3/8/04 1 Methods for Project Evaluation March 8, 2004 Nuclear Energy Economics and Policy Analysis Alternative Methods • Present worth (PW) method • Future worth (FW) method • Annual worth (AW) method • Benefit-cost ratio (BC) method • Internal rate of return (IRR) method Nuclear Energy Economics and 2 Policy Analysis 1 3/8/043/8/04 3 • • • Cost of capital is known • Capital is always available for profitable projects (i.e, access to capital is not restricted.) Nuclear Energy Economics and Policy Analysis Assumptions Future cash flows are known with certainty Analysis is in constant dollars 3/8/04 4 PW Method: NPV = rn - cn + i )n n =1 N Â Decision criterion: FW Method: FV = (rn n =0 N Â - cn + i )N -n Nuclear Energy Economics and Policy Analysis (1Accept if NPV>0; reject if NPV<0 )(1Decision criterion: Accept if FV>0, etc. 23/8/04 5 Pricing a Bond • bond paying 6% per year (payable semi-annually) that is redeemable at par value if the buyer is seeking $1000. N = 10 x 2 = 20 periods r = 6%/2 = 3% per period i = [1.11/2-1]100 = 4.9% per compounding period C = Z = $1000 V(N) = ¥ (P / F + P / A= + = $761.16 Note: bonds. Nuclear Energy Economics and Policy Analysis Example of PW Method: At what price should a buyer purchase a 10-year a 10% per year yield? The face value of the bond is $1000 ,0.049,20) $1000(0.03)( ,0.049,20) 384.1 377.06 The yield typically increases for longer-term 3/8/04 6 Example: • A 10-year U.S. treasury bond that matures in eight years has a face value of $10,000. The bond pays 8% per year (payable quarterly). A prospective buyer of the bond wants to earn 10% per year on her investment (compounded quarterly) because interest rates have risen since the bond was issued. How much should the buyer pay for the bond? V(N ) = ¥ ¥ (P / A + P / F= . I.e., an increase in interest rates causes bond prices to decline. Nuclear Energy Economics and Policy Analysis Influence of interest rates on bond prices $10,000 (0.02) ,0.025,32) $10,000( , 0.025,32) $8,907 33/8/04 Example: Pricing stock •Stock in a company represents a share of ownership, as opposed to a bond, which is essentially a promissory note. • Common stock is more difficult to value than bonds because dividends and prices of common stocks are not constant; investors hope that they will increase over time. • If reliable forecasts of future earnings, dividends, and stock prices could be made, stockvaluation would result from discounting the forecast cash flow. Example (from Riggs and West): An investor is investigating the stock performance of two companies: A and B. Company A has consistently paid dividends that increase 10 cents per year while the selling price of the stock has averaged a 2% annual rise. Company B is a fast growing new company that has paid no dividends because all earnings are retained for expansion, but its market price is expected to increase by $10 per year. Current data about the two companies are summarized below: Company A Company B Dividend $2.25 (10 cent/yr increase) 0 (2% of market price after 5 years) Market Price $28 (2% annual increase) $65 ($10/yr increase) Risk-adjusted 9% 12% discount rate for stock valuation Disregarding tax effects and brokerage commissions to buy or sell, which stock is more attractively priced? Nuclear Energy Economics and 7 Policy Analysis 3/8/04 8 Annual Worth (AW) Method Example: Because of the Assume: Solution: To earn 12% on this project, the annual rental income must equal the AW of the costs: Taxes and insurance/yr = 0.1 x 275000 = $27,500 Upkeep/yr 73,073 ¥ 25)(0.9) = Nuclear Energy Economics and Policy Analysis An investment company is considering building a 25-unit apartment complex in a growing town. long-term growth potential of the town, it is felt that the company could average 90% of full occupancy for the complex each year. If the following items are reasonably accurate estimates, use the AW method to determine the minimum monthly rent that should be charged if a 12% rate of return per year is desired. Land investment cost = $50,000 Building investment cost = $225,000 Study period, N = 20 years Rent per unit per month = ??? Upkeep expense per unit per month = $35 Property taxes and insurance per year = 10% of total initial investment Land cost can be recovered at the end of the 20 year period First determine the equivalent AW of all costs at an interest rate of 12%/yr. Initial investment cost = $50,000 + $225,000 = $35 (12 x 25)(0.9) = $9450 Annual worth of capital costs = $275,000 (A/P,0.12,20) - $50,000 (A/F, 0.12, 20) = $36,123 Equivalent annual worth of costs = -$27000 - $9450 - $36123 = -$73073 Therefore, the minimum annual rental required equals $73,073 to achieve a 12% rate of return, and with annual compounding, the monthly rental amount is given by: (12 $270.64 43/8/04 9 j the IRR, i*, is given by PV i *( ) = Fj 1+ i *( )j j = 0 N Â = 0 Decision criterion: *,, accept the project. *, reject the project. Nuclear Energy Economics and Policy Analysis Internal Rate of Return (IRR) Method For a project with net cash flows, FIf the minimum required rate of return < iIf the minimum required rate of return > i3/8/04 10 th order polynomial in i*. results? Question: Descartes’ Rule of Signs: sequence of coefficients. F0 1X + F2X2 N XN Sign No one solution? Reject Yes No Yes Nuclear Energy Economics and Policy Analysis IRR Method (contd) The equation for the IRR is an N There will in general be more than one root. If more than one of the roots is real and positive, how do we interpret the When is there a unique solution to the IRR problem? For an N-th degree polynomial with real coefficients, the number of real, positive roots is never greater than the number of sign changes in the If we write 1/(1+i*) = X, we can rewrite the IRR equation as + F + . . . . . . . + F = 0 change in F’s >1? Unique IRR; accept if > than minimum acceptable rate of return More than 53/8/04 11 n : *)n £ “PURE INVESTMENT” PROJECTS *)n “MIXED INVESTMENT” PROJECTS Nuclear Energy Economics and Policy Analysis The Project Balance, PB(the amount of money committed to a project at time n) An important distinctionProjects for which PB(i 0 for all n < N Projects for which PB(i > 0 for some n 3/8/04 12Nuclear Energy Economics and Policy Analysis 6Investment flexibility as adecision criterionNuclear Energy Economics andPolicy Analysis$196.5$59.37