Description
This paper examines the appropriate term of the
risk free rate to be used by a regulator in price
control situations, most particularly in the
presence of corporate debt.
Accounting Research Journal
Regulation and the Term of the Risk Free Rate: Implications of Corporate Debt
Martin Lally
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To cite this document:
Martin Lally, (2007),"Regulation and the Term of the Risk Free Rate: Implications of Corporate Debt", Accounting Research
J ournal, Vol. 20 Iss 2 pp. 73 - 80
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Martin Lally, (2007),"Rejoinder: Regulation and the Term of the Risk Free Rate: Implications of Corporate Debt", Accounting
Research J ournal, Vol. 20 Iss 2 pp. 87-88http://dx.doi.org/10.1108/10309610780000693
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Regulation and the Term of the Risk Free Rate: Implications of Corporate Debt
73
Regulation and the Term of the Risk Free
Rate: Implications of Corporate Debt
Martin Lally
School of Economics and Finance
Victoria University of Wellington
Abstract
This paper examines the appropriate term of the
risk free rate to be used by a regulator in price
control situations, most particularly in the
presence of corporate debt. If the regulator
seeks to ensure that the present value of the
future cash flows to equity holders equals their
initial investment then the only choice of term
for the risk free rate that can achieve this is that
matching the regulatory cycle, but it also
requires that the firm match its debt duration to
the regulatory cycle. Failure of the firm to do so
leads to cash flows to equity holders whose net
present value will tend to be negative, and will
also inflict interest rate risk upon equity holders.
This provides the firm with strong incentives to
match its debt duration to the regulatory cycle.
1. Introduction
In regulating the output prices of firms, the
usual process involves periodic reassessment of
prices in the light of prevailing costs, and the
cost of capital is generally a significant
component of these costs. In turn, the risk free
rate is a significant component of the cost of
capital. However, at any given point in time,
there is a range of risk free rates corresponding
to the range of maturity dates for government
bonds, and this gives rise to the question of
which term should be used. This has been a
matter of considerable debate in Australasia,
with debate centering on whether the term
should be matched to the regulatory cycle or
the life of the assets (Australian Competition
& Consumer Commission, 2003; New Zealand
Commerce Commission, 2004).
In assessing the appropriate action by the
regulator, the fundamental principle to be
satisfied is that the present value of the net
cash flows to equity holders should equal their
initial investment (Marshal et al, 1981). If this
principle is not satisfied then equity holders are
either over or under compensated by the
regulator. Following this principle Schmalensee
(1989) shows that the period associated with
this risk free rate should match the regulatory
cycle. In doing so, he assumes that the only
source of uncertainty is over future interest
rates, and that the firm is all equity financed.
Lally (2004) extends the proof to consider
additional sources of uncertainty. In particular,
he considers cost and demand shocks, and risks
arising from depreciation methods in which the
aggregate depreciation allowed by the regulator
may diverge from the cost of the assets.
However, like Schmalensee, the firm is
assumed to be all equity financed. Accordingly,
this paper seeks to consider the implications of
the regulated firm being at least partly debt
financed and the possibility of the firm choosing
a duration for this debt finance that diverges
fromthe length of the regulatory cycle.
Section 2 analyses this problem when only
interest rate risk is present, and shows that the
term of the risk free rate used by the regulator
should continue to match the regulatory cycle.
Furthermore, firms then have strong incentives
to match the duration of their debt to the
regulatory cycle. Section 3 extends the analysis
to consider uncertainty over the firm’s debt
premium, and reaches the same conclusion.
Section 4 concludes.
2. The Analysis Under Only Interest
Rate Risk
We initially suppose that the only source of risk
here is in respect of the risk free rate. Inter alia,
this implies that the firm’s cost of debt equals
the risk free rate. In addition, and in the interests
of focusing upon the central issue, we assume a
regulatory cycle of one year, output prices (and
hence revenues) are set at the beginning of the
year and arise at the end of it, the asset life is two
years, there are no operating costs and the firm
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ACCOUNTING RESEARCH JOURNAL VOLUME 20 NO 2 (2007)
74
maintains a constant leverage ratio of L in book
value terms. The initial investment is denoted C,
with proportion (1 – L) funded by equity
holders. The regulatory depreciation in the first
year is kC and therefore (1 – k)C in the second
year. Interest payments on debt are assumed to
be made annually. So, in respect of one year risk
free rates, the spot rate matches the yield to
maturity. We denote the current one year rate as
R
01
and the one year rate that will arise in one
year as R
12
. By contrast, in respect of two year
rates, the spot rate may differ fromthe yield
to maturity. We denote the current two year
spot rate as
S
R
02
and the two year spot rate that
will arise in one year as
S
R
13
, whilst the
corresponding yields to maturity are denoted
y
R
02
and
y
R
13
. The rates R
12
,
S
R
13
and
y
R
13
are
currently uncertain.
Since the asset has a life of two years then
the feasible debt strategies for the firm are
two-year debt, or one year debt with rollover
after one year.
1
Furthermore, given that the
regulator resets the revenues annually, they
could use the one year risk free rate or the rate
corresponding to the residual life of the asset.
2
So, there are four possible combinations of
actions by the firm and the regulator, and we
address each of them in turn.
Policy 1: The firm uses one year debt and the
regulator uses the one year riskfree rate
The revenues allowed for the second year (set at
the beginning of the year and realised at the end
of it) are the sum of depreciation and the cost of
capital, i.e.,
3
12 2
) 1 ( ) 1 ( R k C k C REV ? + ? = (1)
The cash flow to equity holders at the end of
year 2 is then this revenue net of the debt
repayments and interest payments, i.e.,
1 In this section, the important feature of the debt is its
duration and this could differ fromits termby coupling
suitable swap contracts with debt of a different maturity.
Unless stated otherwise, no swap contracts are assumed.
2 These two possibilities correspond to the alternatives
propounded in the Australasian debate on this issue.
Furthermore, using a rate corresponding to the residual
life of the asset implies use of the two year rate now and
the one year rate in one year.
3 Since leverage is held constant in book value terms, then
debt must be repaid in proportion to the depreciation
allowance embedded in revenues. So, with a time 1
depreciation allowance of Ck, the debt repayment at that
time must be LCk, leaving LC(1-k) to be repaid at time 2.
12 12 2
) 1 ( ) 1 ( ) 1 ( ) 1 ( R k LC k LC R k C k C F ? ? ? ? ? + ? =
) 1 )( 1 )( 1 (
12
R L k C + ? ? =
(2)
This time 2 payoff to equity holders is certain
at time 1, and is therefore present valued at time
1 using the one year risk free rate prevailing at
time 1, i.e.,
) 1 )( 1 (
1
) 1 )( 1 )( 1 (
12
12
1
L k C
R
R L k C
PV ? ? =
+
+ ? ?
=
(3)
The revenues allowed for the first year (set at
the beginning of the year and realised at the end
of it) are the sum of depreciation and the cost of
capital, i.e.,
01 1
CR Ck REV + =
(4)
The cash flow to equity holders at the end of
year 1 is then this revenue net of the debt
repayments and interest payments, i.e.,
01 01 1
LCR LCk CR Ck F ? ? + =
The sum of this cash flow and the residual
value of the equity holders investment at the end
of the first year, with the latter specified in
equation (3), is then as follows.
) 1 )( 1 (
01 01 1 1
L k C LCR LCk CR Ck PV F ? ? + ? ? + = +
) 1 )( 1 (
01
R L C + ? =
This time 1 payoff to equity holders is
certain, and is therefore present valued at time 0
using the one year risk free rate prevailing at
time 0, i.e.,
) 1 (
1
) 1 )( 1 (
01
01
0
L C
R
R L C
PV ? =
+
+ ?
=
(5)
So, the present value of the cash flows to
equity holders equals their initial investment of
C(1 – L). This satisfies the present value principle.
Policy 2: The firm uses one year debt and the
regulator uses the two year riskfree rate
Since the firm is still using one year debt and
the regulatory policy specified here still means
use of the one year rate at time 1, then the
situation in respect of the second year is
unaffected. So, equation (3) is still valid. The
revenues allowed for the first year (set at the
beginning of the year and realised at the end of
it) are the sum of depreciation and the cost of
capital, with the regulator now using the two
year rate (yield to maturity)
4
, i.e.,
4 If a firmengaged in two year borrowing, the relevant rate
would be the yield to maturity. Consistent with this, we
presume that a regulator adopting the two year rate
would adopt the yield to maturity.
