ECE 333
Renewable Energy Systems
Lecture 19: Economics
Prof. Tom Overbye
Dept. of Electrical and Computer Engineering
University of Illinois at Urbana-Champaign
overbye@illinois.edu
Announcements
•
•
•
HW 8 is 5.4, 5.6, 5.11, 5.13, 6.5, 6.19; it should be done
before the 2nd exam but need not be turned in and there
is no quiz on April 9.
Read Chapter 6, Appendix A
Exam 2 is on Thursday April 16); closed book, closed
notes; you may bring in standard calculators and two 8.5
by 11 inch handwritten note sheets
–
In ECEB 3002 (last name starting A through J) or in
ECEB 3017 (last name starting K through Z)
1
Energy Economic Concepts
•
Next several slides cover some general economic
concepts that are useful in evaluating renewable energy
projects
–
–
Useful in general, but quite appropriate for distributed PV
system analysis
Covered partially in Section 6.4 and in Appendix A
2
Energy Economic Concepts
•
The economic evaluation of a renewable energy
resource requires a meaningful quantification of cost
elements
–
–
•
fixed costs
variable costs
We use engineering economics notions for this
purpose since they provide the means to compare on a
consistent basis
–
–
two different projects; or,
the costs with and without a given project
3
Time Value of Money
• Basic notion: a dollar today is not the same as
a dollar in a year
–
–
Would you rather have $10 now or $50 in five
years?
What would a $50,000 purchase you’ll make in
10 years be worth today?
• The convention we use is that payments
occur at the end of each period (e.o.p.)
4
VIDEO TIME!
http://www.investopedia.com/t
erms/t/timevalueofmoney.asp
5
Time Value of Money – Principle
and Interest
•
•
Principle – the initial sum
Interest – productivity of money over time, money
today vs. money tomorrow
–
–
Simple interest – not compounded, interest is only paid
on the principle amount
Compound interest – (what we consider) when interest
is also paid on the interest vs. on the principle only
Difference between the two is greater when: the interest
rate is higher, compounding is more frequent, duration of
payments is longer
EXAMPLE!
P = principal
i = interest value
6
Positive Interest Rate (i > 0)
•
•
•
•
A positive interest rate means that having $1.00
in 10 years is not as good as having one dollar
today
The assumption is that over 10 years, you could do
something better with that $1.00 – you can use the
$1.00 to make more money
You can even put your $1.00 in the bank and earn
interest, which is like the worst case since you
could invest in something better
Hence, i > 0 → Future value > Present value (F > P)
7
Compound Interest
e.o.p.
amount owed
interest for
next period
amount owed for next period
0
P
Pi
P + Pi = P(1+i )
1
P(1+i )
P(1+i ) i
P(1+i ) + P(1+i ) i = P(1+i ) 2
2
P(1+i ) 2
P(1+i ) 2 i
P(1+i ) 2 + P(1+i ) 2 i = P(1+i ) 3
3
P(1+i ) 3
P(1+i ) 3 i
P(1+i ) 3 + P(1+i ) 3 i = P(1+i ) 4
n-1
P(1+i ) n-1
P(1+i ) n-1 i
P(1+i ) n-1 + P(1+i ) n-1 i = P(1+i ) n
n
P(1+i ) n
The value in the last column for the e.o.p. (k-1) provides the value in
the first column for the e.o.p. k (e.o.p. is end of period)
8
Terminology
•
•
We call (1 + i) n the single payment compound
amount factor
1
We define 
1  i 
and    1  i 
is the single payment present worth factor
F is called the future worth; P is called the present
worth or present value at interest i of a future sum F
n
n
•
F  P 1  i 
n
or
P  F 1  i 
n
9
Cash Flows
•
•
A cash flow is a transfer of an amount A t from
one entity to another at end of point (e.o.p.) time t
Each cash flow has (1) amount, (2) time, and (3)
sign
Ex.
I take out a loan
0
1
2
3
4
I make equal repayments for 4 years
10
Cash Flows Diagrams - Overview
End of year 1
0
1
Present
2
3
4
Convention for
cash flows
 inflow
 outflow
Ex.
Take out a loan
Revenue collected
Incoming cash flows
Ex.
Initial purchase
Payments made
Outgoing cash flows11
VIDEO TIME!
http://www.investopedia.com/t
erms/c/cashflow.asp
12
Discount Rate
•
•
•
The interest rate i is typically referred to as the
discount rate d because it is used to “discount” cash
flows to the present
In converting a future amount F to a present worth
P, we can view the discount rate as the interest rate
that can be earned from the best investment
alternative
A postulated savings of $ 10,000 in a project in 5
years is worth at present
P  F5   10,000  1  d 
5
5
13
Discount Rate
•
•
•
For d = 0.1, P = $ 6,201,
while for d = 0.2, P = $ 4,019
In general, the lower the discount factor, the
higher the present worth
The present worth of a set of costs under a given
discount rate is called the life-cycle costs
14
VIDEO TIME!
http://www.investopedia.com/t
erms/d/discountrate.asp
15
Equivalence
•
•
•
•
It can be difficult to tell if a project makes sense or
not just from the cash flow diagram
This is because the payments are in different years,
and the value of money in different years is not
equivalent
n
But, we saw that F  P  1  d 
This means that with an interest rate of i, $P today
is equivalent to $F at the end of year n
16
Equivalence
•
•
Using this notion, we can take any amount kj and
“move” or “discount” it to a future year (j+n1) or to
a past year (j-n2) using the discount rate d
Hence, the following three cash flow sets are
equivalent:
17
Equivalence
•
•
•
Projects can be compared by examining the
equivalence of their cash flow sets
Two cash-flow sets (i.e., for projects)
a
A
 t : t  0,1,2,..., n and
b
A
 t : t  0,1,2,..., n
under a given discount rate d are said to be equivalent
cash-flow sets if their worths, discounted to any point
in time, are identical.
It doesn’t matter which point in time the cash flows are
discounted to, but it is common to discount everything
to the present (called Net Present Value (NPV))
18
Equivalence
•
Common conversion factors
–
–
–
Present Value- (P|A,i%,n) and (P|F,i%,n)
Future Value- (F|A,i%,n) and (F|P,i%,n)
Capital Recovery Factor- (A|P,i%,n)
P = Present value
F = Future value
A = Annual value
19
Equivalence, Example
•
Are these cash-flow sets equivalent?
8,200.40
d = 7%
a
A
 t
b
A
 t
2000 2000 2000 2000 2000
0
1
2
3
4
a
5
6
7
0
1
b
2
20
Equivalence, cont.
•
Let’s move each cash flow set to year 2
Cash flow set a
1
F2  2000(1  i )  2000(1  i )
2
 2000(1  i )3  2000(1  i )4  2000(1  i )5
= 8200.40
Cash flow set b
F2  8200.40
•
Therefore,  A at  and  A bt  are equivalent cash flow sets
under d = 7%
21
Present and Future Value, Example
•
Consider the set of cash flows illustrated below
$ 400
$ 300
$ 200
$ 200
3
0
1
4
2
5
6
7
8
d = 6%
$ 300
22
Example, cont.
•
We compute F 8 at t = 8 for d = 6%
F8  300  1  .06   300  1  .06  
7
Future
Value
•
5
200  1  .06   400  1  .06   200
 $ 951.56
4
2
We next compute P
P  300  1  .06   300  1  .06 
1
Present
Value
•
 200  1  .06 
 $ 597.04
4
3
 400  1  .06 
6
 200  1  .06 
8
We check that for d = 6%
F8  597.04  1  .06   $ 951.56
8
23
Annualized Investment
•
A capital investment, such as a renewable energy
project, requires funds, either borrowed from a bank,
or obtained from investors, or taken from the owner’s
own accounts
•
Conceptually, we may view the investment as a loan
with interest rate i that converts the investment costs
into a series of equal annual payments to pay back
the loan with the interest
24
Annual Payments, Example
What value must A have to make
these cash flows equivalent?
$ 2000
1
2
3
4
5
0
i = 6%
A
A
A
A
A
Solution: Find A such that the NPV is zero
25
Cash Flows, cont.
•
Write down the equation for the net present value of the
cash flow set, set equal to zero, then solve for A
P  A(1.06)1  A(1.06)2  ...  A(1.06) n  0
n
P  A  1  0.06 
t 1
t
n
 A 
t 1
t
(1  d )  1
 

