Description
The analytics
behind this conclusion assume that forward
rates are an unbiased estimate of future spot
rates, an assumption which is inconsistent with
empirical evidence. This has an important
economic implication.
Accounting Research Journal
Comment on Regulation and the Term of the Risk Free Rate: Implications of Corporate Debt
J ason Hall
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To cite this document:
J ason Hall, (2007),"Comment on Regulation and the Term of the Risk Free Rate: Implications of Corporate Debt",
Accounting Research J ournal, Vol. 20 Iss 2 pp. 81 - 86
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Comment on Regulation and the Term of the Risk Free Rate: Implications of Corporate Debt
81
Comment on
Regulation and the Term of the Risk Free
Rate: Implications of Corporate Debt
Jason Hall
UQ Business School
The University of Queensland
Abstract
Lally (2007) concludes that regulators must
estimate the risk-free rate as the yield-to-
maturity on Government debt with a term-to-
maturity equal to the regulatory period, to
ensure that the present value of expected cash
flows equals the investment base. The analytics
behind this conclusion assume that forward
rates are an unbiased estimate of future spot
rates, an assumption which is inconsistent with
empirical evidence. This has an important
economic implication. With the typical case
being that the yield curve is upward-sloping,
adopting a short-term risk-free rate would result
in equityholders being systematically under-
compensated for the actual risk involved in a
long-lived project. If we adopt an alternative
assumption that current rates are an unbiased
estimate of future rates, the regulated rate of
return is a function of the entire forward curve
of interest rates and the accounting depreciation
schedule. For long-lived assets, benchmarking
against the yield-to-maturity on long-dated
Government securities results in a far closer
approximation of the appropriate return than the
use of short-term rates.
1. Introduction
An important contribution of academic research
is to confront conventional wisdom in corporate
practice, and overturn that conventional
thinking where it is theoretically flawed, or
inconsistent with empirical evidence. The
conventional wisdom adopted by regulatory
bodies is to estimate regulated firms’ cost of
capital with reference to long-dated
Government securities. Underlying this basic
premise is the view that the term of the risk-free
rate should approximate the asset life. In
Australian regulatory practice, this is
implemented by estimating the risk-free rate as
the yield-to-maturity on ten-year Government
bonds, the longest dated Government security
available.
Lally (2007) argues that this conventional
wisdom is inconsistent with valuation theory.
His contention is that the present value of
expected cash flows to equityholders can only
be guaranteed to equal the initial equity value
under two conditions. First, the term used to
estimate the risk-free rate equals the regulatory
period. Second, the firm borrows debt with that
same maturity. In practice, this would result in
the estimation of the risk-free rate over a term-
to-maturity of three to five years, and firms
borrowing over the same time period.
I show that this conclusion relies upon the
assumption that forward rates are an unbiased
expectation of future spot rates, a result which
the author agrees does not hold empirically.
However, before considering the theoretical
aspects of the paper, I first consider an
anomalous result which would necessarily flow
from the adoption of the author’s premise.
2. An Anomalous Result
If the author’s premise were implemented in an
Australian regulatory setting. regulated rates of
return would, on average, be systematically
lower than those currently estimated by
regulators, due to the typical case of an upward-
sloping yield curve. The most common outcome
would be the estimation of the risk-free rate
with reference to five-year Government bonds,
as opposed to the standard benchmark of ten-
year Government bonds. The table below
illustrates the potential magnitude of this
change, with reference to Government bond
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ACCOUNTING RESEARCH JOURNAL VOLUME 20 NO 2 (2007)
82
Table 1
Australian Government Bond Rates from 1977 – 2007
Effective annual rate (%) Term premiums (%)
Date 2-year 5-year 10-year 5 minus 2 10 minus 5 10 minus 2
25-Jun-77 10.1 10.5 10.7 0.3 0.2 0.5
25-Jun-82 17.1 17.1 17.1 0.0 0.0 0.0
25-Jun-87 13.6 13.7 13.4 0.1 -0.3 -0.2
25-Jun-92 6.5 7.9 8.9 1.5 1.0 2.5
25-Jun-97 5.6 6.6 7.2 0.9 0.6 1.5
25-Jun-02 5.5 5.9 6.0 0.3 0.2 0.5
25-Jun-07 6.5 6.5 6.3 0.0 -0.1 -0.2
Average 9.3 9.7 9.9 0.4 0.2 0.7
rates prevailing today and for six historical
five-year periods. On average, the ten-year
yield-to-maturity exceeded the five-year yield-
to-maturity by around 20 basis points, and
exceeded the two-year yield-to-maturity by
around 70 basis points.
By itself, this does not present an anomaly.
The author would argue that the current use of
ten-year bond rates provides regulated entities
with an abnormal positive return which should
be corrected. However, what if regulators
decided to reset regulated prices every two
years? On average we would observe a further
reduction in regulated rates of return, purely on
the basis of the Government’s view as to the
appropriate period for review. The regulator’s
objective is to estimate the price which would
prevail were competition present in the
regulated industry. There is no economic reason
why we would observe a relationship between
the competitive market price and the term of the
regulatory period.