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Regulation and the Term of the Risk Free Rate: Implications of Corporate Debt
75
y
CR Ck REV
02 1
+ =
(6)
The cash flow to equity holders at the end of
year 1 is then this revenue net of the debt
repayments and interest payments, with the
latter based on one year debt, i.e.,
01 02 1
LCR LCk CR Ck F
y
? ? + =
(7)
The sum of this cash flow and the residual
value of the equity holder’s investment at the
end of the first year, with the latter specified in
equation (3), is then as follows.
) 1 )( 1 (
01 02 1 1
L k C LCR LCk CR Ck PV F
y
? ? + ? ? + = +
) ( ) 1 )( 1 (
01 02 01
R R C R L C
y
? + + ? =
This time 1 payoff to equity holders is certain,
and is therefore present valued at time 0 using the
one year risk free rate prevailing at time 0, i.e.,
01
01 02
0
1
) (
) 1 (
R
R R C
L C PV
y
+
?
+ ? =
(8)
So, in general, the present value of the cash
flows to equity holders diverges from their
initial investment of C(1 – L). This violates the
present value principle.
Policy 3: The firm uses two year debt and the
regulator uses the one year riskfree rate
Since the regulatory policy specified here
involves use of the one year rate at time 1, then
equation (1) is still valid. Subtraction of the
payments to debt holders at time 2, with the
interest payment based on two-year debt (yield
to maturity), yields the cash flow to equity
holders at time 2 as follows.
y
R k LC k LC R k C k C F
02 12 2
) 1 ( ) 1 ( ) 1 ( ) 1 ( ? ? ? ? ? + ? =
) )( 1 ( ) 1 )( 1 )( 1 (
12 02 12
R R k LC R L k C
y
? ? ? + ? ? =
(9)
This time 2 payoff to equity holders is certain
at time 1, and is therefore present valued at time
1 using the one year risk free rate prevailing at
time 1 (R
12
), i.e.,
12
12 02
1
1
) )( 1 (
) 1 )( 1 (
R
R R k LC
L k C PV
y
+
? ?
? ? ? =
(10)
The revenues allowed for the first year are
still given by equation (4). The cash flow to
equity holders at the end of year 1 is then this
revenue net of the debt repayments and interest
payments, with the latter based on the two year
yield to maturity, i.e.,
y
LCR LCk CR Ck F
02 01 1
? ? + =
(11)
The sum of this cash flow and the residual
value of the equity holders investment at the end
of the first year, with the latter specified in
equation (10), is then as follows.
02 01 1 1
) 1 )( 1 ( L k C LCR LCk CR Ck PV F
y
? ? + ? ? + = +
12
12 02
1
) )( 1 (
R
R R k LC
y
+
? ?
?
01 02 01
) ( ) 1 )( 1 ( R R LC R L C
y
? ? ? + ? =
12
12 02
1
) )( 1 (
R
R R k LC
y
+
? ?
?
The first two terms in this time 1 payoff to
equity holders are certain at time 0, and are
therefore present valued at time 0 using the one
year risk free rate prevailing at time 0. The last
term is not certain, and the present value
process is therefore unclear. Collectively, the
present value at time 0 is as follows.
+
?
? ? =
01
01 02
0
1
) (
) 1 (
R
R R LC
L C PV
y
?
?
?
?
?
?
+
? ?
?
12
12 02
1
) )( 1 (
R
R R k LC
PV
y
(12)
So, again, the present value of the cash flows
to equity holders in general diverges fromtheir
initial investment of C(1 – L). This violates the
present value principle.
Policy 4: The firm uses two year debt and the
regulator uses the two year riskfree rate
The regulatory policy specified here still
involves two-year debt along with the regulator
using the one year rate at time 1. So, equation
(10) is still valid. The revenues allowed for the
first year are given by equation (6). The cash
flow to equity holders at the end of year 1 is
then this revenue net of the debt repayments and
interest payments, with the latter based on two
year debt, i.e.,
y y
LCR LCk CR Ck F
02 02 1
? ? + =
(13)
The sum of this cash flow and the residual
value of the equity holders investment at the end
of the first year, with the latter specified in
equation (10), is then as follows.
12
12 02
02 02 1 1
1
) )( 1 (
) 1 )( 1 (
R
R R k LC
L k C LCR LCk CR Ck PV F
y
y y
+
? ?
? ? ? + ? ? + = +
12
12 02
01 02 01
1
) )( 1 (
) )( 1 ( ) 1 )( 1 (
R
R R k LC
R R L C R L C
y
y
+
? ?
? ? ? + + ? =
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ACCOUNTING RESEARCH JOURNAL VOLUME 20 NO 2 (2007)
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The first two terms in this time 1 payoff to
equity holders are certain at time 0, and are
therefore present valued at time 0 using the one
year risk free rate prevailing at time 0. The last
termis not certain, and the present value process
is therefore unclear. Collectively, the present
value at time 0 is as follows.
+
? ?
+ ? =
01
01 02
0
1
) )( 1 (
) 1 (
R
R R L C
L C PV
y
?
?
?
?
?
?
+
? ?
?
12
12 02
1
) )( 1 (
R
R R k LC
PV
y
(14)
So, again, the present value of the cash flows
to equity holders in general diverges fromtheir
initial investment of C(1 – L). This violates the
present value principle.
In summary, across the four cases considered,
the present values of the cash flows to equity
holders are given in equations (5), (8), (12) and
(14). The only situation in which the present
value principle is in general satisfied is the first
case, in which both the risk free rate used by the
regulator and the duration of the firm’s debt
matches the regulatory cycle. To gain some
sense about the size and sign of the violations of
the present value principle in the remaining
cases, the following analysis is undertaken. We
let C =$10,000, L =.50, k =.50 and R01 =.05,
with the last three justified as typical values and
the first as an arbitrary scalar. Various possible
probability distributions for R12 are then
considered, and the implied values for
y
R
02
are
derived. These implied values are governed by
beliefs about the determinants of the term
structure of interest rates. Suppose that the “local
expectations hypothesis” describes the term
structure of spot interest rates, i.e., the expected
rate of return on all bonds over the next year is
the same.5 The price now of a pure-discount
bond maturing in two years, and paying $1 at
that time, must then be as follows.
?
?
?
?
?
?
+
= =
12
1
0
1
1
05 . 1
1
05 . 1
) (
R
E
P E
P
Expressing P
0
in terms of the current two-
year spot interest rate
S
R
02
, this rate must then
satisfy the following equation
6
.
5 The existence of a liquidity premium in the term
structure will be recognised shortly.
6 Alternativeformulations of theexpectations hypothesis are
presented and analysed in Cox et al (1981). The version
invoked herereadily deals with thepresent valueoperations
arising in thelast termof both equations (12) and (14).
?
?
?
?
?
?
+
=
+
12
2
02
1
1
05 . 1
1
) 1 (
1
R
E
R
S
(15)
We now turn to a coupon-bearing bond that
corresponds to the firmborrowing for two years,
and repaying half of the principal each year
(consistent with k =.50). Letting c denote the
coupon interest rate on this coupon-bearing bond
with face value $1, the two-year yield to
maturity on such a bond must satisfy the
following equation.
2
02
2
02 02
) 1 (
) 1 ( 50 .
05 . 1
50 .
) 1 (
) 1 ( 50 .
1
50 .
S y y
R
c c
R
c
R
c
+
+
+
+
=
+
+
+
+
+
The coupon rate c could take any value.
However, a natural choice would be the yield to
maturity on the bond. With this substitution, the
preceding equation becomes
2
02
02 02
2
02
02
02
02
) 1 (
) 1 ( 50 .
05 . 1
50 .
) 1 (
) 1 ( 50 .
1
50 .
S
y y
y
y
y
y
R
R R
R
R
R
R
+
+
+
+
=
+
+
+
+
+
(16)
Equations (15) and (16) then permit
translation of a probability distribution for R
12
into a value for
y
R
02
. To simplify the analysis,
we assume that R
12
=E(R
12
) ± .01 with equal
probability. For example, suppose E(R
12
) =.07,
so that R
12
is equally likely to be .06 or .08.
