Cost Benefit analysis L5 Discounting Future Impacts · To compute net present value of a project . Assuming that the social discount rate is known · And ignoring uncertainty and risks Discounting · Public projects usually span across multiple years · For a proper comparison, we discount future benefits or costs by presenting everything in terms of their present value · Aggregate the present value of all costs and benefits and propose a project with highest net present value Discounting 1 period - diagram 10000 in t=0 11000 in t=1 cost = 10000 benefit =1000 implement project? 10000 today is not exactly equal to 10000 tomorrow. Opportunity cost of 10000 dollars. Method 1: Future Value Analysis . Compare it with the alternative of investing in risk-free bills . If risk-free interest rate is 5%, the $10, 000 will amount to $10, 500 in a year . This is the future value of $10, 000 · FV (X) = X(1 + i), where i is the annual risk-free rate of interest Method 2: Present Value Analysis . Compare its current value with the current value of the alternative of investing in risk-free bills . If risk-free interest rate is 5%, the $11, 000 at t = 1 is same as having $10, 476 = (11, 000/1.05) now · This is the present value of $11, 000 . PV (Y ) = Y/ 1+i , where i is the annual risk-free rate of interest Y$ you will receive in future. Method 3: Net Present Value Analysis · Compute present value of all benefits and costs and sum them to compute the net present value (NPV) . Choose a project with highest NPV · NPV = PV (benefits) - PV (costs) Adopt project if NPV is positive, or if it has highest NPV Diagram - shows discounting of 11000 to the present value. PV(B)=10476 PV(C)=10000 NPV = 476
Future Value - multiple periods . Compare it with the alternative of investing in risk-free bills · Let - risk-free interest rate, i - number of years, n - initial investment, X · With simple interest: FV (X) = X(1 + ni) · With compound interest: FV (X) = X(1 + i) n · Compound interest factor = (1 + i) n . It is the value of $1 after n years with interest rate i compounded annually Depends on type of interest. X t=0 Xi -> X(1+i) t=1 Xi^2 t=2 To show how to get formula Present Value - multiple periods . Compare its current value with the current value of the alternative of investing in risk-free bills · Let - risk-free interest rate, i - number of years, n - future investment, Y · With compound interest: PV (Y ) = Y /(1 + i) ^n · Present value factor or discount factor = 1 /(1+i) ^n . It is the present value of $1 received after n years with compound interest rate of i Discounting · Discounting is the process of computing