Is the growth potential of stock prices underestimated? A real option approach.
Levyne, Olivier ; Heller, David
I. INTRODUCTION: LIMITS OF THE DISCOUNTED CASH FLOW (DCF) VALUATION
METHOD
From a DCF point of view, the value of the firm (EV) corresponds to
the present value of the future free cash flows (FCF), the discount rate
used being the weighted average cost of capital or WACC (K):
EV = [+[infinity].summation over (t=1)] [FCF.sub.t]/[(1 + K).sup.t]
with
K = k E/E + D + i.(1-[tau]) D/E + D
where E is the equity value and D is the net debt. The embedded
WACC in the firm value calculation is based on the equity value which is
looked for in the DCF approach. For that reason, practitioners include a
loop in their DCF model.
Assuming a perpetuity growth rate g of FCF from year 1 onwards and
a WACC equal to K:
EV = FCF.(1 + g)/K - g
and
K = k E/EV + i.(1 - [tau])D/EV
The reference to the Modigliani and Miller's adjusted cost of
capital enables to get rid of the loop. Indeed, as
K = [rho].(1 - D.[tau]/EV)
where [rho] is the cost of capital of the unleveraged firm with the
same sector risk. In other words, thanks to the Capital Asset Pricing
Model, CAPM,
[rho] = r + [[beta].sup.*].[E([R.sub.M]) - r]
where r is the risk free rate. Then
EV = FCF.(1 + g)/[rho].(1 - D.[tau]/EV) - g
and
EV = FCF.(1 + g) + D.[tau].p/[rho] - g
D.[tau] is the tax shield which is justified by the tax
deductibility of interests which are due on the assumed perpetual
financial debt. Indeed, in the Modigliani and Miller's theory,
D.[tau] results from the simplification of i.D.[tau]/i where i. D.[tau]
is the tax saving on interests and i the corresponding capitalization
rate. In that case, D is obviously the outstanding debt which can be
found in the last available financial statements.
When practitioners deduct D from the EV to obtain the equity value,
a closed form of E can be obtained:
E = FCF.(1 + g) + D.[g - [rho].(1-x)]/[rho] - g
The Modigliani and Miller's theory evidences that the spread
on the risky debt has no impact on the WACC and therefore on the equity
value: the cost of debt does not appear in the two previous formulas and
any increase of the spread of the debt corresponds to an increase of the
risk which is borne by the bondholders and banks. It is therefore
consistent with a decrease of the risk which is borne by the
shareholders. The Table below uses a simple example to show the risk
transfer between stakeholders and the unchanged WACC:
Table 1
The unchanged WACC according to Modigliani and Miller's theory
FCF 100 100
Perpetuity growth = g 3.00% 3.00%
Risk free rate = r 2.00% 2.00%
Market risk premium 7.00% 7.00%
Unleveraged beta = [beta] * 0.90 0.90
Cost of debt
Pretax cost of debt 3.40% 5.50%
Post tax @ 36.1% 2.17% 3.51%
Beta of the debt 0.20 0.50
Leveraged beta = [beta] 1.20 1.07
Cost of equity = k 10.38% 9.49%
WACC = K 7.11% 7.11%
P 8.30% 8.30%
Adjusted cost of capital 7.11% 7.11%
EV 2,509 2,509
Debt 1,000 1,000
Equity 1,509 1,509
For a FCF which is equal to 100, a risk free rate of 2%, a market
risk premium of 7% and an unleveraged beta of 0.9, 2 assumptions
regarding the pretax cost of debt are taken into account: 3.40%, based
on a debt's beta of 0.20 and 5.50% based on debt's beta of
0.50. The corresponding leveraged betas, based on the Hamada's
formula, are respectively 1.20 and 1.07 and the implied costs of equity
are respectively 10.38% and 9.49%. Then both WACC and adjusted costs of
capital are 7.11%. Then the enterprise value is the same in both cases:
2509.
A. Discount Rates
The WACC calculation is a bit subjective as a lot of assumptions
have to be taken into account: the market risk premium depends on a
assumption regarding the perpetuity growth rate of the listed
firms' dividends; when the firm is listed, the cost of equity can
either include its beta (which is different according to the data
provider) or a leveraged beta based on the industry's unleveraged
beta, which depends on the peers which have been included in the sample;
the weighting coefficients can correspond to a target--or
normative--financial structure or be based on an iterative calculation.
In that case, E is the outcome of the DCF valuation approach. In other
terms, each valuator can justify a specific ad-hoc discount rate.
B. Net Debt
The equity value (E) is the difference between the EV and the net
debt (D). The net debt is the difference between the financial debt on
the one hand, cash and cash equivalents on the other hand. The maturity
of the debt is not taken into account. Then if the EV is 100 and the
financial debt is 60, assuming no cash and cash equivalents, the equity
value will be 40, whether the debt matures tomorrow or in 2023. The
reason is that practitioners generally take the book value of the debt
(which corresponds to the face value under the assumption of no
repayment premium) instead of taking the economic value of the debt into
account whereas the financial theory relies on economic values of funds
provided by the firm's stakeholders. If the debt matures tomorrow,
its economic value is its face value but, if it matures in 2023, its
economic value is the present value of the bondholders expected cash
flows, the discount rate reflecting the bankruptcy risk of the firm. In
other terms, the firm's bankruptcy risk is not embedded in the debt
which is deducted from the EV and included in the weighting coefficients
of the WACC.
