Geo.
mean
Cooper
Geo.
mean
Cooper
-5369.202
-0.1801
-0.2246
0.7128
0.7253
0.7100
0.0339
0.0403
-11.4353
0.0011
0.0016
-253.4282
0.0000
0.0001
0.4029
0.3787
1.2197
1.2896
1.2160
7.2855
12.7194
6.2119
53.0788
161.7828
22.2949
386.7068
2057.7771
5.1683
0.1237
0.0947
0.8948
0.9194
0.8893
0.4270
0.5915
0.2260
0.3486
0.8036
-0.7132
0.5863
2.6861
-6.6463
105.6557
0.0823
0.0865
0.0666
0.0744
0.0653
0.4077
0.6736
0.3478
1.3134
3.9976
6.4811
6.9220
36.2497
0.0079
-0.0458
0.4515
0.5632
0.4460
4.3626
5.4786
-7.7279
20.4898
20.0753
-25.0318
39.9962
34.1203
-33.8212
0.0078
0.0514
0.4180
0.6833
0.4203
37.7942
53.3240
250.7433
636.9759
555.3418
765.4217
1987.9010
1481.0919
1339.3430
-0.0566
-0.0980
0.7472
0.7690
0.7454
0.0542
0.0723
-0.5932
0.0029
0.0052
-6.3697
0.0002
0.0004
-27.6286
0.3384
0.3004
1.0600
1.1087
1.0568
1.7911
2.8064
1.4404
3.2080
7.8759
1.1635
5.7458
22.1030
0.2961
0.1243
0.0931
0.8914
0.9171
0.8892
0.3495
0.4729
0.2929
0.1530
0.2933
0.0303
0.0843
0.2425
-0.1200
0.0520
0.0546
0.0415
0.0462
0.0412
0.1757
0.2639
0.1428
0.1863
0.4260
0.1469
0.2057
0.7820
0.6707
0.0179
-0.0064
0.2543
0.3072
0.2532
1.7087
2.0635
1.6523
4.6270
6.2788
-16.8185
10.2360
13.9284
-22.7539
-0.0573
-0.0324
0.0562
0.1570
0.0560
5.0285
7.7756
5.1314
37.7833
68.1855
560.0502
172.7395
282.5067
714.0141
-0.0021
-0.0449
0.7999
0.8134
0.7994
0.1073
0.1267
0.1019
0.0115
0.0161
-0.7086
0.0012
0.0020
-2.2994
0.2501
0.2295
1.0021
1.0470
1.0007
1.0212
1.5833
0.9269
1.0429
2.5067
0.5994
1.0650
3.9687
0.2592
0.1242
0.0924
0.8904
0.9166
0.8894
0.3290
0.4432
0.3033
0.1207
0.2237
0.0765
0.0494
0.1289
-0.0009
0.0367
0.0387
0.0292
0.0326
0.0291
0.1116
0.1651
0.1010
0.0891
0.1869
0.0563
0.0623
0.1939
0.0495
0.0126
-0.0064
0.1720
0.2110
0.1736
1.0730
1.2542
1.0596
2.3848
3.0056
1.8102
4.3273
6.0521
-18.9529
-0.1290
-0.0948
-0.0685
0.0005
-0.0645
1.8040
2.7225
1.7542
9.3687
16.1153
13.1195
31.6779
63.7336
661.1786
0.0128
-0.0356
0.8151
0.8342
0.8145
0.1294
0.1633
0.1239
0.0167
0.0267
-0.0448
0.0022
0.0044
-0.2941
0.2269
0.1987
0.9874
1.0369
0.9861
0.8809
1.4373
0.7982
0.7759
2.0658
0.5200
0.6835
2.9690
0.2305
0.1243
0.0922
0.8901
0.9163
0.8894
0.3223
0.4335
0.3058
0.1117
0.2048
0.0853
0.0417
0.1058
0.0149
0.0299
0.0315
0.0237
0.0265
0.0237
0.0885
0.1301
0.0830
0.0657
0.1341
0.0484
0.0408
0.1195
0.0180
-0.0190
-0.0369
0.1783
0.2124
0.1799
0.9379
1.0733
0.9254
2.0116
2.4169
1.9151
3.6505
4.9265
0.4789
-0.0125
0.0181
0.0364
0.0937
0.0407
1.4739
2.0570
1.4259
7.2130
11.3905
6.3964
25.1682
54.5146
42.2376
0.8930
0.3105
0.0963
0.0298
A Monte Carlo simulation with 10,000 runs for normally distributed rates of one-period return with an expectation value of 12.40 % and a return standard deviation of 26.00 % (CDAX case) is performed. For
different combinations of sample length T (= 10, 25, 50, and 75) and forecasting horizon n (= 1, 10, 20, and 30), distributional parameters of resulting discount factor estimates are presented. In the first two
columns, resulting distributions of arithmetic mean and geometric mean estimators are described. In the last line the correct discount factors are displayed for any forecasting horizon under consideration, i.e.
