ER 100 Lecture 10: Energy Toolkit VI

ER 100 Lecture 10: Energy Toolkit VI Energy Economics (we won t finish today, will return to this Thursday) Daniel M. Kammen Version date: September 29, 2015

Energy Systems Economics Market and technology drivers ( ) Simple Payback Discounting Present/Future Value Uniform payments (Annuities) Capital Recovery Factor Comparing Technologies/Costs

Example Questions How much does energy really cost? (well, that will take longer ) A coal-fired power plant costs $1,200/kW to construct. Coal was forecast to cost $45/ton but is now $90/ton. What is the average and short-term cost of electricity? A PV system will produce electricity for 30 years. What is the cost of each kwh produced during this period? For a financial institution the answer determines how much to charge for this electricity. For homeowner the question may depend critically on not only the balance sheet, but also the risk over time, transaction costs, and the need for up-front payments versus returns over time. If a company makes an investment in energy efficiency improvements, what is the dollar value of the energy that it saves? What are the risks? Meta question: cost is a physical construct or a social construct

Open EnergyINFO Site: http://en.openei.org/apps/tcdb/ 2008-2013 Howard Gruenspecht IEA, April 16, 2012 4

World oil price assumptions 2010 dollars per barrel Source: EIA, AEO2012 and IEO2011; IEA, WEO-2011 Howard Gruenspecht IEA, April 16, 2012 5

Shale gas offsets declines in other U.S. natural gas production sources U.S. dry gas production trillion cubic feet per year History 2010 Projections 23% Shale gas 49% 2% 26% 9% 9% 10% 21% Tight gas Non-associated offshore Coalbed methane Associated with oil Non-associated onshore Alaska 1% 21% 7% 7% 7% 9% Source: EIA, Annual Energy Outlook 2012 Early Release Howard Gruenspecht IEA, 6

Domestic natural gas production grows faster than consumption U.S. dry gas trillion cubic feet per year History 2010 Projections Consumption Domestic supply Net imports Source: EIA, Annual Energy Outlook 2012 Early Release Howard Gruenspecht IEA, 7

Oil to natural gas price ratio remains high over the projection ratio of oil price to natural gas price History 2009 Projections Oil and natural gas prices 2010 dollars per million Btu History 2009 Projections Source: EIA, Annual Energy Outlook 2012 Reference Case Howard Gruenspecht 8

Howard Gruenspecht IEA, April 16, 2012 9

Growth in income and population drive rising energy use; energy intensity improvements moderate increases in energy demand average annual change (2008-2035) percent per year Source: EIA, International Energy Outlook 2011 Howard Gruenspecht 10

Changing oil import needs are set to shift concerns about oil security Net imports of oil mb/d 14 12 10 8 6 4 2 2000 2010 2035 0 China India European Union United States Japan US oil imports drop due to rising domestic output & improved transport efficiency: EU imports overtake those of the US around 2015; China becomes the largest importer around 2020

Electricity Generation Costs Investment ($/kw) (Overnight Capital Costs) + Fixed O&M ($/kw) (e.g., Insurance and Taxes) + Variable O&M ($/kwh) (Fuel + Other O&M)

Economic Feasibility Evaluation Consider first the cost of electricity (COE) compared with the average electricity price (http://www.energy.ca.gov/distgen/economics/decision.html) Total cost of electricity in $/kwh is COE = C&I + O&M + F where C&I is the capital and installation cost, O&M is the operation and maintenance cost, and F is fuel cost. First, assume that there is no inflation and the cost of money is zero. Also to simplify the analysis, only some basic costs are considered. For example, downtime (capacity factor) cost is not included. This cost affects both the COE and the average utility price. Downtime cost can be included by considering the cost of a down minute (or second or hour) in combination with the mean-time between failures and the mean down time.

