Engineering Economics
Energy Economics
Introduction
Most investments involve an initial payment in return for future
income. This is especially true of investments in energy efficiency
and renewable energy systems, both of which typically require an
up-front investment in equipment in order to derive future savings
or future income. In order to evaluate these investments, it is
necessary to understand how the value of money changes over time.
Energy Economics describes the methods used to evaluate investments
which contain cash flows at different times.
Engineers who can clearly and correctly communicate the
financial impacts of energy saving ideas have more influence in the
decision-making process. Those who do not have these skills are
less able to judge the economic merit of an idea or to advocate for
ideas they believe in.
Although economics is important, many important factors are
difficult to translate into dollars. For this reason, economic
analysis should not be the only criteria in accepting or rejecting
a design or investment option.
Simple Payback and Rate of Return
The simplest index of economic feasibility, and one that is very
widely used, is simple payback. Simple payback, SP, is the time
period required for an investment to create a positive cash flow.
Simple payback is:
SP = [1]
Example:
Calculate the simple payback of a lighting retrofit that will
cost $1,000 to implement and will save $250 per year.
SP = = 4 years
The rate of return, ROR, is the reciprocal of the simple
payback. Rate of return represents the annual return on the
investment.
ROR = 1 / SP = [2]
Example:
Calculate the ROR of a lighting retrofit that will cost $1,000
to implement and will save $250 per year.
Rate of Return = = 25% per year
One of the strengths of the simple payback and rate of return
methods for evaluating investments is that the results are
independent of assumptions about the time-value of money. Over
short time periods, the value of money does not change much with
time. Thus, simple payback and rate of return are appropriate
methods to analyze investments with short paybacks.
Time Value of Money
The notion of economic growth, of investing in capital to
generate future profit is a central concept in capitalism.
Entrepreneurs and growing companies are interested in acquiring
money today to make a profit with it tomorrow. Thus, in the right
hands $100 today is worth more than $100 tomorrow; money has a
time-value component.
Example:
Would you rather have $100 now or $100 next year?
In a growing economy, I’d rather have $100 now, because I could
put it in the bank at 5% interest and have $105 next year.
To compare investment options with cash flows that occur at
different times, it is useful to covert all cash flows to a common
time. The most common way to do so is to covert all cash flows into
their “present value”, and then compare the present values to
evaluate alternative investments. Cash flows involving future
amounts of money can be converted to their present values using
three important equations:
· Present Value of a Future Amount
· Present Value of a Series of Annuities
· Present Value of an Escalating Series of Annuities
Present Value of a Future Amount
Consider the common situation of investing a present amount, P,
in an account that pays a rate of interest, i, over n compounding
periods. The future amount, F, can be determined from the following
example. Start with a present amount, P = $100 at a rate of
interest, i = 5% per year. The future amount after n years, Fn,
is:
F0 = 100
F1 = 100 + 100(.05) = P + Pi = P (1+i)
F2 = [100 + 100(.05)] + [100 + 100(.05)](.05) = P(1+i) + P(1+i)i
= P(1+i)2
Fn = P (1+i)n
Equation 3 is the fundamental equation of exponential growth and
can be applied whenever growth is a fixed percentage of the current
quantity.
F = P (1+i)n[3]
Equation 3 can be rearranged to show the present value of a
future amount, as in Equation 4.
P = F (1+i)-n [4]
The factor (1+i)-n is sometimes called the present worth factor,
PWF(i,n). Thus,
P = F(1+i)-n = F PWF(i,n)
Example:
Someone promises to pay you $1,000 in 5 years. If the interest
rate is 10% per year, what amount would you take today that is
equivalent to $1,000 in 5 years (i.e. what is the present value of
$1,000 5 years from now?) ?
P = F(1+i)-n = $1,000 (1+.10)-5 = $621
Present Value of a Series of Annuities
An annuity is a regular payment of income made at the end of a
fixed period. Consider investing an annuity of amount, A, during
each of n compounding periods with an interest rate i. This
situation can be shown graphically in a cash flow diagram. In cash
flow diagrams, income is shown as line extending upward and
payments are shown as lines extending downward. The cash flow
diagram for a series of n investments of amount A is shown
below.
(A)
( 0 1 2 3 n)
The following derivation shows how to calculate the present
value, P, of this series of payments.
Pn = present value of n payments of amount A
= present amount that is equal to a series of payments, A, for n
years
P0= 0
P1 =
P2 =
Pn =
To find closed-form solution, do a little algebra...