The author does not present the regulator’s
objective in terms of the price which would
prevail under competition. He contends that the
regulator’s objective is to set the regulated price
such that the present value of expected
cash flows to equityholders matches the
equityholders’ investment base. This is the
mechanism by which the regulator attempts to
implement its over-riding objective, under the
assumption that the competitive price is that
which allows equityholders to earn a normal
return. Under the proposition of the current
paper, regulated prices would fluctuate
depending upon the Government’s view as to
the most cost-effective and administratively
efficient period for conducting price reviews,
considerations which have nothing to do with
the economic risks facing the firm.
Put another way, if the Government changes
the length of review period, the proposal in the
paper would see the series of expected cash
flows systematically increase or decrease. The
discount rates implied by the Capital Asset
Pricing Model would be unchanged because
there has been no change in their association
with the performance of the economy or market
portfolio, and the yields on Government bonds
have not altered. With a change in expected cash
flows and no change in discount rates the present
value this cash flow series must change. This
stands in direct contrast to the conclusion of the
author that he has identified a mechanism to
ensure that the present value remains constant.
3. Forward Curve and the
Regulated Rate of Return
What is the theoretical reason for these two
opposing conclusions? I show that the author
makes an implicit assumption that forward rates
are an unbiased expectation of future spot rates.
Under this assumption, any increase or decrease
in expected cash flows resulting from a change
in the regulatory period is exactly offset by a
change in the discount rate. Once this
assumption is relaxed, the author’s conclusions
no longer hold. This same assumption is
implicit in Schmalensee (1989) and Lally
(2004). In the latter paper by the same author,
he reaches the same conclusion in the absence
of debt finance. Furthermore, while the
intention of the author is to mitigate against a
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Comment on Regulation and the Term of the Risk Free Rate: Implications of Corporate Debt
83
perceived upward bias in regulated rates of
return, I show that the author’s proposition in
fact creates a downward bias in the regulated
rate of return.
According to the author’s stylised setting, in
setting regulated prices in the first year the
regulator can estimate the risk-free rate as
either (1) the yield-to-maturity on one-year
Government debt; or (2) the yield-to-maturity
on two-year Government debt (matching the
term-to-maturity with asset life). In the
following year, the regulatory period and
remaining asset life are both equal to one year,
so there is only one choice of regulated rate of
return. The second policy choice is analogous to
the real setting in which regulators set prices
every five years, derived from the yield-to-
maturity on ten-year Government debt.
According to the derivations in the paper, the
series of expected cash flows to equityholders
(F
t
) under the first policy is as follows:
( )( )( )
12 2
01 01 1
1 1 1 R L k C F
LCR LCk CR Ck F
+ ? ? =
? ? + =
where:
C = the investment cost;
L = leverage, the proportion of the investment
financed with debt;
k = depreciation in year one, expressed as a
percentage of investment cost;
R
01
= the interest rate on a one-year
Government bond; and
R
12
= the expected interest rate on a one-year
Government bond in one year’s time.
The first expected cash flow (F
1
) is
discounted to time zero using the one-year
interest rate (R
01
). The second expected cash
flow (F
2
) is discounted to time one using the
expected one-year interest rate available at time
one (R
12
) and then discounted to time zero using
the current one-year interest rate (R
01
). Thus, we
have the following expression for the present
value of expected cash flows to equityholders
(PV
0
)
1
:
( )( )
( )( )( )
( )( )
( ) L C
R R
R L k C
R
LCR LCk CR Ck
R R
F
R
F
PV
? =
+ +
+ ? ?
+
+
? ? +
=
+ +
+
+
=
1
1 1
1 1 1
1
1 1 1
12 01
12
01
01 01
12 01
2
01
1
0
(1)
1 Derivations for all equations are available on request. For
brevity, they have been omitted from the paper.
Consider the second term of the series of
equations presented above. The expected cash
flow in year two is discounted by the product of
one plus the one-year interest rate (1 + R
01
) and
one plus the expectation of the one-year rate
available in one year (1 + R
12
). The technically-
correct discount rate for this second year is the
yield-to-maturity on a zero-coupon bond maturing
in two years. This point is made unequivocally by
Damodaran (2001, p. 188) who states:
“the riskless rate is the rate on a zero-coupon bond
that matches the time horizon of the cash flow
being analyzed…this translates into using different
riskless rates for each cash flow on an investment –
the one-year zero-coupon rate for the cash flow in
year 1, the two-year zero-coupon rate for the cash
flow in year 2, and so on.”
Under arbitrage-free pricing of Government
bonds, the following relationship holds, where
R
02
z
is the yield-to-maturity on a two-year zero-
coupon Government bond and R
12
F
is the
forward rate for a one-year Government bond
for investment in one year’s time:
( ) ( )( )
F z
R R R
12 01
2
02
1 1 1 + + = +
(2)
Invoking this discounting assumption for the
second expected cash flow, we have the
following series of equations:
( )( )
( )( )( )
( )( )
( )
( )
( )( )
( )
?
?
?
?
?
?
?
?
?
+ +
?
? ? =
+ +
+ ? ?
