16.4: Hoja de Fórmula
- Page ID
- 110716
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Parte 1: Fundamentos de Matemáticas
- (2.1) Conversión de Porcentaje\[\% = \text{ dec} \times 100 \nonumber \]
- (2.2) Tasa, Porción, Base\[ \text{Rate } = \frac{ \text{Portion}}{ \text{Base}} \nonumber \]
- (3.1) Cambio porcentual:\[ \Delta \% = \frac{\text{New − Old}}{\text{Old}} \times 100 \nonumber \]
- (3.2) Tasa de cambio a lo largo del tiempo\[\text{RoC} = \left( \sqrt[n]{\frac{ \text{New}}{\text{Old}} − 1} \right) \times 100 \nonumber \]
- (3.3) Promedio simple\[ \text{SAvg} = \frac{ \Sigma x}{n} \nonumber \]
- (3.4) Promedio ponderado\[ \text{WAvg} = \frac{\Sigma wx}{\Sigma w} \nonumber \]
- • (3.5) Promedio Geométrico\[\text{GAvg} = \left[ \sqrt[n]{ \left( 1 + \Delta \%_1 \right) \times \left(1 + \Delta \%_2 \right) \times \cdots \times \left(1 + \Delta \%_n \right) − 1} \right] \times 100 \nonumber \]
Parte 2: Aplicaciones de negocios
- (4.1) Salario y Ganancias Brutas Horarias\[ \text{GE } = \text{ Regular Earnings} + \text{Overtime Earnings} + \text{Holiday Earnings} + \text{Statutory Holiday Worked Earnings} \nonumber \]
- (4.2) Impuesto Anual sobre la Renta\[ \text{Income Tax } = \Sigma \text{(Eligible Income In Tax Bracket × Tax Bracket Rate)} \nonumber \]
- (4.3) Números de índice\[ \text{Index Number } = \frac{ \text{Chosen quantity}}{\text{ Base quantity}} \times \text{ Base value} \nonumber \]
- (4.4) Poder de compra de un dólar\[ \text{PPD } = \frac{\$ 1}{\text{CPI}/100} \nonumber \]
- (4.5) Ingresos reales\[ \text{RI } = \frac{ \text{Nominal Income}}{\text{CPI}/100} \nonumber \]
- (5.1) Costo Variable Unitario\[ \text{VC } = \frac{\text{TVC}}{n} \nonumber \]
- (5.2) Ingresos netos mediante un enfoque de ingresos y costos totales\[ \text{NI } = \text{n}(\text{S}) − (\text{TFC} + \text{n}(\text{VC})) \nonumber \]
- (5.3) Margen de Contribución Unitaria\[ \text{CM } = \text{ S} – \text{VC} \nonumber \]
- (5.4) Ingresos netos utilizando el enfoque de margen de contribución total\[ \text{NI } = \text{n}(\text{CM}) – \text{TFC} \nonumber \]
- (5.5) Tasa de contribución si se conoce la información de la unidad\[ \text{CR } = \frac{\text{CM}}{\text{S}} \times 100 \nonumber \]
- (5.6) Tasa de contribución si se conoce información agregada\[ \text{CR } = \frac{\text{ TR − TVC}}{\text{TR}} \times 100 \nonumber \]
- (5.7) Unidad de equilibrio\[n = \frac{\text{TFC}}{\text{CM}} \nonumber \]
- (5.8) Romper par del dólar\[ \text{TR} = \frac{\text{TFC}}{\text{CR}} \nonumber \]
- (6.1) Descuento Individual\[ \text{N} = \text{L} \times (1 - \text{d}) \nonumber \]
- (6.2a y 6.2b) Importe de Descuento\[ \text{D} \$ = \text{L} \times \text{d} \text{ or } \text{D} \$ = \text{L} – \text{N} \nonumber \]
