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)

2

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

3

• 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

Energy Economic Concepts

4

• 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.)

Time Value of Money

5

66

http://www.investopedia.com/terms/t/timevalueofmoney.asp

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• 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

P = principal

i = interest value

Time Value of Money – Principle and Interest

7

EXAMPLE!

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)

8

Compound Interest

e.o.p. amount owedinterest for

next periodamount 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) 9

Terminology

• We call (1 + i) n the single payment compound

amount factor• We define

and

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

11 i

1nn i

1n

F P i 1n

P F i

or

10

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

I take out a loan

I make equal repayments for 4 years

0 1 2 3 4

Ex.

11

Cash Flows Diagrams - Overview

0 1 2 3 4

Present

End of year 1

Incoming cash flows

Initial purchase

Payments made

Take out a loan

Revenue collected

Ex. Ex.

Outgoing cash flows

Convention for cash flows + inflow -

outflow

12

13

http://www.investopedia.com/terms/c/cashflow.asp

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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

555 10,000 1P F d

14

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

15

1616

http://www.investopedia.com/terms/d/discountrate.asp

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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

• But, we saw that • This means that with an interest rate of i, $P today is

equivalent to $F at the end of year n

1n

F P d

17

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:

18

Equivalence

• Projects can be compared by examining the equivalence of their cash flow sets

• Two cash-flow sets (i.e., for projects)

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))

: 0,1,2,..., : 0,1,2,...,a bt tA t n A t n and

19

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

A = Annual value

F = Future value

20

Equivalence, Example

• Are these cash-flow sets equivalent?

0 1 2 3a4 5 6 7

2000 2000 2000 2000 2000

0 1 2b

8,200.40

atA btA

d = 7%

21

Equivalence, cont.

• Let’s move each cash flow set to year 2

• Therefore, are equivalent cash flow sets

under d = 7%

a bt tA Aand

1 22

3 4 5

2000(1 ) 2000(1 )

2000(1 ) 2000(1 ) 2000(1 )

= 8200.40

F i i

i i i

Cash flow set a

Cash flow set b

2 8200.40F

22

Present and Future Value, Example

• Consider the set of cash flows illustrated below

0 1 2

3

4 5 6 7 8

$ 300

$ 300$ 200

$ 400

$ 200

d = 6%

23

Example, cont.

• We compute F 8 at t = 8 for d = 6%

• We next compute P

• We check that for d = 6%

7 5

8

4 2

300 1 .06 300 1 .06

200 1 .06 400 1 .06 200951.56

F

$

1 3

4 6 8

300 1 .06 300 1 .06

200 1 .06 400 1 .06 200 1 .06597.04

P

$

8

8 597.04 1 .06 951.56F $

Future Value

Present Value

24

• 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

Annualized Investment

25

Annual Payments, Example

0

1 2 3 4 5

A

$ 2000

i = 6%A A A A

What value must A have to make these cash flows equivalent?

Solution: Find A such that the NPV is zero26

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

1 1

1 0.06n n

t t

t t

P A A

1

(1 ) 1

(1 )

nnt

nt

d

d d

(1 )2000 (A|P,6%,5)

(1 ) 1

n

n

d dA P

d

$474.79A Annualized Value (A)

1 2(1.06) (1.06) ... (1.06) 0nP A A A

What about asd goes to zero?

27

Annualized Investment

• Then, the equal annual payments are given by

• 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)

(1 )

(1 ) 1

n

n

d dA P

d

Capital Recovery

Factor (CRF)

CRF( , ) (5.20)A P i n

28

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

29

Infinite Horizon Cash-Flow Sets

• Consider a uniform cash-flow set with

• Then,

For an infinite horizon uniform cash-flow set

: , 1, 2, ...tA A t 0

n

1 1n

P A And d

Ad

P d = “ simple rate of return”

1/d = “simple payback”d is also the CRF, since A = dP

30

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

31

Internal Rate of Return

• Consider a cash-flow set

• The value of d for which

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

: , 1, 2, ...tA A t 0

nt

tt 0

P A 0

32

333333

http://www.investopedia.com/terms/i/irr.asp

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Internal Rate of Return Example

8

• Consider the following cash-flow set

0

1 2

$30,000

3 4

$6,000 $6,000 $6,000 $6,000 $6,000

34

Internal Rate of Return

• The present value

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%.

30,000 6,000 (P|A,i%,8)P 0 30,000

(P|A,i%,8)= 56,000

units are years

35

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

or

1,000 200 (P|A,i%,10)0

(P|A,i%,10) 5 The solution of this equation requires either an iterative approach or a value looked up from a table

36

Efficient Refrigerator Example, cont.

•IRR tables show that

and so the IRR is approximately 15%

If the refrigerator has an expected lifetime of 15 years, this value becomes

15(P|A,i%,10) 5.02d %

18.4(P|A,i%,15) 5.00d %

As discussed earlier, the value is 20% if it lasts forever37

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) 38

US Inflation Over Last 350 Years

Source: http://upload.wikimedia.org/wikipedia/commons/2/20/US_Historical_Inflation_Ancient.svg

Historically prices have gone up and gone down. Recently many homeowners found home prices can also fall!

39

Figuring Average Rate of Inflation

• Calculate average inflation rate from 1982 to 2014

32 2341 2.34

100e

ln 2.34ln 1 2.69%

32e e

https://qzprod.files.wordpress.com/2014/11/us-consumer-price-indexes-year-on-year-change-core-cpi-headline-cpi_chartbuilder.png?w=1280

Current(12/2014)basketvalue is about 234 compared to base year of 1982. Annual rate is about 1% in 2014

40

Inflation (Escalation) Rate

• With escalation, an amount worth $1 in year zero becomes $(1+e) in year 1, etc., so

becomes

• We can compare terms to find an equivalent discount rate d’:

2

1 1 1PVF( , ) + ...

1+ 1+ 1+nd n

d d d

2

2

1+ 1+1+PVF( , , ) + ...

1+ 1+ 1+

n

n

e eed e n

d d d

1+ 1

1+ 1+ '

e

d d

41

424242

http://www.investopedia.com/terms/i/inflation.asp

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