ENGR 300 Dept. of Computer Science and Engineering University of Bridgeport, CT 06601

NET PRESENT VALUE - NPV Measures inflows vs. outflows Today’s dollars Cradle to Grave

NET PRESENT VALUE - NPV Not only accounts for the Costs of Inflows and Outflows, but for their timing Sales Revenues Loan Payments Development Costs Ramp Up and Production Costs Marketing and Support Costs Disposal Costs

SENSITIVITY ANALYSIS Answers “What If?” Questions Helps in Making Project Tradeoffs Critical Factor Cost is a Critical Factor in Design ? ? ? ?

SENSITIVITY ANALYSIS Project Parameters can be Varied Development Time Project Loading Interest Rates Sales Price Product Quality

QUALITATIVE ANALYSIS Complex Factors Risks External Factors

DEPRECIATION Only a portion of the cost of an asset can be deducted for tax purposes in one year Tangible Assets decrease in value over time Cars Equipment Buildings

DEPRECIATION METHODS An Asset has an Initial and a Salvage Value at the start and end of its service life The Book Value is the remaining undepreciated value of the asset Straight Line Method (equal amounts) Accelerated Cost Recovery System Modified Accelerated Cost Recovery System

DECISIONS CAN BE INFLUENCED BY THE TIME VALUE OF MONEY One dollar today Will be worth more in the future $ + time = $$$ Present Future

TIMING OF INFLOWS AND OUTFLOWS Present Future cash outflows Future cash inflows time Cash Flow Diagram graphically shows relationships

BASIC TERMINOLOGY P= Present value (NPV in Today's Dollars) F= Future value (Tomorrow’s Dollars) n = Number of compounding periods between “present” and “future” A = uniform Amount received or paid out each compounding period

INTEREST RATE The reward that investors demand for accepting delayed payment Sometimes referred to as the Discount Rate n is the number of periods per year Must convert the yearly percentage rate to its decimal equivalent rate

COMPOUNDING OF INTEREST ANNUAL PERCENTAGE RATE (APR) INCREASES WITH SHORTER PERIODS OF COMPOUNDING 12% Yearly = 12% APR 3% Quarterly = 12.55% APR 1% Monthly = 12.68% APR Continuous = 12.75% APR

COMPOUNDING FORMULAS CONTINUOUS FIXED PERIODS

PRESENT vs. FUTURE VALUE Dollars today are worth more than the same amount of dollars in the future $1000 today will grow to $ in 10 years at 12% compounded monthly

PRESENT vs. FUTURE VALUE Find Present given the Future Value $120 one year from now is worth $ today n=12 periods or monthly Yearly interest rate is 6%, per month is.005

PRESENT vs. FUTURE VALUE How Many Periods? How many years does it take to double your money if the APR=9% SOLVING LOG FUNCTION WORKS TOO

PRESENT vs. FUTURE VALUE What interest rate is needed? What interest rate is needed to make $200 grow to $1000 in ten years, if interest is paid yearly? SOLVING

PRESENT VALUE(P) OF A SERIES OF AMOUNTS n = number of payments of amount A i=interest rate per period (decimal) A= amount of each payment

PRESENT VALUE OF EQUAL PAYMENTS $10 monthly payments for one year interest rate is 6% per year =.005 per month Present Value is greater than one single payment of $120 after a year (in that case, P was $113.03)

AMOUNT OF A LOAN PAYMENT P=$100,000 i=9% per year =.0075 per month n=360 monthly payments Note in 30 years, $289,663 will be paid in payments

MATHEMATICAL BASIS A SERIES OF PAYMENTS IS BROKEN DOWN INTO SUMS OF INFINITE STRINGS OF PAYMENTS etc. P 1 =A/i A etc. A A P=P 1 -P 2 Subtract P 2 =A/(i(1+i) n )

PRESENT VALUE OF INCREASING AMOUNTS A A+B A+2B A+3B Present A=$15 & B=$10 4A+6B=$120 i=.06/4=.015 n=4 (quarterly) Future

ECONOMIC COMPARISON Decisions often include comparisons of economic costs/benefits of alternative actions Inflows/Outflows may occur at several different times Time-value of money must be considered

ALTERNATIVES WITH EQUAL LIVES For each alternative, compute the Net Present Value (NPV) Compute P for all inflows Compute P for all outflows NPV =  P(inflows) -  P(outflows) Alternative with highest NPV is the best choice from an economic viewpoint

ALTERNATIVES WITH UNEQUAL LIVES For each alternative, compute the equivalent uniform cost per period (EUC/P) Assume identical replacement at end of life Compute A for all inflows Compute A for all outflows EUC/P =  A(outflows) -  A(inflows) Alternative with lowest EUC/P is the best choice from an economic viewpoint

DEALING WITH RISK AND UNCERTAINTY Use Expected Value (EV) for inflows and outflows with estimated uncertainties EV = (p 1 )(V 1 ) + (p 2 )(V 2 ) (p n )(V n ) p n is the probability that a value will be V n where p 1 + p p n = 1 Calculate NPV or EUC/P based on expected values

EXPECTED VALUE Saturdays, the following probabilities exist.35 won’t study at all.15 will study for 4 hours.20 will study for 2 hours.30 will study for 1 hour 1.3 IS THE EXPECTED NUMBER OF HOURS OF STUDY ON A TYPICAL SATURDAY

WHAT IF? What If questions can be supported by doing a sensitivity analysis. – Take one variable at a time, holding others fixed, make small changes in that variable observe effect on NPV or on EUC/P – Spreadsheet program is useful for this purpose and doing time value of money calculations