- (6.3) Descuentos Múltiples\[ \text{N} = \text{L} \times \left( 1− \text{d}_1 \right) \times \left( 1− \text{d}_2 \right) \times \cdots \times \left( 1− \text{d}_n \right) \nonumber \]
- (6.4) Descuento Equivalente Individual\[\text{d } = 1 − \left( 1 − \text{d}_1 \right) \times \left( 1 − \text{d}_2 \right) \times \cdots \times \left( 1 − \text{d}_n \right) \nonumber \]
- (6.5) El precio de venta de un producto\[\text{S} = \text{C} + \text{E} + \text{P} \nonumber \]
- (6.6) Importe de Marcado\[\text{M} \$ = \text{E} + \text{P} \nonumber \]
- (6.7) El Precio de Venta de un Producto Usando Monto de Marcado\[ \text{S} = \text{C} + \text{M} \$ \nonumber \]
- (6.8) Porcentaje de Marcado en Costo\[ \text{MoC} \% = \text{M} \$ \text{C} \times 100 \nonumber \]
- (6.9) Marcado en el porcentaje de precio de venta\[ \text{MoS} \% = \frac{\text{M} \$}{\text{S}} \times 100 \nonumber \]
- (6.10) El Precio De Venta De Un Producto\[ \text{S}_{\text{on sale}} = \text{S} \times \left( 1 − \text{d} \right) \nonumber \]
- (6.11a y 6.11b) Importe de rebaja\[\text{D}$ = \text{S} \times \text{d} \text{ or } \text{D} \$ = \text{ S} − \text{S}_{\text{on sale}} \nonumber \]
- (6.12) Porcentaje de rebaja\[ \text{d} = \frac{\text{D} \$}{\text{S}} \times 100 \nonumber \]
- (6.13) Marcado mantenido\[ \text{MM } = \frac{ \text{M} \$ \left( \text{n}_1 \right) + \left( \text{M} \$ − \text{D} \$ \right) \left( \text{n}_2 \right)}{\text{n}_1 + \text{n}_2} \nonumber \]
- (7.1) Precio de Venta con Impuestos Incluidos\[ \text{S}_{\text{tax}} = \text{S} + (\text{S} \times \text{Rate}) \nonumber \]
- (7.2) Remesa GST/HST\[ \text{Remit } = \text{ Tax Collected} − \text{Tax Paid} \nonumber \]
- (7.3) Impuestos a la propiedad\[ \text{Property Tax } = \Sigma ( \text{AV} \times \text{PTR}) \nonumber \]
- (7.4) Cambio de divisas\[ \text{Desired Currency } = \text{ Exchange Rate} \times \text{Current Currency} \nonumber \]
Parte 3: Aplicaciones Financieras de Pago Único
- (8.1) Interés Simple\[\text{I }= \text{ Prt} \nonumber \]
- (8.2) Interés simple para pagos únicos\[ \text{S }= \text{ P(1+rt)} \nonumber \]
- (8.3) Monto de Intereses para Pagos Únicos\[ \text{I } = \text{ S} − \text{P} \nonumber \]
- (9.1) Tasa de Interés Periódica\[ \text{i } = \frac{ \text{IY}}{\text{CY}} \nonumber \]
- (9.2) Número de Períodos Compuestos para Pagos Únicos\[ \text{N } = \text{ CY} \times \text{Years} \nonumber \]
- (9.3) Intereses compuestos para pagos únicos\[ \text{FV } = \text{ PV} (1 + \text{i})^{\text{N}} \nonumber \]
- (9.4) Conversión de tasa de interés\[ \text{i}_{\text{New}} = \left( 1 + \text{i}_{\text{Old}} \right)^{\frac{\text{CY}_{\text{Old}}}{\text{CY}_{\text{New}}}} − 1 \nonumber \]
- (10.1) Importe de interés periódico\[ \text{I } = \text{ PV} \times \text{i} \nonumber \]