1)Pn =
2)Pn =
2-1)Pn =
Pn =
Thus, the present value of a series of n payments of amount A
is:
Pn = [5]
The factor is sometimes called the series present worth factor,
SPWF(i,n). The reciprocal of the series present worth factor is
sometimes called the capital recovery factor, CRF(i,n). Thus,
Pn = = A SPWF(i,n) {SPWF(i,n) = 1 / CRF(i,n)}
Example:
A standard-efficiency furnace costs $100 and consumes $40 per
year in fuel over its 10 year lifetime. A high-efficiency furnace
costs $200 and consumes $20 per year in fuel over its 10 year
lifetime. If interest rates are 10% per year, which is the better
investment?
Standard efficiency High-efficiency
$40 $20
$100
$200
Pstd costs = $100 + $40= $346
Phigh costs = $200 + $20= $323
and
Phigh savings = $346 - $323 = $23
Or you could solve directly for the present value of savings by
setting up your cash flow diagram to reflect ‘savings’ instead of
‘costs’.
($20)
(-$100$100)
Phigh savings = -$100 + $20= $23
It is sometimes easier to understand “annualized savings” than
the “present value” of savings”. To find annual savings, find the
present value of savings, and then annualize that amount by solving
P = A SPWF(i,n) for A.
Example:
Find the annualized savings from investing in the high
efficiency furnace over the standard efficiency furnace:
Ahigh savings = P / SPWF(.10,10) = $23 /
Ahigh savings = $4
Present Value of an Escalating Series of Annuities
Sometimes a recurring annuity, A, is expected to increase over
time at some escalation rate e. For example, as equipment gets
older it often requires more and more maintenance. A cash flow
diagram of an escalating series is shown below.
(A(1+e)n)
(A)
( 0 1 2 3 n)
Using a method similar to the previous derivation, the present
value of an escalating series is:
if e i [6a]
P =
if e = i[6b]
The factor or (depending on whether i = e) is called the
escalating series present worth factor, ESPWF(i,e,n). Thus,
P = A ESPWF(i,e,n)
Note that when e = 0, escalating series present worth factor
ESPWF(i,0,n) is identical to the series present worth factor
SPWF(i,n).
Example:
The maintenance director says that it will cost $50 this year to
maintain an aging piece of equipment and estimates that the
equipment will require 5% more maintenance every year for the next
10 years. New replacement equipment would cost $400 and would
require only $10 of maintenance per year with no expected
escalation over the next 10 years. Which is a better investment if
interest rates are 5%?
Current Equipment:
( 0 1 2 3 10)
($50(1+.05)10)
Pcurrent equip costs = A ESPWF(.05, .05,10)
= = = $476
New Equipment:
( 0 1 2 3 10)
($10)
($400)
Pnew equip costs = $400 + A SPWF(.05,10)
= $400 + $10 = $477
Hence, the two options are expected to cost about the same
amount.
Future Value of a Series of Payments
In addition, it is sometimes useful to calculate the future
value of a series of annuities. Using a derivation similar to that
for the present value of a series of annuities, the future value,
F, of a series of equal annuities, A, that accrue interest at a
rate, i, over n periods is:
Fn = [7]
Example:
The future value of an annual investment of $2,000 per year for
20 years in an IRA that accrues interest at 5% per year is:
F =
Compounding Periods
Typically, interest is paid or payments are due on fixed
intervals rather than continuously. These intervals are called
compounding intervals. Interest rates are typically reported for an
annual compounding period. If interest is paid or payments are due
at other compounding intervals, simply divide the annual interest
rate, i, in the time value of money equations by the number of
compounding periods per year, m, and multiply the number of years,
n, by m.
Example:
Calculate the future value of $100 earning 8% annual interest
compounded quarterly for 10 years.
F = P(1 + i/m)nm = $100 (1 + .08/4)(10*4) = $220
Example:
Calculate the monthly house payment if $100,000 is borrowed at
8% on a 30 year mortgage.
P = A SPWF(i,n)
A = P / SPWF(.08/12, 30*12) = $734 / month
Summary of Time Value of Money Equations
The figure below shows the four time value of money equations
developed so far.
P = F (1+i)-n = F PWF(i,n)
P = = A SPWF(i,n)
Fn =
if e i
P = = A ESPWF(i,e,n)
if e = i
The Discount Rate
So far, we have referred to the rate of growth i as the rate of
interest. More formally, i is the discount rate. The discount rate
is the expected rate of return from an alternative investment. The
alternative investment could be interest from a bank, stock market
appreciation or expected profits from one’s own company. High
discount rates reflect the belief that a large profit can be made
from an alternative investment; thus, money today is very valuable
and future money is less valuable. High discount rates have the
effect of discounting future sums of money or “discounting the
future”. Hence the name “discount” rate. To get a feeling for how
the discount rate affects time value of money, reconsider a
previous example, but compare solutions with two discount
rates.
