ML19250C197
| ML19250C197 | |
| Person / Time | |
|---|---|
| Issue date: | 10/31/1979 |
| From: | Davis S, Nash D, Roberts J Office of Nuclear Reactor Regulation |
| To: | |
| Shared Package | |
| ML19250C191 | List: |
| References | |
| FRN-43FR10370, RULE-PR-30, RULE-PR-40, RULE-PR-50, RULE-PR-70 NUREG-0607, NUREG-0607-DRFT, NUREG-607, NUREG-607-DRFT, NUDOCS 7911210284 | |
| Download: ML19250C197 (22) | |
Text
-
r DRAFT r
NUREG-0607 DISCOUNT RATE FOR ELECTRIC UTILITY INDUSTRY J. O. Roberts S. M. Davis D. A. Nash L. R. Abramson Manuscript Completed:
September 1979 Date Published:
October 1979 Cost-Benefit Analysis Branch Division of Site Safety and Environmental Analysis Office of Nuclear Reactor Regulation and Applied Statistics Branch Division of Technical Support Office of Management and Program Analysis U.S. Nuclear Regulatory Commission Washington D.C. 20555
,2e4 174 7 911210 2 F4,
k CONTENTS p
Page Introduction......................................................
1 Discount Rate.....................................................
1 Present Va1ue...........................................,.........
4 Escalation........................................................
5 levelized Cost....................................................
7 Conclusion.
10 Table 1...........................................................
12 Figure 1..........................................................
13 Appendix 7 *.-J e 6
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PREFACE This report was prepared by the staff of the Cost-Benefit Analysis Branch, Division of Site Safety and Environmental Analysis, Office of Nuclear Reactor Regulation, and the Applied Statistics Branch, Division of Technical Support, Office of Management and Program Analysis, U.S. Nuclear Regulatory Commission.
It reports the results of an analysis of the cost of money to the electric utility industry.
The report discusses discount rate, present value, escalation, and levelized cost.
The rationale for a discount rate that may be used in cost-benefit analyses relating to the production of electricity is presented.
It is hoped that individuals and organizations will find the report useful.
Ccmments and suggestions are invited from Federal and State agencies, industry, and concerned members of the public.
Comments and suggestions should be directed to the Chief, Cost-Benefit Analysis Branch, Division of Site Safety and Environmental Analysis, Office of Nuclear Reactor Regulation, United States Nuclear Regulatory Commission, Washington, D C. 20555.
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DISCOUNT RATE FOR ELECTRIC UTILITY INDUSTRY INTRODUCTION Cost-benefit analyses are carried out in accordance wi'.c the Commission's Regulation, 10 CFR Part 51, which implements the requirements of the National Environmental Policy Act of 1969 (NEPA).
The discount rate used for these analyses and in comparing the et.onomics of alternative generating systems is based on the cost of capital to the utility industry which includes an expecta-tion of inflation.
Hence, economic analysis should include inflation effects.
However, economic analysis could be based on a discount rate equal to the cost of capital without inflation in which case the inflation effects would not be included in the analysis.
As will be shown in later paragraphs, the two methods will produce the same results.
DISCOUNT RATE The relation between the cost of capital with inflation and the cost of capital without inflation is summarized in the following expression:
(1)
(1 + d) = (1 + i)(1 + R)
Where:
d is the cost of capital with inflation (discount rate) i is the annual rate of general inflation and R is the cost of capital without inflation.
= - +.
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=---
i The investor owned utility industry obtains capital by issuing bonds, preferred stock and common stock and by retaining a portion of earnings (in this discussion the retained earnings will be treated as part of the common stock holders equity). While the capitalization ratios will vary from utility to utility, the average for the industry is presented in Table 1 for the period 1955 through 1976.
The industry average capitalization ratios for this period have been nearly constant at about 53% bonds,11% preferred stock and 36% common stock plus retained earnings (retained earnings is about 11%).
The average yield for new issues of utility capital is presented in Table 1 for 1955 through 1976.
Two yields for common stock are presented.
One is based on the average price earning (PE) ratio of utility stock and reflects the yield or return to the investing public.
The other yield is based on the earnings available for common stock hciders expressed as a percent of equity (book value) 'and reflects the State Public Utility Commission's permitted return on the utilities investment.
The yields shown in Table 1 include the investing public's perce.ved rate of inflation.
If the gross national product implicit p i e deflator (IPD) is used as a measure of inflation, the rate of return (R) without inflation can be calculated using formula (1).
The IPD and the calculated yield without inflation are presented in Table 1.
Note that the yield with inflation generally increases as the inflation rate increases over time but that the yield without inflation does not.
