Memorandum on beta relations
In connection with unlevering and relevering of beta, we use a relation to describe the
correlation between beta for an unlevered company β U and beta for the company with
a specific capital structure β L. This memorandum will illustrate the problem that
arises from the use of the common beta relation based on the work of Modigliani and
Miller (M&M) in 1963 1 . We will argue that there is an alternative to solve these
problems based on more realistic assumptions.
Please note that irrespective of the beta relation used, it is an implicit assumption that
the company is able to utilise the tax shield fully. It is often assumed that unutilised
tax shields may be traded freely in the market.
Furtermore, we have, to make the example as simple as possible, assumed that β d
equals zero, meaning that the debt is risk-free and therefore has a required return
equal to the risk-free interest rate, and no investor taxes have been included.
M&M’s beta relation
The use of M&M’s beta relation is based on the implicit assumptions that: 2
•
•
the actual amount of the debt is known and remains constant in perpetuity
future cash flows remain constant in perpetuity.
Our memorandum will illustrate the consequences if the above assumptions are not
met.
The beta relation may be described as:
D

β L = βU ⋅ 1 + (1 − Tc ) ⋅ 
E

With:
βL = Beta for the levered company
βU = Beta for the fully equity- financed company
Tc = Corporation tax rate
D = Market value of debt
E = Market value of equity
We will value a company by means of two valuation methods:
1. The Adjusted Present Value (APV) method, under which the value is split
into two parts. First the company is valued without taking into account the
financing decision, and then the value of the interest tax shields resulting from
the financing decision is calculated. The value estimated under the APV
1
2
See Modigliani & Miller (1963)
M&M’s beta relation is also known as Hamada’s (1969) formula
model is the correct value as the value is estimated based on the assumptions
underlying the use of the beta relation in question.
2. A traditional DCF method, under which future cash flows are discounted at
WACC. The value of the interest tax shield is therefore implicitly included in
the discount rate.
General assumptions:
•
•
•
•
Risk- free interest rate 5% (Rf )
Market risk premium 4% (RP)
Beta unlevered 1 (β U)
Corporation tax rate 30% (Tc)
The fictitious company has a life of 5 years. The free cash flows and the amount of
net interest-bearing debt (NIBD) appear from the following table.
Year
FCF
NIBD
0
300
1
120
250
2
150
200
3
160
150
4
170
100
5
200
0
Re. 1: The APV method
As it is assumed that the amount of the debt is known in perpetuity, the interest tax
shields may be considered risk-free payments and may therefore be discounted at the
risk- free interest rate.
Year
0
Interest tax shield
Required return from a 100% equity-financed company
PV (FCF)
PV(interest tax shield)
Market value from investment decision
610.31
Market value from financing decision
13.30
Market value of the company (EV)
623.62
Market value (EV) ultimo
623.62
1
4.50
9.00%
110.09
4.29
2
3.75
9.00%
126.25
3.40
3
3.00
9.00%
123.55
2.59
4
2.25
9.00%
120.43
1.85
5
1.50
9.00%
129.99
1.18
554.71
450.51
327.80
184.91
0.00
We have calculated below the value of the company by means of the APV model.
The interest tax shield has been calculated as:
Interest-bearing debt (opening) x Rf x Tc
The required return from a fully equity- financed company has been calculated as:
Rf + (β U x RP)
The cash flow is discounted at 9%, and the interest tax shield is discounted at the riskfree interest rate of 5%. This results in an enterprise value of 623.62, with a value
breakdown of 13.30 from the financing decision and 610.31 from the investment
decision.
Re 2: The DCF method
The other method for calculating the company’s value is the traditional DCF model.
As the capital structure does not remain constant over time, a variable required return
must be applied.
The following formula is used for calculating the debt ratio:
Debt structure =
Interest bearing debt primo
Market Value ( EV ) primo
Ex.
Debt structuret =1 =
300.00
= 48.11%
623.62
The following formula is used for calculating WACC:
WACC = Debt structure * (1 − Tc ) * R f + (1 − Debt structure) * ( R f * β L * RP )
Ex.
WACCtid=1 = 48.11% * (1 − 30%) * 5.00 % + (1 − 48.11%) * ( 5.00% + 1.65 * 4.00%) = 7.70%
We have calculated below the value of the company with β L in year 1 calculated as:
β L = 1* (1 + (1 − 30%) * 92.70%) = 1.65
Year
Debt structure
D/E
Beta (L)
WACC
PV(FCF)
Market value (EV) ultimo
0
1
48.11%
92.70%
1.65
7.70%
111.42
2
45.07%
82.05%
1.57
7.78%
129.22
3
44.39%
79.84%
1.56
7.80%
127.86
4
5
45.76% 54.08%
84.36% 117.77%
1.59
1.82
7.76%
7.54%
126.06 137.91
632.46
You will note that the value estimate of the DCF model deviates 8.85 from the correct
value calculated according to the APV model. The deviation is due to assumptions not
being met: the company’s cash flows are not constant and infinite, and the company’s
debt does not remain constant in perpetuity.
The conclusion is that use of the DCF model will result in miscalculation when the
assumptions underlying the beta relation applied are not met. The reason is that the
WACC used for discounting in the DCF model does not treat the interest tax shield
consistent ly with the assumption of risk-free interest tax shields.
In practice, we very rarely see situations where cash flows are an infinite annuity and
the debt remains constant in perpetuity, which must apply in order to use M&M’s beta
relation. We will introduce below an alternative beta relation that is not based on the
assumption of a constant debt level and cash flows being an infinite annuity. The
alternative beta relation has been described by Harris & Pringle (H&P) 3 .
H&P’s beta relation
The use of H&P’s beta relation is based on the implicit assumption that:
•
it is the company’s goal to maintain a constant debt ratio, i.e. that the company
continuously rebalances the debt in order for the debt to constitute a constant
share of the company’s value 4 . The future debt level will therefore not be
known but will depend on operations; accordingly, the correct discount rate
must reflect the operating risk of the company.
On this assumption, the beta relation will look as follows 5 :
D