Example:
A standard-efficiency furnace costs $100 and consumes $40 per
year in fuel over its 10 year lifetime. A high-efficiency furnace
costs $200 and consumes $20 per year in fuel over its 10 year
lifetime. If the discount rate is 10% per year, which is the better
investment? What if the discount rate is 30%?
Cash flow diagrams are:
Standard Efficiency High Efficiency
$40 $20
$100
$200
Savings from high-efficiency furnace
($20)
( 1 2 3 10) (-$100$100)
If the discount rate is 10%, then:
Phigh-eff savings = -$100 + $20= $23
The positive savings indicate that the high-efficiency furnace
is the best investment.
If the discount rate is 30%, then:
Phigh-eff savings = -$100 + $20= $-38
The negative savings indicate that when the discount rate is
30%, the traditional furnace is the better investment.
In summary, a high discount rate reduces the value of future
cash flows, including savings from energy efficiency measures.
In general, renewable energy and energy conservation
technologies have high first capital costs and low future fuel
costs. Thus, high discount rates (which value the present and
‘discount’ the future) work against these technologies. Some people
argue that a discount rate of zero ought to be used in
non-renewable resource decisions. A zero discount rate implies that
the future is equally as important as the present.
Lifecycle Cost and Net Present Value
The most comprehensive way to make investment decisions is to
consider the total cost of a system over its entire life. Because
the costs or revenues during these phases occur at different times,
time-value of money equations can be used to calculate the present
value of all costs and revenues over the lifecycle of a product.
The net present value is the sum of the present values of the costs
and revenues of a system over its lifetime. A positive net present
value indicates that an investment is more cost-effective than
investing the money at the discount rate used in the calculations.
When the net present value is calculated for all costs and revenues
over a product’s lifetime, including manufacturing, operating and
post-use phases, it is called the lifecycle cost.
Example:
Determine the net present value (i.e. lifecycle cost) of a 2 kW
photovoltaic system that costs $8,000 per kW, generates 3,000 kWh
per year that displaces electricity purchased from the utility at
$0.10 /kWh with a projected cost escalation of 2% per year. The
system lifetime is 20 years and the discount rate is 5% per year.
System recycle income or removal costs at the end of the systems
life are negligible.
(A ESPWF(i=.05, e=.02, n=20)) ( 1 2 3 20)
(-$16,000$100=.)
PV_IC = - 2 kW x $8,000 /kW = -$16,000
A = 3,000 kWh/yr x $0.10 /kWh = $300 /yr
NPV = PV_IC + A ESPWF(i=.05, e=.02, n=20)
NPV = -$16,000 + $300 14.665
NPV = -11,600
Thus, this is not a cost-effective investment compared to
alternative investments which return 5% per year.
Return on Investment
Because of the discount rate’s large influence on the results of
time value of money calculations, the discount rate is sometimes
solved for, rather than input, in time value of money calculations.
The discount rate when the present value of an investment is zero
represents the return on an equivalent investment, and is called
the return on investment, ROI. To calculate ROI, set NPV = 0 and
solve for i.
Example:
Reconsider the furnace example. Find the return on investment
for upgrading to a high-efficiency furnace if the enatural gas =
0.84%.
Savings with Real Fuel Price Escalation
(-$100$100) ($20 ESPWF(i, e=.0084, n=10)) ( 1 2 3 10)
NPV = -$100 + $20 ESPWF(i,e=.0084,n=10)
0 = -$100 +$20
$100/$20 = 5 =
By iteration: i = ROI = 16%
In Microsoft Excel, return on investment (ROI) is called
internal rate of return (IRR). The IRR is calculated by an internal
function: IRR(range), where the range is made up of all cash
flows.
Example:
A high efficiency furnace costs $100 more than a low efficiency
furnace and saves $20 per year over a ten year period. Use
Microsoft Excel to calculate the return on investment.
A cash flow diagram of savings is shown below.
($20)
(-$100$100)
In Excel the cash flows below should be entered in separate
cells such as A1: A11.
-100, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20.
Use the IRR function, “=IRR(A1:A11)” to calculate return on
investment.
Evaluating Energy Efficiency Options: The Problem with Simple
Payback
In the industrial sector, the economic criterion used to
evaluate energy saving opportunities is frequently simple payback.
Moreover, many companies demand very short simple paybacks on the
order of 2 years as the criteria for implementation. On the
surface, this appears to be unrational economic decision making,
since a simple payback of 2 years is a rate-of-return of 50%. Thus,
demanding a 2-year simple payback before a project will be funded
appears to indicate that a company has alternative investments that
return 50% per year. Since this is seldom the case, the wisdom of
demanding very short simple paybacks appears to be
questionable.
