# Definitions

## Future Value

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

$\displaystyle CF_T = (1+\frac{r}{m})^{mT}$ 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)

$\displaystyle CF_T = e^{rt}$ where r is the annual interest rate and T is the number of years that the rate is compounded.

### Effective Annual Rate (Non continuous)

$\displaystyle \hat{r} = (1+\frac{r}{m})^m - 1$

### Effective Annual Rate (Continuous)

$\displaystyle \hat{r} = e^r - 1$

### Effective Monthly Rate (Non continuous)

$\displaystyle \hat{r}_M = (1+\frac{r}{m})^{\frac{m}{12}} - 1$

## Discounted Value

### Discount Factor over T Periods

$\displaystyle DF_T = \frac{1}{(1+\frac{r}{m})^{T*m}}$ where r is the interest rate and m is the number of times in the period T that the interest is compounded.

### Present Value with constant R

$\displaystyle PV = \sum_{t=1}^T \frac{C_t}{(1+r)^t}$

## 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 $\displaystyle PV = \frac{C}{r}$ . 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 $\displaystyle PV = \frac{C}{r}*\frac{1}{(1+r)^t}$ . 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 $\displaystyle PV = \frac{C}{r-g}$ . Note that the first time a payment is made is one year after the investment.

## Annuities

### Regular Annuities

The amount of money that has to be invested at an interest rate of r, to yield a constant yearly return of C for T years is given by $\displaystyle PV = \frac{C}{r}*(1-\frac{1}{(1+r)^T})$ . Note that the first annuity is paid out one year from the date of the investment.

### Growing Annuities

The amount of money that has to be invested at an interest rate of r, to yield a yearly return of C which grows by g each year for T years is given by $\displaystyle PV = \frac{C}{r-g}*(1-(\frac{1+g}{1+r})^T)$ . Note that the first annuity is paid out one year from the date of the investment.

## Interest Rate

### Real Interest Rate

The real interest rate given the nominal interest rate r and the inflation rate i is given as $\displaystyle R=\frac{1+r}{1+i}-1$