16.4: Formula Sheet

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Page ID: 22169

Jean-Paul Olivier
Red River College of Applied Arts, Science, & Technology

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Part 1: Mathematics Fundamentals

(2.1) Percentage Conversion \[\% = \text{ dec} \times 100 \nonumber \]
(2.2) Rate, Portion, Base \[ \text{Rate } = \frac{ \text{Portion}}{ \text{Base}} \nonumber \]
(3.1) Percent Change: \[ \Delta \% = \frac{\text{New − Old}}{\text{Old}} \times 100 \nonumber \]
(3.2) Rate Of Change Over Time \[\text{RoC} = \left( \sqrt[n]{\frac{ \text{New}}{\text{Old}} − 1} \right) \times 100 \nonumber \]
(3.3) Simple Average \[ \text{SAvg} = \frac{ \Sigma x}{n} \nonumber \]
(3.4) Weighted Average \[ \text{WAvg} = \frac{\Sigma wx}{\Sigma w} \nonumber \]
• (3.5) Geometric Average \[\text{GAvg} = \left[ \sqrt[n]{ \left( 1 + \Delta \%_1 \right) \times \left(1 + \Delta \%_2 \right) \times \cdots \times \left(1 + \Delta \%_n \right) − 1} \right] \times 100 \nonumber \]

Part 2: Business Applications

(4.1) Salary and Hourly Gross Earnings \[ \text{GE } = \text{ Regular Earnings} + \text{Overtime Earnings} + \text{Holiday Earnings} + \text{Statutory Holiday Worked Earnings} \nonumber \]
(4.2) Annual Income Tax \[ \text{Income Tax } = \Sigma \text{(Eligible Income In Tax Bracket × Tax Bracket Rate)} \nonumber \]
(4.3) Index Numbers \[ \text{Index Number } = \frac{ \text{Chosen quantity}}{\text{ Base quantity}} \times \text{ Base value} \nonumber \]
(4.4) Purchasing Power Of A Dollar \[ \text{PPD } = \frac{\$ 1}{\text{CPI}/100} \nonumber \]
(4.5) Real Income \[ \text{RI } = \frac{ \text{Nominal Income}}{\text{CPI}/100} \nonumber \]
(5.1) Unit Variable Cost \[ \text{VC } = \frac{\text{TVC}}{n} \nonumber \]
(5.2) Net Income Using A Total Revenue And Cost Approach \[ \text{NI } = \text{n}(\text{S}) − (\text{TFC} + \text{n}(\text{VC})) \nonumber \]
(5.3) Unit Contribution Margin \[ \text{CM } = \text{ S} – \text{VC} \nonumber \]
(5.4) Net Income Using Total Contribution Margin Approach \[ \text{NI } = \text{n}(\text{CM}) – \text{TFC} \nonumber \]
(5.5) Contribution Rate If Unit Information Known \[ \text{CR } = \frac{\text{CM}}{\text{S}} \times 100 \nonumber \]
(5.6) Contribution Rate If Aggregate Information Known \[ \text{CR } = \frac{\text{ TR − TVC}}{\text{TR}} \times 100 \nonumber \]
(5.7) Unit Break-even \[n = \frac{\text{TFC}}{\text{CM}} \nonumber \]
(5.8) Dollar Break-even \[ \text{TR} = \frac{\text{TFC}}{\text{CR}} \nonumber \]
(6.1) Single Discount \[ \text{N} = \text{L} \times (1 - \text{d}) \nonumber \]
(6.2a & 6.2b) Discount Amount \[ \text{D} \$ = \text{L} \times \text{d} \text{ or } \text{D} \$ = \text{L} – \text{N} \nonumber \]
(6.3) Multiple Discounts \[ \text{N} = \text{L} \times \left( 1− \text{d}_1 \right) \times \left( 1− \text{d}_2 \right) \times \cdots \times \left( 1− \text{d}_n \right) \nonumber \]
(6.4) Single Equivalent Discount \[\text{d } = 1 − \left( 1 − \text{d}_1 \right) \times \left( 1 − \text{d}_2 \right) \times \cdots \times \left( 1 − \text{d}_n \right) \nonumber \]
(6.5) The Selling Price Of A Product \[\text{S} = \text{C} + \text{E} + \text{P} \nonumber \]
(6.6) Markup Amount \[\text{M} \$ = \text{E} + \text{P} \nonumber \]
(6.7) The Selling Price Of A Product Using Markup Amount \[ \text{S} = \text{C} + \text{M} \$ \nonumber \]
(6.8) Markup On Cost Percentage \[ \text{MoC} \% = \text{M} \$ \text{C} \times 100 \nonumber \]
(6.9) Markup On Selling Price Percentage \[ \text{MoS} \% = \frac{\text{M} \$}{\text{S}} \times 100 \nonumber \]
(6.10) The Sale Price Of A Product \[ \text{S}_{\text{on sale}} = \text{S} \times \left( 1 − \text{d} \right) \nonumber \]
(6.11a & 6.11b) Markdown Amount \[\text{D}$ = \text{S} \times \text{d} \text{ or } \text{D} \$ = \text{ S} − \text{S}_{\text{on sale}} \nonumber \]
(6.12) Markdown Percentage \[ \text{d} = \frac{\text{D} \$}{\text{S}} \times 100 \nonumber \]
(6.13) Maintained Markup \[ \text{MM } = \frac{ \text{M} \$ \left( \text{n}_1 \right) + \left( \text{M} \$ − \text{D} \$ \right) \left( \text{n}_2 \right)}{\text{n}_1 + \text{n}_2} \nonumber \]
(7.1) Selling Price Including Tax \[ \text{S}_{\text{tax}} = \text{S} + (\text{S} \times \text{Rate}) \nonumber \]
(7.2) GST/HST Remittance \[ \text{Remit } = \text{ Tax Collected} − \text{Tax Paid} \nonumber \]
(7.3) Property Taxes \[ \text{Property Tax } = \Sigma ( \text{AV} \times \text{PTR}) \nonumber \]
(7.4) Currency Exchange \[ \text{Desired Currency } = \text{ Exchange Rate} \times \text{Current Currency} \nonumber \]

