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

$\displaystyle DF_T = \frac{1}{(1+r)^T}$

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.