- (10.2) Poder de compra de un dólar (método de interés compuesto)\[ \text{PPD } = \frac{$1}{(1 + \text{i})^{ \text{N}}} \times 100 \nonumber \]
Parte 4: Aplicaciones financieras para pagos de anualidades
- (11.1) Número de pagos de anualidades\[\text{N } = \text{ PY} \times \text{Years} \nonumber \]
- (11.2) Anualidad Ordinaria Valor Futuro\[\text{FV}_{\text{ORD}} = \text{ PMT} \left[ \frac{ \left[ (1 + \text{i})^{\frac{\text{CY}}{\text{PY}}} \right]^\text{N} − 1}{(1 + \text{i} )^{\frac{\text{CY}}{\text{PY}}} − 1} \right] \nonumber \]
- (11.3) Valor futuro adeudado de anualidad\[\text{FV}_{\text{DUE}} = \text{ PMT} \left[ \frac{ \left[ (1 + \text{i})^{\frac{\text{CY}}{\text{PY}}} \right]^\text{N} − 1}{(1 + \text{i} )^{\frac{\text{CY}}{\text{PY}}} − 1} \right] \times (1 + \text{i} )^{ \frac{\text{CY}}{\text{PY}}} \nonumber \]
- (11.4) Valor Actual de Anualidad Ordinaria\[\text{FV}_{\text{ORD}} = \text{ PMT} \left[ \frac{ 1 - \left[ \frac{1}{(1 + \text{i})^{\frac{\text{CY}}{\text{PY}}}} \right]^\text{N}}{(1 + \text{i} )^{\frac{\text{CY}}{\text{PY}}} − 1} \right] \nonumber \]
- (11.5) Valor Presente Vencido de Anualidad\[\text{FV}_{\text{DUE}} = \text{ PMT} \left[ \frac{ 1 - \left[ \frac{1}{(1 + \text{i})^{\frac{\text{CY}}{\text{PY}}}} \right]^\text{N}}{(1 + \text{i} )^{\frac{\text{CY}}{\text{PY}}} − 1} \right] \times (1 + \text{i} )^{ \frac{\text{CY}}{\text{PY}}} \nonumber \]
- (12.1) Valor futuro de una anualidad ordinaria de crecimiento constante\[\text{FV}_{\text{ORD}} = \text{ PMT} (1 + \Delta \%)^{\text{N}−1} \left[ \frac{ \left[ \frac{ (1+ \text{i})^{\frac{\text{CY}}{\text{PY}}}}{1 + \Delta \%} \right]^{\text{N}} - 1}{ \frac{ (1+ \text{i})^{\frac{\text{CY}}{\text{PY}}} }{1 + \Delta \%} - 1} \right] \nonumber \]
- (12.2) Valor futuro de una anualidad de crecimiento constante adeudada\[\text{FV}_{\text{DUE}} = \text{ PMT} (1 + \Delta \%)^{\text{N}−1} \left[ \frac{ \left[ \frac{ (1+ \text{i})^{\frac{\text{CY}}{\text{PY}}}}{1 + \Delta \%} \right]^{\text{N}} - 1}{ \frac{ (1+ \text{i})^{\frac{\text{CY}}{\text{PY}}} }{1 + \Delta \%} - 1} \right] \times (1 + \text{i} )^{ \frac{\text{CY}}{\text{PY}}} \nonumber \]
- (12.3) Valor Presente De Una Anualidad Ordinaria De Crecimiento Constante\[\text{PV}_{\text{ORD}} = \frac{\text{PMT}}{1 + \Delta \%} \left[ \frac{1 - \left[ \frac{1 + \Delta \%}{ (1+ \text{i})^{\frac{\text{CY}}{\text{PY}}}} \right]^{\text{N}}}{ \frac{(1+ \text{i})^{\frac{\text{CY}}{\text{PY}}}}{1 + \Delta \%} - 1} \right] \nonumber \]
- (12.4) Valor Presente de una Anualidad de Crecimiento Constante\[\text{PV}_{\text{DUE}} = \frac{\text{PMT}}{1 + \Delta \%} \left[ \frac{1 - \left[ \frac{1 + \Delta \%}{ (1+ \text{i})^{\frac{\text{CY}}{\text{PY}}}} \right]^{\text{N}}}{ \frac{(1+ \text{i})^{\frac{\text{CY}}{\text{PY}}}}{1 + \Delta \%} - 1} \right] \times (1 + \text{i} )^{ \frac{\text{CY}}{\text{PY}}} \nonumber \]