those discount factors which would be applied in the case of knowing the true return distribution.
36
Table 5: Estimated average discount rates for CDAX and DAX on a total return basis and on a partial return basis (market risk premia) for empirical CDAX and DAX
returns and REXP returns from 1955 to 2003 when discount factor proxies smaller than 0 and greater than 1 are substituted by 0 and 1, respectively (forecasting horizon up to N = 30)
CDAX
n
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Expectedvalue
12.76
12.40
12.44
12.39
12.34
12.29
12.29
12.23
12.24
12.25
12.27
12.27
12.25
12.28
12.29
12.29
12.27
12.27
12.27
12.23
12.23
12.26
12.24
12.28
12.28
12.24
12.23
12.23
12.24
12.25
DAX
Arith.
mean
Geo.
mean
Cooper
12.21
12.15
12.09
12.03
11.97
11.91
11.85
11.79
11.74
11.68
11.62
11.56
11.50
11.44
11.38
11.32
11.26
11.20
11.14
11.08
11.02
10.97
10.91
10.85
10.79
10.73
10.67
10.61
10.55
10.49
9.37
9.32
9.26
9.20
9.14
9.09
9.03
8.97
8.91
8.86
8.80
8.74
8.68
8.63
8.57
8.51
8.45
8.40
8.34
8.28
8.23
8.17
8.11
8.06
8.00
7.94
7.89
7.83
7.78
7.72
12.33
12.33
12.34
12.35
12.36
12.37
12.39
12.40
12.43
12.45
12.48
12.51
12.55
12.60
12.64
12.70
12.77
12.84
12.92
13.01
13.12
13.25
13.39
13.55
13.74
13.96
14.22
14.52
14.86
15.25
Expectedvalue
12.82
12.97
13.02
12.97
12.91
12.85
12.85
12.78
12.79
12.79
12.81
12.81
12.79
12.81
12.83
12.83
12.81
12.81
12.81
12.77
12.78
12.81
12.80
12.84
12.84
12.79
12.77
12.78
12.79
12.80
Market risk premium (CDAX)
Arith.
mean
Geo.
mean
Cooper
12.75
12.68
12.62
12.55
12.48
12.41
12.34
12.28
12.21
12.14
12.07
12.00
11.93
11.87
11.80
11.73
11.66
11.59
11.52
11.46
11.39
11.32
11.25
11.18
11.11
11.05
10.98
10.91
10.84
10.77
9.47
9.41
9.34
9.27
9.21
9.14
9.07
9.01
8.94
8.87
8.81
8.74
8.68
8.61
8.55
8.48
8.41
8.35
8.28
8.22
8.16
8.09
8.03
7.96
7.90
7.84
7.77
7.71
7.65
7.58
12.89
12.90
12.91
12.92
12.93
12.95
12.97
13.00
13.03
13.06
13.10
13.15
13.21
13.27
13.34
13.42
13.52
13.62
13.75
13.89
14.06
14.26
14.49
14.76
15.08
15.45
15.86
16.31
16.80
17.28
Expectedvalue
5.76
5.38
5.48
5.39
5.37
5.34
5.33
5.25
5.28
5.28
5.29
5.30
5.29
5.31
5.31
5.31
5.29
5.30
5.29
5.27
5.27
5.28
5.26
5.29
5.30
5.25
5.24
5.25
5.47
5.48
Arith.
mean
Geo.
