Annotated Equations Sheet


The purpose of this page is both to provide the equations provided in the book and on the given equation sheet, (in nice tabular format) AND to mention equations that are not provided on the sheets that can indeed be useful.

Standard Equation Chart

Flow Type Factor Notation Formula Excel Command Cash Flow Diagram Common Usages / Examples / Explanation
Single Compound Amt (F|P, i, N) (1)
\begin{equation} F=P(1+i)^{N} \end{equation}
=FV(i, N, P,,0) Singleton-1.jpg Single deposit compounded at single rate over time.
Single Present Worth (P|F, i, N) (2)
\begin{equation} P=F(1+i)^{-N} \end{equation}
=PV(i,N,F,,0) Singleton-1.jpg Current deposit amount if result is created by periodic compounding at individual rate.
Equal Payment Compound Amount (F|A, i, N) (3)
\begin{align} F=A\left(\frac{(1+i)^{N}-1}{i}\right) \end{align}
=FV(i,N,A,,0) Equal_payment1.jpg The future value of a series of equal payments, with a specific interest rate
Equal Payment Sinking Fund (A|F, i, N) (4)
\begin{align} A=F\left(\frac{i}{(1+i)^{N}-1}\right) \end{align}
=PMT(i,N,P,F,0) Equal_payment1.jpg The equal payment amount required to save a given future amount, given rate and future amount
Equal Payment Present Worth (P/A, i, N) (5)
\begin{align} P=A\left(\frac{(1+i)^{N}-1}{i(1+i)^{N}}\right) \end{align}
=PV(i,N,A,,0) Equal_payment1.jpg The present value of an amount that will be paid off over time using equivalent payments and a specific rate (EX: Calculate mortgage amount via payments)
Equal Payment Capital Recovery (A/P, i, N) (6)
\begin{align} A=P\left(\frac{i(1+i)^{N}}{(1+i)^{N}-1}\right) \end{align}
=PMT(i,N,P) Equal_payment1.jpg The payment required to pay off a given amount and rate
Gradient Series Linear Gradient (P/G, i, N) (7)
\begin{align} P=G\left(\frac{(1+i)^{N}-iN-1}{i^{2}(1+i)^{N}}\right) \end{align}
Unavailable LinearGradient.jpg The present worth of an amount increasing at a linear rate (every payment = (N-1)*G)
Gradient Series Geometric Gradient (P/A_1,g,i,N) (8)
\begin{align} P=A_{1}\left(\frac{1-(1+g)^{N}(1+i)^{-N}}{i-g}\right) \end{align}


\begin{align} P=A_{1}\left(\frac{N}{1+i}\right)\,(if : i = g) \end{align}
Unavailable GeometricGradient.jpg The present worth of an amount increasing at a geometric rate (every payment = A(1+g)^(N-1))

Internal Rate of Return calculations:

  1. $A_0$, the present worth of the cash flow in period zero.
  2. $A_N\left(1+i\right)^{-N}$, the present worth of the cashflow in period $N$.
\begin{eqnarray} A_{0}+A_{N}\left(1+i\right)^{-N} & = & 0\\ \left(1+i\right)^{-N} & = & -\frac{A_{0}}{A_{N}}\\ i & = & \left(-\frac{A_{N}}{A_{0}}\right)^{\frac{1}{N}}-1. \end{eqnarray}

Tax Calculations:

Actual Rate = Federal Rate + State Rate - (Federal Rate * State Rate)

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