# Difference between revisions of "Introduction to Finance"

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=== Forward Rates === | === Forward Rates === | ||

The interest rate to borrow a sum of money between years t-1 and t is <math>f_t = \frac{(1+r_t)^t}{(1+r_{t-1})^{t-1}}-1</math> | The interest rate to borrow a sum of money between years t-1 and t is <math>f_t = \frac{(1+r_t)^t}{(1+r_{t-1})^{t-1}}-1</math> | ||

+ | |||

+ | === Spot Rate === | ||

+ | The spot rate is defined as the interest rate per year for T years starting today. | ||

+ | |||

+ | === Price of Bond given Spot Rates === | ||

+ | The price of a bond maturing in T years with a semi annual coupon rate of i% and face value of C is given as: | ||

+ | <math>P = \frac{i%*C}{(1+r_0.5)^0.5} + \frac{i%*C}{(1+r_1.0)} + ... + \frac{(1+i%)*c}{(1+r_T)^T}</math> |

## Revision as of 15:10, 16 August 2010

## Contents

# 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}**

### Sum of Regular Deposits

If C amount of money is saved every time period for T time periods with an interest rate of r, the value of the savings after T years 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}*(1-\frac{1}{(1+r)^T})*(1+r)^T}**

## 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+\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

**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 **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-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 **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}*(1-\frac{1}{(1+r)^T})}**
. Note that the first annuity is paid out one year from the date of the investment. Note that the first deposit in the bank is being made in time period 1.

### 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 **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-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 **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 R=\frac{1+r}{1+i}-1}**

## Bond Pricing

### Forward Rates

The interest rate to borrow a sum of money between years t-1 and t 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 f_t = \frac{(1+r_t)^t}{(1+r_{t-1})^{t-1}}-1}**

### Spot Rate

The spot rate is defined as the interest rate per year for T years starting today.

### Price of Bond given Spot Rates

The price of a bond maturing in T years with a semi annual coupon rate of i% and face value of C is given as:
**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 P = \frac{i%*C}{(1+r_0.5)^0.5} + \frac{i%*C}{(1+r_1.0)} + ... + \frac{(1+i%)*c}{(1+r_T)^T}}**