However, the real problem is that simple payback, and hence
rate-of-return, are poor metrics for evaluating cost effectiveness
because they do not take into account project lifetime. Consider
for example a project with a simple payback of 4 years, and hence a
rate-of-return of 25% per year. This appears to be a highly
profitable project. However, if the project lifetime is 3 years,
then the initial investment will never be paid off. In this case,
the 25% per year return is completely misleading. If the project
lifetime is 5 years, the investment is barely cost effective, since
it positive revenue is generated for only one year. If, however,
the project lifetime is 20 years, then the project is highly cost
effective since the project will generate positive revenue for 16
years after it has paid back the initial investment. Thus, to
properly evaluate energy saving investments, the economic analysis
must include project lifetime.
The economic criteria return on investment (ROI) includes
project lifetime, and is thus a much better measure of cost
effectiveness. The return on investment, in contrast to rate of
return, can be directly compared to alternative investments in
order to evaluate economic merit.
Example:
A project has a simple payback of 3 years. Calculate the rate of
return, ROR, and return on investment, ROI, if the project has a
lifetime of 3, 6, 12, 15 and 18 years.
Rate of Return = Annual Savings / Initial Cost
Rate of Return = 1 / Simple Payback = 1 / 3 years = 33%
/year
To evaluate return on investment for a project with a 33% rate
of return, let the initial cost be $1,000. If so the annual savings
would be $333. The net present value of the investment, assuming
that the initial investment has no value at the end of the project
life, is:
NPV = -$1,000 + $333 SPWF(i,n) = -$1,000 + $333
To solve for return on investment ROI, set NPV = 0 and solve for
i. If the project lifetime is 3 years, then:
0 = -$1,000 +$333 By iteration: i = ROI 0%
If the project lifetime is 6 years, then:
0 = -$1,000 +$333 By iteration: i = ROI 24.2%
Graphing the rate of return and return on investment as
functions of project life gives:
Note that ROI calculated in the preceding example is for the
case in which the initial investment has no value at the end of the
project life. If the initial investment had value at the end of the
project life, then that value should be added to the NPV equation
explicitly. However, once ROI is calculated using this method, the
ROI represents the value of an alternative investment with both
capital and interest accumulation. For example, if an alternative
investment returned 24.2% per year for six years, the value of a
$1,000 investment after six years would be:
F = P (1 + i)n = $1,000 (1 + .242)6 = $3,671
The ROI of this alternative investment would be calculated
as:
NPV = 0 = -$1,000 + $3,671 / (1+i)6 and solving for i gives i =
ROI = 24.2%.
Thus, ROI represents the value of an alternative investment with
both capital and interest.
A comparison of rate-of-return and return-on-investment, shows
that return-on- investment is:
· negative when the project life is less than the simple
payback
· much less than rate-of-return for short project lifetimes
· approaches rate-of-return as project life increases.
This shows the importance of project lifetime in determining
project cost effectiveness. It may also explain why some companies
require very short simple paybacks to fund a project. In these
cases, the company may believe that the project lifetime is very
short. In our view, however, the use of return-on-investment makes
these assumptions explicit and is therefore a much more transparent
metric of economic fitness than simple payback.
Inflation and the ‘Real’ Discount Rate
Inflation causes the price of goods and services to rise or,
equivalently, the value of money to decrease. A widely used metric
of the overall rate of inflation in the United States is the
“Implicit Price Deflator”, IPD. Implicit Price Deflator is
determined by the U.S. Department of Commerce. IPDs using year 2000
as the base year, are shown in the table below (Annual Energy
Review 2007, Energy Information Agency, U.S. Department of Energy,
Table D1).
IPD, 1949-2007, 2000 = 1.000
The annual rate of inflation, j, over any period of n years can
be calculated by applying Equation 3.
Example:
Find the general rate of inflation, j, from 1996 to 2006 using
IPDs.
Solution:
F = P (1+j)n
1.16042 = 0.93852 (1 + j)10
j = 0.02145
General inflation rates for various periods are shown in the
table below.
1967-2006
1996-2006
1996-2001
Input
Input
Input
n
39
n
10
n
5
P
0.23893
P
0.93852
P
1.02399
F
1.16042
F
1.16042
F
1.16042
Calculations
Calculations
Calculations
j
0.041354
j
0.02145
j
0.025331
The rate of inflation is important, since inflation erodes the
value of money. For example, if an investment appreciates at an
interest rate of 3% per year and the rate of inflation is also 3%
per year, then the investment does not deliver ‘real’ value.