n
d (1  d )
t 1
n
n
t
d (1  d )
A P
 2000  (A|P,6%,5)
n
(1  d )  1
n
Annualized
Value (A)
A  $474.79
What about as
d goes to zero?
26
Annualized Investment
•
Then, the equal annual payments are given by
d (1  d )n
A  P
n
(1  d )  1
A  P  CRF(i ,n)
•
•
•
Capital Recovery
Factor (CRF)
(5.20)
The capital recovery factor, CRF(i,n), is the inverse
of the present value function PVF
CRF measures the speed with which the initial
investment is repaid
Capital recovery function in Microsoft Excel:
PMT(rate,nper,pv)
27
Mortgage payment example
•
What is the monthly payment for a 100K, 15 year
mortgage with a monthly interest rate of 0.5%?
–
–
–
•
= PMT(0.005,180,100000)
=$843.86 per month
If terms are changed to 20 years payment goes to $716/month
Assume a 100K investment in a PV installation with a
15 year life, monthly interest rate of 0.5%, and no
O&M expenses. What is monthly income needed to
cover the loan?
–
Solution is the same as above
28
Infinite Horizon Cash-Flow Sets
•
Consider a uniform cash-flow set with n  
 At  A : t  0, 1, 2, ... 
•
Then, P  A
n
1