β L = β U ⋅ 1 + 
E

Again the value of the company is calcualted by means of the two methods. The
notation remains the same as previously.
Re 1: The APV method
Year
0
Interest tax shield
Required return from a 100% equity-financed company
PV (FCF)
PV(interest tax shield)
Market value from investment decision
610.31
Market value from financing decision
12.17
Market value of the company (EV)
622.48
Market value (EV) ultimo
622.48
1
4.50
9.00%
110.09
4.13
2
3.75
9.00%
126.25
3.16
3
3.00
9.00%
123.55
2.32
4
2.25
9.00%
120.43
1.59
5
1.50
9.00%
129.99
0.97
554.01
450.12
327.63
184.86
0.00
As it is assumed that both cash flows and interest tax shields carry the operating risk,
both are discounted at 9%. It should be noted that only the value of the financing
decision has been changed from the previous calculation.
3
See Harris & Pringle (1982).
The constant debt ratio is only decisive if a constant WACC is to be used, the crucial thing is that the
amount of the debt is subject to uncertainty.
5
If the beta for loan capital is not zero, the beta relation will be: β L = β U + ( β U − β D ) ⋅ D
E
4
Re 2:The DCF method
As in the previous calculation, we may by using H&P’s beta relation estimate the
required owner’s return and therefore WACC. As the capital structure varies over
time, we will as in the previous calculation operate with a variable WACC.
The formulas for calculating the debt ratio and WACC are identical to those used in
the calculation based on the M&M assumption. The DCF model results in the
following value.
Year
Debt structure
D/E
Beta (L)
WACC
PV(FCF)
Market value (EV) ultimo
0
1
2
3
4
5
48.19% 45.13% 44.43% 45.78% 54.09%
93.03% 82.24% 79.96% 84.45% 117.84%
1.93
1.82
1.80
1.84
2.18
8.28%
8.32%
8.33%
8.31%
8.19%
110.827 127.889 125.921 123.523 134.322
622.48
You will note that there is consistency between the APV and DCF models.
Empirical survey
In September 2001 we asked a number of players in the financial sector and major
consulting firms in Denmark which beta relation they use 6 .
40% of the respondents use a beta relation based on the assumption that the interest
tax shield is risk- free. It should also be noted that 27% do not use any type of beta
relation. The responses are summarised in the following table:
Beta relations
Percentage of responses
Number of responses
Harris &
Pringle
Miles &
Ezzell
Miller &
Modigliani
No
relation
Other
relation
7%
13%
40%
27%
13%
1
2
6
4
2
In addition to the beta relations mentioned above, our survey included another beta
relation described by Miles & Ezzell (1980) (M&E). This beta relation is based on
annual rebalancing of the debt ratio to the effect that the first interest tax shield is
known for certain, and the remaining interest tax shields are unknown. The beta
values arrived at by using M&E are very close to the values reached by using H&P,
and we will therefore not describe this relation in further detail.
Conclusion
It is remarkable that the most commonly used beta relation is based on assumptions
that are so restrictive that they will rarely be met in practice. Furthermore, it
intuitively feels wrong that all future interest tax shields are risk-free. In our opinion,
6
This was one of several questions asked in relation to “cost of capital”.
it is a problem that consistency beetween the APV and DCF models cannot be
achieved when using the M&M beta relation. We will therefore not be able directly to
demonstrate the value that we attribute to the interest tax shield.
Michael B. Hansen/ PricewaterhouseCoopers
Jacob Erhardi/ PricewaterhouseCoopers
Bibliography:
Chambers, D.R. & Harris, R.S. & Pringle, J.J. (1982)
Treatment of Financing Mix in Analyzing Investment Opportunities
Financial Management 11 (Summer 1982)
Genoptrykt i Parum, C. (1999)
Miles, J.A. & Ezzell, J.R. (1980)
The weighted Average Cost of Capital, Perfect Capital Markets and Project
Life
The journal of Financial and Quantitative Analysis, Vol 15, nr. 3
Genoptrykt i Parum, C. (1999)
Modigliani, F. & Miller, M.H. (1963)
Corporate Income taxes and the cost of Capital: a correction
The American Economic Review, Vo l LIII, no. 3
Genoptrykt i Parum, C. (1999)
Taggart, R.A. (1989)
Consistent Valuation and cost of Capital expressions with Corporate and
Personal taxes
NBER Working Paper #3074 August 1989