John F. Kennedy School of Government Harvard University Prof. Robert N. Stavins ENR-201/Econ 1661 Fundamentals of Net Present Value Analysis 1
Fundamentals of Net Present Value Analysis I. Connecting the Overview to Standard Econ Treatment II. Basics of Welfare Economics: Theoretical Foundations of Evaluation Criteria III. Fundamentals of Benefit-Cost and Cost-Effectiveness Analysis A. The Time Domain B. Concept of Present Value C. Net Present Value Analysis D. Alternative to NPV: B/C E. Identifying a Public Sector Discount Rate F. Critique of Benefit-Cost Approach (& Response) IV. Mechanics of NPV Analysis V. NPV Analysis in Practice 2
Benefits of Good or Service Benefits Quantity 3
Cost of Provision of Good or Service Total Cost $ Quantity 4
The Efficient Level of Production & Consumption Total Benefits Total Costs Total Benefits Total Costs Quantity 5
Marginal Benefits, the Demand Function, and the Total Benefits of the Good or Service P (per unit) p q Q(units) 6
Marginal Costs, the Supply Function, and the Total Costs of the Good or Service Cost (per unit) MC q Q(units) 7
Net Benefits Maximized at Quantity where MB=MC, Supply=Demand P $ MC(Supply) p* q MB(Demand) Q 8
Effect of Negative Externality on Efficient Level of Production $ Marginal Social Cost = MPC + Marginal Damages Marginal Private Costs (Supply) Marginal Damages Marginal Benefits (Demand) q* q Quantity of Steel 9
Foundations of Welfare Economics Pareto (1920): a common-sense notion of what it means to make the world better off A (policy) change is an improvement if at least some people are made better off and no one is made worse off. Common-sense appeal; considerable normative standing Is this a good criterion for public policy? Yes (although not exclusive). Is it a useful criterion for public policy? No. Problem: there are virtually always losers; someone is inevitably made worse off. Nicholas Kaldor and John Hicks (1939): world is made better off overall (welfare improvement) if size of gains and size of losses are such that gainers could fully compensate losers (for their losses) and still be better off themselves. So, call this a potential Pareto improvement. Key Aspect of this Kaldor-Hicks criterion: satisfying the Kaldor-Hicks criterion is a necessary condition for satisfying the strict Pareto criterion. Why is this so important? If proposal fails this (weaker) Kaldor-Hicks test, it cannot pass Pareto test. 10
So, at a minimum, we can use Kaldor-Hicks test to weed out really bad policies (policies that can t make world better off in the Pareto efficiency sense) And what is the Kaldor-Hicks test in simple words? Benefits should be greater than Costs But it really says more than that. If Policy #1 has B>C, and another alternative, Policy #2, has even greater (positive) difference between B and C, then the change from Policy 1 to 2 also passes the Kaldor-Hicks test. So, Pareto efficiency and Kaldor-Hicks test do not simply call for any policy for which B>C, but policy for which positive difference between B and C is greatest. This, pure and simple, is the theoretical foundation (the normative justification) for the tool called (properly-done) benefit-cost analysis. [Note: Same point as in search for efficient level of production in figure with total benefits & total costs] 11
Other Economic Criteria Economic efficiency criterion > benefit-cost analysis Difficult, in part because of need for benefit assessment More modest criterion: cost-effectiveness does policy accomplish given purpose in the least costly way? min { q } i C ( q ) st.. q Q Compare with efficiency criterion: max B q C q q { q } Question: i i i i * ( ) ( ) i i i i i * qi? = Q Example: Dean s office, pick-up trucks, & buckets Warning 1: Beware of designing fast trains to the wrong station Warning 2: Beware of confusing C/E analysis with B- C analysis (later) Both criteria (efficiency & cost effectiveness) and both types of analysis focus on aggregate, but who gets the benefits, who pays the costs? Additional key criterion: distributional equity 12