The variation in the calculated yield without inflation provides a measure of how well the investing public was able to perceive among
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- k other considerations the rate of inflation.
This suggests that the market place is better able to take proper account of it than can the State Public Utility Commissions.
When a utility raises capital by issuing common stock, the market place determines the price for the,new issue of common stock and the yield is implied in the price earnings ratio for the stock.
Therefore, the weighted cost of capital based on the common stock yield as determined by the price earnings ratio should be a better indicator of the cost of equity capital to the utility industry than the yield on equity.
The weighted cost of capital (i.e., bond yield X bond capitalization ratio plus preferred stock yield X preferred stock capitalization ratio plus common stock yield X common stock and retained earning capitalization ratio) for both nethods of estimating common stock yields is shown in Table 1.
Note that when 1he weighted cost of capital is based on the PE ratio for common stock yield the cost of capital follows inflation reasonably well, certainly better than when the cost of capital is based on common stock earnings expressed as a rercent of equity.
This is better illustrated by comparing the calculated
.ield without inflation "R".
For the PE case the cost of capital without inflation is nearly constant with an average of 2.93% with a standard deviation of 10.80 where as for the equity case the average is higher and the variability is greater, i.e., average of 4.22 with a standard deviation of 1 1.46.
The calculated yield without inflation for bonds and preferred stock for the years 1974 and 1975 appear to be reflecting unusual conditions and if data for these
~-
-.4 years are excluded the weighted cost of money without inflation is 3.06 + 78 and 4.58 +.88 for the PE case and equity case, respectively.
A statistical analysis of the mean and variance of present value using 3.06%
as the weighted cost of capital without inflation is discussed in the Appendix.
An effective bound for present value is + 11%.
The 3.06% is rounded to 3.00 and will be used as the cost of capital without inflation in following analysis.
So much for historical data.
Our objective is to find a reasonable and appro-priate rate of return for bonds, preferred and common stock and rate of infla-tion for use in economic analysis covering periods up to 40 years (the length of time a nuclear plaat is licensed).
First it should be recognized that in present value analysis the early years of the relevant time frame have the greatest impact on results.
As one moves further into the future the impact on present values diminish.
It will be shown that present value analysis may use a discount rate equal to the cost of capital without general inflation or any combination of inflation rates and discount rates so long as the equality (1 + d) = (1 + i)(1 + R)'is maintained and the present values will be the same as using a zero inflation rate.
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n PRESENT VALUE y
Present value (PV) is defined as the value today of a sum of money in the future, a future value (FV), and expressed as:
(2)
PV = FV/(1 + d)"
Where:
d is the discount rate n is the number of years in the future The present value of a stream of annual payments (A) is:
N (3)
PV = I A /(1 + d)"
n n=1 If A in equation (3) is constnat, equation (3) may be written n
PV = A[(1 + d)"-1]/d(1 + d)"
In the case of a power plant, the cost may be grouped into three general categories:
fixed cost, fuel cost, and operation and maintenance (0&M) cost.
If it is assumed that the plant operates at a constant capacity factor over its life and there is no escalation of cost, then the present value for these s'; reams over the life of the plant may be calculated from the above equation (3).
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k ESCALATION Escalation may be divided into the following two ccmponents:
escalation due to general inflation (i) escalati,on (e) due to real causes such as depletion of high grade U0 and coal resources, new regulatory requirements for safety, 3 s environment, etc., or de-escalation due to cost saving advancements in technology or increased size of operation.
Incorporating these escalation factors into the calculation of present values, we have:
(1 + i)"(1 + e )"
CI + I) (I * '2) j
+A2r.
+...]
PV = n=1 [Ain (4)
(1 + d)"
(1 + d)"
where:
Aln, A2n...
re the annual costs such as the components of power plant operation (i.e., the fixed cost of investment, the variable and fixed cost of operation and maintenance, and fuel cost) or any series with varying annual costs for which a present value is desired.
Note:
e may be different for each A.
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=
+.e-e
==
=
me k
N is the number of years over which present values are desired.
For generating units N is normally taken to be the life of the unit or 30 years for NRC analysis.
that (1 + i)/(1 + d) = 1/(1 + R), from equation (1).
Substituting 1/(1 + R) for (1 + i)/(1 + d) in equation 4 we obtain N
'(1 + e)"
(1 + e)n (5) PV = I [A
+A
+...]
n=1 I" (1 + R)"
2" (1 + R)"
Thus general inflation is not a factor that affects the results of present value analysis.
If we assume a constant capacity factor
- so that Aln' An2'... are constant we can eliminate the summation operation.