mean
Cooper
5.36
5.31
5.25
5.20
5.15
5.09
5.04
4.99
4.94
4.88
4.83
4.78
4.73
4.68
4.63
4.58
4.54
4.49
4.44
4.39
4.35
4.30
4.26
4.21
4.17
4.13
4.08
4.04
4.00
3.96
2.90
2.86
2.82
2.79
2.75
2.71
2.68
2.64
2.61
2.58
2.54
2.51
2.48
2.45
2.42
2.39
2.36
2.33
2.30
2.27
2.24
2.22
2.19
2.16
2.14
2.11
2.09
2.06
2.04
2.02
5.47
5.47
5.47
5.47
5.47
5.47
5.48
5.49
5.49
5.50
5.51
5.53
5.54
5.56
5.58
5.60
5.63
5.65
5.68
5.72
5.75
5.79
5.84
5.89
5.94
5.99
6.05
6.10
6.14
6.17
Market risk premium (DAX)
Expectedvalue
6.35
5.97
6.04
5.99
5.95
5.90
5.88
5.82
5.84
5.83
5.84
5.84
5.82
5.85
5.87
5.87
5.85
5.85
5.85
5.82
5.84
5.87
5.85
5.88
5.89
5.83
5.82
5.82
5.84
5.85
Arith.
mean
Geo.
mean
Cooper
5.90
5.84
5.78
5.72
5.66
5.60
5.54
5.48
5.42
5.36
5.31
5.25
5.19
5.14
5.08
5.03
4.97
4.92
4.87
4.81
4.76
4.71
4.66
4.61
4.56
4.51
4.46
4.42
4.37
4.32
3.05
3.00
2.96
2.92
2.88
2.84
2.80
2.77
2.73
2.69
2.66
2.62
2.59
2.55
2.52
2.49
2.45
2.42
2.39
2.36
2.33
2.30
2.27
2.24
2.21
2.19
2.16
2.13
2.11
2.08
6.02
6.02
6.03
6.03
6.04
6.05
6.06
6.07
6.08
6.10
6.12
6.14
6.16
6.19
6.22
6.26
6.30
6.34
6.39
6.44
6.50
6.57
6.64
6.71
6.79
6.85
6.91
6.95
6.97
6.96
Results of a Monte Carlo simulation with 10,000 runs for 49 rates of return each being drawn with equal probability from the 49 historical return realizations between 1955 and 2003 for the German stock indexes CDAX and DAX are presented. Market risk premia are computed as differences between CDAX or DAX returns on the one side and REXP returns on the other (pre-tax case). For each simulation run,
discount rates according to the arithmetic mean estimator, the geometric mean estimator, and the Cooper estimator are computed. Average discount rate results for these three estimation approaches on a total
return basis for the DAX case and the CDAX case as well as on a partial return basis for these two indexes (under the simplifying assumption of the current term structure of riskless interest rates being identical to zero for all future time periods) are displayed. Thereby, discount factor proxies smaller than 0 and greater than 1 are substituted by 0 and 1, respectively. “Expected value” stands for the corresponding
“true” discount rate which would result in the case of knowing the true return distribution (as implied by the Monte Carlo simulation).
37
Table 6: Typified firm valuation in the CDAX case, the DAX case, and the S&P 500 case
Present value for finite time horizon (30 years)
Firm value for infinite time horizon
(constant cash flow of € 1 bn. per year)
Expected
value
(1) CDAX pre-tax
(total return basis)
Firm value [€ m.]
Relative error
(2) DAX pre-tax
(total return basis)
Firm value [€ m.]
Relative error
(3) CDAX pre-tax
(partial return basis)
Firm value [€ m.]
Relative error
(4) DAX pre-tax
(partial return basis)
Firm value [€ m.]
Relative error
(5) CDAX after tax
(partial return basis)
Firm value [€ m.]
Relative error
(6) DAX after tax
(partial return basis)
Firm value [€ m.]
Relative error
(7) S&P 500 pre-tax
(total return basis)
Firm value [€ m.]
Relative error
(8) S&P 500 pre-tax
(partial return basis)
Firm value [€ m.]