C. Free Cash Flow Computation
The DCF approach is based on FCF which are implicitly looked upon
as deterministic ones. They are discounted which enables to reduce the
weight of the remote ones in the EV. As the cost of equity, which is
embedded in the WACC, is based on the CAPM, the reference to the
firm's leveraged beta enables to refer to its systematic risk.
Moreover, a sensitivity analysis is generally provided by practitioners
in order to underline the uncertainty regarding the achievement of the
underlying business plans. However, the full risk, i.e., the total
volatility ([sigma]) of the FCF is not taken into account.
II. REVIEW OF LITERATURE
A. Black and Scholes
The financial literature dedicated to equity valuation based on
option starts with Black and Scholes' seminal article (1973). This
article presents a company which is financed by common stock and bonds
and whose only asset is shares of common stock in another company. The
bonds are zero-coupon and have a maturity of 10 years. The company plans
to sell all the stocks it holds at the end of 10 years, pay off the
bondholders if possible and pay any remaining money to the stockholders
as a liquidating dividend.
Under these conditions, the stockholders have the equivalent of an
option on the company's assets which has been provided to them by
the bondholders. At the end of 10 years, the equity value, w(x,t), is
the value of the assets, x, less the face value of the bonds or zero,
whichever is greater. Then, the economic value of the bonds is x -
w(x,t). If the company holds business assets instead of financial
assets, and if, at the end of the 10-year period, it issues new common
stock to pay off the bondholders (and pay any money that is left to the
old stockholders to retire their stocks), the economic value of bonds
remains x - w(x,t) where x is the enterprise value. Black & Scholes
underline that an increase in the company's debt, keeping the
enterprise value constant, increases the probability of default and thus
reduces the market value of the bonds. It hurts the existing bond
holders and helps the existing stockholders. Then the bond price falls
and the stock price increases. In this sense, changes in the capital
structure of a firm may affect the price of a common stock, when these
changes become certain, not when the actual changes take place.
B. Merton
Merton (1973) also considers the equity value as a call premium on
the company's assets in the background of the pricing of corporate
liabilities. For that purpose, the dynamics for the enterprise value,
through time, is described by a diffusion-type stochastic process with
the following stochastic differential equation:
dV = ([alpha]V - C).dt + [sigma]V.dz
where [alpha] is the instantaneous expected rate of return on the
firm per unit time, C is the total payouts by the firm per unit time to
either shareholders or liabilities-holders (eg.: dividends or interest
payments) if positive and the cash received by the firm from new
financing if negative, [[sigma].sup.2] is the instantaneous variance of
the return on the firm per unit time, dz is a standard Wiener process.
Moreover, F is the economic value of debt and D is the par value of the
debt, i.e., the amount the firm has promised to pay to the bondholders
on a specified calendar date.
In the event the payment of D is not met, the bondholders take over
the company and the shareholders receive nothing. If there are no
coupons, the PDE applied to D is:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Let F(V, [tau]) be the economic value of debt when the length of
time until maturity is [tau] and then F(V,0) = min(V,D). Let f(V, [tau])
be the economic value of equity when the length of time until maturity
is x and then f(V,0) = max(0;V-D) and: f(V, [tau])= V. [PHI] ([d.sub.1])
- [De.sup.-rt]. [PHI] ([d.sub.2]). As F = V - f:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Let
d = [D.e.sup.-r[tau]]/V
or
V/[D.e.sup.-r[tau]] = 1/d
Then
F = [D.e.sup.-r[tau]] [[PHI]([d.sub.2]) + 1/d.[PHI](-[d.sub.1])]
This formula enables to express the spread on the risky debt. In
that context, let R be the yield to maturity.
Then:
F = [D.e.sup.-R[tau]]
or
F/D = [e.sup.-R[tau]]
and
R = -1/[tau].ln F/D.
Therefore,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Finally,
R - r = spread = -1/[tau]ln [[PHI]([d.sub.2]) +
1/d.[PHI](-[d.sub.1])].
Merton (1973)'s pricing of corporate debt does not include any
enhancement of the enterprise value by the tax shield which is generated
by the tax deductibility of the financial expenses on debt. Such a
principle was pioneered by Modigliani and Miller (1963) who established
that the enterprise value of the leveraged firm is equal to that of the
unleveraged one increased by a tax shield. In that context, the
maximisation of the enterprise value can result from the maximisation of
the level of corporate debt. But, as reminded by Brennan and Schwartz
(1978) such a conclusion leads to the inconsistency between the premise
that management has to maximise the wealth of shareholders and the
empirical observation that most firms do not maximise their indebtness.
This discrepancy is justified by Modigliani and Miller themselves who
remind that retained earnings is a cheaper source of financing than debt
and insist on the need for preserving flexibility. Another explanation
regarding the limitation of the firm's leverage can be found in the
bankruptcy costs which weigh on the enterprise value, as highlighted by
Kraus and Litzenberger (1973).
C. Brennan and Schwartz
Brennan and Schwartz (1978) focus on the optimal capital structure
taking corporate tax and bankruptcy costs into account. They assume that
the enterprise value of the unleveraged firm, U, follows a GBM:
dU/U = [mu].dt + [sigma].dz
where dz is a standard Wiener process. The enterprise value of the
leveraged firm, V, is a function of the enterprise value of the
unleveraged firm (both firms having the same assets) on the one hand, of
the time t until maturity of the debt on the other hand. In other terms,
V=V(U,t). Then, the PDE is:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
On the maturity T of the debt:
V(U,T) = U for U [greater than or equal to] D
V(U,T) = U- C(U) for U < D
where C(U) corresponds to the bankruptcy costs if the firms files
for bankruptcy. Moreover, assuming that [t.sup.-] and [t.sup.+] denote
respectively the instants before and after the dividend payments, d:
V(U,t-) = V(U-d, [t.sup.+]) + d.