Equation (15) then implies that
S
R
02
=.0600,
and equation (16) then implies that
y
R
02
=.0564. We can now assess the present
value of the net cash flows to equity holders
under the four possible policies. Following
policy 1, reflected in equation (5), the present
value of the net cash flows to equity holders is
of course $5000. Following policy 2, the present
value of the net cash flows in (8) is as follows.
5061 $
05 . 1
) 05 . 0564 (. 10000 $
5000 $
0
=
?
+ = PV
Following policy 3, reflected in equation
(12), the last term there requires a present
valuing operation. Consistent with invoking the
“local expectations hypothesis”, the appropriate
present value operation is to discount the
expected payoff in one year at R
01
, to yield the
following result.
05 . 1
) 05 . 0564 (. 5000 $
5000 $
0
?
? = PV
05 . 1
1
) 0564 (. 2500 $
12
12
?
?
?
?
?
?
+
?
?
R
R
E
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Regulation and the Term of the Risk Free Rate: Implications of Corporate Debt
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Since R
12
is equally likely to be .06 or .08, it
follows that PV
0
=$5000. Finally, in respect of
policy 4 reflected in equation (14), the present
value of the cash flows to equity holders is as
follows.
05 . 1
) 05 . 0564 (. 5000 $
5000 $
0
?
+ = PV
5061 $
05 . 1
1
) 0564 (. 2500 $
12
12
=
?
?
?
?
?
?
+
?
?
R
R
E
Table 1 shows these results in the first row,
and the first section of the table considers other
possible values for E(R
12
) along with the
implied values for
y
R
02
.
Consistent with the empirical evidence, we
now acknowledge a “liquidity premium” p in
the term structure of interest rates, i.e., higher
values for
S
R
02
than those implied by the
expectations hypothesis (McCulloch, 1975;
Fama, 1984). In recognition of this, we simply
add p to the value of
S
R
02
arising fromequation
(15).
7
Suppose p =.01. The second section of
Table 1 shows the resulting present values for
policies 1, 2, 3 and 4, reflected in equations (5),
(8), (12) and (14).
Examination of Table 1 reveals the following
conclusions. Firstly, policies 2, 3 and 4 in
general fail the present value principle.
Secondly, if the local expectations hypothesis is
valid, these policies are unbiased, i.e., across the
distribution of possible values for E(R
12
), the net
present value of the net cash flows to equity
holders has expectation zero. Thirdly, in the
presence of a liquidity premium, policies 2 and
4 are biased upwards (the net present value
tends to be positive) and policy 3 is biased
downwards (the net present value tends to be
negative).
In addition to these present value questions,
the firm’s use of two year debt gives rise to
equation (12) or (14), and these involve interest
rate risk to equity holders arising fromshifts in
the one year risk free rate over the first year.
Policies 1 and 2, involving the use of one-year
debt, protect equity holders against this risk.
7 The more conventional formulation of the expectations
hypothesis expresses the forward interest rate as being
equal to the expected future rate, and acknowledgement
of a liquidity premiumthen involves adding the premium
to the expected future rate (see van Horne, 1984, Ch. 5).
The implications of all this for the regulator and
the firm are as follows. The regulator’s goal
should be to satisfy the present value principle.
Given annual resetting of prices and therefore
revenues, use of the two-year risk free rate to
determine the allowed revenues cannot satisfy
the present value principle, regardless of the
debt policy adopted by the firm. By contrast,
use of the one-year rate by the regulator can do
so, providing the firm uses one-year debt. If the
firm elects instead to use two-year debt,
implying policy 3, the resulting future cash
flows to equity holders will have a net present
value that will tend to be negative, under the
plausible condition of a positive liquidity
premium in the term structure (see Table 1). In
addition, the firm’s use of two-year debt inflicts
interest rate risk upon its owners. These two
adverse effects should discourage the firmfrom
acting in this way. So, the regulator should
choose the risk free rate to match the regulatory
cycle and the firm then has strong incentives to
match its debt duration to the regulatory cycle.
The joint result is satisfaction of the present
value principle and avoidance of interest rate
risk to equity holders.
3. Recognition of Re-contracting Risk
The analysis in the previous section is
deliberately simplified; inter alia, it assumes no
information asymmetries and no risks other than
interest rate risk. However, these additional
considerations might lead a regulated firm to
choose a debt term that differed from the
regulatory cycle. For example, a firm might
choose shorter-term debt if it expected its credit
rating to improve (Myers, 1977; Flannery,
1986). Alternatively, a firm might choose
longer-term debt to protect itself against “re-
contracting risk”, i.e., the possibility of the debt
margin (over the risk free rate) rising and not
being compensated for by the regulator
(Diamond, 1991).
8
The presence of this re-
contracting risk presumes that the regulator uses
efficient rather than actual costs, because use of
actual costs would eliminate the firm’s exposure
to re-contracting risk. Any protection that the
8 Diamond also recognises the possibility of the firmnot
being able to re-finance because its financial situation
has deteriorated very significantly. This possibility seems
unlikely in respect of regulated firms, and therefore is not
considered here.
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78
Table 1
Present Value of Cash Flows to Equity Holders
p E(R
12
) Policy 1 Policy 2 Policy 3 Policy 4
0 .07 .0564 $5000 $5061 $5000 $5061
0 .05 .05 $5000 $5000 $5000 $5000
0 .03 .0434 $5000 $4937 $5000 $4937
$5000 $5000 $5000 $5000
.01 .07 .0627 $5000 $5121 $4956 $5077
.01 .05 .0564 $5000 $5061 $4955 $5016
.01 .03 .05 $5000 $5000 $4954 $4954
$5000 $5061 $4955 $5016
This table shows the present value of cash flows to equity holders resulting froman equity investment of $5000, four possible
combinations of regulatory action and debt policy, two possible values for the liquidity premium(p), and three possible values
for the expected one year risk free rate in one year (R
12
) along with the associated current two-year yield to maturity (
y
R
02
). In
all cases, the regulatory cycle is one year and the one year risk free rate is .05. Policy 1 involves the firmusing one year debt
whilst the regulator uses the one year risk free rate in setting allowed revenues. Policy 2 involves the firm’s use of one year
debt and the regulator’s use of the two year risk free rate. Policy 3 involves the firm’s use of two year debt and the regulator’s
use of the one year risk free rate. Policy 4 involves the firm’s use of two year debt and the regulator’s use of the two year risk
free rate. The last row in each section of the table shows the expectation of the present values across the preceding three
possible scenarios.
firm gains from re-contracting risk will come at
the expense of interest rate risk (unless interest
rate swaps are used to match the duration of the
debt to the regulatory cycle), i.e., borrowing for
the longer period removes the firm’s re-
contracting risk but at the cost of exposing
equity holders to interest rate risk, whereas
matching the borrowing term to the regulatory
cycle removes interest rate risk but incurs re-
contracting risk.
The issue of information asymmetry is not
likely to be particularly significant for regulated
firms, and we therefore do not further consider
it here. By contrast, re-contracting risk may be
significant, and is therefore examined in this
section. To do so, we augment the framework of
the previous section. In particular, the regulator
is assumed to allow a corporate debt premium
of p and this premium matches that actually
incurred now by the firm, for both one and two
year debt. After one year, the premium allowed
by the regulator is unchanged but that actually
incurred by the firm(on any newly issued debt)
may differ from it (and is denoted p
1
). We now
investigate the implications of this for the four
policies examined in the previous section.
Policy 1: The firm uses one year debt and the
regulator uses the one year riskfree rate
In respect of the net cash flow to equity holders
at time 2, equation (2) requires addition of an
allowance for p to the revenues and addition of
the allowance p
1
to the interest payment to debt
holders, yielding the following net cash flow at
time 2 to equity holders.
) 1 ( ) )( 1 ( ) 1 (
12 2
k LC Lp R k C k C F ? ? ? + ? + ? =
) )( 1 (
1 12
p R k LC + ? ?
) )( 1 ( ) 1 )( 1 )( 1 (
1 12
p p k CL R L k C ? ? + + ? ? =
Present valuing this at time 1 yields the
following.
12
1
1
1
) )( 1 (
) 1 )( 1 (
R
p p k CL
L k C PV
+
? ?