Relative error
Arith.
mean
Geo.
mean
Cooper
(constant cash flow of € 1 bn. per year)
Change
after 20
years
Change
after 25
years
Expected
value
Arith.
mean
Geo.
mean
Cooper
Change
after 20
years
Change
after 25
years
7,888.01
0.00%
8,416.70
6.70%
10,728.92
36.02%
7,627.92
-3.30%
7,995.81
1.37%
7,811.87
-0.97%
8,142.95
0.00%
12,978.00
59.38%
14,393.27
76.76%
10,096.54
23.99%
12,557.10
54.21%
12,373.16
51.95%
7,581.28
0.00%
8,143.37
7.41%
10,760.13
41.93%
7,259.68
-4.24%
7,667.98
1.14%
7,460.75
-1.59%
7,791.67
0.00%
9,117.21
17.01%
14,697.54
88.63%
7,521.06
-3.47%
8,641.82
10.91%
8,434.58
8.25%
9,514.32
0.00%
10,018.48
5.30%
12,270.54
28.97%
9,244.09
-2.84%
9,595.29
0.85%
9,416.00
-1.03%
9,949.75
0.00%
10,962.90
10.18%
14,191.99
42.64%
9,768.56
-1.82%
10,539.71
5.93%
10,360.41
4.13%
9,103.78
0.00%
9,640.07
5.89%
12,149.61
33.46%
8,787.61
-3.47%
9,166.55
0.69%
8,969.06
-1.48%
9,484.42
0.00%
10,474.42
10.44%
14,033.51
47.96%
9,229.97
-2.68%
10,000.90
5.45%
9,803.42
3.36%
9,860.96
0.00%
10,526.23
6.75%
13,296.09
34.84%
9,536.66
-3.29%
10,021.73
1.63%
9,782.16
-0.80%
10,421.31
0.00%
12,014.64
15.29%
16,817.42
61.38%
10,293.56
-1.23%
11,510.15
10.45%
11,270.57
8.15%
9,536.60
0.00%
10,242.31
7.40%
13,314.49
39.61%
9,155.41
-4.00%
8,497.29
-1.99%
9,417.08
-1.25%
10,050.53
0.00%
11,619.32
15.61%
16,918.80
68.34%
9,814.38
-2.35%
11,058.30
10.03%
10,794.09
7.40%
8,184.54
0.00%
8,461.34
3.38%
9,897.79
20.93%
8,170.93
-0.17%
8,298.77
1.40%
8,233.09
0.59%
8,477.41
0.00%
8,905.82
5.05%
10,887.00
28.42%
8,454.44
-0.27%
8,743.25
3.14%
8,677.57
2.36%
8,897.93
0.00%
9,288.13
4.39%
11,060.34
24.30%
8,920.60
0.25%
9,088.37
2.14%
9,003.86
1.19%
9,278.94
0.00%
9,914.76
6.85%
12,452.02
34.20%
9,316.00
0.40%
9,714.99
4.70%
9,630.48
3.79%
Firm value estimates and corresponding relative estimation errors are presented for different situations (varying according to their distributional assumptions – CDAX, DAX, and S&P case –, the consideration
of personal taxes – “pre-tax” vs. “after tax” –, and the estimation of discount rates on a total or a partial return basis). In the left columns, an expected future cash flow of € 1 bn. per period from n = 1 to n = 30
is assumed. For the right columns an expected future cash flow of € 1 bn. per period from n = 1 on until infinity is assumed. A bootstrapping approach for different distributional assumptions is applied in order
to compute average firm value estimates when utilizing an arithmetic mean estimator, a geometric mean estimator, and the corresponding Cooper estimator. Moreover, as a fourth and a fifth estimation procedure, a switch from the Cooper estimator to the arithmetic mean estimator after 20 or 25 years, respectively, is considered. The results of these five estimation approaches are compared to the “true” net present
value which would result in the case of knowing the true return distribution (denoted as “Expected value” in Table 6) and relative estimation errors are computed. Discount factor estimates are restricted to the
interval [0, 1].
38
Table Ad1: Return estimation errors when assuming a binomial, normal or lognormal distribution in the
CDAX case (sample length: T = 75)
Binomial distribution
n
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Arith.
mean
-0.08
-0.12
-0.16
-0.20
-0.24
-0.28
-0.32
-0.36
-0.40
-0.44
-0.48
-0.52
-0.57
-0.61
-0.65
-0.69
-0.73
-0.77
-0.81
-0.85
-0.89
-0.93
-0.97
-1.02
-1.06
-1.10
-1.14
-1.18
-1.22
-1.26
Geo.