Also, if we simplify the mathematics by letting (1 + e)/(1 + R) = 1/(1 + r) then equation (5) may be reduced to:
(6)
PV = A)[(1 + r )"-1]/r (1 + r )"+ A ECI + "2)" I3/"2(1 + r )"+ A 2
2 3*'
j j
j
- A comparison of variable vs constant capacity over the life of a generating unit is made in Appendix D, Table D-1 of NUREG-0480.
The comparison indicates that a constant capacity factor of 65% produced about the same present value cost over 30 years of operation as the following scenario for varying the capacity factor over the life of the plant.
1st year of operation 40%
2nd year of operation 65%
3rd year of operation 70%
4th year thru 15th year 70%
16th to 30th year dropping 2 percentage points per year, to 40% in 30th year.
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k Note:
if e = o then 1 + r = 1 + R = 1.03 LEVELIZED COST While there is general agreement on the methodology for present value calcula-tions, there is less agreement en the values for discount and escalation rates and essentially no agreement on the methodology and parameters for calculating levelized cost.
In the following paragraphs, two methods for levelizing costs are discussed, each based on a rational approach but producing'quite different levelized values.
It should be noted that the levelized cost is not required for making comparisons unless the alternatives being compared produce different amounts of energy.
In such cases other methods may be used to normalize energy production cost to a common basis.
If levelized values are used for comparing alternative generating systems, some caution is in order.
To calculate the "levelized cost" of electricity generation, the present value of the required revenue stream is set equal to the present value of the cost of generating the electricity.
The following expres= ion summarizes this:
Present value of revenues = Present Value of Production cost (1+i)"
N (1+i)"(1+e)",
g 3
(7)
_j (P)(C) (1+d)n = PV = I ln (1+d)"
A n=1
=-
u-
- - - = ::
=
-9_
Where:
P is the annual power delivered (fWh) which is a constant for constant s
capacity factor.
C is the levelized unit cost of power ($/FMh) i, e, d, N and A are explained above.
$/FMh = mills /kWh Since C is a constant the above expression may be rearranged as:
gj{1+i)"(1+e)"
Abl+I)(I+'2)"
A j
2
,,,, )
(8)
C =
"*I (I+d)
(I*d)"
p[l+i)"
n=1 (1+d)"
Note that i and e for the fixed charge component of electricity production is zero.
Also note that if the real escalation rate (e) for the other terms is zero or sufficiently small, the levelized unit cost (C) is less than the first year cost of generation because the capital cost of the plant does not escalate with inflation.
If the plant capacity factor is constant over the life of the plant, equation (8) may be reduced to:
A [1 + r))"- 1]/r (1 + r )" + A2 [(1 + r )"- 1]/r (I * "2)"
j j
j 2
2 P[(1 + R)"- 1]/R(1 + R)"
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- n.: n r_-
s Remember that:
(1 + r) = (1 + d)/(1 + i)(1 + e)
(1 + R) = (1 + d)/(1 + i) thus when e = 0 (1 + r) = (1 + R) and note th'at for the fixed charge term i and e are zero.
An alternate method for levelizing present value cost over the life of the plant is to exclude the inflation factor (1 + i) from the revenue side of equation (7).
This method is the more typical and results in the levelized costs being in nominal dollars versus the first method which is in terms of constant dollars.
The differences between the two methods can be identified when the two methods are reduced to a mathematical form, i.e., equation (8) above for the first method (constant dollars) and equation (10) below for the second method (Nominal dollars). Also, the differences are presented graphically in Figure 1 f'or zero and five percent inflation.
g fl+i)"(1+e )"
Afl+f)"(1+e)"
N A
j 2
+... ]
I
+
(10)
C = ""I fl+d)"
II+d)
N I
Pn
"*I (1+d)"
For constant capacity factor the expression reduces to:
A [1 + r )N-1]/r (1 + r )N +.A [(1 + r )"~ I3/I2(I + # )
j j
j 2
2 1
(11) C =
,2
+...
P[(1 + d)"- 1]/d(1 + d)"
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L When the inflatten rate (i) is zero, the two methods (equations 8 and 10) produce the same levelized cost.
The levelized cost is shown in Figure 1 for zero and five percent inflation.
Note that:
If the levelized cost in constant dollars were inflated at 5% the cost would follow the curved line and the present value of the area under the curve is equal to the present value of the area under the levelized cost in nominal dollars.
The constant dollar and nominal dollar levelized cost lines would move closer together for inflation rates lower than 5% and farther apart with higher inflation rates.
If i and e are zero the first year's cost of generation is the levelized cost.