Economic Feasibility Evaluation C&I cost calculation: C & I ($/ kwh) ( TIC per kw )( FCR) ( CF)(8760 h / year) where TIC is the total installed cost, FCR is the fixed charge rate and CF is the capacity factor. FCR is calculated with FCR($/ kwh) where Ay is the amortization period in years. ( TIC) /( Ay) ( TIC) The capacity factor CF represents how much is the system utilized and it is calculated with CF (total operated hours per year) (8760 h / year)

Basics Over Time Money is a dynamic variable. i.e. money value changes in time. When money is invested, its amount changes with the return. When money is transferred into assets, its value change due to depreciation and/or inflation. Discounted cash flow is a technique that allow us to evaluate values as they change over time. Let s define how we call things first: P is the present value. i is the interest for a given period of time, i.e. the cost of money during the period under study. F is the future value. n is the number of periods of time we consider. Fundamental concept: time is money. Hence, as time ticks forward, so does the value of money.

Simple Payback Simple payback is the time to recover an investment, through savings, without discounting. Example: A compact fluorescent light, CFL, costs $6 and uses 20 Watts instead of 75 W. The savings is 55 W for the 4 hours a day that it is operated. Electricity costs 12 cents/kwh The Savings Energy: (55 watts)(4hr/day)(365day/year) = 80kWh/year Money: (80kWh/year)(0.12$/kWh) = $9.6/year Time (payback) : $6 bulb = 0.62 year = 7.5 months $9.6 savings/year

Discounting Is a dollar bill today worth the same as a dollar bill a year ago? Well, not really. The value of the same dollar bill today is not the same as a year ago If a year ago I had deposited the dollar bill in a savings account in the bank, today that dollar bill would worth more. Conversely, if a year ago I bought a chocolate bar for $1, today I would need that same dollar plus inflation to buy the same chocolate bar (of course, if I consider that that precisely chocolate bar is as good as a year ago). Hence, in most simple cases, present value and future value are related by F = (1+i) n P The factor (1+i) n is the compound interest.

The changing value of money Basic Example Transactions over time can be represented with cash flow diagrams. What is the value of something that I bought a year ago for $1 if the annual inflation rate was 5%? $ 1.05 0 1 t $ 1 The values are obtained using F = (1+i) n P, or: $1.05 = (1+0.05) 1 ($1)

More Complex: Inflation and the Value of Money First, a couple of notes about inflation. Inflation creates an additional cost for money. Circulating money in an economy can come from different sources: - loans (i.e. government bonds) - printing bills without any real treasury income of equivalent value. Hence, in this last case there are more bills without any income to back them up, each of them is then worth less. So inflation can be seen as an additional hidden tax that affects money value. Over time all money regardless the source needs to be adjusted by inflation. Indexed cash flows do not consider inflation. The real interest rate for a loan is (Fisher equation): (1 + i) = (1+ r)(1 + d); i =nominal interest rate; r = real interest Hence, d = inflation rate (1 + r) = (1 + i)/(1 + d) For common small interest rate values the above equations can be replaced by rates sum or rates difference, respectively.

Utility rates are divided among; Fundamentals Types of customers Energy consumption (by layers or bands) Time-of-Use Industrial and commercial customers also usually receive an extra charge, called demand charge, based on the highest amount of power consumed by the facility. Hence, local generation or EE programs may reduce electric bills even when there is not enough capacity for the entire facility by reducing the demand charge. Peak power consumption can be characterized with the load factor Average power Load factor (%)= 100% Peak power Real-time pricing is a rate structure in which prices are set based on the different energy cost during periods every day

The Meaning & Art of Discounting Financial decisions involve the evaluation of the potential costs and benefits of the stream of returns over time, so that the value of money changes over time. Your valuation of risk, and future generations will determine your discount rate (or your banker will). An interest rate compares values at different times. It represents both the anxiety (risk tolerance) of consumers to get a product and the marginal productivity of invested capital. Inflation also changes money s value over time. Usually discount rates reflect both interest rates and inflation. That is: discount rate (r) = interest rate + inflation Discount rates are prices, specifically the relative price, used to compare future and present goods and services in benefit-cost analyses.

