EECE 450 — Engineering Economics — Formula Sheet

EECE 450 ¡ª Engineering Economics ¡ª 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 ¡Ù 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 ¡Á 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 ¡Ù 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 ) ¡Á (1 + i ¡ä) n = FV(pos CFs, e inv )

i ¡ä = 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 ) = ? X = expected value of X =

¡Æ P( x j ) x j

all j

Var ( X ) = variance of X =

¡Æ P( x j )( x j ?? 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 ?

¡Æ di

i =1

Straight-Line (SL):

Annual charge: dt = (B ¨C S)/N

Book value at end of period t: BVt = B ? t¡Ád

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¨CD)n¨C1

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¨Cd/2)(1¨Cd)n¨C2 for n ¡Ý 2

UCC at end of period n: UCCn = B(1¨Cd/2)(1¨Cd)n¨C1

? 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 ¡Á(1?T)

Before-tax cash flow (BTCF) = I+CCA+taxable income

After-tax cash flow (ATCF) = Net profit + CCA + I

= (Taxable income)¡Á(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 ¨C I ¨C 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¡ä)(1+f)

i = i¡ä + f + (i¡ä)(f)

i= market interest rate; i¡ä= real interest rate

f= inflation rate

Weighted Average Cost of Capital (WACC):

WACC =

D

E

¡Á (1 ? TC )id + ¡Á 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¨CT)

ie= cost of equity

7-Feb-08

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download