# Introduction to Finance

# Definitions

## Future Value

### Compounding Factor for T Periods (Non continuous)

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle CF_T = (1+\frac{r}{m})^{mT}}**
where r is the annual interest rate, m is the number of time per year the rate is compounded, and T is the number of years that the rate is compounded.

### Compounding Factor for T Periods (Continuous)

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle CF_T = e^{rt}}**
where r is the annual interest rate and T is the number of years that the rate is compounded.

### Effective Annual Rate (Non continuous)

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{r} = (1+\frac{r}{m})^m - 1}**

### Effective Annual Rate (Continuous)

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{r} = e^r - 1}**

### Effective Monthly Rate (Non continuous)

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{r}_M = (1+\frac{r}{m})^{\frac{m}{12}} - 1}**

## Discounted Value

### Discount Factor over T Periods

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle DF_T = \frac{1}{(1+r)^T}}**

### Present Value with constant R

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle PV = \sum_{t=1}^T \frac{C_t}{(1+r)^t}}**

## Perpetuities

### Regular Perpetuities

The amount of money that has to be invested at an interest rate of r, to yield a constant yearly return of C is **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle PV = \frac{C}{r}}**
. Note that the first time C is paid out one year after the investment.

### Deferred Perpetuities

The amount of money that has to be invested at an interest rate of r, to yield a constant yearly rate of return of C in t years from now is **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle PV = \frac{C}{r}*\frac{1}{(1+r)^t}}**
. Note that the first time C is paid out is in year t+1.

### Growing Perpetuities

The amount of money that has to be invested at an interest rate of r, to yield a yearly rate return of C that grows by g every year is <math>PV = \frac{C}{r-g}. Note that the first time a payment is made is one year after the investment.