Cosepp Technology Online - Financial Mathematics

1. Simple Interest. The calculation of simple interest is usually based on the following formula:

Where "I" is the simple interest to be calculated; "P" is the principal amount invested or borrowed; "R" is the rate of interest; and "T" is the time period, in number of days. Sometimes, in lieu of ( T / 365 ), the less accurate ( T / 12 ) is employed, where "T" is the time period, in months.

For example, the simple interest calculated for the principal amount of $ 1,000 invested or borrowed at 5% per annum for 60 days, would be as follows:

Rounding And Decimal Remainder. Hence, because our monetary unit of 1 dollar consists of 100 cents, the calculation of interest, in cases like the above example, is usually given as either $8.21 or $8.22, depending on the financial institution involved.

A deposit taking institution is more likely to pay $8.21 interest, whereas a lending institution is more apt to charge $8.22. Only financial institutions with sophisticated computer software systems are able to pay or charge the exact amount, and carry forward the remainder of the decimal calculation.

Admittedly, while the above rounding remainder is insignificant in respect of the above example, such decimal remainders can nonetheless easily add up to form a very significant sum if the calculations involve the principal amounts of many investors and/or borrowers. Or if the calculations are performed daily over a number of years.

Tricks. In the past, there were some financial institutions that fudged the above formula in secret. For example, some lending institutions, instead of using 365 days as the denominator, used 360, thereby resulting in an otherwise higher interest amount being charged to a borrower, and could still claim that they were charging interest at 5% for 60 days.

And there's also the problem of leap years, which have 366 days, the extra day of which may or may not be ignored. Increasing the denominator, of course, reduces the interest amount otherwise payable on a borrowed amount. This extra day may not be so relevant in relation to a borrowed sum of, say, $100; but it would become remarkably so if the sum were to be in the order of, say, $100 million.

Since mathematics is normally an exact science, it may well be prudent to be precise whenever and if reasonably possible in this imprecise and imperfect world of ours.

2. Compound Interest - Yearly. The calculation of interest, compounded yearly, is usually based on the following formulae:

Where "F" is the future value of the principal amount, "P" is the principal amount, "R" is the rate of interest, and "n" is the number of years that the principal amount is to compounded, and "I" is the compound interest amount.

I have decided to present the method for calculating interest, compounded yearly, as a two step process, rather than just one complicated formula, in order to allow the logic of the calculations to be more readily ascertainable.

For example, the yearly compound interest calculated for the principal amount of $1,000 invested or borrowed at 6% per annum for 3 years, would be as follows:

Rounding and Decimal Remainder. Here, the rounding and decimal remainder problems are similar, in the main, to those as discussed above with respect to Simple Interest.

3. Compound Interest - Periods. The calculation of interest, compounded on specified periods, is usually based on the following formulae:

Where "F" is the future value of the principal amount, "P" is the principal amount, "R" is the rate of interest, "k" is the number of compounding periods within a year, and "n" is the number of years that the principal amount is to compounded, and "I" is the compound interest amount.

I have decided to present the method for calculating interest, compounded by periods, as a two step process, rather than just one complicated formula, in order to allow the logic of the calculations to be more readily ascertainable.

For example, the periods compound interest calculated for the principal amount of $1,000 invested or borrowed at 12% per annum, compounded quarterly for 1 year, would be as follows:

Rounding and Decimal Remainder. Here, the rounding and decimal remainder problems are similar, in the main, to those as discussed above with respect to Simple Interest.

4. Present Value. The calculation of present value is usually based on the following formula:

Where "P" is the present value, "F" is the future value, "R" is the rate of interest that is to be used as the discount rate, and "n" is the number of years to be discounted (i.e. the number of years from the present date that "F" is expected to be received).

As can be seen, the above formula for present value is really a juxtaposition of the formula used to calculate compound interest, using reverse logic.

For example, the present value of $ 5,000 which is expected to be received in 2 years' time, assuming a discount rate of 5% per annum, would be as follows:

Discount Rate. In the above example, we used the interest rate of 5% in our calculation. When used in this way, such a rate is generally known as the discount rate, as its discounts a future value amount to the present.

The higher the discount rate employed, the lower would be the present value. And the lower the discount rate employed, the higher would be the present value. It is beyond the scope of this elementary explanation to elaborate further about the various theories or opinions on discount rates.

5. Internal Rate of Return - Yearly. The calculation of the internal rate of return, in respect of one amount, is usually based on the present value formula, as follows:

Where "P" is the present value, "F" is the future value, "R" is the internal rate of return, and "n" is the number of years to be discounted (i.e. the number of years from the present date that "F" is expected to be received.

For example, the internal rate of return on $5,000 invested at the beginning of the year and receiving $20,000 at the end of 2 years, would be as follows:

6. Internal Rate of Return - Complex. The calculation of the internal rate of return, in respect of a complex financial cash flow, is usually based on the following formula:

Where "O" is the initial outlay or investment, "F₁ ... F_n" are the cash flows in periods "1 ... n", and "R" is the internal rate of return that is to be calculated, expressed as a percentage per annum.

Of course, the above formula can be extremely complex if the future cash flows compound more than once yearly, are many, and occur at irregular intervals. For practical purposes, the usual method employed to calculate "R " is by trial and error. That is, numerous values for "R" are chosen and evaluated until such time as the above formula equates. With computers, such calculations are a lot more manageable.

In the above example, the internal rate of return is 7.59%, a figure arrived at by the use of a computer program, compounding yearly.