Aligning costs and values? Generally, businesses don t value it if they don t measure it

Exponential Examples 12% interest rate 7% interest rate 230 yrs. (e-folding time for CO2) isotope with 30 years half life (Strontium-90) 10 5 0 0 10 20 30 40 50 time elapsed (years)

DISCOUNT RATES FOR COST-EFFECTIVENESS, LEASE PURCHASE, AND RELATED ANALYSES Effective Dates. This appendix is updated annually around the time of the President's budget submission to Congress. This version of the appendix is valid through the end of January 2002. Nominal Discount Rates. Nominal interest rates based on the economic assumptions from the budget are presented below. These nominal rates are to be used for discounting nominal flows, which are often encountered in lease-purchase analysis. Nominal Interest Rates on Treasury Notes and Bonds of Specified Maturities 3-Year 5-Year 7-Year 10-Year 30-Year 5.4 % 5.4 % 5.4 % 5.4 % 5.3 % Real Discount Rates. Real interest rates based on the economic assumptions from the budget are presented below. These real rates are to be used for discounting real (constant-dollar) flows, as is often required in cost-effectiveness analysis. Real Interest Rates on Treasury Notes and Bonds of Specified Maturities (in percent) 3-Year 5-Year 7-Year 10-Year 30-Year 3.2 % 3.2 % 3.2 % 3.2 % 3.2 % Analyses of programs with terms different from those presented above may use a linear interpolation. For example, a four-year project can be evaluated with a rate equal to the average of the three-year and five-year rates. Programs with durations longer than 30 years may use the 30-year interest rate.

Continuous and Discrete Time Analysis Discount rates are set for different time frames. Most discounting is annual, but we can use any interval, such as monthly discounts or daily discounts. If we start with a monthly discount rate (r M ) and we want to find out the correspondent annual discount rate (r A ): (1+ r A ) = (1+ r M ) 12 or, (1+ r A ) 1/12 = (1+ r M ) Suppose you keep $10 in the (piggy) bank. Assuming a 7% annual discount rate, after 10 years this amount is worth $5 You can find this value using discrete discounting or you can approximate using continuous discounting. discrete discounting 10 ( 1+ 0.07) t continuous discounting 10 rt e

Present and Future Value The future amount of money, F, is related to the present amount by after one time period. And again after two time periods So in general Which, for any n, is trivially the compound interest formula (1+r) F 2 P = n F 1 = P(1+ r) = F r 2 1 (1+ r) = P(1+ ) F = P (1+ r) n F n F(1+ r)-n (1+r) is called the present value factor

Net Present Value The Net Present Value method of analysis simply places all costs and benefits of a technology or service as equivalent costs or benefits at the present time. It is commonly used in engineering and financial comparisons of energy and other investment options. In general, the Net Present Value, is the discounted series of costs and benefits: å n NPV= B t - C t t =0 (1+ r) t

Uniform Series Amounts + + + + + - r r U P r r r r P r r r P U n n n n n - - ) (1 1 - : payments future of series a of P, lue, present va the yields also This ns. calcualtio mortgage in used equation the is This. factor charge fixed or the, factor recovery capital the as known is ) (1 1 - Where ) (1 1 - -1 ) (1 ) (1 : by lue present va the to related is then this P, borrowed, amount initial an on interval each made is payment U a Assume : payments of series annual) or monthly (e.g. uniform a in costs compare To

Cash Flow Evaluation This method is based on discrete discounting Suppose there is a energy firm. For instance, an IPP IPP = Independent Power Producer R: revenue = energy sales C: costs = operational expenses - profit = R - C In order to find out how much the business is worth we want to know its Net Present Value (NPV), assuming an operational life time (t). Start with a steady business, which produces the same revenue every interval (year).