+ ? ? =
(17)
In respect of the net cash flow to equity
holders at time 1, this comprises the revenue
allowed by the regulator net of principal and
interest payments to debt holders, with both the
revenue allowance and the interest payments
based on the debt premium p as follows.
) ( ) (
01 01 1
p R LC LCk Lp R C Ck F + ? ? + + =
01
) 1 ( ) 1 ( R L C L Ck ? + ? =
Adding this to the present value in (17)
yields the following.
01 1 1
) 1 )( 1 ( ) 1 ( ) 1 ( L k C R L C L Ck PV F ? ? + ? + ? = +
12
1
1
) )( 1 (
R
p p k CL
+
? ?
+
12
1
01
1
) )( 1 (
) 1 )( 1 (
R
p p k CL
R L C
+
? ?
+ + ? =
y
R
02
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Regulation and the Term of the Risk Free Rate: Implications of Corporate Debt
79
The first termhere is certain at time 0, and is
therefore present valued at time 0 using R
01
. In
respect of the second term, this is uncertain at
time 0 but mean zero and the risk arising fromp
1
would be largely unsystematic. So, the present
value would be approximately zero. Thus, the
present value of the last equation is as follows.
?
?
?
?
?
?
+
? ?
+
+
+ ?
=
12
1
01
01
0
1
) )( 1 (
1
) 1 )( 1 (
R
p p k CL
PV
R
R L C
PV
) 1 ( L C ? =
(18)
So, the present value of the cash flows to
equity holders still equals their initial investment
of C(1 – L), and this satisfies the present value
principle.
Policy 2: The firm uses one year debt and the
regulator uses the two year riskfree rate
In respect of the debt premium, the analysis here
parallels that of policy 1 above, with debt
premiumrisk arising in the time 2 cash flows to
equity holders and netting out of the time 1 cash
flows. This debt premiumrisk in the time 2 cash
flows generates the same final termas in equation
(17). Otherwise equation (8) applies. So, the
resulting present value at time 0 is as follows.
?
?
?
?
?
?
+
? ?
+
+
?
+ ? =
12
1
01
01 02
0
1
) )( 1 (
1
) (
) 1 (
R
p p k CL
PV
R
R R C
L C PV
01
01 02
1
) (
) 1 (
R
R R C
L C
+
?
+ ? =
(19)
This present value matches that derived for
policy 2 in the previous section and diverges
fromthe equity holders initial investment.
Policy 3: The firm uses two year debt and the
regulator uses the one year riskfree rate
Since the firm does not re-contract with debt
holders, the corporate debt premiumreflected in
the interest payments to debt holders at times 1
and 2 matches the regulator’s allowances at
those times for the debt premium. So, there is no
effect upon the net cash flows to equity holders
and the present value of their cash flows is then
as given in equation (12), i.e.,
+
?
? ? =
01
01 02
0
1
) (
) 1 (
R
R R LC
L C PV
?
?
?
?
?
?
+
? ?
?
12
12 02
1
) )( 1 (
R
R R k LC
PV
(20)
Again, the present value of the cash flows to
equity holders diverges from their initial
investment of C(1 – L).
Policy 4: The firm uses two year debt and the
regulator uses the two year riskfree rate
Paralleling the situation regarding policy 3,
there is no effect from a corporate debt
premium upon the net cash flows to equity
holders. The present value of these net cash
flows is then as given in equation (14), i.e.,
+
? ?
+ ? =
01
01 02
0
1
) )( 1 (
) 1 (
R
R R L C
L C PV
?
?
?
?
?
?
+
? ?
?
12
12 02
1
) )( 1 (
R
R R k LC
PV
(21)
Again, the present value of the cash flows to
equity holders diverges from their initial
investment of C(1 – L).
In summary, and in the presence of a
corporate debt premium, the present values of
the cash flows to equity holders across the four
policy scenarios are given in equations (18),
(19), (20) and (21). These results are identical to
those in the previous section, in which a
corporate debt premium was absent, and the
numerical results in Table 1 therefore still
apply. As before, the only policy combination
that satisfies the present value principle is the
first, in which the regulator uses the risk free
rate matching the regulatory cycle and the firm
chooses debt with the same term. In addition,
the firm’s use of debt with that term protects
equity holders from interest rate risk and
exposes them to re-contracting risk. By contrast,
the use of longer-term debt can never satisfy the
present value principle, exposes equity holders
to interest rate risk and protects them from re-
contracting risk.
The implications of all this for the firm and
the regulator are as follows. As before, the
regulator’s goal should be to satisfy the present
value principle. Using the two-year risk free rate
cannot satisfy this principle, regardless of the
debt policy adopted by the firm. By contrast,
use of the one-year rate will achieve this, so
long as the firm uses one-year debt. If the firm
elects instead to use two-year debt, implying
policy 3, the resulting future cash flows to
equity holders will have a net present value that
will tend to be negative, will expose them to
interest rate risk and protect them from re-
contracting risk. The adverse effect upon the
present value of equity holders cash flows, and
exposure to interest rate risk, should discourage
the firm from acting in this way, but the
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ACCOUNTING RESEARCH JOURNAL VOLUME 20 NO 2 (2007)
80
protection from re-contracting risk will exert a
contrary effect. If this contrary effect induces
the firmto use longer-term debt, and it fails to
use swaps to match the duration of its debt to
the regulatory cycle, then the present value
principle will no longer be satisfied. However,
this cannot justify the regulator switching to the
use of the two-year risk free rate (switching
frompolicy 3 to 4) because the only effect from
doing so will be to exchange a net present value
on the cash flows to equity holders that tends to
be negative for one that tends to be positive (see
Table 1).
4. Conclusions
This paper examines the appropriate term of the
risk free rate to be used by a regulator in price
control situations, and in the presence of debt. If
the only source of risk is interest rate risk, then
the regulator should choose the term of the risk
free rate to satisfy the present value principle.
The only choice of termfor the risk free rate
that can achieve this is that matching the
regulatory cycle, but it also requires that the
firm match its debt duration to the regulatory
cycle. Failure of the firmto do so leads to cash
flows to equity holders whose net present value
will tend to be negative, and will also inflict
interest rate risk upon equity holders. This
provides the firm with strong incentives to
match its debt duration to the regulatory cycle.
If the issue of re-contracting risk arises,
because the regulator invokes efficient rather
than actual costs, the regulator should still seek
to satisfy the present value principle. Again, the
only choice of termfor the risk free rate that can
achieve this is that matching the regulatory
cycle, and again this requires that the firm
match its debt duration to the regulatory cycle.
If the firm chooses a longer duration on its debt,
in an effort to eliminate or reduce re-contracting
risk, the net present value of the resulting cash
flows to equity holders will tend to be negative,
and the firm will inflict interest rate risk upon
equity holders. Furthermore, the firm’s choice
of the longer duration on its debt cannot justify
the regulator lengthening the term of the risk
free rate used in setting prices because the only
effect from doing so will be to exchange a net
present value on the cash flows to equity
holders that tends to be negative for one that
tends to be positive.
References
Australian Competition & Consumer Commission. (2003),
East Australian Pipeline Limited Access Arrangement
for the Moomba to Sydney Pipeline System
(www.accc.gov.au).
Cox, J., Ingersoll, J. and Ross, S. (1981), ‘A Re-examination
of Traditional Hypotheses about the Term Structure
of Interest Rates’, Journal of Finance, vol. 36,
pp. 769–799.
Diamond, D. (1991), ‘Debt Maturity Structure and Liquidity
Risk’, Quarterly Journal of Economics, vol. 106(3),
pp. 709–737.
Fama, E. (1984), ‘The Information in the TermStructure’,
Journal of Financial Economics, vol. 13(4),
pp. 509–528.
Flannery, M. (1986), ‘Asymmetric Information and Risky
Debt Maturity Choice’, Journal of Finance, vol. 41, pp.
19–37.
Lally, M. (2004), Regulation and the Choice of the Risk
Free Rate, Accounting Research Journal, vol. 17(1),
pp. 18–23.
Marshal, W., Yawitz, J. and Greenberg, E. (1981), ‘Optimal
Regulation Under Uncertainty’, The Journal of Finance,
vol. 36, pp. 909–922.
McCulloch, J. (1975), ‘An Estimate of the Liquidity
Premium’, Journal of Political Economy, vol. 83,
pp. 95–119.