mean
-3.09
-3.13
-3.17
-3.21
-3.25
-3.29
-3.33
-3.37
-3.41
-3.45
-3.49
-3.53
-3.57
-3.61
-3.65
-3.69
-3.73
-3.77
-3.81
-3.85
-3.89
-3.93
-3.97
-4.01
-4.05
-4.09
-4.13
-4.17
-4.21
-4.25
Cooper
0.00
0.01
0.01
0.02
0.03
0.03
0.05
0.06
0.07
0.09
0.11
0.13
0.16
0.19
0.22
0.25
0.29
0.34
0.39
0.44
0.50
0.57
0.65
0.74
0.84
0.95
1.07
1.22
1.38
1.58
Normal distribution
Arith.
mean
-0.37
-0.14
-0.06
-0.09
-0.08
-0.05
-0.12
-0.23
-0.32
-0.42
-0.49
-0.48
-0.52
-0.55
-0.61
-0.65
-0.69
-0.72
-0.76
-0.83
-0.86
-0.90
-0.96
-1.00
-1.06
-1.09
-1.15
-1.18
-1.25
-1.25
Geo.
mean
-2.85
-3.08
-3.17
-3.15
-3.31
-3.29
-3.37
-3.48
-3.58
-3.69
-3.76
-3.76
-3.81
-3.84
-3.91
-3.95
-4.00
-4.03
-4.08
-4.16
-4.20
-4.24
-4.31
-4.36
-4.42
-4.46
-4.52
-4.57
-4.64
-4.65
Cooper
0.46
0.28
0.25
0.32
0.21
0.30
0.28
0.24
0.21
0.17
0.17
0.26
0.29
0.34
0.36
0.42
0.48
0.56
0.63
0.68
0.79
0.90
0.99
1.13
1.28
1.47
1.67
1.93
2.22
2.64
Lognormal distribution
Arith.
mean
-0.01
-0.17
-0.12
-0.02
-0.14
-0.29
-0.26
-0.29
-0.33
-0.35
-0.43
-0.51
-0.56
-0.62
-0.62
-0.67
-0.71
-0.77
-0.80
-0.84
-0.89
-0.93
-1.00
-1.07
-1.11
-1.18
-1.21
-1.29
-1.32
-1.34
Geo.
mean
-2.84
-2.66
-2.71
-2.81
-2.97
-3.12
-3.09
-3.11
-3.15
-3.17
-3.24
-3.32
-3.37
-3.44
-3.43
-3.47
-3.51
-3.57
-3.60
-3.64
-3.69
-3.72
-3.79
-3.86
-3.90
-3.96
-4.00
-4.07
-4.10
-4.12
Cooper
0.07
0.29
0.28
0.23
0.11
0.00
0.08
0.10
0.11
0.15
0.12
0.10
0.11
0.10
0.17
0.19
0.22
0.24
0.28
0.33
0.36
0.42
0.45
0.49
0.56
0.62
0.72
0.80
0.93
1.08
For binomially, normally, and lognormally distributed one-period returns, average errors (in percentage points deviation from
the correct discount rate) in estimating expected one-period returns for discounting purposes by the arithmetic mean estimator, the geometric mean estimator, and the Cooper estimator are presented. Thereby, forecasting horizons from n = 1 to n =
30 are considered. The sample length is assumed to be T = 75 in all cases. Expectation value of one-period returns (12.40 %)
and corresponding return standard deviation (26.00 %) are assumed to be identical to those parameter values as estimated for
the German stock index CDAX based on historical return data (CDAX case). For computations regarding normal and lognormal one-period return distributions, a Monte Carlo simulation with 10,000 runs has to be applied.
39
CDAX case: Normal distribution
(T = 10 periods)
35%
30%
25%
20%
15%
10%
5%
0%
-5%
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
(T = 25 periods)
35%
30%
25%
20%
15%
10%
5%
0%
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
(T = 50 periods)
25%
20%
15%
10%
5%
0%
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
(T = 75 periods)
25%
20%
15%
10%
5%
0%
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Forecasting horizon
Expected value
Figure Ad1:
Arithmetic mean
Geometric mean
Cooper estimator
Average estimated discount rates for normally distributed one-period returns in the CDAX
case (expected return: 12.40 %; return standard deviation: 26.00 %) for sample lengths of
T = 10, 25, 50, and 75 periods
Based on a Monte Carlo simulation with 10,000 runs, average estimated discount rates according to the arithmetic mean estimator, the geometric mean estimator, and the corresponding Cooper estimator for normally distributed one-period returns in
the CDAX case (i.e. for a one-period expected rate of return of 12.40 % and corresponding return standard deviation of 26.00
%) for sample lengths of T = 10, 25, 50, and 75 periods are presented as a function of forecasting horizon n = 1, …, 30. “Expected value” stands for the “true” discount rate which would result in the case of knowing the true return distribution.