NPV Example: Pollution Controls An automotive repair shop purchases a system to capture emissions of VOCs. The purchase price is $15,000 (from company cash - today s money) The equipment will be used for 7 years, and will cost $1,000/year to operate. After 7 years, the manufacturer will buy it back for $2,000. What is the total present value of owning and operating the system using a 4% annual interest rate? $2,000 4% discount rate 0 1 2 3 4 5 6 7 $15,000 $1,000 each Net Present Value = PV(capital cost) PV(annual costs) + PV(salvage)

NPV Pollution Controls Example Con t: P cap is already in NPV terms (t=0). P cap = $15,000 P O&M = uniform series of costs: U 1 - (1 + r r) n = ($1,000) 1 - (1.04) 0.04 7 = $6,002 P salvage = F(1+r) -t = $2000(1+0.04) -7 = $1520 NPV = - P cap - P O&M + P resale = $15,000 -$6,002 + $1,520 = -$19,482

Capital Recovery Factor (CRF) PV B æ r 1-1 ç è 1 + r ( ) t ö ø Here we have PV as a function of B (= payment) but the converse is also useful. How much is PV worth in the future in terms of annual payments invested now? Suppose you buy a Compact Fluorescent Light (CFL). The cost of the CFL is the PV of the investment. You know that per month you can save X kwh, based on the difference between the power of the CFL and regular light bulbs, the period of use, and the electricity cost. The (CRF) (PV) shows how much your investment is worth per month, over the life time of the equipment. (In the case of light bulbs, of course, the lifetime is a function of its usage.) PV B æ r 1-1 ö ç è ( 1 + r) t Þ (PV)r B Þ B PV ø æ ö ç 1 - è 1 ( 1 + r) t ø CRF r 1- ( 1 + r) -t r 1- ( 1+ r) -t CRF

Example: Comparing Light Bulbs incandescent light versus CFL power: 75 W 20 W = 80% less energy lifetime: 750 hrs. 10,000 hrs. cost: $0.75 $7.50 assume electricity cost $0.12/kWh and annual interest rate of 7% Þ the key is what is the average daily use of the light bulb 2 hours per day 730 hours per year timeframe r CRF - 1- ( 1+ r) t 750 730 1.03 years 10000 730 13.7 years CRF = 1.04 CRF = 0.12 annual bulb cost: 1.04 x $0.75 = $0.78 0.12 x $7.5 = $0.90 annual energy cost: 730h x 75W = 54,750 Wh 54.75 kwh, 730h x 20W = 14,600 Wh 14.6 kwh, 54.75 kwh x $0.12/kWh = $6.57 14.6 kwh x $0.12/kWh = $1.75 Total levelized cost $6.57 + $0.78 = $7.35 $1.75+ $0.90 = $2.65

Summary of Definitions Benefit-Cost Analysis -- A systematic quantitative method of assessing the desirability of government projects or policies when it is important to take a long view of future effects and a broad view of possible side-effects. Cost-Effectiveness -- A systematic quantitative method for comparing the costs of alternative means of achieving the same stream of benefits or a given objective. Net Present Value -- The difference between the discounted present value of benefits and the discounted present value of costs. Discount Rate -- The interest rate used in calculating the present value of expected yearly benefits and costs. Real Interest Rate -- An interest rate that has been adjusted to remove the effect of expected or actual inflation. Real interest rates can be approximated by subtracting the expected or actual inflation rate from a nominal interest rate. (A precise estimate can be obtained by dividing one plus the nominal interest rate by one plus the expected or actual inflation rate, and subtracting one from the resulting quotient.) Capital Asset -- Tangible property, including durable goods, equipment, buildings, installations, and land. Opportunity Cost -- The maximum worth of a good or input among possible alternative uses. Sunk Cost -- A cost incurred in the past that will not be affected by any present or future decision. Sunk costs should be ignored in determining whether a new investment is worthwhile. Fixed charge rate -- The fixed charge rate is the mechanism that distributes the cost of the plant and capital over the life of the asset. The rate incorporates the time value of money, federal and state income taxes, property taxes, and depreciation expenses. Annual Energy Output -- Annual energy output is a function of the capacity factor and the capacity of the generating unit. Internal Rate of Return -- The discount rate that sets the net present value of the stream of net benefits equal to zero. The internal rate of return may have multiple values when the stream of net benefits alternates from negative to positive more than once. IRR> 0 is generally required for any project.