If the coupon payment, iD, is included and if [tau] is the
corporate tax rate:
V(U, [t.sup.-]) = V(U, [t.sup.+]) - i.(1 - [tau]).D+ i.D
where i.(1 - [tau]).D corresponds to the required capital increase
to restore the enterprise value of the leveraged firm after the coupon
payment. The development of the brackets and the simplification by i.D
provides:
V(U, [t.sup.-]) = V(U, [t.sup.+]) + i.D. [tau].
If the dividend and the coupon are paid on the same day:
V(U, [t.sup.-]) = V(U-d, [t.sup.+]) + d + i.D. [tau].
Finally, taking bankruptcy costs, C(U), into account:
V(U, [t.sup.-]) = V(U-d, [t.sup.+]) + d + i.D. [tau]. for U
[greater than or equal to] D V(U, [t.sup.-]) = [U.sub.-] C(U) for U <
D
These two last formulas correspond to boundary conditions to solve
the abovementioned PDE. But such an equation does not have a closed form
solution. This is why Brennan & Schwartz utilize numerical
techniques to determine an optimal leverage.
D. Leland
Leland (1994) solves the Brennan and Schwartz (1978)'s PDE
assuming that the firm has run into debt to perpetuity. Such an assumed
time independent environment is fully justified when the debt is rolled
over by a new one. F being the economic value of the claim, as in Merton
(1973)'s seminal paper, V the enterprise value and C the coupon
paid, per instant of time when the firm is solvent:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Then, if the claim has no time dependency:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The solving of such a PDE requires taking firstly the homogeneous
equation corresponding equation (ie without C) into account:
1/2 [[sigma].sup.2][V.sup.2]F"(V) + r.V.F'(V) - r.F(V) =
0
This reminds Dixit and Pindyck (1994)'s PDE--which is:
1/2.[[sigma].sup.2][V.sup.2]F"(V) +(r-[delta])V.F'(V) -
r.F(V> = 0] -for [delta] = 0.
In that case, the solutions of the characteristic equations are Pi
and P2 is that:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Then
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Then
[[beta].sub.2] = -2r/[[sigma].sup.2]
The solution of the homogeneous equation is therefore:
F(V) = [A.sub.1].[V.sup.1] + [A.sub.2].[V.sup.-X]
where X = -2r/[[sigma].sup.2].
Taking C into account, the general solution of the PDE is:
F(V) = [A.sub.0] + [A.sub.1].[V.sup.1] + [A.sub.2].[V.sup.-X]
where the constants of which (ie, [A.sub.0], [A.sub.1] and
[A.sub.2]) are determined by boundary conditions.
Let [alpha] be the fraction of value which will be lost to
bankruptcy costs, leaving debtholders with value (1 - [alpha]).[V.sub.b]
and stockholders with nothing, [V.sub.B] being the level of enterprise
value at which bankruptcy is declared. Then, the value of debt, D(V) is
such that:
If V = [V.sub.B], then D(V) = (1-[alpha]).[V.sub.B] (1)
If V [right arrow] + [infinity], then D(V) = C/r (2)
But, if V [right arrow] +[infinity], [V.sup.-X] = 0; then the
condition (1) requires that [A.sub.0] = C/r and [A.sub.1] = 0.
Moreover taking the condition (2) into account:
C/r + [A.sub.2].[V.sup.-X.sub.B] = (1 - [alpha]).[V.sub.B]
Then
[A.sub.2] = (1-[alpha]).[V.sub.B] - C/r/[V.sup.-X.sub.B]
and
D(V) = C/r + [(1-[alpha]).[V.sub.B] - C/r] [[V/[V.sub.B]].sup.-X]
Regarding the bankruptcy costs, BC:
If V = [V.sub.B], then BC(V) = [alpha].[V.sub.B] (3)
If V [right arrow] +[infinity], then BC(V) = 0 (4)
As, if V [right arrow] +[infinity], [V.sup.-X] = 0; then the
condition (4) requires that [A.sub.0] = 0 and [A.sub.1] = 0. Moreover
taking the condition (3) into account:
[A.sub.2.] [V.sup.-X.sub.B] = [alpha].[V.sub.B].
Then
[A.sub.2]. = [alpha]. [V.sub.B]/[V.sup.-X.sub.B]
and
BC(V) = [alpha].[V.sub.B]. [(V/[V.sub.B])(V/[V.sub.B]).sup.-X]
Regarding the tax benefits, TB:
If V = [V.sub.B], then TB(V) = 0 (5)
If V [right arrow] +[infinity], then TB(V) = [tau]C/r (6)
As, if V [right arrow] +[infinity], [V.sup.-X] = 0; then the
condition (6) requires that [A.sub.0] = [tau].C/r and [A.sub.1] = 0.
Moreover taking the condition (5) into account:
[tau].C/r + [A.sub.2].[V.sup.-X.sub.B] = 0.
Then
[A.sub.2] = -[tau].C/r/[V.sup.-X.sub.B]
and
TB(V) = [tau].C/r - [[tau].C/r].[[V/[V.sub.B]].sup.-X]
Finally, the enterprise value EV, taking bankruptcy costs and tax
benefits into account, is:
EV = V + TB(V) - BC(V) = V + [tau].C/r [1 -
[(V/[V.sub.B]).sup.-X]]-[alpha].[V.sub.B].[(V/[V.sub.B]).sup.-X]
And the equity value,
E(V) = EV - D(V)
Then
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Finally,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Furthermore, the expression of EV evidences that the asset value is
maximized by setting [V.sub.B] as low as possible, assuming it is not
imposed by a covenant. The value of [V.sub.B] which enables to maximize
the equity value, E(V), is such that:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
For V = [V.sub.B]:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and
VB = (1 - [tau]).C/r + [[sigma].sup.2]/2
Therefore, [V.sub.B] is independent from V and [alpha]. Moreover,
if r, [sigma] or [tau] increases, [V.sub.B] decreases; if C increases,
[V.sub.B] increases too.