Part 3: Single Payment Financial Applications

(8.1) Simple Interest \[\text{I }= \text{ Prt} \nonumber \]
(8.2) Simple Interest For Single Payments \[ \text{S }= \text{ P(1+rt)} \nonumber \]
(8.3) Interest Amount For Single Payments \[ \text{I } = \text{ S} − \text{P} \nonumber \]
(9.1) Periodic Interest Rate \[ \text{i } = \frac{ \text{IY}}{\text{CY}} \nonumber \]
(9.2) Number of Compound Periods For Single Payments \[ \text{N } = \text{ CY} \times \text{Years} \nonumber \]
(9.3) Compound Interest For Single Payments \[ \text{FV } = \text{ PV} (1 + \text{i})^{\text{N}} \nonumber \]
(9.4) Interest Rate Conversion \[ \text{i}_{\text{New}} = \left( 1 + \text{i}_{\text{Old}} \right)^{\frac{\text{CY}_{\text{Old}}}{\text{CY}_{\text{New}}}} − 1 \nonumber \]
(10.1) Periodic Interest Amount \[ \text{I } = \text{ PV} \times \text{i} \nonumber \]
(10.2) Purchasing Power Of A Dollar (Compound Interest Method) \[ \text{PPD } = \frac{$1}{(1 + \text{i})^{ \text{N}}} \times 100 \nonumber \]

Part 4: Annuity Payments Financial Applications

(11.1) Number Of Annuity Payments \[\text{N } = \text{ PY} \times \text{Years} \nonumber \]
(11.2) Ordinary Annuity Future Value \[\text{FV}_{\text{ORD}} = \text{ PMT} \left[ \frac{ \left[ (1 + \text{i})^{\frac{\text{CY}}{\text{PY}}} \right]^\text{N} − 1}{(1 + \text{i} )^{\frac{\text{CY}}{\text{PY}}} − 1} \right] \nonumber \]
(11.3) Annuity Due Future Value \[\text{FV}_{\text{DUE}} = \text{ PMT} \left[ \frac{ \left[ (1 + \text{i})^{\frac{\text{CY}}{\text{PY}}} \right]^\text{N} − 1}{(1 + \text{i} )^{\frac{\text{CY}}{\text{PY}}} − 1} \right] \times (1 + \text{i} )^{ \frac{\text{CY}}{\text{PY}}} \nonumber \]
(11.4) Ordinary Annuity Present Value \[\text{FV}_{\text{ORD}} = \text{ PMT} \left[ \frac{ 1 - \left[ \frac{1}{(1 + \text{i})^{\frac{\text{CY}}{\text{PY}}}} \right]^\text{N}}{(1 + \text{i} )^{\frac{\text{CY}}{\text{PY}}} − 1} \right] \nonumber \]
(11.5) Annuity Due Present Value \[\text{FV}_{\text{DUE}} = \text{ PMT} \left[ \frac{ 1 - \left[ \frac{1}{(1 + \text{i})^{\frac{\text{CY}}{\text{PY}}}} \right]^\text{N}}{(1 + \text{i} )^{\frac{\text{CY}}{\text{PY}}} − 1} \right] \times (1 + \text{i} )^{ \frac{\text{CY}}{\text{PY}}} \nonumber \]
(12.1) Future Value Of A Constant Growth Ordinary Annuity \[\text{FV}_{\text{ORD}} = \text{ PMT} (1 + \Delta \%)^{\text{N}−1} \left[ \frac{ \left[ \frac{ (1+ \text{i})^{\frac{\text{CY}}{\text{PY}}}}{1 + \Delta \%} \right]^{\text{N}} - 1}{ \frac{ (1+ \text{i})^{\frac{\text{CY}}{\text{PY}}} }{1 + \Delta \%} - 1} \right] \nonumber \]
(12.2) Future Value Of A Constant Growth Annuity Due \[\text{FV}_{\text{DUE}} = \text{ PMT} (1 + \Delta \%)^{\text{N}−1} \left[ \frac{ \left[ \frac{ (1+ \text{i})^{\frac{\text{CY}}{\text{PY}}}}{1 + \Delta \%} \right]^{\text{N}} - 1}{ \frac{ (1+ \text{i})^{\frac{\text{CY}}{\text{PY}}} }{1 + \Delta \%} - 1} \right] \times (1 + \text{i} )^{ \frac{\text{CY}}{\text{PY}}} \nonumber \]
(12.3) Present Value Of A Constant Growth Ordinary Annuity \[\text{PV}_{\text{ORD}} = \frac{\text{PMT}}{1 + \Delta \%} \left[ \frac{1 - \left[ \frac{1 + \Delta \%}{ (1+ \text{i})^{\frac{\text{CY}}{\text{PY}}}} \right]^{\text{N}}}{ \frac{(1+ \text{i})^{\frac{\text{CY}}{\text{PY}}}}{1 + \Delta \%} - 1} \right] \nonumber \]
(12.4) Present Value Of A Constant Growth Annuity Due \[\text{PV}_{\text{DUE}} = \frac{\text{PMT}}{1 + \Delta \%} \left[ \frac{1 - \left[ \frac{1 + \Delta \%}{ (1+ \text{i})^{\frac{\text{CY}}{\text{PY}}}} \right]^{\text{N}}}{ \frac{(1+ \text{i})^{\frac{\text{CY}}{\text{PY}}}}{1 + \Delta \%} - 1} \right] \times (1 + \text{i} )^{ \frac{\text{CY}}{\text{PY}}} \nonumber \]
(12.5) Ordinary Perpetuity Present Value \[ \text{PV}_{\text{ORD}} = \frac{\text{PMT}}{(1 + \text{i})^{\frac{\text{CY}}{\text{PY}}} − 1} \nonumber \]
(12.6) Perpetuity Due Present Value \[ \text{PV}_{\text{DUE}} = \text{PMT} \left( \frac{1}{(1+ \text{i})^{\frac{\text{CY}}{\text{PY}}} − 1} + 1 \right) \nonumber \]