+
+
? ? +
=
+ +
+
+
=
12 12
12 01
12 01
12
01
01 01
12 01
2
01
1
0
1 1
1
1 1
1 1
1 1 1
1
1 1 1
R R
R R
k
L C
R R
R L k C
R
LCR LCk CR Ck
R R
F
R
F
PV
F
F
F
F
(3)
According to this equation, the following
conclusions hold under positive interest rates
and depreciation:
• If the forward rate for investment in one year
exceeds the expectation of the spot rate in
one year (that is, if R
12
F
> R
12
), the present
value of expected cash flows will be less
than the initial equity investment [that is, PV
0
< C(1-L)];
• If the forward rate for investment in one year
is less than the expectation of the spot rate in
one year (that is, if R
12
F
< R
12
), the present
value of expected cash flows will be greater
than the initial investment [that is, PV
0
>
C(1-L)]; and
• The present value of expected cash flows to
equityholders will equal the investment base
only in the case where the forward rate and
the expected future spot rate are equal.
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ACCOUNTING RESEARCH JOURNAL VOLUME 20 NO 2 (2007)
84
The key point is that determining the period
one regulated rate of return is affected in a
material way by an assumption regarding the
relationship between spot rates, forward rates
and expected changes in spot rates. If we form
an expectation that forward rates are an
unbiased expectation of future spot rates, then it
would be reasonable to set regulated rates of
return with reference to the regulatory period.
When presented with an upward-sloping yield
curve, lower regulated rates of return in the first
period would be offset by the expectation of
higher regulated rates of return in a subsequent
period.
However, the empirical evidence is that
forward rates are no better at predicting future
spot rates than current spot rates (Fama, 1976)
and we typically observe an upward-sloping
yield curve. Hence, setting regulated rates of
return with reference to near-term interest rates
is likely to result in regulated rates of return
which do not allow the firm to earn its cost of
capital. The author’s attempt to mitigate against
a perceived upward bias in regulated rates is
actually likely to cause a downward bias in
those rates.
The series of equations below demonstrate
how to estimate the regulated rate of return in
period one (labelled r
1
) with reference to any
expected rate of return in period two (r
2
):
( ) ( )
( )( )
( )( ) ( ) ( )
F
F F
F
R
r r R k R R
r
R R
C k r k C
R
kC Cr
C
12
2 2 12 12 01
1
12 01
2
01
1
1
1 1 1
1 1
1 1
1
+
+ ? ? ? + +
=
+ +
? + ?
+
+
+
=
(4)
Now suppose we make the alternative
assumption that current spot rates are an
unbiased expectation of future spot rates. We
will also assume that the regulated rate of return
in the second year is set with reference to the
remaining asset life of one year. Together, these
assumptions imply that r
2
= R
12
= R
01
. The
equation for the regulated rate of return in year
one becomes a weighted average of the one-
year rate (R
01
) and the one-year forward rate
(R
12
F
):
( ) ( )
F
F F
R
R R k R R
r
12
01 12 01 12
1
1
1
+
? ? +
=
(5)
In contrast, if we make the author’s implicit
assumption that the expected one-year rate
is equal to the forward rate (that is,
r
2
= R
12
= R
12
F
), then the equation for the
regulated rate of return in year one is the current
one-year rate (R
01
) as demonstrated below:
( )( ) ( ) ( )
( )( ) ( ) ( )
( ) ( ) [ ]
01
12
01 12
12
12 12 12 12 01
12
2 2 12 12 01
1
1
1 1 1
1
1 1 1
1
1 1 1
R
R
R R
R
R R R k R R
R
r r R k R R
r
F
F
F
F F F F
F
F F
=
+
? + +
=
+
+ ? ? ? + +
=
+
+ ? ? ? + +
=
(6)
Equation 6 shows that it is only under this
latter assumption that the expectations
hypothesis holds that the regulated rate of return
in period one is equal to the current one-year
rate.
4. Numerical Example
A numerical example confirms these
conclusions. I use the same expositional data
presented in the paper, whereby the investment
base is $10,000 (C = $10,000), accounting
depreciation is spread equally over the two-year
asset life (k = 0.50); the one-year risk-free rate
is 5 percent (R
01
= 0.05); and the one-year
forward rate is 7.01 percent
2
(R
12
F
= 0.0701). If
we expect the one-year rate next period to
remain at its current level of 5 percent, the
appropriate regulated rate of return is 5.94
percent, computed as follows:
( ) ( )
( ) ( )
% 94 . 5
0701 . 1
05 . 0 0701 . 0 5 . 0 05 . 1 0701 . 0
1
1
12
01 12 01 12
1
=
? ?
=
+
? ? +
=
F
F F
R
R R k R R
r
(7)
The series of expected cash flows available
to the firm are as follows:
• Year 1 expected cash flows are $5,594
(10,000 × 0.0594 + 5,000 = $5,594); and
• Year 2 expected cash flows are $5,250
(5,000 x 0.0500 + 5,000) = $5,250).
The present value of this series of expected
cash flows is $10,000 as shown below:
2 This is derived from the assumption that the two-year
spot rate is 6 percent; that is, (1.06)
2
/1.05 – 1 = 0.0701.
