Annotated Equations Sheet

Purpose

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{align} \[F=P(1+i)^{N}\] \end{align}	=FV(i, N, P0)	not included	Single deposit compounded at single rate over time.
Single	Present Worth (P|F, i, N)	(2) \begin{align} \[P=F(1+i)^{-N}\] \end{align}	=PV(i,N,F0)	not included	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,A0)	not included	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} \[ A=F\left(\frac{1}{(1+i)^{N}-1}\right)\] \end{align}	=PMT(i,N,P,F,0)	not included	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,A0)	not included	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)	not included	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	not included	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} OR (9) \begin{align} \[ P=A_{1}\left(\frac{N}{1+i}\right)\,(if : i = g) \end{align}	Unavailable	not included	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 cashflow in period zero.
2. $A_N\left(1+i\right)^{-N}$, the present worth of the cashflow in period $N$.

(10)

\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)

To Do:

Rebuild images in book for insertion
Add secondary sheet