-------------------------- ENGR 300
Dept. of Computer Science and EngineeringUniversity of Bridgeport, CT 06601
NET PRESENT VALUE - NPV
Measures inflows vs. outflowsToday’s dollarsCradle 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
Cost is a Critical FactorCritical Factor in Design
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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
iinPeriodYearly
PerYear
% .01
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
APR i
APR ePeriod
n
i
per year
Yearly
( ) _
% .
1 1
101CONTINUOUS
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 $3300.39 in 10 years at 12% compounded monthly
PF
iPeriodnTotal
( )1
PRESENT vs. FUTURE VALUE Find Present given the Future Value
$120 one year from now is worth $113.03 today n=12 periods or monthly Yearly interest rate is 6%, per month is .005
P
$120
( . )$113.
1 0050312
PRESENT vs. FUTURE VALUE How Many Periods?
How many years does it take to double your money if the APR=9%
PP
n years
n
21 09
2109
8 043
( . )ln
ln( . ).
SOLVING
LOG FUNCTION WORKS TOO
PRESENT vs. FUTURE VALUEWhat interest rate is needed?
What interest rate is needed to make $200 grow to $1000 in ten years, if interest is paid yearly?
$200$1000( )
. .ln( )
1
1 1746 17 46%
10
1000200
10
i
i e
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
PA i
i
n
( ( ) )1 1
PRESENT VALUE OF EQUAL PAYMENTS
$10 monthly payments for one year interest rate is 6% per year = .005 per month
P
$10 ( ( . ) )
.$116.
1 1 005005
0912
Present Value is greater than one single paymentof $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
A
$100000 .( ( . ) )
$804.0075
1 1007562360
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.P1=A/iA
etc.A
A
P=P1-P2
Subtract
P2=A/(i(1+i) n)
PRESENT VALUEOF INCREASING AMOUNTS
PA i
i
B i n i
i
P
nk
nk n
k
k
( ( ) ) ( ( ) ( ) ( ) )
$15 ( ( . ) ).
$10 ( ( . ) ( . ) )
.$114.
1 1 1 1 1
1 1015015
1015 3 1015
01591
1
1
41
34
A A+BA+2B A+3B
Present
A=$15 & B=$104A+6B=$120i=.06/4=.015n=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 lifeCompute A for all inflowsCompute A for all outflowsEUC/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 uncertaintiesEV = (p1)(V1) + (p2 )(V2) + .......+ (pn)(Vn)
pn is the probability that a value will be Vn
where p1 + p2 +.......+ pn = 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
EV . . . . .35 0 15 4 20 2 30 1 131.3 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