Taking Time into Account Costs may be incurred this year, benefits in future (typical investment) or benefits this year, costs in future How can we avoid comparing apples & oranges? Question: Putting aside uncertainty & inflation, would you prefer to receive $10K today or $10K one year from now? How about $10K today or $20K next year? How about $10K today or $15K next year, etc,...? Indeed, what amount a year from now makes you indifferent with $10K today? Take that number, FV, divide by PV (10K), subtract 1, and that s your consumption rate of interest, your personal discount rate FV 1 r PV = Same reasoning if we re talking about a C, instead of a B So, discounting converts monetary amounts to ones for a standardized year (not about uncertainty or inflation later) Now, let s generalize for t years... 13
Present Value The future value of money invested in the present at rate r: FV = (1 + r) t PV t What s maximum amount you d be willing to pay for a guaranteed payment of $100 t years from today? Why? Answer is present value: PV = FVt (1 + r) For stream of B s and C s across several years (terminal year = T), define net present value (NPV) as: t T B1 C1 B2 C2 BT CT Bt C NPV = ( B0 C0) + + + + = 2 T (1 + r) (1 + r) (1 + r) (1 + r) t = 0 t t B B C C B NPV = B + + + C + + + = T T 1 T 1 T t t 0 T 0 T t t (1 + r) (1 + r) (1 + r) (1 + r) t= 0 (1 + r) t= 0 (1 + r) C NPV B C B C T T T t t t t T = = t t t t= 0 (1 + r) t= 0 (1 + r) t= 0 (1 + r) 14
Inflation Nominal amounts are valued according to prices in years in which returns occur. Real amounts are valued according to prices in a single year. Nominal interest rates are approximately equal to sum of real interest rate and expected rate of inflation. Remember: Need to be internally consistent; everything real or everything nominal 15
Net Present Value Analysis Present Value Criteria comes from notion of potential Pareto improvements. Recall definition of net present value: discounted present value of the (possibly infinite) streams of expected benefits and costs: NPV B C B C = = t t t t t t t t= 0 (1 + r) t= 0 (1 + r) t= 0 (1 + r) Project is admissible (do it if no alternatives) if net present value is positive. Project with greatest NPV is preferred. 16
Can choice of discount rate affect the decision? Yes. Costs Hydroelectric Gas Turbine Fixed Costs Very High Low Operating Costs Very Low High With low discount rate: prefer hydro With high discount rate: prefer gas turbine Does higher discount rate favor or disfavor public policy interventions? It s ambiguous. Will choice of discount rate necessarily affect analytical result? No. Bottom Line: Do Sensitivity Analysis with Alternative Discount Rates! 17
Alternative to NPV Benefit-Cost Ratio (B/C): discounted present value of benefits divided by discounted present value of costs Project is admissible if B/C > 1.0 (OK, because same as B - C > 0) Project is preferred if it has greatest B/C ratio (?) Problem: Doesn t always give the right answer (where right refers to Kaldor-Hicks test) 18
Benefit-Cost Ratio B C PV PV B1 B2 B3 = B0 + + + + 2 3 (1 + r) (1 + r) (1 + r) C1 C2 C3 = C0 + + + + 2 3 (1 + r) (1 + r) (1 + r) Decision Rule 1. If B/C>1 then do the project. (If both B and C are positive, then this is the same as the NPV criterion, and is therefore ok for deciding whether to do a single project). 2. When choosing among projects, do projects in order of benefit/cost ratios (this is not ok). 19
Problems with Benefit-Cost Ratios 1. Scale Ignored: projects of different sizes (again) Project 1 Project 2 Benefits (pv) 100 10,000 Costs (pv) 2 9,000 NPV 98 1,000 B/C 50 1.11 For graphic example, see figure of total B s and C s 20
2. Can be manipulated by shifting benefits to costs (or vice versa) particularly important with externalities. Project 1 Project 2 (millions) (millions) Benefits (pv) 5 4 (3) Costs (pv) 2 2 (1) NPV 3 2 (2) B/C 2.5 2.0 (3.0) Suppose we re-classify $1 million in costs in project 2 as negative benefits. (This means that benefits are 3 and costs are 1.) The NPV does not change. B/C becomes 3. 21