CONCLUSION It is concluded that NRC use a discount rate equal to 3% in economic studies pertaining to the generation of electricity.
Such studies include comparisons of alternative energy sources, optimum timing analysis, decommissioning analysis, radioactive waste disposal studies, value impact analysis of safety issues, alternative plant designs and any other studies where cost and benefits occur at different times.
'. studies where it is desirable to show cost and benefits in current or future year dollars, the discount rate (d) and inflation rate (i) should be
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Thus a choice of i based on a forecast of inflation determines d.
The simplest such forecast might be to assume that the implicit price deflator for a recent year or average of several recent years would remain constant in.the future.
While it is not necessary to levelize cost in order to compare alternatives, there may be situations where levelized costs are desirable.
In such instances it is suggested that the first method for levelizing cost be used, i.e.,
levelized cost would be shown in constant dollars.
Note that if zero inflation is assumed (i.e., 3% discount rate) both levelizing methods produce the same levelized cost.
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5 10 15 20 25 30 Years FIGURE 1
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LEVELIZED COST IN CONSTANT AND NOMINAL DOLLARS i
APPENDIX y
The Mean and Variance of Present Value The basic formula for the present value of a string of annual payments A over a period of N years car. be written as N
j,9) n PV = A I 1 + d)'
n=1 I
(I)
AI
=
n=1 (1 + R)"
where (1 + i)/(1 + d) = 1/(1 + R) and i = rate of inflation, d = cost of capital with inflation, R = cost of capital without inflation.
This formulation assumes that R is constant over time.
In reality, R varies over time, so that a more realistic formula is N
1 (2)
PV = A I n=1 (1 + R )
n where R, is the cost of capital without inflation in year n.
A more convenient form for (2) is
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A-2 N
PV = A I X"
(3) n=1 where X
- n 1
Rn A statistical analysis of 20 values of the weighted cost of capital based on the P/E ratio without inflation from Table 1 for the period 1955-1977 (no data was available for 1965; 1974 and 1975 were omitted as years with extraordinarily high inflation) indicates that these 20 values are approximately normally distributed with no apparent trend over time.
Accordingly, we assume that X,j X,...,X are independent random variables with common mean p and variance 2
N 2
o.
(For our purpose, there is no need to assume that the X are n rmally n
distributed.)
For the 20 values of X fr m Table 1, the observed mean and standard deviation n
are p =.9704 g
o, =.0073 Accordingly, in the derivation which follows, we shall assume that o & p < l.
(Note that R > 0 implies that X # I*)
n n
We will now derive approximate bounds for the mean and variance of PV from (3). We write X in the normalized form:
n
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~
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n' (4) where E(Y ) = 0 and Var (Y ) = 1.
Thus (3) becomes n
n N
PV = A I (p + oY ).
(5) n n=1 Since a << p, an excellent approximation to PV is given by the first three ter.ns in the binomial expansion of (p + oY )".
We have n
PV D A (p" + np"'I oY +
9 Y ).
(6) n n
n=1 Taking expected values of both sides yields (p"+"f"'I)p-2,2).
(7)
E(PV) %
A n
n=1 The sum of the geometric series N
" N+1 p" = "I ~ P I
(8) n=1 Since p < 1, substituting (8) into (7) yields the approximate bound
A-4 k
E(PV)gA[M^g+1 +3N g--
I n(n - 1)]
n=1 N+1*3
=A[Y^"
g--
N(N2 _ j )),
(g)
This is the desired bound on E(PV).
Note that the first term in (9) is exactly equal to PV if Xn " E' I***'
II is constant.
Thus the second term n
2 2
8 s A "-- N(N - 1)
(10) 6 is an approximate bound on the expected value of the increase in PV due to the randomness of the R
- n To get an approximate bound for the variance of PV, following (6) we write PV % A (p" + nu"'I oY )
n n=1 and take the variance of both sides to get 2 2 2n-2 2 Var (PV) % A g
n (jj) n=1 Again using p < 1, we have
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l l 'l
A-5 4
2 2 2
Var (PV) { A a
n n=1 2 2 "f"
- I)(2"
- I)
(I2)
=A a
6 As an example, consider the case where N = 30 and p and o are given by the observed values from Table 1.
p=p =.9704 g
o=c =.0073 N = 30 Then E(,'V) ( 19. 71 A B =.24 A
[ Var (PV)]b (.71 A Effective upper and lower bounds for PV can be taken as E(PV) + 3 [ Var (PV)]b.
For this example these bounds are equal to 19.71 A + 2.13 A or about ; 11% of the PV.
__