Myers, S. (1977), ‘Determinants of Corporate Borrowing’,
Journal of Financial Economics, vol. 5, pp. 147–175.
New Zealand Commerce Commission. (2004), Gas Control
Inquiry Final Report (www.comcom.govt.nz).
Schmalensee, R. (1989), ‘An Expository Note on
Depreciation and Profitability Under Rate-of-Return
Regulation’, Journal of Regulatory Economics, vol. 1,
pp. 293–298.
Van Horne, J. (1984), Financial Market Rates and Flows,
2
nd
edition, Prentice-Hall, Inc., Englewood Cliffs.
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This article has been cited by:
1. Kevin Davis. 2014. The Debt Maturity Issue in Access Pricing. Economic Record 90:10.1111/ecor.2014.90.issue-290, 271-281.
[CrossRef]
2. Jason Hall. 2007. Comment on Regulation and the Term of the Risk Free Rate: Implications of Corporate Debt. Accounting
Research Journal 20:2, 81-86. [Abstract] [PDF]
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doc_955933271.pdf
This paper examines the appropriate term of the
risk free rate to be used by a regulator in price
control situations, most particularly in the
presence of corporate debt.
Accounting Research Journal
Regulation and the Term of the Risk Free Rate: Implications of Corporate Debt
Martin Lally
Article information:
To cite this document:
Martin Lally, (2007),"Regulation and the Term of the Risk Free Rate: Implications of Corporate Debt", Accounting Research
J ournal, Vol. 20 Iss 2 pp. 73 - 80
Permanent link to this document:http://dx.doi.org/10.1108/10309610780000691
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Accounting Research J ournal, Vol. 20 Iss 2 pp. 81-86http://dx.doi.org/10.1108/10309610780000692
Martin Lally, (2007),"Rejoinder: Regulation and the Term of the Risk Free Rate: Implications of Corporate Debt", Accounting
Research J ournal, Vol. 20 Iss 2 pp. 87-88http://dx.doi.org/10.1108/10309610780000693
Ping-Sheng Koh, (2005),"Institutional Ownership and Income Smoothing: Australian Evidence", Accounting Research
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Regulation and the Term of the Risk Free Rate: Implications of Corporate Debt
73
Regulation and the Term of the Risk Free
Rate: Implications of Corporate Debt
Martin Lally
School of Economics and Finance
Victoria University of Wellington
Abstract
This paper examines the appropriate term of the
risk free rate to be used by a regulator in price
control situations, most particularly in the
presence of corporate debt. If the regulator
seeks to ensure that the present value of the
future cash flows to equity holders equals their
initial investment then the only choice of term
for the risk free rate that can achieve this is that
matching the regulatory cycle, but it also
requires that the firm match its debt duration to
the regulatory cycle. Failure of the firm to do so
leads to cash flows to equity holders whose net
present value will tend to be negative, and will
also inflict interest rate risk upon equity holders.
This provides the firm with strong incentives to
match its debt duration to the regulatory cycle.
1. Introduction
In regulating the output prices of firms, the
usual process involves periodic reassessment of
prices in the light of prevailing costs, and the
cost of capital is generally a significant
component of these costs. In turn, the risk free
rate is a significant component of the cost of
capital. However, at any given point in time,
there is a range of risk free rates corresponding
to the range of maturity dates for government
bonds, and this gives rise to the question of
which term should be used. This has been a
matter of considerable debate in Australasia,
with debate centering on whether the term
should be matched to the regulatory cycle or
the life of the assets (Australian Competition
& Consumer Commission, 2003; New Zealand
Commerce Commission, 2004).
In assessing the appropriate action by the
regulator, the fundamental principle to be
satisfied is that the present value of the net
cash flows to equity holders should equal their
initial investment (Marshal et al, 1981). If this
principle is not satisfied then equity holders are
either over or under compensated by the
regulator. Following this principle Schmalensee
(1989) shows that the period associated with
this risk free rate should match the regulatory
cycle. In doing so, he assumes that the only
source of uncertainty is over future interest
rates, and that the firm is all equity financed.
Lally (2004) extends the proof to consider
additional sources of uncertainty. In particular,
he considers cost and demand shocks, and risks
arising from depreciation methods in which the
aggregate depreciation allowed by the regulator
may diverge from the cost of the assets.
However, like Schmalensee, the firm is
assumed to be all equity financed. Accordingly,
this paper seeks to consider the implications of
the regulated firm being at least partly debt
financed and the possibility of the firm choosing
a duration for this debt finance that diverges
fromthe length of the regulatory cycle.
Section 2 analyses this problem when only
interest rate risk is present, and shows that the
term of the risk free rate used by the regulator
should continue to match the regulatory cycle.
Furthermore, firms then have strong incentives
to match the duration of their debt to the
regulatory cycle. Section 3 extends the analysis
to consider uncertainty over the firm’s debt
premium, and reaches the same conclusion.
Section 4 concludes.
2. The Analysis Under Only Interest
Rate Risk
We initially suppose that the only source of risk
here is in respect of the risk free rate. Inter alia,
this implies that the firm’s cost of debt equals
the risk free rate. In addition, and in the interests
of focusing upon the central issue, we assume a
regulatory cycle of one year, output prices (and
hence revenues) are set at the beginning of the
year and arise at the end of it, the asset life is two
years, there are no operating costs and the firm
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ACCOUNTING RESEARCH JOURNAL VOLUME 20 NO 2 (2007)
74
maintains a constant leverage ratio of L in book
value terms. The initial investment is denoted C,
with proportion (1 – L) funded by equity
holders. The regulatory depreciation in the first
year is kC and therefore (1 – k)C in the second
year. Interest payments on debt are assumed to
be made annually. So, in respect of one year risk
free rates, the spot rate matches the yield to
maturity. We denote the current one year rate as
R
01
and the one year rate that will arise in one
year as R
12
. By contrast, in respect of two year
rates, the spot rate may differ fromthe yield
to maturity. We denote the current two year
spot rate as
S
R
02
and the two year spot rate that
will arise in one year as
S
R
13
, whilst the
corresponding yields to maturity are denoted
y
R
02
and
y
R
13
. The rates R
12
,
S
R
13
and
y
R
13
are
currently uncertain.
Since the asset has a life of two years then
the feasible debt strategies for the firm are
two-year debt, or one year debt with rollover
after one year.
1
Furthermore, given that the
regulator resets the revenues annually, they
could use the one year risk free rate or the rate
corresponding to the residual life of the asset.
2
So, there are four possible combinations of
actions by the firm and the regulator, and we
address each of them in turn.
Policy 1: The firm uses one year debt and the
regulator uses the one year riskfree rate
The revenues allowed for the second year (set at
the beginning of the year and realised at the end
of it) are the sum of depreciation and the cost of
capital, i.e.,
3
12 2
) 1 ( ) 1 ( R k C k C REV ? + ? = (1)
The cash flow to equity holders at the end of
year 2 is then this revenue net of the debt
repayments and interest payments, i.e.,
1 In this section, the important feature of the debt is its
duration and this could differ fromits termby coupling
suitable swap contracts with debt of a different maturity.
Unless stated otherwise, no swap contracts are assumed.
2 These two possibilities correspond to the alternatives
propounded in the Australasian debate on this issue.
Furthermore, using a rate corresponding to the residual
life of the asset implies use of the two year rate now and
the one year rate in one year.
3 Since leverage is held constant in book value terms, then
debt must be repaid in proportion to the depreciation
allowance embedded in revenues. So, with a time 1
depreciation allowance of Ck, the debt repayment at that
time must be LCk, leaving LC(1-k) to be repaid at time 2.
12 12 2
) 1 ( ) 1 ( ) 1 ( ) 1 ( R k LC k LC R k C k C F ? ? ? ? ? + ? =
) 1 )( 1 )( 1 (
12
R L k C + ? ? =
(2)
This time 2 payoff to equity holders is certain
at time 1, and is therefore present valued at time
1 using the one year risk free rate prevailing at
time 1, i.e.,
) 1 )( 1 (
1
) 1 )( 1 )( 1 (
12
12
1
L k C
R
R L k C
PV ? ? =
+
+ ? ?
