EECE 450 — Engineering Economics — Formula Sheet
Cost Indexes:
Ordinary Geometric Gradient Annuity:
Cost at time A Index value at time A
=
Cost at time B Index value at time B
1 − (1 + g ) n (1 + i ) − n 
P = A1 
; i ≠ g
i−g


Power sizing:
Cost of asset A  Size (capacity) of asset A 
=

Cost of asset B  Size (capacity) of asset B 
x = power - sizing exponent
x
Learning Curve:
TN = Tinitial × N b
log(learning curve rate)
log 2
TN = time to make Nth unit
Tinitial = time to make first unit
b=
N = number of finished units
b = learning curve exponent
Simple Interest:
P=
nA1
;i = g
(1 + i )
 (1 + i ) n − (1 + g ) n 
F = A1 
; i ≠ g
i−g


F = nA1 (1 + i ) n −1 ; i = g
A1 = payment in first period (end)
g = periodic rate of growth
P, F , i, n as above for compound interest
Simple Annuity Due:
1 − (1 + i ) − n 
P = A
 (1 + i )
i


 (1 + i ) n − 1 
F = A
 (1 + i )
i


A = cash amount (beginning of period)
P, F , i, n as above for compound interest
Interest earned on amount P : I = Pin
Maturity value : F = P (1 + in)
i = interest rate per time period
n = number of time periods
Nominal, Periodic, Effective Interest Rates:
Compound Interest:
i=
F = P(1 + i ) n
F = future value
P = present value
i = periodic interest rate
n = number of periods
Ordinary Simple Annuity:
1 − (1 + i ) − n 
P = A

i


 (1 + i ) n − 1 
F = A

i


A = periodic payment (end of period)
P, F , i, n as above for compound interest
Ordinary Arithmetic Gradient Annuity:
r
m
(
)m
(1 + ieff ) = 1 + mr
r = nominal interest rate per year
m = number of compounding periods per year
ieff = effective interest rate (compounded annually)
i = periodic interest rate
Equivalent Interest Rates:
(1 + i p ) p = (1 + ic ) c
i p = interest rate for payment period
p = number of payment periods per year
ic = interest rate for compounding period
c = number of compounding periods per year
Ordinary General Annuity:
1 − (1 + i p ) − n 

P = A
ip


1

n
Aeq = G  −

n
 i (1 + i ) − 1 
 (1 + i ) n − in − 1 
P = G

2
n
 i (1 + i )

Aeq = equivalent periodic payment
 (1 + i p ) n − 1 

F = A
ip


i p = interest rate for payment period
G = gradient amount (periodic increment)
n = number of payment periods
P, i, n as above for compound interest
P, F , A as above for annuities
Prepared by Ron Mackinnon, University of British Columbia, © 2008.
7-Feb-08
Perpetual Annuities:
Ordinary : P =
A
i
A
A
(1 + i ) = + A
i
i
A
Geometric Growth : P =
;i > g
i−g
P, A, i, g as above for annuities
Due : P =
Investment Criteria:
CF1
CF2
CFn
+
+ ... +
1
2
(1 + r ) (1 + r )
(1 + r ) n
NPV = net present value
NPV = CF0 +
NFV = CF0 (1 + r ) n + CF1 (1 + r ) n −1 + ... + CFn
NFV = net future value
EACF = equivalent annual cash flow =
NPV
 1−(1+ r ) − n 
 r 
CF j = cash flow at time j
n = lifetime of investment
r = MARR = minimum acceptable rate of return
CF1
CF2
CFn
0 = CF0 +
+
+ ... +
1
2
(1 + i ) (1 + i )
(1 + i ) n
i = IRR = internal rate of return
PV(neg CFs, e fin ) × (1 + i ′) n = FV(pos CFs, e inv )
i ′ = MIRR = modified internal rate of return
e fin = financing rate of return
e inv = reinvestment rate of return
Benefit - cost ratio, BCR =
PV(positive cash flows)
PV(negative cash flows)
Probability:
E( X ) = Weighted average =
w1S1 + L + wk S k
w1 + L + wk
wi = weight for Scenario i
Si = value of X for Scenario i
E( X ) = µ X = expected value of X =
∑ P( x j ) x j
all j
Var ( X ) = variance of X =
∑ P( x j )( x j −µ X ) 2
all j
P ( x j ) = Probability( X = x j )
Depreciation:
B= initial (purchase) value or cost basis
S= estimated salvage value after depreciable life
dt= depreciation charge in year t
N= number of years in depreciable life
t
Book value at end of period t: BVt = B −
∑ di
i =1
Straight-Line (SL):
Annual charge: dt = (B – S)/N
Book value at end of period t: BVt = B − t×d
Prepared by Ron Mackinnon, University of British Columbia, © 2008.
Sum-of-Years’-Digits (SOYD):
SOYD = N(N+1)/2
Annual charge: dt = (B − S)(N − t + 1)/SOYD
Declining balance (DB):
D= proportion of start of period BV that is depreciated
Annual charge: dn = BD(1–D)n–1
Book value at end of period n: BVn = B(1-D)n
Capital Cost Allowance (CCA):
d= CCA rate
UCCn= Undepreciated capital cost at end of period n
Annual charge: CCA1 = B(d/2) for n = 1;
CCAn = Bd(1–d/2)(1–d)n–2 for n ≥ 2
UCC at end of period n: UCCn = B(1–d/2)(1–d)n–1
 BdTC  1 + i 2 
PV(CCA tax shields gained) = 


 i + d  1+ i 
 SdTC   1 
PV(CCA tax shields lost) = 


 i + d   (1 + i )N 
TC = firm' s tax rate; i = discount rate
Investment Project Cash Flows:
Taxable income = OR−OC−CCA−I
Net profit = taxable income ×(1−T)
Before-tax cash flow (BTCF) = I+CCA+taxable income
After-tax cash flow (ATCF) = Net profit + CCA + I
= (Taxable income)×(1−T) + CCA + I
= (BTCF − I − CCA)(1 −T) + CCA + I
= (OR − OC)(1 −T) + I(T) + CCA(T)
Net cash flow from operations
= ATCF – I – DIV
= (OR − OC)(1−T) + I(T) + CCA(T) − I − DIV
= (OR − OC − I)(1−T) + CCA(T) − DIV
= Net profit + CCA − DIV
OR= operating revenue; OC= operating cost
I= interest expense; DIV = dividends; T= tax rate
Net cash flow = Net cash flow from operations
+ New equity issued + New debt issued
+ Proceeds from asset disposal − Repurchase of equity
− Repayment of debt (principal) − Purchase of assets
 dT 1 + i 2 
Net capital investment = B 1 − C

 i + d 1+ i 
 dT   1 
Net salvage value = S 1 − C  

 i + d   (1 + i )N 
Inflation:
(1+i) = (1+i′)(1+f)
i = i′ + f + (i′)(f)
i= market interest rate; i′= real interest rate
f= inflation rate
Weighted Average Cost of Capital (WACC):
WACC =
D
E
× (1 − TC )id + × ie
V
V
V = D+E
D= market value of debt; E= market value of equity
V= market value of firm
id= cost of (rate of return on) debt
after-tax cost of debt: idt = id(1–T)
ie= cost of equity
7-Feb-08