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Comment on Regulation and the Term of the Risk Free Rate: Implications of Corporate Debt
85
( )( )
000 , 10 $
672 , 4 328 , 5
0701 . 1 05 . 1
250 , 5
05 . 1
594 , 5
0
=
+ =
+ = V
Once we introduce leverage, we can verify
that the series of expected cash flows available
to equityholders is equal to their investment
base. This conclusion will hold, provided the
firm borrows in a way which matches debt
repayments to the series of expected cash flows
– a standard cash flow matching strategy for
minimising interest rate risk. Suppose that
leverage is 50 percent (L = 0.50), allocated
amongst a zero-coupon bond maturing in one
year and a zero-coupon bond maturing in two
years. Each of these bonds has a future value
equal to L times the expected cash flow in the
relevant year. So the firm’s borrowing position
of $5,000 is as follows:
• Borrow $2,664 at 5 percent per annum over
one year, requiring repayment of $2,797; and
• Borrow $2,336 at 6 percent per annum over
two years, requiring repayment of $2,625;
Under these borrowing terms, the present
value of expected cash flows to equityholders is
exactly $5,000 as shown below:
( )( )
000 , 5 $
336 , 2 664 , 2
0701 . 1 05 . 1
625 , 2 250 , 5
05 . 1
797 , 2 594 , 5
0
=
+ =
?
+
?
= PV
5. Comparison to Existing
Regulatory Practice
How does this stylised example compare to
current practice of setting the regulated rate of
return with reference to the yield-to-maturity on
ten-year Government bonds? Equation 4 shows
that the appropriate regulated return in period
one is a weighted average of the current one-
year rate (R
01
) and the one-year forward rate for
investment in one year (R
12
F
) where the weight
placed on each component is directly related to
accounting depreciation. At very high rates of
accounting depreciation in period one (suppose
that k ? 1), the regulated rate of return
approximates the current one-year rate, as
shown below:
( ) ( )
( )
01
12
12 01
12
01 12 01 12 12
12
01 12 01 12
1
1
1
1
1
1
R
R
R R
R
R R R R R
R
R R k R R
r
F
F
F
F F F
F
F F
=
+
+
=
+
+ ? +
=
+
? ? +
=
(8)
At the other extreme, where there is almost
no depreciation in the first period (suppose that
k ? 0), the regulated return in period one
becomes:
( )
F
F
R
R R
r
12
01 12
1
1
1
+
+
=
(9)
Using the figures from the numerical example,
the depreciation assumption could result in a
regulated rate of return in period one anywhere
from 5 percent (at k = 1) to 6.88 percent
(at k = 0). Under the same zero-coupon rate
assumptions, the coupon rate and yield-
to-maturity on a two-year coupon-paying
Government bond trading at par would be 5.97
percent, computed as follows:
( )
% 97 . 5
06 . 1
1
05 . 1
1
2
=
+
+ =
c
c c
This yield-to-maturity is approximately the
same as the regulated rate of return of 5.94
percent under the assumption that k = 0.50.
However, the magnitude of any difference will
depend on the particular shape of the yield
curve at one moment in time, and the particular
depreciation rate. In drawing a practical
implication from this analysis, it is prudent to
ask, what set of assumptions are most consistent
with the depreciation of regulated assets and
observations about changes in interest rates?
Regulated assets have lives which are
significantly greater than ten years. So the
actual amount of depreciation in any given year
is a small percentage. By extrapolation from
equation 4, the smaller the depreciation rate, the
more weight is applicable to forward rates in
determining the near-term regulated rate of
return. Under these reasonable assumptions,
while the yield-to-maturity on ten-year
Government bonds is not the technically-correct
estimate of the regulated rate of return, it is a far
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ACCOUNTING RESEARCH JOURNAL VOLUME 20 NO 2 (2007)
86
closer approximation than the rate estimated
with reference to the short-term regulatory
period. Given that we observe an upward-
sloping yield curve more than half the time, the
use of the short-term rate is likely to create a
downward bias in the regulated rate of return.
Hence, the author’s proposal to eliminate a
perceived upward bias in the regulated rate of
return would, in fact, create a downward bias in
that return.
6.Conclusion
The author concludes that regulators must
estimate the risk-free rate as the yield-to-
maturity on Government debt with a term-to-
maturity equal to the regulatory period, to
ensure that the present value of expected cash
flows equals the investment base. The analytics
behind this conclusion assume that forward
rates are an unbiased estimate of future spot
rates, an assumption which is inconsistent with
empirical evidence. This has an important
economic implication. With the typical case
being that the yield curve is upward-sloping,
adopting a short-term risk-free rate would result
in equityholders being systematically under-
compensated for the actual risk involved in a
long-lived project. If we adopt an alternative
assumption that current rates are an unbiased
estimate of future rates, the regulated rate of
return is a function of the entire forward curve
of interest rates and the accounting depreciation
schedule. For long-lived assets, benchmarking
against the yield-to-maturity on long-dated
Government securities results in a far closer
approximation of the appropriate return than the
use of short-term rates.
References
Damodaran, A. (2001), Corporate Finance: Theory and
Practice, 2nd edition: John Wiley and Sons, New York.
Fama, E.F. (1976), “Forward rates as predictors of future
spot rates,” Journal of Financial Economics, vol. 3,
pp. 361–377.
Lally, M. (2004), “Regulation and the choice of the risk
free rate,” Accounting Research Journal, vol. 17(1),
pp. 18–23.
Lally, M. (2007), “Regulation and the term of the risk-free
rate: Implications of corporate debt,” Accounting
Research Journal, vol. 20(2), pp. 73–81.
Schmalensee, R. (1989), “An expository note on
depreciation and profitability under rate-of-return
regulation,” vol. 1, pp. 293–298.
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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]
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doc_729558713.pdf
The analytics
behind this conclusion assume that forward
rates are an unbiased estimate of future spot
rates, an assumption which is inconsistent with
empirical evidence. This has an important
economic implication.