=
(3)
The revenues allowed for the first year (set at
the beginning of the year and realised at the end
of it) are the sum of depreciation and the cost of
capital, i.e.,
01 1
CR Ck REV + =
(4)
The cash flow to equity holders at the end of
year 1 is then this revenue net of the debt
repayments and interest payments, i.e.,
01 01 1
LCR LCk CR Ck F ? ? + =
The sum of this cash flow and the residual
value of the equity holders investment at the end
of the first year, with the latter specified in
equation (3), is then as follows.
) 1 )( 1 (
01 01 1 1
L k C LCR LCk CR Ck PV F ? ? + ? ? + = +
) 1 )( 1 (
01
R L C + ? =
This time 1 payoff to equity holders is
certain, and is therefore present valued at time 0
using the one year risk free rate prevailing at
time 0, i.e.,
) 1 (
1
) 1 )( 1 (
01
01
0
L C
R
R L C
PV ? =
+
+ ?
=
(5)
So, the present value of the cash flows to
equity holders equals their initial investment of
C(1 – L). This satisfies the present value principle.
Policy 2: The firm uses one year debt and the
regulator uses the two year riskfree rate
Since the firm is still using one year debt and
the regulatory policy specified here still means
use of the one year rate at time 1, then the
situation in respect of the second year is
unaffected. So, equation (3) is still valid. The
revenues allowed for the first year (set at the
beginning of the year and realised at the end of
it) are the sum of depreciation and the cost of
capital, with the regulator now using the two
year rate (yield to maturity)
4
, i.e.,
4 If a firmengaged in two year borrowing, the relevant rate
would be the yield to maturity. Consistent with this, we
presume that a regulator adopting the two year rate
would adopt the yield to maturity.
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y
CR Ck REV
02 1
+ =
(6)
The cash flow to equity holders at the end of
year 1 is then this revenue net of the debt
repayments and interest payments, with the
latter based on one year debt, i.e.,
01 02 1
LCR LCk CR Ck F
y
? ? + =
(7)
The sum of this cash flow and the residual
value of the equity holder’s investment at the
end of the first year, with the latter specified in
equation (3), is then as follows.
) 1 )( 1 (
01 02 1 1
L k C LCR LCk CR Ck PV F
y
? ? + ? ? + = +
) ( ) 1 )( 1 (
01 02 01
R R C R L C
y
? + + ? =
This time 1 payoff to equity holders is certain,
and is therefore present valued at time 0 using the
one year risk free rate prevailing at time 0, i.e.,
01
01 02
0
1
) (
) 1 (
R
R R C
L C PV
y
+
?
+ ? =
(8)
So, in general, the present value of the cash
flows to equity holders diverges from their
initial investment of C(1 – L). This violates the
present value principle.
Policy 3: The firm uses two year debt and the
regulator uses the one year riskfree rate
Since the regulatory policy specified here
involves use of the one year rate at time 1, then
equation (1) is still valid. Subtraction of the
payments to debt holders at time 2, with the
interest payment based on two-year debt (yield
to maturity), yields the cash flow to equity
holders at time 2 as follows.
y
R k LC k LC R k C k C F
02 12 2
) 1 ( ) 1 ( ) 1 ( ) 1 ( ? ? ? ? ? + ? =
) )( 1 ( ) 1 )( 1 )( 1 (
12 02 12
R R k LC R L k C
y
? ? ? + ? ? =
(9)
This time 2 payoff to equity holders is certain
at time 1, and is therefore present valued at time
1 using the one year risk free rate prevailing at
time 1 (R
12
), i.e.,
12
12 02
1
1
) )( 1 (
) 1 )( 1 (
R
R R k LC
L k C PV
y
+
? ?
? ? ? =
(10)
The revenues allowed for the first year are
still given by equation (4). The cash flow to
equity holders at the end of year 1 is then this
revenue net of the debt repayments and interest
payments, with the latter based on the two year
yield to maturity, i.e.,
y
LCR LCk CR Ck F
02 01 1
? ? + =
(11)
The sum of this cash flow and the residual
value of the equity holders investment at the end
of the first year, with the latter specified in
equation (10), is then as follows.
02 01 1 1
) 1 )( 1 ( L k C LCR LCk CR Ck PV F
y
? ? + ? ? + = +
12
12 02
1
) )( 1 (
R
R R k LC
y
+
? ?
?
01 02 01
) ( ) 1 )( 1 ( R R LC R L C
y
? ? ? + ? =
12
12 02
1
) )( 1 (
R
R R k LC
y
+
? ?
?
The first two terms in this time 1 payoff to
equity holders are certain at time 0, and are
therefore present valued at time 0 using the one
year risk free rate prevailing at time 0. The last
term is not certain, and the present value
process is therefore unclear. Collectively, the
present value at time 0 is as follows.
+
?
? ? =
01
01 02
0
1
) (
) 1 (
R
R R LC
L C PV
y
?
?
?
?
?
?
+
? ?
?
12
12 02
1
) )( 1 (
R
R R k LC
PV
y
(12)
So, again, the present value of the cash flows
to equity holders in general diverges fromtheir
initial investment of C(1 – L). This violates the
present value principle.
Policy 4: The firm uses two year debt and the
regulator uses the two year riskfree rate
The regulatory policy specified here still
involves two-year debt along with the regulator
using the one year rate at time 1. So, equation
(10) is still valid. The revenues allowed for the
first year are given by equation (6). The cash
flow to equity holders at the end of year 1 is
then this revenue net of the debt repayments and
interest payments, with the latter based on two
year debt, i.e.,
y y
LCR LCk CR Ck F
02 02 1
? ? + =
(13)
The sum of this cash flow and the residual
value of the equity holders investment at the end
of the first year, with the latter specified in
equation (10), is then as follows.
12
12 02
02 02 1 1
1
) )( 1 (
) 1 )( 1 (
R
R R k LC
L k C LCR LCk CR Ck PV F
y
y y
+
? ?
? ? ? + ? ? + = +
12
12 02
01 02 01
1
) )( 1 (
) )( 1 ( ) 1 )( 1 (
R
R R k LC
R R L C R L C
y
y
+
? ?
? ? ? + + ? =
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The first two terms in this time 1 payoff to
equity holders are certain at time 0, and are
therefore present valued at time 0 using the one
year risk free rate prevailing at time 0. The last
termis not certain, and the present value process
is therefore unclear. Collectively, the present
value at time 0 is as follows.
+
? ?
+ ? =
01
01 02
0
1
) )( 1 (
) 1 (
R
R R L C
L C PV
y
?
?
?
?
?
?
+
? ?
?
12
12 02
1
) )( 1 (
R
R R k LC
PV
y
(14)
So, again, the present value of the cash flows
to equity holders in general diverges fromtheir
initial investment of C(1 – L). This violates the
present value principle.
In summary, across the four cases considered,
the present values of the cash flows to equity
holders are given in equations (5), (8), (12) and
(14). The only situation in which the present
value principle is in general satisfied is the first
case, in which both the risk free rate used by the
regulator and the duration of the firm’s debt
matches the regulatory cycle. To gain some
sense about the size and sign of the violations of
the present value principle in the remaining
cases, the following analysis is undertaken. We
let C =$10,000, L =.50, k =.50 and R01 =.05,
with the last three justified as typical values and
the first as an arbitrary scalar. Various possible
probability distributions for R12 are then
considered, and the implied values for
y
R
02
are
derived. These implied values are governed by
beliefs about the determinants of the term
structure of interest rates. Suppose that the “local
expectations hypothesis” describes the term
structure of spot interest rates, i.e., the expected
rate of return on all bonds over the next year is
the same.5 The price now of a pure-discount
bond maturing in two years, and paying $1 at
that time, must then be as follows.
?
?
?
?
?
?
+
= =
12
1
0
1
1
05 . 1
1
05 . 1
) (
R
E
P E
P
Expressing P
0
in terms of the current two-
year spot interest rate
S
R
02
, this rate must then
satisfy the following equation
6
.
5 The existence of a liquidity premium in the term
structure will be recognised shortly.
6 Alternativeformulations of theexpectations hypothesis are
presented and analysed in Cox et al (1981). The version
invoked herereadily deals with thepresent valueoperations
arising in thelast termof both equations (12) and (14).
?
?
?
?
?
?
