Applied welfare econ cost benefit analysis ch6

Applied Welfare Econ & Cost Benefit Analysis Chapter Discounting Future Benefits and Costs Discounting Purpose: Some practical issues one must know in order to compute the net present value (NPV) of a project It assumes the social discount rate is given, which is reasonable as the rate is often set by an public agency The chapter covers: the basics of discounting (two-periods); compounding and discounting over multiple periods (years); the timing of benefits and costs; horizon (terminal) values; comparing projects with different time frames; inflation and the difference between nominal and real dollars; relative price changes; and sensitivity analysis in discounting Discounting Appendix 6A provides shortcut formulas for calculating the present value of annuities and perpetuities You are responsible for going through this appendix yourselves BASICS OF DISCOUNTING Projects with Lives of One Year  Discounting takes place over periods not years However, for simplicity, assume that each period is a year First we discuss projects that last one year  There are three possible methods to evaluate potential projects: future value analysis, present value analysis and net present value analysis Each gives the same answer PV = FV (1 + i) BASICS OF DISCOUNTING  Future Value Analysis – Choose the project with the largest future value, FV, where the future value in one year of an amount X invested at interest rate i is: FV = X (1 + i) (6.1)  Present Value Analysis – Choose the project with the largest present value, PV, where the present value of an amount Y received in one year is: PV = Y/(1 + i) (6.2)  If the PV of a project equals X, and the FV of a project equals Y, both equations (6.1) and (6.2) imply: PV=FV /(1 + i)  This equation shows that discounting (the process of calculating the present value of future amounts) is simply the opposite of compounding (the process of calculating future values) BASICS OF DISCOUNTING  Net Present Value Analysis – Choose the project with the largest net present value, which calculates the sum of the present values of all the benefits and costs of a project (including the initial investment):  NPV = PV(benefits) – PV(costs) (6.3) BASICS OF DISCOUNTING Usually projects are evaluated relative to the status quo:  If there is only one new potential project and its impacts are calculated relative to the status quo, it should be selected if its NPV > 0, and should not be selected if its NPV <  If the impacts of multiple, mutually exclusive alternative projects are calculated relative to the status quo, one should choose the project with the highest NPV, as long as this project’s NPV >  If the NPV < for all projects, one should maintain the status quo COMPOUNDING AND DISCOUNTING OVER MULTIPLE YEARS    Interest is compounded when an amount is invested for a number of years and the interest earned each period is reinvested Interest on reinvested interest is called compound interest The future value, FV, of an amount X invested for n years with interest compounded annually at rate i is: FV= X (1+i)n (6.4) Present value again     Present value: the amount a specified money sum received at a future date is worth today $1 today can turn into $1(1+r) in a year’s time right? Therefore, we know that the present value (PV) of $1(1+r) of next year is $1 of today This is because $1 is equal to $ 1+r 1+r Present value vs Future Value  When we translate the future into the present, we discount the future value (we make it look smaller to get the PV)  When we translate the present into the future, we compound the present value (we make it look bigger to get the FV) Other Considerations   Shorter projects also have an additional benefit (not included in EANB) because one does not necessarily have to roll-over the shorter project when it is finished A better option might be available at that time This additional benefit is called quasi-option value and is discussed further in Chapter NET PRESENT VALUE  If the NPV of a prospective project is positive, it should be accepted, but if it is negative then the project should be rejected  OK, I will believe it…  but then, what should I with my savings? NET PRESENT VALUE  what should I with my savings?  Invest them in any project that promises returns at a rate r  That is the key: your project was not good if you used a discount rate r  Because you would lose money in PV if the discount rate was to be r NET PRESENT VALUE  In that case, instead of going for the project, put you money into something that pays an interest rate r  like a bank account Internal rate of return  Another way to look at it is to find the particular value of the discount rate that would make the NPV=0 for your project  That is called the internal rate of return of your project Internal rate of return  Mathematically:  IRR = r*  such that  NPV(r*)=0 D NPVÝr Þ = > t Bt Ý1+r DÞt?1 ? Ct Ý1+r Ý Ý tÝ =0 Internal rate of return  Therefore, if you can find an alternative project that pays more than r*, you should put your money there!!! Not in your original project!  Think about it: if the IRR (r*) is lower than what you can get anywhere else, then you are not getting a benefit from the project to cover the opportunity cost of your money! Internal rate of return  Think about it in a different way: if the IRR (r*) is lower than the rate of interest at which you need to repay your loan for the funds for the project, you should not undertake the project! Example of the power of compounding  The Dutch allegedly bought Manhattan in 1626 for about $24 worth of beads and trinkets  if Native Americans had invested in tax-free bonds 7% APR bonds, it would now be worth over $2.0 trillion > assessed value of Manhattan  if US had invested $7.2 million it paid Russia in 1867 in tax-free 7% APR bonds  money worth only $50.9 billion < Alaska's current value REAL VERSUS NOMINAL DOLLARS    Conventional private sector financial analysis measures monetary amounts in nominal dollars (sometimes called current dollars) But, due to inflation, one cannot buy as many goods and services with a dollar today as one could one, two or more years previously It is important to control for inflation We control for inflation by converting nominal dollars to real dollars (sometimes called constant dollars) We usually use the consumer price index (CPI), but sometimes use the gross national product (GNP) Estimates of Inflation and Problems with CPI The CPI is the most commonly used measure of inflation Prior to 1998, the CPI overstated inflation by about 0.8% to 1.6% per annum Four reasons for the overstatement are:  1) Commodity substitution effect: The CPI did not accurately reflect changes in consumer purchases, such as switching to lowerpriced substitutes;  2) New goods: The “basket” of goods did not include some new products, e.g., new (cheaper) generic drugs;  3) Quality improvements: The CPI did not accurately reflect changes in product quality, e.g more safe or reliable cars;  4) Discount store effect: Consumers are shopping more at discount stores, which have less expensive products Real rates of interest  In a world with inflation  What is the difference between the real and the nominal interest rate?  Imagine that prices actually grow at a rate:  We are going to label the real interest rate  i* and the nominal interest rate i E Real rates of interest  If we want to put next year’s prices in real terms, we want to discount the inflation  we want to adjust for inflation, dividing by  1+  Using this idea, we can find the real rate of interest  i* is more or less equal to i ones we normally face Ý for low levels of inflation, like the E Real rates of interest  The proof is given in Varian, for example  Check that it makes sense, and then just remember that the results is a good approximation in most cases NEXT Existence Values READ CHAPTER ... projects: future value analysis, present value analysis and net present value analysis Each gives the same answer PV = FV (1 + i) BASICS OF DISCOUNTING  Future Value Analysis – Choose the project... DISCOUNTING  Net Present Value Analysis – Choose the project with the largest net present value, which calculates the sum of the present values of all the benefits and costs of a project (including... discounting (two-periods); compounding and discounting over multiple periods (years); the timing of benefits and costs; horizon (terminal) values; comparing projects with different time frames; inflation