### 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

### Market or Investment Portfolio, Investors, Securities, Time Value of Money Concepts

Posted by Awais Ahmad in ACADEMIC on October 19, 2012

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

**Investment/Market Portfolio:**

When an investor invests in multiple

stocks/securities, it is called Investment Portfolio. Maintaining a Portfolio

is a very important step taken by investors. By maintaining a Portfolio, Risk

can be mitigated / minimized by maintaining a portfolio and higher margins of

profits can be earned. In this case, if one stock/security defaults, it does

not necessarily mean Investor is also in loss. Instead, investor can compensate

the loss of one stock from other stocks/securities. Fig. 5 shows how

maintaining a Portfolio minimizes the Portfolio Risk. Fig shows that Portfolio

size is taken on x-axis and portfolio risk on y-axis, which results a curved

graph.

**Classification of Investors:**

Investors can be classified on the basis of their risk-taking/bearing capacity. How much risk an investor bears, depends on investor’s personal capacity, attitude, interest and behavior. For example:

**1.****Risk Seekers**

Risk seekers seek for riskier investment. They are capable of assuming a higher risk and have strong and healthy financial position.

**2.****Risk Avoiders**

They avoid riskier investments, because they have not strong and healthy financial position. They choose those instruments, which have less variation in returns.

**3.****Risk Bearers**

Risk bearers fall in between the above categories. They choose moderate levels of risk they can bear according to their capacity.

**Hedging:**

Risk reduction is known as Hedging. They do it by using *Derivative Instruments*.

**Security:**

A Security refers to a publicly traded financial instrument, as opposed to a privately placed instrument. Securities have greater liquidity than otherwise similar instruments, which are not traded in Open Market. Security is considered to be an insurance against an emergency, according to banking definitions.

**Classification of Securities:**

The securities have been classified according to the functional operation aspects as under:

**1.****Intangible Securities**

These are personal exclusive undertakings by a party to pay the amount of advances outstanding against a borrower. Examples of such securities are Demand Promissory Note, bill of exchange or a Bond, Guarantee and Indemnity etc.

**2.****Tangible Securities**

These are the securities which can be realized from sale or transfer. Examples of such securities are Shares, Stock, Land, Building and Goods.

**3.****Prime Securities**

These are also called Primary Securities. Such securities are main covers for an advance and are deposited by the borrower himself. When a depositor of term deposits offers his *Term Deposit Receipt* to cover and advance, it is the Primary Security according to banking term.

**4.****Collateral Securities**

These are the securities provided as an additional cover for an advance, where either he security is not very stable in value, or where the realization of the security to cover the outstanding amount of balance is difficult. In case of the default by borrower, bank has the authority to sell these shares of security and adjust the advance.

**5.****Movable Securities**

These are the securities, which are legally and physically both in possession of the lending bank. Examples are Term Deposit Receipts, Goods, Vehicles and Merchandise etc.

**6.****Immovable Securities**

These are the securities, where the legal possession or right to takeover is entrusted to the lending bank, but the physical possession remains with borrowers.

**7.****Government Securities**

These are the long-term securities issued by the government for financing social programs. They are perceived as Risk-free, are highly liquid and carry attractive coupon rates. Like T-bills (Treasury Bills), government securities are sold through auctions and are actively traded in secondary markets.

**Time Value of Money:**

The theory of Time Value of Money states that the value of money decreases with the passage of time. This concept can be described as “A Dollar in hand today is more worth of a Dollar tomorrow”. This happens because of *Inflation*. Inflation is a situation, where the prices as a whole are increasing. The rate at which the prices are increase is known as *Inflation Rate*. Two terms are necessary to explain while discussing Inflation and theory of Time Value of Money:

**1.****Nominal Interest Rate/Quoted Interest Rate:**

Nominal Interest Rate is a rate at which money invested grows. Banks generally offer Nominal Rate of Interest to the depositors.

**2.****Real Interest Rate**

Real Interest is a rate, at which the purchasing power of an investment increases. Market Interest Rates are Nominal Interest Rates.

**Relationship of Inflation, Nominal and Real Interest Rates:**

Real Interest Rates, Nominal Interested Rates and Inflation Rates have strong relationship with each other, which can be expressed in the form of an equation:

The above equation shows that if Inflation Rate increases, then Real Interest Rate decreases and vice versa.

Another approximate relationship also exists between the three rates:

It means, by subtracting Inflation Rate from Nominal Interest Rate, the approximate Real Interest Rate can be calculated.

**References:**

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

Management of Banking and Financial Services by *Padmalatha Suresh & Justin Paul*

Handouts Investment Analysis and Portfolio Management

Lectures on Financial Investment and Portfolio

### 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*

### 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