+
=
+
12
2
02
1
1
05 . 1
1
) 1 (
1
R
E
R
S
(15)
We now turn to a coupon-bearing bond that
corresponds to the firmborrowing for two years,
and repaying half of the principal each year
(consistent with k =.50). Letting c denote the
coupon interest rate on this coupon-bearing bond
with face value $1, the two-year yield to
maturity on such a bond must satisfy the
following equation.
2
02
2
02 02
) 1 (
) 1 ( 50 .
05 . 1
50 .
) 1 (
) 1 ( 50 .
1
50 .
S y y
R
c c
R
c
R
c
+
+
+
+
=
+
+
+
+
+
The coupon rate c could take any value.
However, a natural choice would be the yield to
maturity on the bond. With this substitution, the
preceding equation becomes
2
02
02 02
2
02
02
02
02
) 1 (
) 1 ( 50 .
05 . 1
50 .
) 1 (
) 1 ( 50 .
1
50 .
S
y y
y
y
y
y
R
R R
R
R
R
R
+
+
+
+
=
+
+
+
+
+
(16)
Equations (15) and (16) then permit
translation of a probability distribution for R
12
into a value for
y
R
02
. To simplify the analysis,
we assume that R
12
=E(R
12
) ± .01 with equal
probability. For example, suppose E(R
12
) =.07,
so that R
12
is equally likely to be .06 or .08.
Equation (15) then implies that
S
R
02
=.0600,
and equation (16) then implies that
y
R
02
=.0564. We can now assess the present
value of the net cash flows to equity holders
under the four possible policies. Following
policy 1, reflected in equation (5), the present
value of the net cash flows to equity holders is
of course $5000. Following policy 2, the present
value of the net cash flows in (8) is as follows.
5061 $
05 . 1
) 05 . 0564 (. 10000 $
5000 $
0
=
?
+ = PV
Following policy 3, reflected in equation
(12), the last term there requires a present
valuing operation. Consistent with invoking the
“local expectations hypothesis”, the appropriate
present value operation is to discount the
expected payoff in one year at R
01
, to yield the
following result.
05 . 1
) 05 . 0564 (. 5000 $
5000 $
0
?
? = PV
05 . 1
1
) 0564 (. 2500 $
12
12
?
?
?
?
?
?
+
?
?
R
R
E
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Since R
12
is equally likely to be .06 or .08, it
follows that PV
0
=$5000. Finally, in respect of
policy 4 reflected in equation (14), the present
value of the cash flows to equity holders is as
follows.
05 . 1
) 05 . 0564 (. 5000 $
5000 $
0
?
+ = PV
5061 $
05 . 1
1
) 0564 (. 2500 $
12
12
=
?
?
?
?
?
?
+
?
?
R
R
E
Table 1 shows these results in the first row,
and the first section of the table considers other
possible values for E(R
12
) along with the
implied values for
y
R
02
.
Consistent with the empirical evidence, we
now acknowledge a “liquidity premium” p in
the term structure of interest rates, i.e., higher
values for
S
R
02
than those implied by the
expectations hypothesis (McCulloch, 1975;
Fama, 1984). In recognition of this, we simply
add p to the value of
S
R
02
arising fromequation
(15).
7
Suppose p =.01. The second section of
Table 1 shows the resulting present values for
policies 1, 2, 3 and 4, reflected in equations (5),
(8), (12) and (14).
Examination of Table 1 reveals the following
conclusions. Firstly, policies 2, 3 and 4 in
general fail the present value principle.
Secondly, if the local expectations hypothesis is
valid, these policies are unbiased, i.e., across the
distribution of possible values for E(R
12
), the net
present value of the net cash flows to equity
holders has expectation zero. Thirdly, in the
presence of a liquidity premium, policies 2 and
4 are biased upwards (the net present value
tends to be positive) and policy 3 is biased
downwards (the net present value tends to be
negative).
In addition to these present value questions,
the firm’s use of two year debt gives rise to
equation (12) or (14), and these involve interest
rate risk to equity holders arising fromshifts in
the one year risk free rate over the first year.
Policies 1 and 2, involving the use of one-year
debt, protect equity holders against this risk.
7 The more conventional formulation of the expectations
hypothesis expresses the forward interest rate as being
equal to the expected future rate, and acknowledgement
of a liquidity premiumthen involves adding the premium
to the expected future rate (see van Horne, 1984, Ch. 5).
The implications of all this for the regulator and
the firm are as follows. The regulator’s goal
should be to satisfy the present value principle.
Given annual resetting of prices and therefore
revenues, use of the two-year risk free rate to
determine the allowed revenues cannot satisfy
the present value principle, regardless of the
debt policy adopted by the firm. By contrast,
use of the one-year rate by the regulator can do
so, providing the firm uses one-year debt. If the
firm elects instead to use two-year debt,
implying policy 3, the resulting future cash
flows to equity holders will have a net present
value that will tend to be negative, under the
plausible condition of a positive liquidity
premium in the term structure (see Table 1). In
addition, the firm’s use of two-year debt inflicts
interest rate risk upon its owners. These two
adverse effects should discourage the firmfrom
acting in this way. So, the regulator should
choose the risk free rate to match the regulatory
cycle and the firm then has strong incentives to
match its debt duration to the regulatory cycle.
The joint result is satisfaction of the present
value principle and avoidance of interest rate
risk to equity holders.
3. Recognition of Re-contracting Risk
The analysis in the previous section is
deliberately simplified; inter alia, it assumes no
information asymmetries and no risks other than
interest rate risk. However, these additional
considerations might lead a regulated firm to
choose a debt term that differed from the
regulatory cycle. For example, a firm might
choose shorter-term debt if it expected its credit
rating to improve (Myers, 1977; Flannery,
1986). Alternatively, a firm might choose
longer-term debt to protect itself against “re-
contracting risk”, i.e., the possibility of the debt
margin (over the risk free rate) rising and not
being compensated for by the regulator
(Diamond, 1991).
8
The presence of this re-
contracting risk presumes that the regulator uses
efficient rather than actual costs, because use of
actual costs would eliminate the firm’s exposure
to re-contracting risk. Any protection that the
8 Diamond also recognises the possibility of the firmnot
being able to re-finance because its financial situation
has deteriorated very significantly. This possibility seems
unlikely in respect of regulated firms, and therefore is not
considered here.
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Table 1
Present Value of Cash Flows to Equity Holders
p E(R
12
) Policy 1 Policy 2 Policy 3 Policy 4
0 .07 .0564 $5000 $5061 $5000 $5061
0 .05 .05 $5000 $5000 $5000 $5000
0 .03 .0434 $5000 $4937 $5000 $4937
$5000 $5000 $5000 $5000
.01 .07 .0627 $5000 $5121 $4956 $5077
.01 .05 .0564 $5000 $5061 $4955 $5016
.01 .03 .05 $5000 $5000 $4954 $4954
$5000 $5061 $4955 $5016
This table shows the present value of cash flows to equity holders resulting froman equity investment of $5000, four possible
combinations of regulatory action and debt policy, two possible values for the liquidity premium(p), and three possible values
for the expected one year risk free rate in one year (R
12
) along with the associated current two-year yield to maturity (
y
R
02
). In
all cases, the regulatory cycle is one year and the one year risk free rate is .05. Policy 1 involves the firmusing one year debt
whilst the regulator uses the one year risk free rate in setting allowed revenues. Policy 2 involves the firm’s use of one year
debt and the regulator’s use of the two year risk free rate. Policy 3 involves the firm’s use of two year debt and the regulator’s
use of the one year risk free rate. Policy 4 involves the firm’s use of two year debt and the regulator’s use of the two year risk
free rate. The last row in each section of the table shows the expectation of the present values across the preceding three
possible scenarios.
firm gains from re-contracting risk will come at
the expense of interest rate risk (unless interest
rate swaps are used to match the duration of the
debt to the regulatory cycle), i.e., borrowing for
the longer period removes the firm’s re-
contracting risk but at the cost of exposing
equity holders to interest rate risk, whereas
matching the borrowing term to the regulatory
cycle removes interest rate risk but incurs re-
contracting risk.
The issue of information asymmetry is not
likely to be particularly significant for regulated
firms, and we therefore do not further consider
it here. By contrast, re-contracting risk may be
significant, and is therefore examined in this
section. To do so, we augment the framework of
the previous section. In particular, the regulator
is assumed to allow a corporate debt premium
of p and this premium matches that actually
incurred now by the firm, for both one and two
year debt. After one year, the premium allowed
by the regulator is unchanged but that actually
incurred by the firm(on any newly issued debt)
may differ from it (and is denoted p
1
). We now
investigate the implications of this for the four
policies examined in the previous section.
