# Posts Tagged compounding

### ANNUITY AND ITS TYPES

Posted by Awais Ahmad in ACADEMIC on November 4, 2012

*Author: Awais Ahmad (comsian027@gmail.com)*

**Annuity:**

An Annuity is a series of equal payments made at fixed intervals for a specified number of periods. These equal payments are denoted by the PMT and can occur at either the beginning or the end of each period. Future and Present Values of Annuity: Future Value of an Annuity can be calculated, where a series of equal payments are made at a fixed intervals for a specific number of periods. The principle applied here is just like Compounding. However, method of calculating Future Value of Annuity differs in Ordinary Annuity and Annuity Due. Similarly, Present Value of an Annuity can also be calculated by using the principle of Discounting, but the method of calculating Present Value of Annuity differs in Ordinary Annuity and Annuity Due. Types of Annuity: Annuity has following types depending on the period of payment.

**1. Ordinary/Deferred Annuity:**

If the payments of Annuity occur at the end of each period, it is called Ordinary or Deferred Annuity.

**Future Value of Ordinary Annuity:**

If equal payments PMT is made at the end of n periods, providing a saving of i, then Future Value of Annuity (FVa or FVAn) can be calculated as:

**Present Value of Ordinary Annuity:**

If equal payment PMT is made at the end of n periods, providing a saving of *i*, then Present Value of Annuity PVA_{n} can be calculated as:

**2. ****Annuity Due**

If the payments of Annuity occur at the beginning of each period, such Annuity is called Annuity Due.

**Future Value of Annuity Due:**

If equal payment PMT is made at the beginning of n periods, providing a saving of *i*, then Future Value of such Annuity *FVA _{n}* can be calculated as:

The only difference between Future Value of Deferred Annuity and Annuity Due is that every term of Future Value of Annuity Due is compounded for one extra period, reflecting the fact that each payment for an Annuity Due occurs one period earlier than Ordinary Annuity.

**Present Value of Annuity Due:**

If equal PMT is made at the beginning of n periods, providing a saving of *i*, then Present Value of such Annuity PVA_{n} can be calculated as:

The only difference between Present Value of Deferred Annuity and Annuity Due is that every term of Present Value of Annuity Due is discounted for one extra period, reflecting the fact that each payment for an Annuity Due occurs one period earlier than for Ordinary Annuity.

**3. ****Perpetuity**

Some Annuities go on indefinitely, or perpetually, and are called Perpetuities. The Present Value of such Annuities is simple to calculate.

**REFERENCES:**

Financial Management – Theory & Practice by *Eugene F. Brigham *&* Michael C. Ehrhardt*

Notes on Investment Analysis and Portfolio Management

Lectures of Respectable Teahers

### Compounding, Discounting and Effective Annual Rate

Posted by Awais Ahmad in ACADEMIC on October 20, 2012

*Author: Awais Ahmad (comsian027@gmail.com)*

**Compounding:**

The process of going from today’s value (Present Value; denoted by PV) to Future Value (denoted by FV) is called Compounding. If *i *is the Interest Rate, then Interest Amount (INT) can be calculated as:

INT ($) = PV x *i*

The Future Value will be the Present Value plus the amount of Interest, so:

FV = PV + INT

FV = PV + PV x *i*

FV = PV (1 + *i*)

If the amount is deposited or invested for *n* periods, the same formula can be written as:

FV_{n} = PV_{n} (1 + *i*) ^{n}

The term (1 + *i*)^{n} is known as *Future Value Interest Factor* and is denoted by FVIF_{i,n}, so:

FV_{n} = PV_{n} (1 + *i*)^{n} = PV (FVIF_{i,n})

In some cases, Interest is paid semiannually, which means Interest is paid twice a year. Similarly Interest payment 4 times a year means Interest is paid quarterly. For such cases, the above formula can be more generalized:

Where *m *is the number of times Interest Payment is made in a year. However, in such case, *i* is taken as *Nominal Rate of Interest*.

**Discounting:**

The process of calculating Present Value (PV) from Future Value (FV) is called Discounting. As we know:

FV_{n} = PV_{n} (1 + *i*) ^{n}

Solving for PV, we have:

Or we can write it as under:

PV_{n} = FV_{n} (1 + *i*)^{-n}

The term (1 + *i*)-^{n} is called *Present Value Interest Factor*, and is denoted by PVIF_{i,n}; therefore:

PV_{n} = FV_{n} (1 + *i*)^{-n} = FV (PVIF_{i,n})

Same as previous case, if the Interest is paid semiannually or quarterly, a more general formula is applicable:

Where *i* is taken as *Nominal Rate of Interest*.

**Effective Annual Rate (EAR):**

Effective Annual Rate is defined as the rate which would produce the same Future Value, if annual Compounding had been used. It is also called *Equivalent Annual Rate*, and can be calculated as:

As we have taken Annual Compounding, therefore n is not shown in the formula, and *i* will be taken as *Nominal Rate of Interest*.

**References:**

Financial Management – Theory & Practice by *Eugene F. Brigham & Michael C. Ehrhardt*

Investment Analysis & Portfolio Management – Lectures