Assumptions: $450 per kw of installed capacity = fixed cost 80% capacity factor 98% of operational costs is fuel 1 kwh = 3414 Btu natural gas price $3/million BTU life time of the power plant is 20 years, =0.4 Analysis: Example: Electricity cost for a Natural Gas power plant ( /kwh) Total cost PV = fixed cost + PV{of recurring operational costs (O&M)} Annual Energy Output (AEO) E = (capacity factor) (8,760 hrs/yr) (1kW of capacity/yr) = (0.8) (8,760 hrs) (1kW) = 7,008 kwh Annual (O&M) = AEO fuel cost conversion efficiency 98% Life Cycle Cost -- The overall estimated cost for a particular program alternative over the time period corresponding to the life of the program, including direct and indirect initial costs plus any periodic or continuing costs of operation and maintenance. Discount rate = 7% Annual (O&M) = {7,008 kwh (3414 Btu/kWh)} ($3/MMBtu) = $183 0.4 0.98

Example con t: Electricity cost for a Natural Gas power plant ( /kwh) Initial investment + Operational costs = Total cost $ $ $ $ $ $ $ TIMELINE 0 = Total energy output Total revenues electricity cost = PV (Total Energy output) PV $450 + $183 æ 0.07 1-1 ö ç è ( 1 + 0.07) 20 ø cost = $2388 140 =1.7,160kWh /kwh total output = 7,008 x 20 = 140,160 kwh taxes, profits, externalities

Example: Flood Lights for a Business Incandescent light versus CFL power: 150 W 60 W lifetime: 2 years @ 3,000 hrs/yr 8 years @ 3,000 hrs/year cost: $4.50 $14.50 Assume non-california electricity cost $0.07/kWh and annual interest (capital) rate of 8% Annualized Annual Annual capital Total energy operating cost levelized = use cost U = Current = 150 450 W kwh r = P 1 - (1 + cost = $31.5 lamp 3,000 : 0.07 r) - n + $2.52 hrs/yr $ $ 4.50 * ( 0.08 1 - (1.08 ) - 2 = / = kwh 450 = $34.02/yea kwh $31.5/yr ) r $ 2.52 /year Energy Efficient Lamp : Annual energy Annual operating use cost = = 60 W 3,000 hrs/yr 180 kwh 0.07 $ / = 180 kwh = kwh $12.6/yr Annualized capital Total cost = levelized U = P cost r 1 - (1 + = $12.6 r) - + n $2.52 $ 14.50 * ( 0.08 1 - (1.08 ) - 8 = $15.12/yea ) r $ 2.52 /year The energy efficient lamp is $18.90 cheaper/ye ar

Cost-Effectiveness: Comparing Policies Say we want to compare two policies that reduce CO 2 emissions 1) Flood lights to CFL 2) Installing rooftop PV To compare them in terms of their costeffectiveness, we would look at their incremental costs divided by their incremental CO 2 emission reductions

Flood Lights Cost. $18.90/yr savings in levelized terms. Emissions. We reduced electricity use by 450-180 = 270 kwh/yr. Using a US average emission factor of 0.63 kgco 2 /kwh, we ve reduced CO 2 emissions by 170 kgco 2 /yr. Abatement Cost. -$18.9/170 kgco 2 = -$110/tCO 2

Rooftop PV Cost. For PV, the incremental cost is vis-a-vis a retail price (~$0.10/kWh for the US), so the added cost of PV @ $0.30/kWh is $0.30/kWh - $0.10/kWh = $0.20/kWh. Emissions. For each kwh generated, 0.63 kgco 2 are reduced. Abatement Cost. ($0.20/kWh)/(0.63 kgco 2 /kwh) = $320/tCO 2

Technology Comparison

The Learning Curve

The Learning Curve

Learning Curves Compared

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Gas Turbine Learning Curve Innovation" stage (20% per doubling): learning enhanced through applied RD&D (R&D). Commercialization phase: slower rate (10% per doubling) when the technology was commercialized in niche markets Incremental investment for this technology - from first application of the technology to electricity generation until the technology was fully competitive: US$ 5 billion. Source: data adapted from MacGregor et al. (1991; data from GE only)

Average and min/max reactor construction costs per year of completion date for U.S. and France versus cumulative capacity completed.