A Digression on Budgetary Constraints In many policy contexts, costs are typically borne by private sector, not by public sector, but... In some public policy domains, costs may be borne by public sector, and subject to budgetary constraints. Does this change the Kaldor-Hicks decision criteria? No. Criterion remains: in any choice situation, select the alternative that produces the greatest net benefits. But, does this still hold if task is to choose one or more among a set of projects, where there is a constraint (on total cost budget or managerial hours, or acre-feet of water, whatever)? Kaldor-Hicks criterion is: choose the set of project(s) that yield the maximum net benefit subject to the resource constraint. Does using B/C ratio achieve this? No. What s an operational approach? For each project, calculate net benefit per unit of constraint (cost), rank according to this index, and select projects with positive NPV until constraint is met. 22
Identifying a Public Sector Discount Rate For private sector analysis, relevant discount rate: opportunity cost of funds to specific firm (also relevant in some public policy contexts) For public sector, there s no unanimously-accepted, conceptually-correct approach (an area of active research stimulated by new consideration of very long time-horizon problems) Thinking about the social discount rate Define SDR: relative valuation placed by society on future consumption that is presently sacrificed Aggregation of individual consumption rates of interest Question: why should this be any different from the private rates of return? 1. Concern for Future Generations: private market may focus unduly on short term. But... Individuals are concerned about future generations Governments may have shorter, not longer (political) time horizons than firms 2. Paternalism: people systematically discount the future excessively, even for their own well being. 23
What about uncertainty in the benefit and/or cost streams? Add a risk premium to discount rate? No. First, risk is not a big issue for many types of policies. Why? Risk of project is determined by covariance of its returns with returns of complete portfolio Hence, many projects (policies) are riskless, because.. b s and c s of projects are widely dispersed and uncorrelated with other aspects of welfare 24
Second, some public policies can be risky, but adding risk premium to social discount rate is not correct Affects b s and c s differently Risky benefit is worth less than expected value; so, upward adjustment of discount rate Uncertain cost is valued more (by risk-averse individuals) than expected value of cost; so, downward adjustment of discount rate Also, increasing discount rate has ambiguous effects on NPV (depends on time-pattern of Bs and Cs) Correct approaches: [Substitute certainty-equivalents for all benefits and for all costs, and use risk-free discount rate] Produce probability distributions, not point estimates; Monte Carlo analysis (see figure) So, do Sensitivity Analysis What about a simple range of results? Problem: implies uniform distribution Typical advice from economists for social discount rate: about 2 to 3% (real) 25
Probability Distributions of Annualized Net Benefits of a Hypothetical Proposed Regulation and of an Alternative Regulation with the Same Expected Value of Net Benefits Probability Probability Distribution of Annualized Net Benefits of Proposed Regulation Expected Value: $1 Billion Probability that Annualized Net Benefits are Actually Negative: 20% Probability Distribution of Annualized Net Benefits of Alternative Regulation Expected Value: $1 Billion Probability that Annualized Net Benefits are Actually Negative: <1% -1 0 1 2 3 4 5 6 Annualized Net Benefits (Billions of Dollars) 26
What about very long time horizons, in particular, intergenerational policies? Not an issue for the vast majority of policies, but important for some (GCC, radioactive waste disposal, and others) Obvious ethical and economic issues (cannot consult preferences of all affected parties) One approach two components from optimal growth models: pure time preference (treat it as zero; i.e. do not favor present generation s consumption over that of future gen s) adjustment to reflect fact that marginal utility of consumption will decline over time as consumption per capita increases (estimates of elasticity of marginal utility and rate of growth of consumption) Result: as low as 0.5%, as high as 3% More radical approach hyperbolic discounting (some empirical basis, but time inconsistent) 27