E. Geske
Geske (1977) proposed a valuation formula of corporate liabilities
which includes n-1 individual coupon payments due before the principal
plus interest must be repaid. Such a valuation is based on the
generalisation of a compound options pricing model.
Geske (1979) used a compound options pricing model in which the
stock can be viewed as an option on the value of the firm. In this
setting, a call on the common stock is an option on an option. Let V be
the value of the firm (or its enterprise value), S the value of stock, D
the face value of debt and K the strike price of the call on equity. Let
[t.sup.*] be the expiration date of the call on equity and T the
maturity date of the debt. The following graphs illustrate Greske's
principles which drive to the compound option's premium
[TABLE 2 OMITTED]
At the intermediate date [t.sup.*], the call holder exercises his
option on the stock if the call is in the money, ie if [S.sub.t*]>K.
Otherwise, if [S.sub.t*]=K (or if [S.sub.t*]<K), the call on the
stock remains unexercised. As the value of the stock, S, depends on the
value of the firm's assets, V, such a situation happens when the
enterprise value V is equal to (or lower than) [V.sup.*]. Then,
[V.sup.*] is the value of V such that [S.sub.[tau]] - K = 0. In other
terms, the call holder pays K on t = [t.sup.*] if, at this date, V>V*
in order to keep the possibility to pay M on t=T in order to get the
firm's assets. In that case:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and
[b.sub.2] = [b.sub.1] - [[sigma].sub.V]. [square root of
[[tau].sub.2]]
where N(.) and [PHI] are respectively the bivariate and univariate
cumulative normal distribution functions.
F. Charitou and Trigeorgis
Charitou and Trigeorgis (2004) transpose Geske's compound
option pricing model when a coupon interest I comes due at an
intermediate date [t.sup.*], while the debt's maturity is T. If, on
t=[t.sup.*], V is lower than V* such that E([V.sup.*],
[[tau].sub.1])-I=0, the stockholders voluntarily default on the interest
payment, I. In that case, based on Geske's notations, K is replaced
by I and C is replaced by E which corresponds to the equity value.
Indeed, the stockholders have the option to pay I, at the intermediate
date [t.sup.*], to keep the possibility to pay M on t=T in order to get
the firm's assets. In that case:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and the probability of default on the intermediate date T is
P[E<I] = P[V<V*] = [PHI](-[a.sub.2]) where [a.sub.1], [a.sub.2],
[b.sub.1], [b.sub.2] and [rho] have the values which have been defined
by Geske (1979).
III. EMPIRICAL STUDY
A. Database
A sample of 40 firms belonging to the French CAC 40 has been set
up. For each firm, 4 data as at 22/02/2013 have been extracted from the
Facset financial data base: the market capitalization, the brokers'
consensus on EV, the brokers' consensus on the target price and the
volatility of the shares which corresponds to the standard deviation of
the return over one year. For most brokers, the enterprise value is the
output of a DCF valuation. Each daily volatility has then been
multiplied by [square root of 252] in order to turn it into a yearly
one. The French risk free rate, paid on 10-year T-Bonds, is 2.20% which
corresponds to 2.18% in continuous time. Moreover, the financial last
available financial debt has been found in each firm's balance
sheet as at 31/12/2011.
The financial institutions (Axa, Credit Agricole, BNP Paribas and
Societe Generate) can be valued by brokers taking a cash flow approach
into account. But in that case, the cash flow corresponds to the excess
equity which could be paid out to the stockholders taking solvency
constraints into account. Then the cash flow is a theoretical dividend
(hence the "dividend discount model" which is given to such a
valuation approach) and the sum of the present value of the forecasted
free cash flows enables to get directly the equity value, the discount
rate being the cost of equity. For insurance companies, like Axa, the
target Solvency 1 ratio, which is consistent with the company's
targeted rating, is taken into account. For banks, the target common
equity tier 1 ratio, which is required by the Basel 3 regulation, is
taken into account. Anyway, for insurance companies as for banks, no
enterprise value is embedded in the valuation.
When the firm has a negative net debt, the DCF approach changes
from a broker to another: some calculate the present value of free cash
flows based on the cost of equity, others calculate algebraically the
WACC with a negative net debt. In order to get rid of such possible
discrepancies from a methodology point of view, the 5 firms with a
negative net debt as at 31/12/2011--Cap Gemini, EADS, L'Oreal,
STMicroelectronics, Technip--have been excluded. Moreover, Renault,
whose weight of the consumer credit business in the accounts is
significant, has also been excluded from the sample. At the end of the
day, the empirical study is based on 30 firms.
From a Black & Scholes-Merton's perspective, the
brokers' consensus on enterprise value corresponds to the price of
the underlying asset and the strike price to the amount of the debt in
the accounts. Beyond that, the application of the option pricing models
requires 2 other parameters: the time to expiration and the underlying
asset's volatility. In the Black & Scholes and Merton's
seminal papers, the debt is a zero coupon. Then, the option's time
to expiration corresponds to the residual maturity of the bond. For most
firms, the debt is made of bonds with coupons and financial borrowings
from banks. From a theoretical point of view, a compound option with
several maturities should be taken into account. However, in order to
apply the Black-Scholes-Merton's pricing model, an average residual
maturity of each company's debt has been calculated as a proxy of
the time to expiration [tau].