Part 5: Amortization & Special Financial Concepts

(13.1) Interest Portion Of An Ordinary Single Payment \[ \text{INT } = \text{ BAL} \times ((1+ \text{i})^{\frac{\text{CY}}{\text{PY}}} − 1) \nonumber \]
(13.2) Principal Portion Of A Single Payment \[ \text{PRN } = \text{ PMT} − \text{INT} \nonumber \]
(13.3) Principal Portion For A Series Of Payments \[ \text{PRN } = \text{BAL}_{\text{P}1} − \text{BAL}_{\text{P}2} \nonumber \]
(13.4) Interest Portion For A Series Of Payments \[ \text{INT } = \text{ N} \times \text{PMT} − \text{PRN} \nonumber \]
(13.5) Interest Portion Of A Due Single Payment \[ \text{INT}_{\text{DUE}} = ( \text{BAL} − \text{PMT}) \times ((1+ \text{i})^{\frac{\text{CY}}{\text{PY}}} − 1) \nonumber \]
(14.1) The Cash Price For Any Bond \[ \text{Cash Price } = \text{ PRI} + \text{AI} \nonumber \]
(14.2) Bond Coupon Annuity Payment Amount \[ \text{PMT}_{\text{BOND}} = \text{ Face Value} \times \frac{\text{CPN}}{\text{CY}} \nonumber \]
(14.3) Bond Price On An Interest Payment Date \[ \text{Date Price } = \frac{\text{FV}}{(1+\text{i})^{\text{N}}} + \text{PMT}_{\text{BOND}} \left[ \frac{1-\frac{1}{[1+\text{i}]^\text{N}}}{\text{i}} \right] \nonumber \]
(14.4) Bond Premium or Discount \[\text{Premium or Discount } = \text{ PRI} − \text{Face Value} \nonumber \]
(14.5) Bond Cash Price On A Non-Interest Payment Date \[ \text{Cash Price } = (\text{Date Price})(1 + \text{i})^{\text{t}} \nonumber \]
(14.6) Accrued Interest On A Non-Interest Payment Date \[\text{AI} = \text{PMT}_{\text{BOND}} \times \text{t} \nonumber \]
(14.7) Interest Portion Of A Sinking Fund Single Payment Due \[ \text{INT } = ( \text{BAL} + \text{PMT}) \times ((1 + \text{i})^{\frac{\text{CY}}{\text{PY}}} − 1) \nonumber \]
(14.8) The Annual Cost Of The Bond Debt \[ \text{ACD} = (\text{Face Value} \times \text{CPN}) + (\text{PMT} \times \text{PY}) \nonumber \]
(14.9) The Book Value Of The Bond Debt \[ \text{BVD} = \text{Bonds Outstanding} − \text{BAL} \nonumber \]
(15.1) Net Present Value \[ \text{NPV } = (\text{Present Value Of All Future Cash Flows}) − (\text{Initial Investment}) \nonumber \]
(15.2) Net Present Value Ratio \[\text{NPV}_{\text{RATIO}} = \frac{\text{NPV}}{\text{CFo}} \nonumber \]