Accounting Research Journal
Comment on Regulation and the Term of the Risk Free Rate: Implications of Corporate Debt
J ason Hall
Article information:
To cite this document:
J ason Hall, (2007),"Comment on Regulation and the Term of the Risk Free Rate: Implications of Corporate Debt",
Accounting Research J ournal, Vol. 20 Iss 2 pp. 81 - 86
Permanent link to this document:
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Martin Lally, (2007),"Rejoinder: Regulation and the Term of the Risk Free Rate: Implications of Corporate Debt", Accounting
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Comment on Regulation and the Term of the Risk Free Rate: Implications of Corporate Debt
81
Comment on
Regulation and the Term of the Risk Free
Rate: Implications of Corporate Debt
Jason Hall
UQ Business School
The University of Queensland
Abstract
Lally (2007) concludes that regulators must
estimate the risk-free rate as the yield-to-
maturity on Government debt with a term-to-
maturity equal to the regulatory period, to
ensure that the present value of expected cash
flows equals the investment base. The analytics
behind this conclusion assume that forward
rates are an unbiased estimate of future spot
rates, an assumption which is inconsistent with
empirical evidence. This has an important
economic implication. With the typical case
being that the yield curve is upward-sloping,
adopting a short-term risk-free rate would result
in equityholders being systematically under-
compensated for the actual risk involved in a
long-lived project. If we adopt an alternative
assumption that current rates are an unbiased
estimate of future rates, the regulated rate of
return is a function of the entire forward curve
of interest rates and the accounting depreciation
schedule. For long-lived assets, benchmarking
against the yield-to-maturity on long-dated
Government securities results in a far closer
approximation of the appropriate return than the
use of short-term rates.
1. Introduction
An important contribution of academic research
is to confront conventional wisdom in corporate
practice, and overturn that conventional
thinking where it is theoretically flawed, or
inconsistent with empirical evidence. The
conventional wisdom adopted by regulatory
bodies is to estimate regulated firms’ cost of
capital with reference to long-dated
Government securities. Underlying this basic
premise is the view that the term of the risk-free
rate should approximate the asset life. In
Australian regulatory practice, this is
implemented by estimating the risk-free rate as
the yield-to-maturity on ten-year Government
bonds, the longest dated Government security
available.
Lally (2007) argues that this conventional
wisdom is inconsistent with valuation theory.
His contention is that the present value of
expected cash flows to equityholders can only
be guaranteed to equal the initial equity value
under two conditions. First, the term used to
estimate the risk-free rate equals the regulatory
period. Second, the firm borrows debt with that
same maturity. In practice, this would result in
the estimation of the risk-free rate over a term-
to-maturity of three to five years, and firms
borrowing over the same time period.
I show that this conclusion relies upon the
assumption that forward rates are an unbiased
expectation of future spot rates, a result which
the author agrees does not hold empirically.
However, before considering the theoretical
aspects of the paper, I first consider an
anomalous result which would necessarily flow
from the adoption of the author’s premise.
2. An Anomalous Result
If the author’s premise were implemented in an
Australian regulatory setting. regulated rates of
return would, on average, be systematically
lower than those currently estimated by
regulators, due to the typical case of an upward-
sloping yield curve. The most common outcome
would be the estimation of the risk-free rate
with reference to five-year Government bonds,
as opposed to the standard benchmark of ten-
year Government bonds. The table below
illustrates the potential magnitude of this
change, with reference to Government bond
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ACCOUNTING RESEARCH JOURNAL VOLUME 20 NO 2 (2007)
82
Table 1
Australian Government Bond Rates from 1977 – 2007
Effective annual rate (%) Term premiums (%)
Date 2-year 5-year 10-year 5 minus 2 10 minus 5 10 minus 2
25-Jun-77 10.1 10.5 10.7 0.3 0.2 0.5
25-Jun-82 17.1 17.1 17.1 0.0 0.0 0.0
25-Jun-87 13.6 13.7 13.4 0.1 -0.3 -0.2
25-Jun-92 6.5 7.9 8.9 1.5 1.0 2.5
25-Jun-97 5.6 6.6 7.2 0.9 0.6 1.5
25-Jun-02 5.5 5.9 6.0 0.3 0.2 0.5
25-Jun-07 6.5 6.5 6.3 0.0 -0.1 -0.2
Average 9.3 9.7 9.9 0.4 0.2 0.7
rates prevailing today and for six historical
five-year periods. On average, the ten-year
yield-to-maturity exceeded the five-year yield-
to-maturity by around 20 basis points, and
exceeded the two-year yield-to-maturity by
around 70 basis points.
By itself, this does not present an anomaly.
The author would argue that the current use of
ten-year bond rates provides regulated entities
with an abnormal positive return which should
be corrected. However, what if regulators
decided to reset regulated prices every two
years? On average we would observe a further
reduction in regulated rates of return, purely on
the basis of the Government’s view as to the
appropriate period for review. The regulator’s
objective is to estimate the price which would
prevail were competition present in the
regulated industry. There is no economic reason
why we would observe a relationship between
the competitive market price and the term of the
regulatory period.
