# Financial Formulas

A ready reference for financial formulas, enables to look-up the formula and it’s key input variables for ease in calculation and reference.

Following key financial formulas are covered in this post.

- Rate of return conversion for different periodicity
- Discounting
- Loan Amortization
- Derivative related formulas

Note:

- This post is work in progress and other relevant formulas and details periodically updated.
- All decimal figures are rounded up to 4 decimal places

## 1. Rate Conversion

### 1. Periodic Rate of Return (PRR)

Annual rate is given and the objective is to convert to periodic rate i.e. convert *annual rate* of return to either *monthly, quarterly or half-yearly* return. This cannot be simply achieved by dividing the annual rate by the number of periods. The following formulas has to be used.

\[monthly \space / \space quarterly \space / \space biannually = (1+r)^{\frac{1}{n}}-1\] Where

- r = annual rate
- n = shorter period for which rate has to be obtained. Thus
- n = 12 for monthly
- n = 2 for bi-annually
- n = 4 for quarterly

**Examples**

- Convert 17% annual rate to quarterly rate equivalent.

- Solution: Using the formula \((1+r)^{\frac{1}{n}}-1\) wherein r = 0.17 and n = 4, the quarterly rate is 0.0400

### 2. Effective Annual Rate (EAR)

*When compounding is done multiple times in a year and we are interested in finding EAR.*

Following formula gives the EAR when compounding is done n number of times.

\[EAR = (1+\frac{r}{n})^n -1\] Where

- r = interest rate / compounding period
- n = number of compounding periods
- n = 12 for monthly
- n = 2 for bi-annually
- n = 4 for quarterly

**Examples**

- Annual rate of return is 17% which is compounded twice a year. Calculate EAR.

- Solution: Using the formula \((1+\frac{r}{n})^n -1\) the EAR is 0.1772

*Effective interest rate per compounding period (similar to PRR, however, in this case compounding is done multiple times in the year)*

This second formula give the effective interest rate per compounding period.

\[R = (1+\frac{r}{n})^\frac{n}{p} -1\] Where

- p = No of payment periods per year
- r = nominal annual interest rate
- n = No. of compounding periods per year
- R = Rate per payment period

**Example**

- Annual rate of return is 17% which is compounded twice a year and payments are made quarterly. Calculate PRR.

- Solution: Using the formula \((1+\frac{r}{n})^\frac{n}{p} -1\) the PRR (with multiple compoundings in a year) is 0.0416

## 2. Discounting

### 1. Present Value (PV)

Describe present value.

\[PV = (1 + r)^{-t}\]

Where

- r = Annual rate of return
- t = Number of years

**Examples**

- Calculate the present value of 3 year maturity instrument when the rate per annum is 18%.

- Solution: Using the formula, \(PV = (1 + r)^{-t}\) the PV discount value is 0.6086

### 2. Present Value (compounded periodically)

In this case the formula is just the putting a negative in the exponent i.e. -nt

\[PV \space factor = (1+\frac{r}{n})^{-nt}\] Where

- r = Annual rate of return
- n = Number of times compounding is done per year
- t = Number of years

- Calculate the present value of 3 year maturity instrument when the rate per annum is 18% and compounding is done quarterly.

- Solution: Using the formula, \(PV = FV.(1+\frac{r}{n})^{-nt}\) the PV discount value is 0.1121

### 3. Annuity Formula

Describe annuity.

\[Annuity \space factor = [1 - \frac{(1+r)^{-t}}{r}]\]

Where

- r = Annual rate of return
- n = Number of times compounding is done per year
- t = Number of years

### 3. Future value (compounded once a year)

\[PV = (1 + r)^{t}\]

Where

- r = Annual rate of return
- t = Number of years

### 4. Future Value (compounded periodically)

Compounding Interest Rate (when compounding more frequent than on yearly basis)

Examples are monthly, quarterly, bi-annually.

\[FV \space factor = (1+\frac{r}{n})^{nt}\]

Where

- r = Annual interest rate
- n = number times compounding is done on a yearly basis
- t = time in years

Examples