- (12.5) Perpetuidad Ordinaria Valor Presente\[ \text{PV}_{\text{ORD}} = \frac{\text{PMT}}{(1 + \text{i})^{\frac{\text{CY}}{\text{PY}}} − 1} \nonumber \]
- (12.6) Valor Presente Debido a Perpetuidad\[ \text{PV}_{\text{DUE}} = \text{PMT} \left( \frac{1}{(1+ \text{i})^{\frac{\text{CY}}{\text{PY}}} − 1} + 1 \right) \nonumber \]
Parte 5: Amortización y Conceptos Financieros Especiales
- (13.1) Porción de Interés de un Pago Único Ordinario\[ \text{INT } = \text{ BAL} \times ((1+ \text{i})^{\frac{\text{CY}}{\text{PY}}} − 1) \nonumber \]
- (13.2) Porción principal de un pago único\[ \text{PRN } = \text{ PMT} − \text{INT} \nonumber \]
- (13.3) Porción principal para una serie de pagos\[ \text{PRN } = \text{BAL}_{\text{P}1} − \text{BAL}_{\text{P}2} \nonumber \]
- (13.4) Porción de interés para una serie de pagos\[ \text{INT } = \text{ N} \times \text{PMT} − \text{PRN} \nonumber \]
- (13.5) Porción de Interés de un Pago Único Vencido\[ \text{INT}_{\text{DUE}} = ( \text{BAL} − \text{PMT}) \times ((1+ \text{i})^{\frac{\text{CY}}{\text{PY}}} − 1) \nonumber \]
- (14.1) El precio en efectivo de cualquier bono\[ \text{Cash Price } = \text{ PRI} + \text{AI} \nonumber \]
- (14.2) Cupón de Bonos Monto de Pago de Anualidad\[ \text{PMT}_{\text{BOND}} = \text{ Face Value} \times \frac{\text{CPN}}{\text{CY}} \nonumber \]
- (14.3) Precio del bono en una fecha de pago de intereses\[ \text{Date Price } = \frac{\text{FV}}{(1+\text{i})^{\text{N}}} + \text{PMT}_{\text{BOND}} \left[ \frac{1-\frac{1}{[1+\text{i}]^\text{N}}}{\text{i}} \right] \nonumber \]
- (14.4) Bono Premium o Descuento\[\text{Premium or Discount } = \text{ PRI} − \text{Face Value} \nonumber \]
- (14.5) Precio en Efectivo del Bono en una Fecha de Pago Sin Intereses\[ \text{Cash Price } = (\text{Date Price})(1 + \text{i})^{\text{t}} \nonumber \]
- (14.6) Intereses devengados en una fecha de pago sin intereses\[\text{AI} = \text{PMT}_{\text{BOND}} \times \text{t} \nonumber \]
- (14.7) Porción de Intereses de un Fondo de Hundimiento Pago Único Vencido\[ \text{INT } = ( \text{BAL} + \text{PMT}) \times ((1 + \text{i})^{\frac{\text{CY}}{\text{PY}}} − 1) \nonumber \]
- (14.8) El costo anual de la deuda de bonos\[ \text{ACD} = (\text{Face Value} \times \text{CPN}) + (\text{PMT} \times \text{PY}) \nonumber \]
- (14.9) El valor contable de la deuda de bonos\[ \text{BVD} = \text{Bonds Outstanding} − \text{BAL} \nonumber \]
- (15.1) Valor Presente Neto\[ \text{NPV } = (\text{Present Value Of All Future Cash Flows}) − (\text{Initial Investment}) \nonumber \]
- (15.2) Relación de Valor Presente Neto\[\text{NPV}_{\text{RATIO}} = \frac{\text{NPV}}{\text{CFo}} \nonumber \]