The author does not present the regulator’s
objective in terms of the price which would
prevail under competition. He contends that the
regulator’s objective is to set the regulated price
such that the present value of expected
cash flows to equityholders matches the
equityholders’ investment base. This is the
mechanism by which the regulator attempts to
implement its over-riding objective, under the
assumption that the competitive price is that
which allows equityholders to earn a normal
return. Under the proposition of the current
paper, regulated prices would fluctuate
depending upon the Government’s view as to
the most cost-effective and administratively
efficient period for conducting price reviews,
considerations which have nothing to do with
the economic risks facing the firm.
Put another way, if the Government changes
the length of review period, the proposal in the
paper would see the series of expected cash
flows systematically increase or decrease. The
discount rates implied by the Capital Asset
Pricing Model would be unchanged because
there has been no change in their association
with the performance of the economy or market
portfolio, and the yields on Government bonds
have not altered. With a change in expected cash
flows and no change in discount rates the present
value this cash flow series must change. This
stands in direct contrast to the conclusion of the
author that he has identified a mechanism to
ensure that the present value remains constant.
3. Forward Curve and the
Regulated Rate of Return
What is the theoretical reason for these two
opposing conclusions? I show that the author
makes an implicit assumption that forward rates
are an unbiased expectation of future spot rates.
Under this assumption, any increase or decrease
in expected cash flows resulting from a change
in the regulatory period is exactly offset by a
change in the discount rate. Once this
assumption is relaxed, the author’s conclusions
no longer hold. This same assumption is
implicit in Schmalensee (1989) and Lally
(2004). In the latter paper by the same author,
he reaches the same conclusion in the absence
of debt finance. Furthermore, while the
intention of the author is to mitigate against a
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Comment on Regulation and the Term of the Risk Free Rate: Implications of Corporate Debt
83
perceived upward bias in regulated rates of
return, I show that the author’s proposition in
fact creates a downward bias in the regulated
rate of return.
According to the author’s stylised setting, in
setting regulated prices in the first year the
regulator can estimate the risk-free rate as
either (1) the yield-to-maturity on one-year
Government debt; or (2) the yield-to-maturity
on two-year Government debt (matching the
term-to-maturity with asset life). In the
following year, the regulatory period and
remaining asset life are both equal to one year,
so there is only one choice of regulated rate of
return. The second policy choice is analogous to
the real setting in which regulators set prices
every five years, derived from the yield-to-
maturity on ten-year Government debt.
According to the derivations in the paper, the
series of expected cash flows to equityholders
(F
t
) under the first policy is as follows:
( )( )( )
12 2
01 01 1
1 1 1 R L k C F
LCR LCk CR Ck F
+ ? ? =
? ? + =
where:
C = the investment cost;
L = leverage, the proportion of the investment
financed with debt;
k = depreciation in year one, expressed as a
percentage of investment cost;
R
01
= the interest rate on a one-year
Government bond; and
R
12
= the expected interest rate on a one-year
Government bond in one year’s time.
The first expected cash flow (F
1
) is
discounted to time zero using the one-year
interest rate (R
01
). The second expected cash
flow (F
2
) is discounted to time one using the
expected one-year interest rate available at time
one (R
12
) and then discounted to time zero using
the current one-year interest rate (R
01
). Thus, we
have the following expression for the present
value of expected cash flows to equityholders
(PV
0
)
1
:
( )( )
( )( )( )
( )( )
( ) L C
R R
R L k C
R
LCR LCk CR Ck
R R
F
R
F
PV
? =
+ +
+ ? ?
+
+
? ? +
=
+ +
+
+
=
1
1 1
1 1 1
1
1 1 1
12 01
12
01
01 01
12 01
2
01
1
0
(1)
1 Derivations for all equations are available on request. For
brevity, they have been omitted from the paper.
Consider the second term of the series of
equations presented above. The expected cash
flow in year two is discounted by the product of
one plus the one-year interest rate (1 + R
01
) and
one plus the expectation of the one-year rate
available in one year (1 + R
12
). The technically-
correct discount rate for this second year is the
yield-to-maturity on a zero-coupon bond maturing
in two years. This point is made unequivocally by
Damodaran (2001, p. 188) who states:
“the riskless rate is the rate on a zero-coupon bond
that matches the time horizon of the cash flow
being analyzed…this translates into using different
riskless rates for each cash flow on an investment –
the one-year zero-coupon rate for the cash flow in
year 1, the two-year zero-coupon rate for the cash
flow in year 2, and so on.”
Under arbitrage-free pricing of Government
bonds, the following relationship holds, where
R
02
z
is the yield-to-maturity on a two-year zero-
coupon Government bond and R
12
F
is the
forward rate for a one-year Government bond
for investment in one year’s time:
( ) ( )( )
F z
R R R
12 01
2
02
1 1 1 + + = +
(2)
Invoking this discounting assumption for the
second expected cash flow, we have the
following series of equations:
( )( )
( )( )( )
( )( )
( )
( )
( )( )
( )
?
?
?
?
?
?
?
?
?
+ +
?
? ? =
+ +
+ ? ?
