## Assignment on Simple Interest

Simple Interest:

Simple Interest, = ∗ ∗ I = Simple Interest

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P = Principal (amount borrowed or lent)

Final Amount, = + = + n = number of interest periods

i = simple interest rate

Compound Interest Factors: i = interest rate

n = number of interest (compounding) periods

P = present sum (initial principal amount)

A = an equal periodic amount (uniform series)

F = a future sum

(F/P): Algebraic Notation: = (1 + ) Factor Notation: = (/ , )

(P/F): Algebraic Notation: = (1 + )− Factor Notation: = (/ , )

(F/A): Algebraic Notation: = { (1+)−1

}

Factor Notation: = (/ , )

(A/F): Algebraic Notation: = {

(1+)−1 }

Factor Notation: = (/ , )

(P/A): Algebraic Notation: = { (1+)−1

(1+) }

Factor Notation: = (/ , )

(A/P): Algebraic Notation: = { (1+)

(1+)−1 }

Factor Notation: = (/ , )

(A/G): Factor Notation: = 1 ± (/ , ) (A/G): Algebraic Notation: ( 1

(1+)−1 )

1= Constant amount = Gradient amount

(P/G): Factor Notation: 1(/ , ) ± (/ , ) (P/G): Algebraic Notation: ( (1+)−−1

2(1+) )

Interest Rate Conversion:

Effective Interest Rate, =

Effective Annual Interest rate, = (1 +

)

− 1

Nominal Interest Rate, = {(1 + ) (

1

)

− 1}

Continuous Interest Factors:

(F/P): Algebraic Notation: = Factor Notation: = [/ , ]

(P/F): Algebraic Notation: = ( 1

)

Factor Notation: = [/ , ]

(F/A): Algebraic Notation: = (−1)

( −1)

Factor Notation: = [/ , ]

(A/F): Algebraic Notation: = (−1)

( −1)

Factor Notation: = [/ , ]

(P/A): Algebraic Notation: = (1− −)

( −1)

Factor Notation: = [/ , ]

(A/P): Algebraic Notation: = ( −1)

(1− −)

Factor Notation: = [/ , ]

Continuous Interest Rate Conversion:

Effective Annual Interest rate, = − 1 Nominal Interest Rate, = ln(1 + )

Bonds:

=

(/

, ) + (/

, ) Where: = ℎ , = , = ,

= # , = # , =

Depreciation Methods or Models:

In General: = −1 − = − ∑ P = Initial Cost = 0; = Book Value at EOY t; = Depreciation for year t; L = Salvage Value; n = Depreciation Life

Straight Line:

= ( − )

= − ∑ = − { ( − )

}

Sum of Years Digits:

= { ( − + 1)

(+1)

2

} ( − )

= ( ( − )

) (

( − + 1)

( + 1) ) ( − ) +

Usage:

= ( − )

= − ∑

Sinking Fund:

= ( − )(/ , )(/ , − 1) = − ∑ = − ( − )(/ , )(/ , )

Declining Balance: = (−1), = (1 − ) −1, = (1 − )

where = 1 − √

, Switch to SL if :

−1−

−(−1) > −1

Double Declining Balance: = (−1), = (1 − ) −1

, = − ∑ = (1 − ) where =

2

Equivalent Annual Cost of Capital Recovery Plus Return:

= ( − )(/ , ) + ()

Taxes:

= − − − G = Gross Income C = Cost of Goods Sold

= Tax Depreciation Allowance I = Interest Paid on Debt Obligations

Tax rates:

Taxable income Tax rate

0-\$50,000 15% over \$0

\$50,000 – \$75,000 \$7,500 + 25% over \$50,000

\$75,000 – \$100,000 \$13,750 + 34% over \$75,000

\$100,000 – \$335,000 \$22,250 + 39% over \$100,000

\$335,000 – \$10 million \$113,900 + 34% over \$335,000

\$10 million – \$15 million \$3,400,000 + 35% over \$10 million

\$15 million – \$18,333,333 \$5,150,000 + 38% over \$15 million

>= \$18,333,333 35%

Effective tax rate =

MACRS Depreciation:

* Year to switch from declining balance to straight line

Measures of Merit:

NPV=∑ =0 (/ , ) or: NPV=∑

(1+) =0

IRR: 0 = ∑ =0 (/ , ) or: 0 = ∑

(1+) =0

Payback Period: 0 = ∑ ′

=0

Linear Break-Even Models:

= ( − − − − ) − ( − − − − ) ( ) = ( − − − − ) ( ) ′ = + + = → = ( − −

′)(1 − ) ≠ → = ( − −

′)(1 − ) + ( − ) Where:

P = after-tax profit per unit of time V = volume of sales per unit of time

s = selling price per unit c = variable cost per unit (raw material, direct labor, direct supplies, etc.)

F = fixed costs per unit of time = book depreciation per unit of time I = debt interest expense per unit of time = tax depreciation per unit of time T = tax rate s = gross income (revenues) per unit of time cV = variable costs per unit of time

= ++

− ( − )

= + ( ++

)

( − )

Cost Comparisons:

Equivalent Annual: Equivalent Annual Cost of Capital Recovery and Return (ECR): = ( − )(/ , ) + + )

Capitalized Cost () = ⁄ ;Where A=equivalent annual amount, i=interest rate

Present Worth () = (/ , ); ℎ = (when infinite service life and lifetimes are different), if lifetime the same, discount all cash flows for each alternative to present

Benefit-Cost Analysis: Project acceptable if: B – C ≥ 0 or B/C ≥ 1

Inflation:

= ′ + + ′ ; ℎ : , ′: , =

′ = ( − )

(1 + )

\$ → () \$ → ( ′)

Price Index: = −−1

−1 100% ; where cost index would be given

: ℎ ℎ &

DPMO (Defects Per Million Opportunities):

=

[ ] 1,000,000

Process Capability:

Measure of Potential Capability: = −

6 ; Where UTL=Upper Tolerance Limit, LTL=Lower Tolerance Limit,

=Standard deviation

Measure of Actual Capability:  

 

 

 3

X-UTL ,

3

LTLX min=C

pk ; Where �̅� = mean (average of samples)

Cp and Cpk ≥ 1 in order for the process to be capable If Cp≠ Cpk  process not centered

Kanban:

C

SDL k

)(1  ; Where:

k=Number of Kanban card sets (a set is a card) D=Average number of units demanded over some time period L=lead time to replenish an order S=Safety stock expressed as a percentage of demand during lead time C=Container size (in number of units per container)