The discount rate with the effect of inflation removed is called
the “real” discount rate and indicates the “real” return from an
investment. To calculate the real discount rate, i’, one needs to
explicitly remove the effect of inflation, j. To do so, consider
the following derivation.
Interest Adds Value to MoneyInflation Devalues Money
F = F =
If both are active:
F =
Find i’, the “effective” or “real” discount rate, such that:
F = =
i’ = [8]
Example:
Returns from an S&P index fund indicate that a single share
of the fund was worth $14.90 in 1976 and $124.56 in 1994. What is
the market” rate of return from this investment?
Using F = P (1+i)n, we can solve for i, the average annual rate
of return. We assume that the investment began with a present value
P = $14.90 and ended with a future value F = $124.56 after 18
years.
i = market discount rate
i = = = 12.52 %
If the GDP implicit price deflator (an indication of general
inflation) was 52.3 in 1976 and 126.1 in 1994, what is the annual
rate of inflation over this period?
j = annual rate of inflation
j = = = 5.01%
What is the real annual rate of return (discount rate) over the
period?
i’ = real discount rate = = = 7.15%
Energy Cost Escalation
In energy economics, many calculations include cost of energy,
which frequently changes over time. In most cases, the rate of
escalation of energy prices is different than the general rate of
inflation. One way to estimate energy price escalation rates in the
future is to consider past rates of energy price escalation. The
tables below show historical U.S. electricity and natural gas
prices (Annual Energy Review 2006, Energy Information Agency, U.S.
Department of Energy).
Using this data, the energy price escalation rate, e, over any
period can be calculated by applying Equation 3.
Example:
Find the energy price escalation rate, e, for natural gas for
the residential sector from 1996 to 2006 using nominal cost data
from the Annual Energy Review.
Solution:
F = P (1+e)n
13.76 = 6.34 (1 + e)10
e = 0.081
Residential natural gas price escalation rates for various
periods are shown in the table below.
Residential Nominal
Residential Nominal
Residential Nominal
1967-2006
1996-2006
2001-2006
Input
Input
Input
n
39
N
10
n
5
P
1.04
P
6.34
P
9.63
F
13.76
F
13.76
F
13.76
Calculations
Calculations
Calculations
e
0.068461
e
0.08057
e
0.073986
Residential electricity price escalation rates for various
periods are shown in the table below.
Residential
Residential
Residential
1967-2006
1996-2006
2001-2006
Input
Input
Input
n
39
n
10
n
5
P
2.3
P
8.36
P
8.58
F
10.4
F
10.4
F
10.4
Calculations
Calculations
Calculations
e
0.039448
e
0.022075
e
0.039224
The ‘real’ energy price escalation rate, e’, which represents
the energy price escalation rate with the effect of inflation
removed, can be determined in two ways. The first is by removing
the effect of inflation, j, from the nominal energy escalation rate
e using Equation 9. The derivation for Equation 9 is analogous to
the derivation for ‘real’ discount rate in Equation 8.
e’ = [9]
Example: Find the real energy price escalation rate, e, for
residential natural gas from 1996 to 2006.
Solution:
The nominal annual energy escalation rate, e, from 1996 to 2006
was e = 0.081. The annual rate of inflation over this period was
0.021. The real annual energy escalation rate e’ is:
e’ =
e’ = = 0.058
Alternately, ‘real’ energy escalation rates, e’, could be
determined by applying Equation 3 to energy costs reported in
“constant dollars”.
Example:
Find the real energy escalation rate, e’, for natural gas for
the residential sector from 1996 to 2006 using constant dollar cost
data from the Annual Energy Review.
Solution:
F = P (1+e)n
11.86 = 6.76 (1 + e)10
e = 0.058
Choice of Real or Nominal Discount and Escalation Rates
It is important to use either real rates or nominal rates in
time-value of money calculations and to avoid mixing real and
nominal rates in the same calculation. For example, either use
nominal discount rate, i, and nominal energy price escalation rate,
e, or use real discount rate, i’, and real energy price escalation
rates e’. When money is borrowed or invested at interest, the
interest is typically represents a nominal rate; thus, to avoid
mixing nominal and real rates, it is common practice to use nominal
rates in time value of money calculations.
Tax Deductions for Fuel, Interest and Equipment Depreciation
Tax laws sometimes allow businesses to deduct fuel, interest and
equipment depreciation expenses for tax purposes. Because combined
local, state and federal corporate income taxes are often nearly
50% of profits, it is often necessary to consider these deductions
when evaluating investment options.
