Equations Sheet Beta

## Annotated Equations Sheet

The purpose of this page is both to provide the equations provided in the book and on the given equation sheet, AND to mention equations that are not provided on the sheets that can indeed be useful.
Here is a link for a printable PDF of the equations sheet

What you are finding = factor notation = formula = Excel/Google Docs Command

1. Compounded Amount - Single Payment: Commonly used to calculate the FW of a single deposit compounded at a single rate over time.(1)
$$FW=(F|P,i,N)=P(1+i)^N=FV(i,N,P_0)$$

2. Present Worth - Single: The amount required to deposit now in order to have a FW in N time periods with interest, i.

(2)
\begin{align} PW=(P|F,i,N)=\frac{F}{(1+i)^N}=PV(i,N,F_0) \end{align}

3. Compound Amount - Equal Payment: : The future value of a series of equal payments with a specific interest rate.

(3)
\begin{align} F=(F|A,i,N)=A\left(\frac{(1+i)^N-1}{i}\right)=FV(i,N,A_o) \end{align}

4. Sinking Fund - Equal Payment: The amount required to deposit every time period at an interest rate for N time periods for a future value.

(4)
\begin{align} A=(A|F,i,N)=F\left(\frac{1}{(1+i)^N-1}\right)=PMT(i,N,P,F,0) \end{align}

5. Present Worth - Equal Payment: Finds the present worth of a series of equal payment

(5)
\begin{align} P=(P|A,i,N)=A\left(\frac{(1+i)^N-1}{i(1+i)^N}\right)=PV(i,N,A_0) \end{align}

6. Capital Recover - Equal Payment: Finds the payment requred to pay off a balance with a given interest rate and time periods N

(6)
\begin{align} A=(A|P,i,N)=P\left(\frac{i(1+i)^N}{(1+i)^N-1}\right)=PMT(i,N,,P) \end{align}
###### A constant series cash flow diagram.

7. Linear Gradient: The present worth of an amount increasing at a linear rate. (Every Payment = G(n-1))(7)
\begin{align} P=(P|G,i,N)=G\left(\frac{(1+i)^N-iN-1}{i^2 (1+i)^N}\right) \end{align}
###### A Linear Gradient Series. Notice how the cash flow starts in time period 2.

8. Geometric Gradient: The present worth of an amount increasing at a geometric rate. (Every Payment = A(1+g)^(N-1))(8)
\begin{align} P+(P|A_1,g,i,N)=A_1\left(\frac{1-(1+g)^N\left(\frac{1}{(1+i)^N}\right)}{i-g}\right) or P=A_1\left(\frac{N}{1+i}\right)|g=i \end{align}

#### 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 $N$.
(9)
\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)

page revision: 3, last edited: 29 Aug 2016 22:33