What rates are actually used by government agencies? General answer: considerable range For many years, U.S. OMB required use of 7% real discount rate for Regulatory Impact Analyses. Why? One possible rationale: agencies don t have information necessary to estimates b s and c s (including certainty equivalents) correctly, so used ad hoc adjustment of higher discount rate Pluralistic system: Agencies want new policies, programs, and projects, and so may exaggerate benefits relative to costs; OMB counteracted this by using higher discount rate Changes in OMB Guidelines (September, 2003): (1) use 3% & 7% for intra-gen, lower for inter-gen (2) calculate NPV for several levels of policy (3) use formal probability analysis (Monte Carlo) (4) expanded use of C/E analysis, along with NPV (5) VSL and VSLY (QALY) later 28
Choices for the Analyst When Carrying Out a Net Present Value Analysis 1. Defining the Limits of the Analysis Geographically Temporally Primary, secondary impacts 2. Jobs Generally, labor belongs on the cost side ( tie one hand behind their backs ) Politicians typically view job-creation as a benefit of public projects But, if workers are otherwise involuntarily unemployed, real cost is less than wage rate (forgone leisure), and... There can be positive externalities (social benefits) of reducing unemployment. There are, of course, issues regarding the distribution of costs, and distribution of benefits... 29
3. Beware of Double-Counting Example of agricultural irrigation project: a. Present value of increased stream of net revenue from farming b. Increased value of land Not to be added; indeed, they are the same 4. Comparing Projects of Different Lengths (in Time) Gas turbine vs. Coal/steam plant Two methods (identical results): 1. Replicate shorter project; re-calculate 2. Equivalent Annual Net Benefit (later) 30
5. Consider Real Costs and Real Benefits (soon in detail) Correct concept of real costs 1. Opportunity cost is what matters (e.g. value of land whether purchased or condemned) 2. Do not include transfers (e.g. taxes, turnpike tolls) only real resource costs (capital & labor) Correct concept of real benefits 6. Uncertainty 1. Total WTP/WTA for relevant goods or services (area under demand curve) 2. Cost of an alternative is just that a cost, not a benefit (c/e does not imply b>c) LATER 3. Not just financial benefits, but all true benefits Can uncertainty be quantified? How should we deal with it? Sensitivity analysis [Full Probabilistic Characterization (Monte Carlo)] 7. Broader Issues: Necessary and Sufficient Conditions? (Arrow,et al.) 31
BUSWAY SYSTEM Two Proposals to Improve Houston s Public Transportation System Basic Information Limited exclusive lanes: 1-year construction period, PV cost $1 billion, 50-year life (include initial equipment) Equipment: 2,000 buses @ $250,000/bus, 6-year lifetime (i.e. $500 million every 6 years) Operation & maintenance costs: $280 million/year Travel-time savings valued at $9.1 billion (pv) Net air-pollution reduction valued at $456 million (pv) RAIL TRANSIT SYSTEM Surface rail lines & stations: 1-year construction period (hire these guys!), PV cost of $3 billion, 75-year lifetime (includes initial equipment) Equipment: 200 electric rail cars @ $2 million/each, 10- year lifetime; plus 1,200 buses @ $250,000/bus, 10- year lifetime (i.e. $700 million every 10 years) Operation & maintenance costs: $260 million/year Travel-time savings valued at $9.7 billion (pv) Net air-pollution reduction valued at $967 million (pv) 32
