# Posts Tagged notes

### 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

### PROBABILITY OF A FAIR COIN (Experimental Proof)

Posted by Awais Ahmad in ACADEMIC on October 18, 2012

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

**PROVING THE PROBIBILITY OF A FAIR COIN TO BE 0.5 OR 50%**

Question: Prove with experiment that the Probability of a fair coin is 0.5 or 50%.

Experiment:

**Step 1:** Take a fair coin and toss it 200 times and record each observation and the outcomes thereof. The following observations were recorded by tossing a fair coin 200 times:

For the time being, for our convenience, we have classified the observations of the experiment with a difference of 25.

Here:

Head of the Fair Coin = N

Tail of the Fair Coin = M

As we see as per Table, First column shows the number of observations taken from the experiment, classified with the difference of 25. Second column shows the appearance of Head (N) or Tail (M) during the experiment.

**Step 2:** Now we perform calculations of Probability in further details by the table given below:

Colum 1 of Table shows the Number of Experiments’ Lower and Upper Limits (The data of 200 observations is classified with a difference of 5 for detailed analysis). Second column shows outcomes of appearing Head (N) or Tail (M), and Cumulative Outcomes. TOTAL Colum shows the number of experiments during a particular time. Next column shows the probabilities of appearing N or M of the coin and their Cumulative Probabilities. These probabilities are calculated by the following formula:

P (N) = N / (TOTAL)

P (M) = M / (TOTAL)

**Step 3:** Then P (N) and P (M) showed the cumulative probabilities of N and M respectively. The point should be noted that, before experiment performed, the Probabilities of both N and M were Zero. Last column shows the number of Cumulative Observations/Outcomes of the experiment. Last Row TOTAL showed that the experiment was repeated for 200 times, among which 98 times, N appeared and M appeared for 102 times.

Now we plot these values on the line graph, and can show the probabilities of N, M and both N & M on the same plot area.

**Conclusion:**

From the experiment Data and Graphical Representations, we can see that the Probabilities of Head (N) and Tail (M) are about equal to 0.5 (50%) when we repeat the same experiment for 200 times. So it is proved that the **Probability of a fair coin is 0.5 (50%).**

**Note:** If the same experiment is provided for more than 200 times, we can get more accurate results.

**References:**

Question from Statistics for Business and Economics – by David R. Anderson (Author), Dennis J. Sweeney (Author), Thomas A. Williams (Author), Jeffrey D. Camm (Author), James James J. Cochran (Author)

Experiment on Fair Coin

### ECONOMIC DEVELOPMENT AND THE ROLE OF FINANCIAL ASSETS

Posted by Awais Ahmad in ACADEMIC on October 16, 2012

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

An efficient and strong Financial System leads to the Economic Development of a nation. A country is said to be economically developed, when it has strong Financial Markets and competent Financial Intermediaries.

One of the essential criteria for the assessment of Economic Development is the quality and quantity of assets in a nation at a specific time. These assets can be classified based on their distinct characteristics. Classification of Assets is shown through a diagram chart (Fig. 2):

As we are concerned with Financial Markets, we will focus on Financial Assets.

**Financial Assets:**

In macro sense, Financial Assets are regulated by the government of an economy. Financial Assets smoothen the trade and transactions of an economy and give the society a standard measure of valuation. Financial Assets also represent the current/future value of physically and intangibly held assets. They show a right on another asset and include Currency Instruments (Cash, Foreign Currency etc.) and Claim Instruments (Debentures, Shares, Deposits, Unit Certificates, Tax Saving Investment etc.)

Properties of Financial Assets:

The following are the properties of Financial Assets, which distinguish them from Physical and Intangible Assets:

**1. ****Currency:**

Financial Assets are exchange documents with an attached value. Their values are dominated in currency units determined by the government of an economy.

**2. ****Divisibility**

Financial Instruments are divisible into smaller units. The total value is represented in terms of divisions that can be handled in a trade. The divisibility characteristic of Financial Assets enables all players, small or big, to participate in the market.

**3. ****Convertibility**

Financial Assets are convertible into any other type of asset. This characteristic of convertibility gives flexibility to financial instruments. Financial Instruments need not necessary be converted into another form of Financial Asset; they can also be converted into Physical/Tangible and Intangible Assets.

**4. ****Reversibility**

This implies that a financial instrument can be exchanged for any other asset and logically, the so formed asset may be transferred back into the original financial instrument.

**5. ****Liquidity**

Liquidity implies that the present need for other forms of asset prevails over holding the financial instrument. The financial asset can be exchanged for currency with another market participant who does not have immediate cash need, but expects future benefits.

**6. ****Cash Flow**

The holding of the financial instrument results in a stream of cash flows that are the benefits accruing to the holder of the financial instrument. However, a financial instrument by itself does not create a cash flow.

**References:**

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

Management of Banking and Financial Services – 2e, by *Padmalatha Suresh & Justin Paul*

### FINANCIAL SYSTEM

Posted by Awais Ahmad in ACADEMIC on October 16, 2012

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

The Financial System of any country has two important segments:

- Financial Markets 2. Financial Intermediaries

**1. ****Financial Marktes:**

A Financial Market is a place/system, where Financial Instruments are exchanged. Such markets enhance the unique characteristics of Financial Instruments (Stocks, Bonds, Mortgages, Auto Loans, and Certificates of Deposits etc.)

**2. ****Financial Intermediaries:**

Financial Intermediaries create assets out of the surpluses of the economy. They ensure liquidity of savings by surplus units. They also reduce information costs, mitigate and evaluate risk tied to the surplus units.

The Prime objective of Financial System is to channel Surpluses arising in the economy through the activities of households, corporate houses and the government into deficit units in the economy, again in the form of households, corporate houses and the government. However, this flow of funds differ between Banking Institutions and Non-banking Financial Institutions in a Financial System. This flow of funds through Financial System, and the difference between the two is shown as in Fig. 1:

References:

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

2. *Management of Banking & Financial Services – *2e by *Padmalatha Suresh & Justin Paul*