In many cases, taxes have the effect of equaling reducing both
initial costs and operating savings. Initial costs are reduced
since the initial cost can depreciated over the lifetime of the
equipment and deducted from income. Similarly, annual savings are
reduced by the tax rate applied to profit. Hence, when considering
taxes, the initial cost and annual savings are both reduced by the
tax rate, however the simple payback and ROI remain unchanged.
Example:
Find the NPV and annualized NPV for an energy investment which
costs $1,000 and returns $500 per year for 10 years if the discount
rate is 5% per year and the energy cost escalation rate is 2% per
year. Do so without considering taxes and with considering taxes
assuming the total tax rate is 40%.
Solution Without Taxes:
Solution With Taxes:
Interest Expenses
If equipment is purchased with a mortgage, the interest is
usually deductible. Because energy saving equipment usually has a
higher first cost than traditional equipment, interest deductions
usually enhance the cost effectiveness of energy saving
investments. Following the fuel savings example:
Tax Savings = Tax1 - Tax2 = TR(Interest2 - Interest1) = TR(I2 -
I1)
The example below shows how to calculate the interest and
principle components of a mortgage payment.
Example:
Show the interest and principle components of loan payments for
a $10 loan borrowed at an interest rate of 10% for 5 years.
P = A SPWF(.10,5)
A = P / SPWF(.10,5) = $10 / 3.79 = $2.64 per year
The components are shown in the table below.
At end
of year
Payment
amount
Interest part
of payment
Principle part
of payment
Principle
remaining
1
2.64
.1(10) = 1.00
2.64 - 1.00 = 1.64
10 - 1.64 = 8.36
2
2.64
.1(8.36) = .84
2.64 - .84 = 1.80
8.36 - 1.80 = 6.56
3
2.64
.1(6.56) = .66
2.64 - .66 = 1.98
6.56 - 1.98 = 4.58
4
2.64
.1(4.58) = .46
2.64 - .46 = 2.18
4.58 - 2.18 = 2.40
5
2.64
.1(2.40) = .24
2.64 - .24 = 2.40
2.40 - 2.40 = 0
Depreciation Expenses
Because equipment wears out over time, many tax laws allow
businesses to deduct the cost of equipment wear from income taxes.
The annual amount of wear is called the depreciation, D. Several
methods are usually allowed by the tax laws to calculate the
depreciation. Straight-line depreciation calculates D as:
D = (Purchase Cost - Salvage Value) / Equipment Lifetime
As in the following examples,
Tax Savings = TR(D1 - D2)
The higher costs of energy conserving equipment usually
increases the depreciation deduction and make energy saving
investments more attractive.
Uncertainty in Economic Analyses
Studies have shown that the results of economic analyses are
most sensitive to i, e and n. Because of this, it is recommended
that one determine and report the sensitivity of the economic
analyses results using analytical or substitutional methods.
Example:
Determine the sensitivity of the present value of a future
amount of $100 in year 10 if the discount rate is 6% 2%.
By calculus...
P = F (1+i) -n
dP =
if F and n are known exactly, then dn = dF = 0 and
dP =
= F (-n) (1+i)-n-1 di = $100 (-10) (1+.06)-10-1 (.02)
=
Or by substitution...
P = F (1+i) -n at limits of i
Plow = $100 (1+.08)-10 = $46.3
Phigh = $100 (1+.04)-10 = $67.5
Corporations and Energy Investments
The Congressional Office of Technology Assessment concluded that
the biggest factor affecting industrial energy efficiency is the
will to invest in new technology. In normal course of business,
worn out and obsolete technologies are replaced with better and
usually more energy efficient technologies.
Will to invest influenced by:
· Maturity of industry: Young, high-growth industries tend to
invest heavily, while mature industries with price based
competition and low profit margins have little incentive.
· Business climate: Growth and competition encourage
investment.
· Corporate climate: “If it ain’t broke don’t fix it” versus
“continual improvement”
· Shortage of technical personnel, especially in “lean”
companies.
· Regulations (mainly environmental)
· Energy’s fraction of production costs.
Many corporations demand a simple payback of 2 or 3 years for
energy efficiency investments. This extremely high investment
hurdle causes the “efficiency gap”, which economists have
identified as “chronic underinvestment” in energy efficiency. Some
corporations report that the reason for the high economic hurdle is
that energy efficiency projects must compete for in-house capital
and management time. Some analysts have observed that mandatory
projects (such as regulatory compliance, replacement of essential
equipment, and maintenance of product quality) and strategic
projects (such as those which increase market share or new product
development) have higher corporate priority than discretionary
(energy efficiency) projects. Another commonly cited reason for
high discount rates for energy efficiency projects is the risk
associated with those projects. Although risk means different
things to different organizations, when risk is measured in terms
of volatility, the risk of energy efficiency investments is about
the same as U.S. T-bills, while the return on investment is greater
than that of small company stocks.