Learning Curve Analytics Costs (C) are observed to decline with production volume (V). From an initial cost C(V 0 ) and volume V 0, we find that: C(V)/C(V 0 ) = (V/V 0 ) b Increase production by a factor of two, then the cost declines by: Progress ratio, R = = C(2V 0 )/C(V 0 ) = (2V 0 /V 0 ) b =2 b R is typically 70% - 90% across a wide range of technologies. b gives the slope of the learning curve

Learning Curve Analytics This tells us two things: 1) If we know the value of b we can calculate technology costs at a given volume (a is a scaling parameter) C(V) = av -b 2) From R = C(2V 0 )/C(V 0 ) we know that a doubling in volume will give us a cost of C(2V 0 ) = R * C(V 0 )

Using the Learning Curve Progress Ratio (R) = 2 -b If R is 0.8 1. Substitute 0.8 for R 2. Take the ln of both sides 3. Divide by ln2

How Can We Use This? Learning Curve Example: For a power plant, capital costs (CC) at time t are a function of generating capacity (GNCP) at time t, times a scaling parameter (a), raised to the power of negative b. The parameter a is calibrated at time t = 0 (solve the above for a at time t = 0)

Learning Curve Example Con t Let s say that initially we have 20 GW of wind that costs $1500/kW Now we can calculate capital costs for any future amount of generating capacity So if we decide that there will be 40 GW of wind in 2015, capital costs will have fallen to

Distributions of Learning Rates Distribution of observed learning rates (bars) and fit with a normal distribution (solid curve) based on the mean (μ) and standard deviation (σ) of an observed set of learning rates. Data from Dutton and Thomas (1984).

Distributions of Learning Rates (extra information)

The McKinsey Curves

Evolving Market Fundamentals New energy sources are often compared with the option of powering from the grid. Utility rates are divided among; Types of customers Energy consumption (by layers or bands) Time-of-Use Industrial and commercial customers also usually receive an extra charge, called demand charge, based on the highest amount of power consumed by the facility. Hence, DG may reduce electric bills even when there is not enough capacity for the entire facility by reducing the demand charge. Peak power consumption can be characterized with the load factor Average power Load factor (%)= 100% Peak power Real-time pricing is a rate structure in which prices are set based on the different energy cost during periods every day

Supplemental Data / Sources

Howard Gruenspecht IEA, April 16, 2012 59

Emerging economies continue to drive global energy demand Growth in primary energy demand Mtoe 4 500 4 000 3 500 3 000 2 500 2 000 1 500 1 000 500 0 2010 2015 2020 2025 2030 2035 China India Other developing Asia Russia Middle East Rest of world OECD Global energy demand increases by one-third from 2010 to 2035, with China & India accounting for 50% of the growth

National Petroleum Council Study Source: National Petroleum Council Howard Gruenspecht 61

Tremendous prospects for natural gas Largest natural gas producers in 2035 Russia United States China Iran Qatar Canada Algeria Australia India Norway Conventional Unconventional 0 200 400 600 800 1 000 Unconventional natural gas supplies 40% of the 1.7 tcm increase in global supply, but best practices are essential to successfully address environmental challenges bcm

Coal won the energy race in the first decade of the 21st century Growth in global energy demand, 2000-2010 Mtoe 1 600 1 400 1 200 1 000 800 600 400 200 Nuclear Renewables Oil Natural gas 0 Total non-coal Coal Coal accounted for nearly half of the increase in global energy use over the past decade, with the bulk of the growth coming from the power sector in emerging economies