+
+
? ? +
=
+ +
+
+
=
12 12
12 01
12 01
12
01
01 01
12 01
2
01
1
0
1 1
1
1 1
1 1
1 1 1
1
1 1 1
R R
R R
k
L C
R R
R L k C
R
LCR LCk CR Ck
R R
F
R
F
PV
F
F
F
F
(3)
According to this equation, the following
conclusions hold under positive interest rates
and depreciation:
• If the forward rate for investment in one year
exceeds the expectation of the spot rate in
one year (that is, if R
12
F
> R
12
), the present
value of expected cash flows will be less
than the initial equity investment [that is, PV
0
< C(1-L)];
• If the forward rate for investment in one year
is less than the expectation of the spot rate in
one year (that is, if R
12
F
< R
12
), the present
value of expected cash flows will be greater
than the initial investment [that is, PV
0
>
C(1-L)]; and
• The present value of expected cash flows to
equityholders will equal the investment base
only in the case where the forward rate and
the expected future spot rate are equal.
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ACCOUNTING RESEARCH JOURNAL VOLUME 20 NO 2 (2007)
84
The key point is that determining the period
one regulated rate of return is affected in a
material way by an assumption regarding the
relationship between spot rates, forward rates
and expected changes in spot rates. If we form
an expectation that forward rates are an
unbiased expectation of future spot rates, then it
would be reasonable to set regulated rates of
return with reference to the regulatory period.
When presented with an upward-sloping yield
curve, lower regulated rates of return in the first
period would be offset by the expectation of
higher regulated rates of return in a subsequent
period.
However, the empirical evidence is that
forward rates are no better at predicting future
spot rates than current spot rates (Fama, 1976)
and we typically observe an upward-sloping
yield curve. Hence, setting regulated rates of
return with reference to near-term interest rates
is likely to result in regulated rates of return
which do not allow the firm to earn its cost of
capital. The author’s attempt to mitigate against
a perceived upward bias in regulated rates is
actually likely to cause a downward bias in
those rates.
The series of equations below demonstrate
how to estimate the regulated rate of return in
period one (labelled r
1
) with reference to any
expected rate of return in period two (r
2
):
( ) ( )
( )( )
( )( ) ( ) ( )
F
F F
F
R
r r R k R R
r
R R
C k r k C
R
kC Cr
C
12
2 2 12 12 01
1
12 01
2
01
1
1
1 1 1
1 1
1 1
1
+
+ ? ? ? + +
=
+ +
? + ?
+
+
+
=
(4)
Now suppose we make the alternative
assumption that current spot rates are an
unbiased expectation of future spot rates. We
will also assume that the regulated rate of return
in the second year is set with reference to the
remaining asset life of one year. Together, these
assumptions imply that r
2
= R
12
= R
01
. The
equation for the regulated rate of return in year
one becomes a weighted average of the one-
year rate (R
01
) and the one-year forward rate
(R
12
F
):
( ) ( )
F
F F
R
R R k R R
r
12
01 12 01 12
1
1
1
+
? ? +
=
(5)
In contrast, if we make the author’s implicit
assumption that the expected one-year rate
is equal to the forward rate (that is,
r
2
= R
12
= R
12
F
), then the equation for the
regulated rate of return in year one is the current
one-year rate (R
01
) as demonstrated below:
( )( ) ( ) ( )
( )( ) ( ) ( )
( ) ( ) [ ]
01
12
01 12
12
12 12 12 12 01
12
2 2 12 12 01
1
1
1 1 1
1
1 1 1
1
1 1 1
R
R
R R
R
R R R k R R
R
r r R k R R
r
F
F
F
F F F F
F
F F
=
+
? + +
=
+
+ ? ? ? + +
=
+
+ ? ? ? + +
=
(6)
Equation 6 shows that it is only under this
latter assumption that the expectations
hypothesis holds that the regulated rate of return
in period one is equal to the current one-year
rate.
4. Numerical Example
A numerical example confirms these
conclusions. I use the same expositional data
presented in the paper, whereby the investment
base is $10,000 (C = $10,000), accounting
depreciation is spread equally over the two-year
asset life (k = 0.50); the one-year risk-free rate
is 5 percent (R
01
= 0.05); and the one-year
forward rate is 7.01 percent
2
(R
12
F
= 0.0701). If
we expect the one-year rate next period to
remain at its current level of 5 percent, the
appropriate regulated rate of return is 5.94
percent, computed as follows:
( ) ( )
( ) ( )
% 94 . 5
0701 . 1
05 . 0 0701 . 0 5 . 0 05 . 1 0701 . 0
1
1
12
01 12 01 12
1
=
? ?
=
+
? ? +
=
F
F F
R
R R k R R
r
(7)
The series of expected cash flows available
to the firm are as follows:
• Year 1 expected cash flows are $5,594
(10,000 × 0.0594 + 5,000 = $5,594); and
• Year 2 expected cash flows are $5,250
(5,000 x 0.0500 + 5,000) = $5,250).
The present value of this series of expected
cash flows is $10,000 as shown below:
2 This is derived from the assumption that the two-year
spot rate is 6 percent; that is, (1.06)
2
/1.05 – 1 = 0.0701.