Source: Laitner, J., Ehrhardt-Martinez, K. and Prindle, W.,
2007, “American Energy Efficiency Investment Market”, Energy
Efficiency Finance Forum, American Council or and Energy Efficient
Economy .
Some of these barriers can be lessened by energy service
companies offering shared savings and the linking of energy
efficiency to pollution prevention and a good corporate image.
Some Good References
Bartlett, A.A., “Forgotten Fundamentals of the Energy Crisis",
The American Journal of Physics, Volume 46, September 1978, pages
876 to 888.)
Duffie, J. and Beckman, W., 1991. Solar Engineering of Thermal
Processes, John Wiley and Sons, Inc., New York, NY.
Stoecker, W., Design of Thermal Systems, 1989. Design of Thermal
Systems, McGraw-Hill, Inc., New York, NY.
Thuesen, G. and Fabrycky, W., 1993. Engineering Economy,
Prentice Hall, Englewood Cliffs, NJ.
U.S. Congress, Office of Technology Assessment, Industrial
Energy Efficiency, OTA-E-560, 1993.
15
EnergyEconomics.doc
$250/year
$1,000
n
n
j)
(1
i)
P(1
+
+
j
1
j
i
+
-
1
e
P
F
ln
n
1
-
ú
û
ù
ê
ë
é
÷
ø
ö
ç
è
æ
1
e
14.90
124.56
ln
18
1
-
ú
û
ù
ê
ë
é
÷
ø
ö
ç
è
æ
1
e
P
F
ln
n
1
-
ú
û
ù
ê
ë
é
÷
ø
ö
ç
è
æ
1
e
52.3
126.1
ln
18
1
-
ú
û
ù
ê
ë
é
÷
ø
ö
ç
è
æ
j
1
j
i
+
-
.0501
1
.0501
.1252
+
-
Table 8.10 Average Retail Prices of Electricity, 1960-2006
(Cents per Kilowatthour, Including Taxes)
Year
Residential
Commercial 1
Industrial 2
Nominal
Real
Nominal
Real
Nominal
Real
1960
2.6
12.4
2.4
11.4
1.1
5.2
1961
2.6
12.2
2.4
11.3
1.1
5.2
1962
2.6
12.1
2.4
11.1
1.1
5.1
1963
2.5
11.5
2.3
10.6
1
4.6
1964
2.5
11.3
2.2
9.9
1
4.5
1965
2.4
10.7
2.2
9.8
1
4.4
1966
2.3
9.9
2.1
9.1
1
4.3
1967
2.3
9.6
2.1
8.8
1
4.2
1968
2.3
9.2
2.1
8.4
1
4
1969
2.2
8.4
2.1
8
1
3.8
1970
2.2
8
2.1
7.6
1
3.6
1971
2.3
8
2.2
7.6
1.1
3.8
1972
2.4
8
2.3
7.6
1.2
4
1973
2.5
7.9
2.4
7.5
1.3
4.1
1974
3.1
8.9
3
8.6
1.7
4.9
1975
3.5
9.2
3.5
9.2
2.1
5.5
1976
3.7
9.2
3.7
9.2
2.2
5.5
1977
4.1
9.6
4.1
9.6
2.5
5.9
1978
4.3
9.4
4.4
9.6
2.8
6.1
1979
4.6
9.3
4.7
9.5
3.1
6.3
1980
5.4
10
5.5
10.2
3.7
6.9
1981
6.2
10.5
6.3
10.7
4.3
7.3
1982
6.9
11
6.9
11
5
8
1983
7.2
11
7
10.7
5
7.7
1984
7.15
10.57
7.13
10.54
4.83
7.14
1985
7.39
10.6
7.27
10.43
4.97
7.13
1986
7.42
10.41
7.2
10.11
4.93
6.92
1987
7.45
10.18
7.08
9.67
4.77
6.52
1988
7.48
9.88
7.04
9.3
4.7
6.21
1989
7.65
9.74
7.2
9.17
4.72
6.01
1990
7.83
9.6
7.34
9
4.74
5.81
1991
8.04
9.52
7.53
8.92
4.83
5.72
1992
8.21
9.5
7.66
8.87
4.83
5.59
1993
8.32
9.41
7.74
8.76
4.85
5.49
1994
8.38
9.28
7.73
8.56
4.77
5.28
1995
8.4
9.12
7.69
8.35
4.66
5.06
1996
8.36
8.91
7.64
8.14
4.6
4.9
1997
8.43
8.84
7.59
7.95
4.53
4.75
1998
8.26
8.56
7.41
7.68
4.48
4.64
1999
8.16
8.34
7.26
7.42
4.43
4.53
2000
8.24
8.24
7.43
7.43
4.64
4.64
2001
8.58
8.38
7.92
7.73
5.05
4.93
2002
8.44
8.1
7.89
7.57
4.88
4.68
2003
8.72
8.2
8.03
7.55
5.11
4.8
2004
8.95
8.18
8.17
7.47
5.25
4.8
2005
9.45
8.38
8.67
7.69
5.73
5.08
2006
10.4
8.96
9.36