Policy 1: The firm uses one year debt and the
regulator uses the one year riskfree rate
In respect of the net cash flow to equity holders
at time 2, equation (2) requires addition of an
allowance for p to the revenues and addition of
the allowance p
1
to the interest payment to debt
holders, yielding the following net cash flow at
time 2 to equity holders.
) 1 ( ) )( 1 ( ) 1 (
12 2
k LC Lp R k C k C F ? ? ? + ? + ? =
) )( 1 (
1 12
p R k LC + ? ?
) )( 1 ( ) 1 )( 1 )( 1 (
1 12
p p k CL R L k C ? ? + + ? ? =
Present valuing this at time 1 yields the
following.
12
1
1
1
) )( 1 (
) 1 )( 1 (
R
p p k CL
L k C PV
+
? ?
+ ? ? =
(17)
In respect of the net cash flow to equity
holders at time 1, this comprises the revenue
allowed by the regulator net of principal and
interest payments to debt holders, with both the
revenue allowance and the interest payments
based on the debt premium p as follows.
) ( ) (
01 01 1
p R LC LCk Lp R C Ck F + ? ? + + =
01
) 1 ( ) 1 ( R L C L Ck ? + ? =
Adding this to the present value in (17)
yields the following.
01 1 1
) 1 )( 1 ( ) 1 ( ) 1 ( L k C R L C L Ck PV F ? ? + ? + ? = +
12
1
1
) )( 1 (
R
p p k CL
+
? ?
+
12
1
01
1
) )( 1 (
) 1 )( 1 (
R
p p k CL
R L C
+
? ?
+ + ? =
y
R
02
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The first termhere is certain at time 0, and is
therefore present valued at time 0 using R
01
. In
respect of the second term, this is uncertain at
time 0 but mean zero and the risk arising fromp
1
would be largely unsystematic. So, the present
value would be approximately zero. Thus, the
present value of the last equation is as follows.
?
?
?
?
?
?
+
? ?
+
+
+ ?
=
12
1
01
01
0
1
) )( 1 (
1
) 1 )( 1 (
R
p p k CL
PV
R
R L C
PV
) 1 ( L C ? =
(18)
So, the present value of the cash flows to
equity holders still equals their initial investment
of C(1 – L), and this satisfies the present value
principle.
Policy 2: The firm uses one year debt and the
regulator uses the two year riskfree rate
In respect of the debt premium, the analysis here
parallels that of policy 1 above, with debt
premiumrisk arising in the time 2 cash flows to
equity holders and netting out of the time 1 cash
flows. This debt premiumrisk in the time 2 cash
flows generates the same final termas in equation
(17). Otherwise equation (8) applies. So, the
resulting present value at time 0 is as follows.
?
?
?
?
?
?
+
? ?
+
+
?
+ ? =
12
1
01
01 02
0
1
) )( 1 (
1
) (
) 1 (
R
p p k CL
PV
R
R R C
L C PV
01
01 02
1
) (
) 1 (
R
R R C
L C
+
?
+ ? =
(19)
This present value matches that derived for
policy 2 in the previous section and diverges
fromthe equity holders initial investment.
Policy 3: The firm uses two year debt and the
regulator uses the one year riskfree rate
Since the firm does not re-contract with debt
holders, the corporate debt premiumreflected in
the interest payments to debt holders at times 1
and 2 matches the regulator’s allowances at
those times for the debt premium. So, there is no
effect upon the net cash flows to equity holders
and the present value of their cash flows is then
as given in equation (12), i.e.,
+
?
? ? =
01
01 02
0
1
) (
) 1 (
R
R R LC
L C PV
?
?
?
?
?
?
+
? ?
?
12
12 02
1
) )( 1 (
R
R R k LC
PV
(20)
Again, the present value of the cash flows to
equity holders diverges from their initial
investment of C(1 – L).
Policy 4: The firm uses two year debt and the
regulator uses the two year riskfree rate
Paralleling the situation regarding policy 3,
there is no effect from a corporate debt
premium upon the net cash flows to equity
holders. The present value of these net cash
flows is then as given in equation (14), i.e.,
+
? ?
+ ? =
01
01 02
0
1
) )( 1 (
) 1 (
R
R R L C
L C PV
?
?
?
?
?
?
+
? ?
?
12
12 02
1
) )( 1 (
R
R R k LC
PV
(21)
Again, the present value of the cash flows to
equity holders diverges from their initial
investment of C(1 – L).
In summary, and in the presence of a
corporate debt premium, the present values of
the cash flows to equity holders across the four
policy scenarios are given in equations (18),
(19), (20) and (21). These results are identical to
those in the previous section, in which a
corporate debt premium was absent, and the
numerical results in Table 1 therefore still
apply. As before, the only policy combination
that satisfies the present value principle is the
first, in which the regulator uses the risk free
rate matching the regulatory cycle and the firm
chooses debt with the same term. In addition,
the firm’s use of debt with that term protects
equity holders from interest rate risk and
exposes them to re-contracting risk. By contrast,
the use of longer-term debt can never satisfy the
present value principle, exposes equity holders
to interest rate risk and protects them from re-
contracting risk.
The implications of all this for the firm and
the regulator are as follows. As before, the
regulator’s goal should be to satisfy the present
value principle. Using the two-year risk free rate
cannot satisfy this principle, regardless of the
debt policy adopted by the firm. By contrast,
use of the one-year rate will achieve this, so
long as the firm uses one-year debt. If the firm
elects instead to use two-year debt, implying
policy 3, the resulting future cash flows to
equity holders will have a net present value that
will tend to be negative, will expose them to
interest rate risk and protect them from re-
contracting risk. The adverse effect upon the
present value of equity holders cash flows, and
exposure to interest rate risk, should discourage
the firm from acting in this way, but the
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80
protection from re-contracting risk will exert a
contrary effect. If this contrary effect induces
the firmto use longer-term debt, and it fails to
use swaps to match the duration of its debt to
the regulatory cycle, then the present value
principle will no longer be satisfied. However,
this cannot justify the regulator switching to the
use of the two-year risk free rate (switching
frompolicy 3 to 4) because the only effect from
doing so will be to exchange a net present value
on the cash flows to equity holders that tends to
be negative for one that tends to be positive (see
Table 1).
4. Conclusions
This paper examines the appropriate term of the
risk free rate to be used by a regulator in price
control situations, and in the presence of debt. If
the only source of risk is interest rate risk, then
the regulator should choose the term of the risk
free rate to satisfy the present value principle.
The only choice of termfor the risk free rate
that can achieve this is that matching the
regulatory cycle, but it also requires that the
firm match its debt duration to the regulatory
cycle. Failure of the firmto do so leads to cash
flows to equity holders whose net present value
will tend to be negative, and will also inflict
interest rate risk upon equity holders. This
provides the firm with strong incentives to
match its debt duration to the regulatory cycle.
If the issue of re-contracting risk arises,
because the regulator invokes efficient rather
than actual costs, the regulator should still seek
to satisfy the present value principle. Again, the
only choice of termfor the risk free rate that can
achieve this is that matching the regulatory
cycle, and again this requires that the firm
match its debt duration to the regulatory cycle.
If the firm chooses a longer duration on its debt,
in an effort to eliminate or reduce re-contracting
risk, the net present value of the resulting cash
flows to equity holders will tend to be negative,
and the firm will inflict interest rate risk upon
equity holders. Furthermore, the firm’s choice
of the longer duration on its debt cannot justify
the regulator lengthening the term of the risk
free rate used in setting prices because the only
effect from doing so will be to exchange a net
present value on the cash flows to equity
holders that tends to be negative for one that
tends to be positive.
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This article has been cited by:
1. Kevin Davis. 2014. The Debt Maturity Issue in Access Pricing. Economic Record 90:10.1111/ecor.2014.90.issue-290, 271-281.
[CrossRef]
2. Jason Hall. 2007. Comment on Regulation and the Term of the Risk Free Rate: Implications of Corporate Debt. Accounting
Research Journal 20:2, 81-86. [Abstract] [PDF]
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doc_955933271.pdf