The underlying asset's volatility corresponds, from a
corporate finance point of view, to the enterprise value's
volatility. But the assets are rarely listed, except for holding
companies which are not represented in the CAC 40 index. If they are not
listed, their value (ie spot EV) and their volatility have to be
estimated.
The EV and its volatility is based on the methodology which is
proposed by Hull, Nelken and White (2004) and commonly used by
Moody's rating agency. Based on Ito's lemma:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Then with F = E (for equity), x=V (for enterprise value), a(x,t) =
m.V and b(x,t) = [[sigma].sub.V]V
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
In that, case:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and
[[sigma].sub.V] V. [partial derivative]E/[partial derivative]V =
[[sigma].sub.E]E
Finally,
[[sigma].sub.E]E = [[sigma].sub.V]V.[PHI]([d.sub.1])
Moreover, thanks to the Merton's formula:
E = V. [PHI] ([d.sub.1]) - [De.sup.-rt]. [PHI] ([d.sub.2]).
The values of V and [[sigma].sub.V] can be obtained thanks to
Excel's solver applied to the following nonlinear system:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Only the obtained value of [[sigma].sub.V] is taken into account,
as the enterprise value is based on the DCF approach.
The following table provides the detailed calculation of
Merton's economic values of debt and equity based on the previous
example assumptions regarding the EV (2509) and risk free rate (2%). The
assets' volatility is supposed to be 30% and the average maturity
of the financial debt is 5 years.
The table below evidences that the DCF's equity value (1509)
is obtained only if the time to expiration, i.e., the average residual
maturity of the financial debt is nil. Otherwise, the longer the
maturity is, the higher the time premium and therefore the equity value
is. Moreover, the higher the volatility is, the more important the
likelihood of an increase in the share price is which implies a higher
equity value. Such calculations have been made for the 30 firms
belonging to the [sample.sup.12].
B. Empirical Models
The empirical study is focused on the growth potential of the stock
price of healthy listed firms which belong to the French CAC 40. Such a
growth potential can be based on brokers' target prices which can
be compared to the listed prices of stocks. In that case, the target
price is the enterprise value, which corresponds to the present value of
future free cash flows, as determined by brokers, reduced by the net
debt that can be found in the accounts. But such a net debt, which is
based on its face value without taking its maturity into account and
therefore the probability of bankruptcy, may be overestimated. The
growth potential of the stock price may increase, should the equity
value be based on Black & Scholes-Merton in order to include the
bankruptcy risk which depends on the debt's face value but also on
its maturity and the assets' volatility. The Black &
Scholes-Merton approach provides a new breakdown of the DCF enterprise
value (V) between equity and net debt economic values. In that case:
Equity value = Brokers' EV - [V. [PHI] (-[d.sub.1]) + [De.sup.-rt].
[PHI] ([d.sub.2])-cash and equivalents] where:
[d.sub.1] = 1n(V/D) + (r +
[[sigma].sup.2.sub.V]/2).[tau]/[[sigma].sub.V].[square root of [tau]]
and
[d.sub.2] = [d.sub.1] - [[sigma].sub.v].[square root of [tau]]
The comparison between both growth potentials may be explained by
the corresponding leverage ratios. For that reason, the net debt to EV
is calculated based on the net debt which is in the accounts on the one
hand, on the economic value of the net debt which is given by the Black
and Scholes-Merton's model on the other hand. These ratios are
respectively noted D/EV and B/E[V.sup.1]. An alternative to
Merton's debt economic value, B, is the following breakdown:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and
B = [D.e.sup.-r[tau]] - [phi](-[d.sub.2]). [[D.e.sup.-r[tau]] -
[phi](-[d.sub.1])/[phi](-[d.sub.2]).V]
where [phi](-[d.sub.1])/[phi](-[d.sub.2]).EV is the amount of debt
which will be recovered by the bondholders should the firm file for
bankruptcy. Then [phi](-[d.sub.1])/[phi](-[d.sub.2]) is the recovery
rate and [D.e.sup.-r[tau]] - [phi](-[d.sub.1])/[phi](-[d.sub.2]).V is
the expected discounted loss which will be borne by the bondholders
given the assumed default of the firm. As [phi](-[d.sub.2]) is the
probability of bankruptcy, [phi](-[d.sub.2]). [[D.e.sup.-rt] -
[phi](-[d.sub.1])/[phi](-[d.sub.2]).V] is the expected discounted
shortfall. Finally, as used by Moody's KMV and the risk departments
of banks in the background of risk weighted assets calculations:
Value of debt = par value of debt--probability of default x
expected discounted LGD, where LGD means "Loss Given Default".
Gemalto and Legrand, which have a probability of default of 0%, are
excluded of the sample for this part of the analysis that is therefore
limited to 28 firms.
The 3 main parameters of the economic value of the net debt seem to
be its maturity ([tau]), the recovery rate given default
([phi](-[d.sub.1])[phi](-[d.sub.2]), which includes the probability of
default and the weight of its face value which be expressed as a
percentage of the enterprise value (D/EV). In that context, a multiple
regression is tested in order to explain the growth potential based on
the Black & Scholes Merton's equity [value.sup.12].
C. Empirical Results
1. Equality test of assets' and equities' volatilities
The means of the stocks and assets volatilities are respectively
28% and 22%. The significance of the 6% discrepancy can be tested using
the data provided in the following tables. The table is dedicated to the
equality test of variances.
