# Posts Tagged formula

### 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

### Investments, Risk and Return

Posted by Awais Ahmad in ACADEMIC on October 18, 2012

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

**Investment:**

Investment may be defined as an activity that commits funds in any financial/physical form in the present with an expectation of receiving additional return in the future. The expectation brings with it a probability that the quantum of return may vary from a minimum to maximum. This possibility of variation in the actual return is known as *Investment Risk*. Thus every investment involves a *Return* and *Risk*.

Investment is an activity that is undertaken by those who have *Savings*. *Savings* can be defined as the excess of Income over Expenditure. Two important characteristics of Investment are *Return* and *Risk*.

*Return:*

Every Investment has and expectation of Return, which is the margin earned by investor in future by investing at present. The margin earned reflects the concept of *Time Value of Money*, which describes that a Dollar currently held will be less worth tomorrow. Investors take advantage from this saving, which is called Return (denoted by ** k**or

**)**

*r**Risk:*

Risk is the chance of uncertainty that may rise after investment. It is the event that may arise by affecting the Return of the Investment. Every investor should forecast the Return and Risk while investing and should keep into account all possible events that may occur in future. It is denoted by ** d** and is the

*Standard Deviation*, which reflects the deviation from the

*Expected Rate of Return*.

The formulae for Risk and Return of and Investments are:

*Relationship between Risk and Return:*

Risk and Return have positive relation to each other (Direct Relationship):

Risk **µ** Return

The relationship shows that if an investment is tied with high expectation of Return, it will have high Risk and vice versa. This relationship is also shown in the diagram:

**Types of Risk:**

There are various types of Risk, among which, following are the most important to describe:

**1.****Business Risk:**

Business Risk is one which is tied to a particular business. Such risk arises when the investor is not sure whether the business will successful or not in which he/she has invested.

**2.****Financial Risk:**

Financial Risk is the additional risk placed on the common stockholders as a result of the decisions to finance with debt. The use of debt, also called *Financial Leverage* also concentrates the firm’s business risk on its stockholders. Financial Risk arises, when to decide whether the firm should be *Levered* (50% debt and a portion of equity) or *Unlevered* (All Equity). In this way, it makes the *Capital Structure* of a firm.

**3.****Country Risk:**

Country Risk is one that arises from a particular country. A country’s environment may be favourable for investors, who want to invest in that country. Investors need to forecast such risk before making the investment decision.

**4.****Diversifiable Risk/Unsystematic/Idiosyncratic:**

The part of a stock’s risk, that can be minimized/neglected is called Diversifiable Risk. Such Risk is caused by some random events like lawsuits, strikes, successful and unsuccessful marketing programs etc. This type of risk can be minimized while investing in multiple stocks; a phenomenon known as *Diversification Principle*.

**5.****Non-diversifiable/Systematic/Market Risk:**

The part of stock’s risk, that cannot be minimized or neglected is known as Non-diversifiable Risk. This type of risk stems from factors that systematically affect most firms; such as war, inflation, recessions and high interest rates. Such risk remains constant even when investing in multiple stocks. It is measured in terms of Beta (β).

**6.****Inflation Risk:**

Technically speaking, this type of risk is a subtype of Systematic Risk. This risk stems from variation in inflation rate. When inflation rate increases, it reduces the *Real Interest Rate* and vice versa. More detail about the relationship between Inflation Rate and Real Interest Rate will be explained in related articles.

**7.****Interest Rate Risk:**

Interest Rate Risk can be said to be a subtype of Systematic Risk. This type of risk arises with the fluctuation on Interest Rate in market. The more is the market Interest Rate, the more will be Interest Rate Risk tied to a particular stock (will be explained in later articles).

**8.****Exchange Rate Risk:**

Exchange Rate Risk stems from changes in Exchange Rate or currency fluctuations. The more is the currency fluctuation in market, the more will be the Exchange Rate Risk for and Investor.

**9.****Liquidity/Marketability Risk:**

Liquidity Risks generally arise when one business acquires another business. *Liquidity* refers to a condition when a firm is not generating enough profit from its operations, that can meet the overall cost of a business. By doing so, investors of one organization purchase the stocks of *Liquidating Firm*, but are not sure whether their decision to purchase the stocks of such firm will benefit at a future date or not. Such uncertain factors are called Liquidity Risk, when combine together.

**10.****Political Risk:**

Political Risk arise from the political environment of an economy. If a country’s political environment is uncertain and unfavourable, the investors are called facing Political Risk while investing in such country.

**11.****Stand-alone Risk:**

If an investor has only one investment, the risk tied to that particular investment is known as Stand-alone Risk. If an investor has only one investment, it is highly risky for him/her, as he/she cannot compensate his/her loss from alternative stocks, in case the stock hald by him/her is losing its value.

**12.****Portfolio Risk:**

If an investor has invested in multiple stocks/securities, the risk tied to all of those stocks/securities (combined together) is called Portfolio Risk. If a security defaults, the investor still has a chance to earn profit from other stocks/securities held by him/her. Portfolio Risk can be measured by the following formula.

*References:*

Investment Analysis and Portfolio Management (handouts)

Lectures by Mr. Wasim Anwar

Financial Management – Theory & Practice; 10e by *Eugene F. Brigham & Michael C. Ehrhardt*