More elaborate example: capital costs with financing

Capital Costs with Financing / 1 Consider that you need to determine how to power a load that will double after 3 years of the initial operation. You have the option of buying enough DG units to power the initial load and the load increase in 3 years for $ 2M or you can buy enough for the initial load by $1M and buy the rest of the DG units in 5 years. Consider that the annual inflation rate is 3 %. You are financing your capital investment with a loan. Loans interest rate are a couple of points above a savings account interest rate which, in turn, is a couple of points above the inflation rate. So the savings account rate is 5 % and the loan s rate is 7 %. What should you do? You have 3 options: 1) Get a loan today for $ 2 M for all the capacity including the load increase in year 3. 2) Get a loan for $ 1M now to power the initial load and after 3 years get more money for the additional load. 3) Get a loan today for $ 2M, use only what it is needed initially ($1M) and invest the rest until using it in year 3

Capital Costs With Financing / 2 Case #1: Get a loan today capacity including the load increase in year 3. $ 2.25 M 0 1 2 3 t $ 2 M Since the loan s interest rate is 7 % - 3 % of inflation, the cost of that money (or the equipment that I bought with it without considering depreciation) is: $2.25M = (1+0.04) 3 ($2M) Note: I could have considered the effect of equipment depreciation with a higher interest rate. Other methods can also be used.

Capital Costs with Financing Investment.. 3 Case #2: Get a loan for $ 1M now to power the initial load and after 3 years get more money for the additional load. $ 2.217 M 0 1 2 3 t $ 1 M $ 1.093 M Now I need to consider the cost of the initial loan plus the cost of the additional investment adjusted by inflation. $2.217M = (1+0.04) 3 ($1M) + (1+0.03) 3 ($1M) Note: As with any thing, money can have different values.

Capital Cost with Financing / 4 Case #3: Get a loan today for $ 2M, use only what it is needed initially ($1M) and invest the rest until using it in year 3. $ 2.28 M 0 1 2 3 t $ 1 M$ 1 M $ 1.093 M $ 1.06 M Now I also need to consider the cost of the initial loan plus the cost of the additional investment adjusted by inflation. But after 3 years the investment to buy the equipment to power the additional load comes from the $1 M I initially deposited in the bank. Hence $2.28 M = (1+0.04) 3 ($2M) + [(1+0.03) 3 ($1M) - (1+0.02) 3 ($1M)]

Present Value Derivation (extra) What is the Present Value (PV) of an infinite series of equal annual Profits (), assuming a fixed annual discount rate (r)? PV + + +... + 1 r 1 1+ r 2 1+ r 3 1+ r n - + 1 1+ r ( ) ( ) ( ) ( ) ( ) n multiplying both sides of the equation by (1+r): (PV ) 1+ r ( ) + ( 1 + r) + 1 1 + r ( ) +... 2 ( 1 + r) + n -1 ( 1 + r) n PV ( 1+ r) + PV PV + PV r + PV PV r PV r PV r This illustrates the relationship between the PV and the interest rate: The higher the discont rate the lower the PV

Annuity (supplement: derivation, not required) What is the Present Value (PV) of a finite series of equal annual Profits (B), assuming a fixed annual interest rate (r)? B B B B B B B PV + + +... - - -... + PV ( ) 1 ( ) 2 ( ) 3 ( ) t ( ) t + 1 ( ) t + 1+ r 1+ r 1+ r 1+ r 1+ r 1+ r 2 ( 1+ r) t n B ( 1+ r) 1 + B ( 1 + r) 2 + B ( 1 + r) 3 +... 1 ( 1 + r) t - æ 1 ç è ( 1+ r) t PV B æ r 1-1 ç è 1 + r ( ) t ö ø ö æ B ø ( 1+ r) 1 + B ( 1+ r) 2 +... B ö ç è ( 1+ r) n ø æ Bö ç æ Bö è r ø ç è r ø PV B r - æ 1 ö ç è ( 1+ r) t B ø r Þ PV B æ r 1-1 ö ç è ( 1 + r) t ø This provides a method to determine the PV of the operational costs of an investment, such as a power plant, over its life cyce.