COSTS Interest Rate 0.05 BUSWAY RAPID TRANSIT Year Op/Maint (pv) Equip (pv) Op/Maint (pv) Equip (pv) 1 280 267 0 0 260 248 0 0 2 280 254 0 0 260 236 0 0 3 280 242 0 0 260 225 0 0 4 280 230 0 0 260 214 0 0 5 280 219 0 0 260 204 0 0 6 280 209 500 373 260 194 0 0 7 280 199 0 0 260 185 0 0 8 280 190 0 0 260 176 0 0 9 280 180 0 0 260 168 0 0 10 280 172 0 0 260 160 700 430 11 280 164 0 0 260 152 0 0 12 280 156 500 278 260 145 0 0 13 280 148 0 0 260 138 0 0 14 280 141 0 0 260 131 0 0 15 280 135 0 0 260 125 0 0 16 280 128 0 0 260 119 0 0 17 280 122 0 0 260 113 0 0 18 280 116 500 208 260 108 0 0 19 280 111 0 0 260 103 0 0 20 280 106 0 0 260 98 700 264 21 280 101 0 0 260 93 0 0 22 280 96 0 0 260 89 0 0 23 280 91 0 0 260 85 0 0 24 280 87 500 155 260 81 0 0 25 280 83 0 0 260 77 0 0 26 280 79 0 0 260 73 0 0 27 280 75 0 0 260 70 0 0 28 280 71 0 0 260 66 0 0 29 280 68 0 0 260 63 0 0 30 280 65 500 116 260 60 700 162 31 280 62 0 0 260 57 0 0 32 280 59 0 0 260 55 0 0 33 280 56 0 0 260 52 0 0 34 280 53 0 0 260 49 0 0 35 280 51 0 0 260 47 0 0 36 280 48 500 86 260 45 0 0 37 280 46 0 0 260 43 0 0 38 280 44 0 0 260 41 0 0 39 280 42 0 0 260 39 0 0 40 280 40 0 0 260 37 700 99 41 280 38 0 0 260 35 0 0 42 280 36 500 64 260 33 0 0 43 280 34 0 0 260 32 0 0 44 280 33 0 0 260 30 0 0 45 280 31 0 0 260 29 0 0 46 280 30 0 0 260 28 0 0 47 280 28 0 0 260 26 0 0 48 280 27 500 48 260 25 0 0 49 280 26 0 0 260 24 0 0 50 280 24 0 0 260 23 700 61 51 260 22 0 0 52 260 21 0 0 53 260 20 0 0 54 260 19 0 0 55 260 18 0 0 56 260 17 0 0 57 260 16 0 0 58 260 15 0 0 59 260 15 0 0 60 260 14 700 37 61 260 13 0 0 62 260 13 0 0 63 260 12 0 0 64 260 11 0 0 65 260 11 0 0 66 260 10 0 0 67 260 10 0 0 68 260 9 0 0 69 260 9 0 0 70 260 9 700 23 71 260 8 0 0 72 260 8 0 0 73 260 7 0 0 74 260 7 0 0 75 260 7 0 0 Non-discounted Sum 14000 4000 19500 4900 Present Value 5112 1329 5066 1076
Busways vs. Rapid Transit Cost-Benefit Analysis (all values are PV, $ millions) Busway RT Costs Construction (incl. init. equip) 1,000 3,000 Equipment 1,329 1,076 Operation and Maintenance 5,112 5,066 Total Costs 7,440 9,143 Benefits Travel Time Savings 9,128 9,671 Reduced Air Pollution 456 967 Total Benefits 9,584 10,638 Evaluation Criteria Benefit/Cost Ratio 1.29 1.16 (B/C) Net Present Value 2,144 1,496 (B-C) Equivalent Annual Net Benefits 111 73 Interest Rate: 0.05 34
Equivalent Annual Net Benefits Purpose: To compare projects/policies of different duration; and helpful for conveying results to non-experts, in any case Definition: The Equivalent Annual Net Benefit is the constant net benefit (in every year of the project) that sums in present value terms to the net present value of the project. So, EANB > simple average. Equivalent Annual Net Benefit of project with NPV, interest rate r, and lifetime T, is: EANB = NPV r 1 + r (1 + r) T Note: For infinite planning horizon, EANB = NPV [r/(1+r)] Example: r = 0.05, NPV = 100 Simple Avg [100/(T+1)] EANB T=0 = 100.0 100.0 EANB T=4 = 22.0 20.0 EANB T=9 = 12.3 10.0 EANB T=49 = 5.2 2.0 EANB T=99 = 4.8 1.0 EANB T=inf = 4.8 0.0 This is for annuity due, i.e., convention of b s & c s at beginning of each time period. 35
Equivalent Annual Net Benefits for Ordinary Annuity Uses alternative convention: assume benefits and costs occur at end of each year, with first payment occurring one year from now. Equivalent Annual Net Benefit of project with NPV, interest rate r, and lifetime T, is: EANB = NPV r 1 (1 + r) T Note: For infinite planning horizon, EANB = NPV r Example: r = 0.05, NPV = 100 Simple Avg [100/T] EANB T=1 = 105.0 100.0 EANB T=5 = 23.1 20.0 EANB T=10 = 13.0 10.0 EANB T=50 = 5.5 2.0 EANB T=100 = 5.0 1.0 EANB T= inf = 5.0 0.0 36
Busways vs. Rapid Transit (10%) Cost-Benefit Analysis Costs (all values are PV, $ millions) Busway RT Construction 1,000 3,000 Equipment 641 439 Operation and Maintenance 2,776 2,598 Total Costs 4,418 6,037 Benefits Travel Time Savings 4,957 4,994 Reduced Air Pollution 248 499 Total Benefits 5,205 5,493 Evaluation Criteria Benefit/Cost Ratio 1.18 0.91 (B/C) Net Present Value 788-544 (B-C) Equivalent Annual Net Benefits 72-49 Interest Rate: 0.10 37
Busways vs. Rapid Transit (3%) Cost-Benefit Analysis Costs (all values are PV, $ millions) Busway RT Construction 1,000 3,000 Equipment 1,953 1,778 Operation and Maintenance 7,204 7,722 Total Costs 10,157 12,501 Benefits Travel Time Savings 12,865 14,562 Reduced Air Pollution 643 1,456 Total Benefits 13,508 16,018 Evaluation Criteria Benefit/Cost Ratio 1.33 1.28 (B/C) Net Present Value 3,351 < 3,517 (B-C) Equivalent Annual Net Benefits 125 > 115 Interest Rate: 0.03 38