If the variances are equal, the ratio of the standard variances
obeys a Fisher-Snedecor's distribution:
[S.sup.2.sub.x]/[S.sup.2.sub.Y] [right arrow] F ([n.sub.p] - 1;
[n.sub.Q] - 1)
where [n.sub.P]=30 and [n.sub.Q]=30. Hence:
T = [S.sup.2.sub.x]/[S.sup.2.sub.Y] [right arrow] F(29; 29)
The Fischer-Snedecor's table provides: P[T>1.86] = 5%. In
other words, if the variances are equal, T has a 5% probability to be
higher than 1.86. By experimentation, [t.sup.*.sub.0] = 1.41 < 1.86.
Hence, with a 5% error risk, the variances of the volatilities of the
stocks on the one hand, of the assets on the other hand, are equal. Then
a Student's test enables to know whether the stocks' and
assets' volatilities are significantly different. The table below
is dedicated to such a test:
If the means are equal, the following ratio obeys a Student's
distribution:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where [n.sub.P]=30 and [n.sub.Q]=30. Hence, T [right arrow] S(58).
The Student's table provides: P[-2.00<T<2.00]=95%. In other
words, if the means are equal, T has a 95% probability to be in a
[-2.00; 2.00] range. By experimentation, [t.sub.0] = 3.76 > 2.00.
Then, with a 5% error risk, the means of the volatilities of the stocks
on the one hand, of the assets on the other hand, are different. Such a
conclusion justifies the determination of the assets' volatilities
to use the Black & Scholes-Merton's approach of equity
valuation.
2. Equality test of stock prices' potential growth based on
brokers' and Black and Scholes-Merton's approach
The means of the growth potential based on brokers' target
prices and Black & Scholes-Merton's approach of equity
valuation are respectively 7.5% and 13.7%. The significance of the 6.2%
discrepancy can be tested thanks to data provided in the following
tables. The table bellows is dedicated to the equality test of
variances.
As in the former equality test of variances, if the variances of
growth potentials are equal, T = [S.sup.2.sub.x]/[S.sup.2.sub.Y] has a
5% probability to be higher than 1.86.
By experimentation, [t.sup.*.sub.0] = 0.20 < 1.86. Hence, with a
5% error risk, the variances of the potential growth based on brokers on
the one hand, on the Black and Scholes-Merton's approach on the
other hand, are equal. Then a Student's test enables to know
whether the average growth potentials are significantly different. The
table below is dedicated to such a test.
If the means are equal, the following ratio obeys a Student's
distribution: T = [right arrow] S(58)as in the former equality test of
means. The Student's table provides: P[-2.00 < T < 2.00]=95%.
By experimentation, [t.sup.*.sub.0] = -1.37. Then [t.sup.*.sub.0] is
obviously in the [-2.00; 2.00] range. Hence, with a 5% error risk, the
means of the potential growth based on brokers on the one hand, on the
Black & Scholes-Merton's approach on the other hand, are equal.
The difference between the brokers' and the Black &
Scholes-Merton's approaches corresponds to the net debt's
amount which is deducted from the DCF's enterprise value. The
reason of such a result is that the firms belonging to the CAC 40 index
are healthy ones. Then, their probability of default is very low,
narrowing 0%. In that case, [PHI] (-[d.sub.2]) = 0 which means that
[PHI] ([d.sub.2]) = 1 which happens when [d.sub.2]=+[infinity]. Then
d1=+[infinity] too which implies that [PHI] ([d.sub.1]) = 1. Based on
the Black & Scholes-Merton formula that: E=S - [D.e.sup.-rt]. As
[tau] is relatively low, E is not far from S - D.
The explanation of the equality of growth potentials can be
completed by a statistical test of equality of leverage ratios which
correspond to net debt / enterprise value.
3. Equality test of leverage ratios based on the net debts in the
firms' accounts and on recalculated net debts including Black and
Scholes-Merton's approach
The means of the leverage ratios based on brokers' target
prices and Black & Scholes-Merton's approach of equity
valuation are respectively 25% and 18%. The significance of the 7%
discrepancy can be tested thanks to data provided in the following
tables. The table bellows is dedicated to the equality test of
variances.
As in the former equality tests of variances, if variances of the
leverage ratios are equal, T = [S.sup.2.sub.x]/[S.sup.2.sub.Y] has a 5%
probability to be higher than 1.86. By experimentation, [t.sup.*.sub.0]
= 2.08 > 1.86. Hence, with a 5% error risk, the variances of the
potential growth based on brokers on the one hand, on the Black &
Scholes-Merton approach on the other hand, are different. Then an Aspin
Welch's test enables to know whether the average leverage ratios
are significantly different. Table 11 is dedicated to such a test.
If the means are equal, the following ratio obeys a Student's
distribution:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The Student's table provides: P[-2.01<T<2.01]=95%. By
experimentation, [t.sup.*.sub.0] = 1.47. Then [t.sup.*.sub.0] is
obviously in the [-2.01 ; 2.01] range. Hence, with a 5% error risk, the
means of the leverage ratios based on brokers on the one hand, on the
Black & Scholes-Merton approach on the other hand, are equal.
4. Multiple regression to explain the growth potential of the stock
price
Let g be the listed stock's growth potential, RRGD the
recovery rate given default, D/EV the net debt in accounts to enterprise
value and [tau] the maturity of the financial debt. The second table
below indicates that: g = 2.08.RRGD - 0.85.D/EV + 0.05. [tau] - 0.47.