8.07
6.09
5.25
Table 6.8 Natural Gas Prices by Sector, 1967-2006
(Dollars per Thousand Cubic Feet)
YearResidential SectorCommercial Sector 1Industrial Sector 2
PricesPricesPrices
Nominal RealNominal RealNominal Real
19671.044.350.743.100.341.42
19681.044.170.732.930.341.36
19691.054.020.742.830.351.34
19701.093.960.772.800.371.34
19711.153.980.822.840.411.42
19721.214.010.882.920.451.49
19731.294.050.942.950.501.57
19741.434.121.073.080.671.93
19751.714.501.353.550.962.53
19761.984.931.644.081.243.08
19772.355.502.044.771.503.51
19782.565.592.234.871.703.72
19792.986.012.735.511.994.02
19803.686.813.396.272.564.74
19814.297.264.006.773.145.31
19825.178.244.827.683.876.17
19836.069.295.598.574.186.41
19846.129.055.558.204.226.24
19856.128.785.507.893.955.67
19865.838.185.087.133.234.53
19875.547.574.776.522.944.02
19885.477.234.636.122.953.90
19895.647.184.746.032.963.77
19905.807.114.835.922.933.59
19915.826.894.815.702.693.19
19925.896.824.885.652.843.29
19936.166.975.225.913.073.47
19946.417.105.446.033.053.38
19956.066.585.055.482.712.94
19966.346.765.405.753.423.64
19976.947.275.806.083.593.76
19986.827.075.485.683.143.25
19996.696.845.335.453.123.19
20007.767.766.596.594.454.45
20019.639.408.438.235.245.12
20027.897.576.636.364.023.86
20039.639.058.407.895.895.54
200410.759.829.438.626.535.97
200512.8411.3911.5910.288.567.59
200613.7611.8611.9710.327.896.80
j
1
j
e
+
-
÷
ø
ö
ç
è
æ
$
$/year
t
InitialCos
Year
SavingsPer
j
1
j
e
+
-
0.021
1
0.021
0.081
+
-
Input Data
AS ($/yr) 500
IC ($) 1,000
n (yrs) 10
i 0.05
e 0.02
Calculations
ESPWF = (1-((1+e)/(1+i))^n)/(i-e) 8.388106
Ps ($)= AS x ESPWF 4,194
NPV ($) = Ps - IC 3,194
SPWF = (1-(1-i)^(-n))/i 7.721735
A_NPV ($/yr) = NPV / SPWF 414
Input Data
AS ($/yr) 500
IC ($) 1000
n (yrs) 10
i 0.05
e 0.02
tr 0.5
Calculations
ASAT ($/yr) = AS x (1-tr) 250
ICAT ($) = IC x (1-tr) 500
ESPWF = (1-((1+e)/(1+i))^n)/(i-e) 8.388106
Ps ($) = ASAT x ESPWF 2,097
NPV ($) = Ps - ICAT 1,597
SPWF = (1-(1-i)^(-n))/i 7.721735
A_NPV ($/yr) = NPV / SPWF 207
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6
Year IPDYear IPDYear IPD
19490.1635219700.2753419910.84444
19500.1653119710.2891119920.86385
19510.1771819720.3016619930.88381
19520.1802219730.3184919940.90259
19530.1824319740.3472519950.92106
19540.1841719750.3800219960.93852
19550.1874319760.4019619970.95414
19560.1939319770.4275219980.96472
19570.2003819780.4575719990.97868
19580.2049819790.4954820001.00000
19590.2075119800.5404320011.02399
19600.2104119810.5911920021.04187
19610.2127819820.6272620031.06404
19620.2156919830.6520720041.09462
19630.2179819840.6765520051.13000
19640.2213119850.6971320061.16567
19650.2253519860.7125020071.19664
19660.2317619870.73196
19670.2389319880.75694
19680.2491319890.78556
19690.2614919900.81590
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P(1
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