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Comment on Regulation and the Term of the Risk Free Rate: Implications of Corporate Debt
85
( )( )
000 , 10 $
672 , 4 328 , 5
0701 . 1 05 . 1
250 , 5
05 . 1
594 , 5
0
=
+ =
+ = V
Once we introduce leverage, we can verify
that the series of expected cash flows available
to equityholders is equal to their investment
base. This conclusion will hold, provided the
firm borrows in a way which matches debt
repayments to the series of expected cash flows
– a standard cash flow matching strategy for
minimising interest rate risk. Suppose that
leverage is 50 percent (L = 0.50), allocated
amongst a zero-coupon bond maturing in one
year and a zero-coupon bond maturing in two
years. Each of these bonds has a future value
equal to L times the expected cash flow in the
relevant year. So the firm’s borrowing position
of $5,000 is as follows:
• Borrow $2,664 at 5 percent per annum over
one year, requiring repayment of $2,797; and
• Borrow $2,336 at 6 percent per annum over
two years, requiring repayment of $2,625;
Under these borrowing terms, the present
value of expected cash flows to equityholders is
exactly $5,000 as shown below:
( )( )
000 , 5 $
336 , 2 664 , 2
0701 . 1 05 . 1
625 , 2 250 , 5
05 . 1
797 , 2 594 , 5
0
=
+ =
?
+
?
= PV
5. Comparison to Existing
Regulatory Practice
How does this stylised example compare to
current practice of setting the regulated rate of
return with reference to the yield-to-maturity on
ten-year Government bonds? Equation 4 shows
that the appropriate regulated return in period
one is a weighted average of the current one-
year rate (R
01
) and the one-year forward rate for
investment in one year (R
12
F
) where the weight
placed on each component is directly related to
accounting depreciation. At very high rates of
accounting depreciation in period one (suppose
that k ? 1), the regulated rate of return
approximates the current one-year rate, as
shown below:
( ) ( )
( )
01
12
12 01
12
01 12 01 12 12
12
01 12 01 12
1
1
1
1
1
1
R
R
R R
R
R R R R R
R
R R k R R
r
F
F
F
F F F
F
F F
=
+
+
=
+
+ ? +
=
+
? ? +
=
(8)
At the other extreme, where there is almost
no depreciation in the first period (suppose that
k ? 0), the regulated return in period one
becomes:
( )
F
F
R
R R
r
12
01 12
1
1
1
+
+
=
(9)
Using the figures from the numerical example,
the depreciation assumption could result in a
regulated rate of return in period one anywhere
from 5 percent (at k = 1) to 6.88 percent
(at k = 0). Under the same zero-coupon rate
assumptions, the coupon rate and yield-
to-maturity on a two-year coupon-paying
Government bond trading at par would be 5.97
percent, computed as follows:
( )
% 97 . 5
06 . 1
1
05 . 1
1
2
=
+
+ =
c
c c
This yield-to-maturity is approximately the
same as the regulated rate of return of 5.94
percent under the assumption that k = 0.50.
However, the magnitude of any difference will
depend on the particular shape of the yield
curve at one moment in time, and the particular
depreciation rate. In drawing a practical
implication from this analysis, it is prudent to
ask, what set of assumptions are most consistent
with the depreciation of regulated assets and
observations about changes in interest rates?
Regulated assets have lives which are
significantly greater than ten years. So the
actual amount of depreciation in any given year
is a small percentage. By extrapolation from
equation 4, the smaller the depreciation rate, the
more weight is applicable to forward rates in
determining the near-term regulated rate of
return. Under these reasonable assumptions,
while the yield-to-maturity on ten-year
Government bonds is not the technically-correct
estimate of the regulated rate of return, it is a far
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ACCOUNTING RESEARCH JOURNAL VOLUME 20 NO 2 (2007)
86
closer approximation than the rate estimated
with reference to the short-term regulatory
period. Given that we observe an upward-
sloping yield curve more than half the time, the
use of the short-term rate is likely to create a
downward bias in the regulated rate of return.
Hence, the author’s proposal to eliminate a
perceived upward bias in the regulated rate of
return would, in fact, create a downward bias in
that return.
6.Conclusion
The author concludes that regulators must
estimate the risk-free rate as the yield-to-
maturity on Government debt with a term-to-
maturity equal to the regulatory period, to
ensure that the present value of expected cash
flows equals the investment base. The analytics
behind this conclusion assume that forward
rates are an unbiased estimate of future spot
rates, an assumption which is inconsistent with
empirical evidence. This has an important
economic implication. With the typical case
being that the yield curve is upward-sloping,
adopting a short-term risk-free rate would result
in equityholders being systematically under-
compensated for the actual risk involved in a
long-lived project. If we adopt an alternative
assumption that current rates are an unbiased
estimate of future rates, the regulated rate of
return is a function of the entire forward curve
of interest rates and the accounting depreciation
schedule. For long-lived assets, benchmarking
against the yield-to-maturity on long-dated
Government securities results in a far closer
approximation of the appropriate return than the
use of short-term rates.
References
Damodaran, A. (2001), Corporate Finance: Theory and
Practice, 2nd edition: John Wiley and Sons, New York.
Fama, E.F. (1976), “Forward rates as predictors of future
spot rates,” Journal of Financial Economics, vol. 3,
pp. 361–377.
Lally, M. (2004), “Regulation and the choice of the risk
free rate,” Accounting Research Journal, vol. 17(1),
pp. 18–23.
Lally, M. (2007), “Regulation and the term of the risk-free
rate: Implications of corporate debt,” Accounting
Research Journal, vol. 20(2), pp. 73–81.
Schmalensee, R. (1989), “An expository note on
depreciation and profitability under rate-of-return
regulation,” vol. 1, pp. 293–298.
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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.
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