d
n 
1
A
d
For an infinite horizon uniform cash-flow set
A
 d
P
d = “ simple rate of return”
1/d = “simple payback”
d is also the CRF, since A = dP
29
Internal Rate of Return
•
•
•
Until now, we have always specified the interest
rate or discount rate
Now we’ll “solve for” the rate at which it makes
sense to do the project
This is called the internal rate of return, also called
the “break-even interest rate”
–
•
Higher is better because a higher IRR means that even if
the interest rate gets higher, the project still makes sense
to do
Note there is no closed form solution - use a table
(or Excel, etc.) to look it up
30
Internal Rate of Return
•
Consider a cash-flow set
 A  A : t  0, 1, 2, ...
t
•
The value
of d for which
n
P 
t
A

 t 0
t 0
•
is called the internal rate of return (IRR)
The IRR is a measure of how fast we recover an
investment or stated differently, the speed with which
the returns recover an investment
31
VIDEO TIME!
http://www.investopedia.com/t
erms/i/irr.asp
32
Internal Rate of Return Example
•
Consider the following cash-flow set
$6,000 $6,000 $6,000 $6,000
$6,000
0
1
$30,000
2
3
4
8
33
Internal Rate of Return
•
The present value
P   30, 000  6, 000 (P|A,i%,8)  0
30, 000
(P|A,i%,8)=
5
6, 000
units are years
has the (non-obvious) solution of d equal to about 12%.
–
•
From Table 5.4: rows= n, values= (P|A, i%, n), cols= IRR
The interpretation is that with a 12% discount rate, the
present value of the cash flow set is 0 and so 12% is the
IRR for the given cash- flow set
–
The investment makes sense as long as other investments yield
less than 12%.
34
Efficient Refrigerator Example
•
•
A more efficient refrigerator incurs an investment of
additional $ 1,000 but provides $ 200 of energy savings
annually
For a lifetime of 10 years, the IRR is computed from the
solution of
0   1, 000  200 (P|A,i%,10)
or (P|A,i%,10)  5
The solution of this equation
requires either an iterative
approach or a value looked
up from a table
35
Efficient Refrigerator Example, cont.
•IRR tables show that
(P|A,i%,10) d  15%  5.02
and so the IRR is approximately 15%
If the refrigerator has an expected lifetime of 15 years,
this value becomes
(P|A,i%,15) d  18.4%  5.00
As discussed earlier, the value is 20% if it lasts forever
36
Impacts of Inflation
•
•
•
Inflation is a general increase in the level of prices in
an economy; equivalently, we may view inflation as a
general decline in the value of the purchasing power
of money
Inflation is measured using prices: different products
may have distinct escalation rates
Typically, indices such as the CPI – the consumer
price index – use a market basket of goods and
services as a proxy for the entire U.S. economy
–
reference basis is the year 1967 with the price of $ 100 for
the basket (L 0); in the year 1990, the same basket cost $ 374
(L 23)
37
US Inflation Over Last 350 Years
Historically prices have gone up and gone down. Recently
many homeowners found home prices can also fall!
Source: http://upload.wikimedia.org/wikipedia/commons/2/20/US_Historical_Inflation_Ancient.svg
38
Figuring Average Rate of Inflation
•
Calculate average inflation rate from 1982 to 2014
234
32
Current
1

e


2.34


100
(12/2014)
basket
ln  2.34 
ln  1  e  
 e  2.69%
value is
32
about 234
compared to
base year of
1982. Annual
rate is about 1%
in 2014
https://qzprod.files.wordpress.com/2014/11/us-consumer-price-indexes-year-on-year-change-core-cpi-headline-cpi_chartbuilder.png?w=1280
39
Inflation (Escalation) Rate
•
With escalation, an amount worth $1 in year zero
becomes $(1+e) in year 1, etc., so
PVF(d ,n) 
1
1
1
+

...

n
1+d 1+d 2
1+d 
becomes
PVF(d ,e,n) 
•
1+e 
2
1+e 
n
1+e
+
 ... 
2
n
1+d 1+d 
1+d 
We can compare terms to find an equivalent
discount rate d’:
1+e
1

1+d 1+d '
40
VIDEO TIME!
http://www.investopedia.com/t
erms/i/inflation.asp
41