The coefficient of determination, [R.sup.2], is around 0.8 which is high
but such a regression is justified only if the 4 coefficients are
significantly different from 0.
The first table below enables to test whether the 4 coefficients
are simultaneously equal to 0. Under the assumption RRGD = D/EV = [tau]
= 0, the F stat obeys a Fisher-Snedecor distribution F(k;n-k-1) with k=3
and n=28. Then F [right arrow] F(3;24). The Fisher-Snedecor's table
provides: P[F>3.01] = 5%. In other words, if the 4 coefficients are
simultaneously equal to 0, F has a 5% probability to be higher than
3.01. By experimentation, [t.sup.*] = 34.62 > 3.01. Hence, with a 5%
error risk, the 4 coefficients are not simultaneously equal to 0.
The second table enables to test whether each of the 4 coefficients
is equal to 0. For each coefficient a, if a=0 then the T stat obeys a
Student distribution with n-k-1 degrees of freedom. Here, k=3 and n=28.
Then T [right arrow] S(24). The Fischer-Snedecor's table enables to
get P[-2.06<T<2.06] = 95%. In other words, if a coefficient is
equal to 0, T has a 95% probability to be in a [-2.06; 2.06] range. By
experimentation: [t.sup.*](c) = 5.76 < -2.06 where c is a constant,
[t.sup.*](RRGD) = 4.71 > 2.06, [t.sup.*](D/EV) = -2.44 < -2.06,
[t.sup.*](t) = 3.51>2.06. Hence, with a 5% error risk, none of the 4
coefficients is equal to 0.
IV. CONCLUSION
The Discounted Cash Flow Valuation Method, DCF approach seems to
undervalue the stock prices as the net debt, which is deducted from the
firm value, can be found in the accounts of the firms, whereas it should
be an economic value. For the CAC 40 non-financial firms, the growth
potential based on a DCF including the economic value of the net debt is
in average not meaningfully different. The economic value of the net
debt is based on the Black & Scholes-Merton's model, which
enables to take the probability of default, the maturity of the debt and
the assets' volatility into account. But, in the case of CAC 40
companies, the probability of default is very low and the debt's
average maturity is relatively limited. In that case, the
Black-Scholes-Merton's additional value is not significant. This
result is confirmed by the comparison of the leverage ratios which are
not meaningfully different when based on the net debt in the accounts
and on the economic value of the net debt. However, the growth potential
is explained by the main parameters and an output of the
Black-Scholes-Merton's model, namely the face value of debt, its
maturity and the recovery rate given default. Finally, the growth
potential which can be explained is not underestimated. Our analysis
provides a basis for future research and can be used in other financial
markets.
ENDNOTE
(1.) For detailed information, please contact the authors.
REFERENCES
Black, F., and M. Scholes, 1973, "The Pricing of Options and
Corporate Liabilities", Journal of Political Economy, Vol. 81, No.3
(May-June), pp. 637-654.
Brennan, J., and E. S. Schwartz, 1978, "Corporate Income
Taxes, Valuation and the Problem of Optimal Capital Structure",
Journal of Business, Volume 51, Issue 1, pp. 103-114.
Charitou, A., and L. Trigeorgis, 2004, "Explaining Bankruptcy
Using Option Theory", working paper.
Dixit A., and R. Pindyck, 1994, Investment under Uncertainty,
Princeton University Press.
Geske, R., 1979, "The Valuation of Compound Options",
Journal of Financial Economics, 7, pp. 63-81.
Geske, R., 1977, "The Valuation of Corporate
Liabilities", Journal of Financial and Quantitative Analysis,
November, pp. 541-552.
Hull, J. C., I. Nelken, and A. White, 2004, "Merton's
Model, Credit Risk and Volatility Skews", Journal of Credit Risk,
Vol. 1, No1, pp. 3-28.
Kraus, A., and R.H. Litzenberger, 1973, "A State Preference
Model of Optimal Financing Leverage", Journal of Finance, Volume
28, pp. 911-922.
Leland, H.E., 1994, "Corporate Debt Value, Bond Covenants and
Optimal Capital Structure", Journal of Finance, Volume 49, Issue 4,
pp. 1213-1252.
Levyne, O., and J.M. Sahut, 2008, "Options Reelles",
Dunod.
Merton, R.C., 1973, "On the Pricing of Corporate Debt: The
Risk Structure of Interest
Rates", presentation at the American Finance Association
Meetings, New York.
Modigliani, F., and M.H. Miller, 1963, "Taxes and the Cost of
Capital: a Correction", American Economic Review, Volume 53, Issue
1, pp. 433-443.
Olivier Levyne (a) and David Heller (b)
(a) Professor of Finance at ISC Paris, Ph.D. in Economics, Advanced
Ph.D. (French HDR) in Management Science, ISC Paris Business School,
France Olivier.levyne@iscparis.com
(b) Ph.D. student, ISC Paris Business School, France David.