40
CDAX case: Lognormal distribution
(T = 10 periods)
30%
25%
20%
15%
10%
5%
0%
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
(T = 25 periods)
30%
25%
20%
15%
10%
5%
0%
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
(T = 50 periods)
20%
15%
10%
5%
0%
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
(T = 75 periods)
20%
15%
10%
5%
0%
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Forecasting horizon
Expected value
Figure Ad2:
Arithmetic mean
Geometric mean
Cooper estimator
Average estimated discount rates for lognormally distributed one-period returns in the
CDAX case (expected return: 12.40 %; return standard deviation: 26.00 %) for sample
lengths of T = 10, 25, 50, and 75 periods
Based on a Monte Carlo simulation with 10,000 runs, average estimated discount rates according to the arithmetic mean estimator, the geometric mean estimator, and the corresponding Cooper estimator for lognormally distributed one-period returns in the CDAX case (i.e. for a one-period expected rate of return of 12.40 % and corresponding return standard deviation
of 26.00 %) for sample lengths of T = 10, 25, 50, and 75 periods are presented as a function of forecasting horizon n = 1, …,
30. “Expected value” stands for the “true” discount rate which would result in the case of knowing the true return distribution..
41
CDAX case without extreme results: Normal distribution
(T = 10 periods)
20%
15%
10%
5%
0%
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
(T = 25 periods)
20%
15%
10%
5%
0%
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
(T = 50 periods)
20%
15%
10%
5%
0%
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
(T = 75 periods)
20%
15%
10%
5%
0%
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Forecasting horizon
Expected value
Figure Ad3:
Arithmetic mean
Geometric mean
Cooper estimator
Average estimated discount rates for normally distributed one-period returns in the CDAX case
(expected return: 12.40 %; return standard deviation: 26.00 %) for sample lengths of T = 10, 25,
50, and 75 periods when discount factor proxies smaller than 0 and greater than 1 are substituted
by 0 and 1, respectively
Based on a Monte Carlo simulation with 10,000 runs, average estimated discount rates according to the arithmetic mean estimator, the geometric mean estimator, and the corresponding Cooper estimator for normally distributed one-period returns in
the CDAX case (i.e. for a one-period expected rate of return of 12.40 % and corresponding return standard deviation of 26.00
%) for sample lengths of T = 10, 25, 50, and 75 periods are presented as a function of forecasting horizon n = 1, …, 30. Thereby, discount factor proxies smaller than 0 and greater than 1 are substituted by 0 and 1, respectively. “Expected value”
stands for the “true” discount rate which would result in the case of knowing the true return distribution.
42
CDAX case without extreme results: Lognormal distribution
(T = 10 periods)
20%
15%
10%
5%
0%
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
(T = 25 periods)
20%
15%
10%
5%
0%
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
(T = 50 periods)
20%
15%
10%
5%
0%
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
(T = 75 periods)
20%
15%
10%
5%
0%
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Forecasting horizon
Expected value
Figure Ad4:
Arithmetic mean
Geometric mean
Cooper estimator
Average estimated discount rates for normally distributed one-period returns in the CDAX case
(expected return: 12.40 %; return standard deviation: 26.00 %) for sample lengths of T = 10, 25 50,
and 75 periods when discount factor proxies smaller than 0 and greater than 1 are substituted by 0
and 1, respectively
Based on a Monte Carlo simulation with 10,000 runs, average estimated discount rates according to the arithmetic mean estimator, the geometric mean estimator, and the corresponding Cooper estimator for lognormally distributed one-period returns in the CDAX case (i.e. for a one-period expected rate of return of 12.40 % and corresponding return standard deviation
of 26.00 %) for sample lengths of T = 10, 25, 50, and 75 periods are presented as a function of forecasting horizon n = 1, …,
30. Thereby, discount factor proxies smaller than 0 and greater than 1 are substituted by 0 and 1, respectively. “Expected value” stands for the “true” discount rate which would result in the case of knowing the true return distribution.
43