heller@iscparis.com
Table 3
Firms characteristics
Name Market Cap Target
equity value
Accor S.A. 6,414 6,599
Air Liquide S.A. 28,942 31,507
Alstom S.A. 10,170 10,846
ArcelorMittal SA 19,185 23,427
Bouygues S.A. 6,517 7,362
Carrefour S.A. 14,894 15,209
Compagnie St-Gobain 16,720 18,169
Danone S.A. 33,932 32,517
Electricite de France S.A. 27,030 29,788
Essilor International S.A. 16,261 16,381
France Telecom 19,922 26,652
GDF Suez S.A. 35,444 41,789
Gemalto N.V. 6,001 6,197
Lafarge S.A. 14,114 14,891
LeGrand S.A. 9,229 8,398
LVMH 67,241 77,707
Michelin 12,124 14,427
Pernod Ricard S.A. 25,969 25,025
PPR S.A. 21,635 20,855
Publicis Groupe S.A. 10,601 10,151
SAFRAN S.A. 14,260 15,000
Sanofi S.A. 96,611 103,893
Schneider Electric S.A. 32,727 31,138
Solvay S.A. 9,292 9,454
Total S.A. 90,047 105,145
Unibail-Rodamco SE 16,464 16,812
Vallourec S.A. 5,291 5,000
Veolia Environnement S.A. 4,774 5,472
Vinci S.A. 20,640 25,363
Vivendi 20,449 24,258
Name EV Equity's
Consensus volatility
[[sigma].sub.E]
Accor S.A. 7,074 29%
Air Liquide S.A. 34,666 20%
Alstom S.A. 12,777 36%
ArcelorMittal SA 33,957 39%
Bouygues S.A. 11,541 32%
Carrefour S.A. 20,306 33%
Compagnie St-Gobain 24,221 34%
Danone S.A. 39,300 21%
Electricite de France S.A. 67,684 28%
Essilor International S.A. 16,356 20%
France Telecom 51,448 27%
GDF Suez S.A. 79,613 25%
Gemalto N.V. 5,567 28%
Lafarge S.A. 24,324 33%
LeGrand S.A. 10,157 23%
LVMH 70,663 26%
Michelin 13,998 31%
Pernod Ricard S.A. 33,425 19%
PPR S.A. 22,942 25%
Publicis Groupe S.A. 10,372 19%
SAFRAN S.A. 15,092 24%
Sanofi S.A. 103,138 22%
Schneider Electric S.A. 35,684 34%
Solvay S.A. 12,156 33%
Total S.A. 108,524 21%
Unibail-Rodamco SE 28,495 20%
Vallourec S.A. 6,653 41%
Veolia Environnement S.A. 17,728 39%
Vinci S.A. 33,963 28%
Vivendi 34,146 30%
Table 4
Merton approach based on DCF
Check EV (no loop on K) 2,509
D 1,000
[sigma] of assets 30%
T 5.00
R 2.00%
d1 1.86
d2 1.18
F(d1) 0.97
F(d2) 0.88
F(-d1) 0.03
Economic value of debt 878
Equity 1,631
Table 5
Sensitivity of equity value
t Volatility of assets
1631 0% 10% 20% 30% 40% 50%
0 1,509 1,509 1,509 1,509 1,509 1,509
5 1,604 1,604 1,606 1,631 1,684 1,754
10 1,690 1,690 1,703 1,764 1,856 1,958
15 1,768 1,768 1,793 1,876 1,986 2,098
20 1,838 1,839 1,872 1,968 2,087 2,199
25 1,902 1,903 1,942 2,046 2,165 2,272
Table 6
Equality test of variances (F-test)
[[sigma].sub.E] [[sigma].sub.V]
Mean 28.0% 22.2%
Variance 0.4% 0.3%
Observations 30 30
Degrees of freedom 29 29
F 1.41
P(F<=f) unilateral 0.18
Critical value for F 1.86
(unilateral)
Table 7
Equality test of means: 2 observations with equal variances
[[sigma].sub.E] [[sigma].sub.V]
Mean 28.0% 22.2%
Variance 0.4% 0.3%
Observations 30 30
Weighted variance 0.4%
Hypothetical means difference 0
Degrees of freedom 58
Stat t 3.76
P(T<=t) bilateral 0.00
Critical value for F 2.00
(bilateral)
Table 8
Equality test of variances (F-test)
g brokers g vs BS' E
Mean 7.5% 13.7%
Variance 1.1% 5.1%
Observations 30 30
Degrees of freedom 29 29
F 0.20
P(F<=f) unilateral 0.00
Critical value for 1.86
F (unilateral)
Table 9
Equality test of means: 2 observations with equal variances
g brokers g vs BS' E
Mean 7.5% 13.7%
Variance 1.1% 5.1%
Observations 30 30
Weighted variance 3.1%
Hypothetical means difference 0
Degrees of freedom 58
Stat t -1.37
Critical value for 2.00
F (bilateral)
Table 10
Equality test of variances (F-test)
D/EV B/EV
Mean 25.0% 18.5%
Variance 3.9% 1.9%
Observations 30 30
Degrees of freedom 29 29
F 2.08 0
P(F<=f) unilateral 0.03 0
Critical value for F (unilateral) 1.86
Table 11
Equality test of means: 2 observations with different variances
D/EV B/EV
Mean 25.0% 18.5%
Variance 3.9% 1.9%
Observations 30 30
Hypothetical means difference 0
Degrees of freedom 52
Stat t 1.47
Critical value for F (bilateral) 2.01
Table 12
Variance analysis
Degrees of Sum of Mean of F F
freedom squared squared critical
value
Regression 3 1.14 0.38 34.62 0.00
Residual figures 24 0.26 0.01
Total 27 1.41
Table 13
T-test of the four coefficients
Coefficients Error t stat =
type [t.sup.*]
Constant -0.47 0.08 -5.76
RRGD 2.08 0.44 4.71
D/EV -0.85 0.35 -2.44
T 0.05 0.01 3.51
Probability Lower limit Lower limit
for 95% for 95%
confident confident
threshold threshold
Constant 0.00 -0.64 -0.30
RRGD 0.00 1.17 2.99
D/EV 0.02 -1.58 -0.13
T 0.00 0.02 0.07