The above cash flows also represent a series of share trading transactions. So, the question arises, just what sort of performance do the transactions all represent. The answer is supplied by calculating the internal rate of return in respect of the whole cash flow.

Again, in the above example, many amateurs would simply add up the outflows [$10,079.99], add up the inflows [$13,801.15], calculate a profit of $3,721.16, and then calculate and publish a profit of 36.92% [by dividing $3,721.16 by $10,079.99]. Clearly, this amateurish method is absolutely wrong because it ignores the passage of time, i.e. the opportunity cost of capital.

Problems With Calculating the Internal Rate of Return. The are a number of problems in relation to calculating the internal rate of return in respect of complex cash flows. Unfortunately, it is beyond the scope of this elementary explanation to elaborate further.

7. Dividend Yield. The calculation of the dividend yield is usually based on the following formula:

Where "Y" is the dividend yield in percentage terms per annum, "D" is the total dividends amount paid over the year, and "T" is the total number of shares issued.

For example, the dividend yield with respect to a company that has paid out $6,000,000 in dividends over the year, and having 100,000,000 shares on issue, would be as follows:

Complications. Most published figures on dividend yields are usually based on the latest paid out annual dividends. Some analysts, though, choose to emphasise prospective dividend yields. And then there are the problems associated with share options, and with shares issued during the period that the dividends are paid. Unfortunately, it is beyond the scope of this elementary explanation to elaborate further.

8. Earnings Ratio. The calculation of the earnings ratio is usually based on the following formula:

Where "ER" is the earnings ratio, i.e. the earnings per share, "E" is the total earnings over the year, and "T" is the total number of shares issued.

For example, the earnings ratio with respect to a company that derived a net profit of $9,000,000 over the year, and having 100,000,000 shares on issue, would be as follows:

Complications. Most published figures on earnings ratios are usually based on the latest profit reports. Some analysts, though, emphasise prospective earnings ratios. And then there are the problems associated with share options, and with shares issued during the period that the profits are derived. Unfortunately, it is beyond the scope of this elementary explanation to elaborate further.

9. Price Earnings Ratio. The calculation of the price earnings ratio is usually based on the following formulae:

Where "PE" is the price earnings ratio, "MP" is the market price of the company's shares, "ER" is the earnings ratio, and "E" is the total earnings over the year.

For example, the price earnings ratio with respect to a company deriving a net profit of $9,000,000 over the year, having a current market price per share of $1.35, and having 100,000,000 shares on issue, would be as follows:

In the above example, the price earnings ratio would be 15. That is, the above company would take 15 years in order to earn the current market price of its shares.

Importance. The price earnings ratio is an extremely important tool in shares analyses. It gives a quick pointer to a company's ranking vis-à-vis other companies' in the same industry, and sometimes in other industries.

Complications. Most published figures on price earnings ratios are usually based on the latest profit reports. Some analysts, though, emphasise prospective price earnings ratios. And then there are the problems associated with share options, and with shares issued during the period that the price earnings ratio is calculated. Unfortunately, it is beyond the scope of this elementary explanation to elaborate further.

10. Balance Sheet. The balance sheet in respect of an entity is usually based on the following formula:

Where "Assets" are all the assets of the entity, "Liabilities" are all its liabilities, and "Equity" is the surplus remaining after deducting all liabilities from the entity's total assets. In essence, then, a balance sheet is really a list of balances remaining at any given point in time. This list usually gets categorised in a meaningful or useful way.

For example, a company having, say, $400,000 goes to a bank and borrows, say, $100,000 in order to buy a taxi business for $500,000, comprising $450,000 for the taxi licence plate and $50,000 for the motor vehicle.

Immediately after this purchase, the company's balance sheet would therefore be as follows:

			$
ASSETS

	Taxi Licence Plate		450,000
	Motor Vehicle		50,000
			–––––––
			500,000
LESS: LIABILITIES

	Bank Loan		100,000
			–––––––
EQUITY

	Issued Capital
		400,000 Shares at $1 each	400,000
			=======

In practice, the above formula is usually expanded and adapted to meet the nature of the business as well as the legal structure of the entity concerned. Assets, liabilities, and capital are also usually categorised into their main constituent subcategories.

Generally Accepted Accounting Principles (GAAP). As can be seen from the above example, the balance sheet is represented by the amounts as actually incurred, a practice which is patently logical and completely consistent with the historical cost accounting principle. In other words, the carrying amounts are their actual incurred costs.

Some public companies, though, play around with some of these carrying amounts, occasionally citing rather esoteric reasons for doing so. Most of the corporate scandals and fraud on Wall Street that came to light during 2002 had their origins to this playing around. A small minority of auditors, similarly fraudulent, who should have signalled these games, didn't, as they did not seemingly want to have their consulting, auditing, and other fees jeopardised. The main culprits involved in these scandals and fraud went out of business. Some were indicted for fraud and sent to jail.

There are, of course, many other accounting principles which need to be complied with. All these principles are referred to as the generally accepted accounting principles (GAAP), and they all govern how a balance sheet should be structured and calculated. Departures from GAAP are usually frowned upon by the accounting profession, and would result in a qualified audit statement if the balance sheet were subject to auditing requirements. However, in practice, not all departures from GAAP necessarily lead to a qualified audit statement.

Legislative Requirements. In addition to the generally accepted accounting principles, there are also certain legislative requirements that may need to be adhered to in drawing up a balance sheet. And of course, things can get quite tricky if these requirements conflict with GAAP, as some do.

Complications. Unfortunately, it is beyond the scope of this elementary explanation to elaborate further.

11. Net Profit. The net profit of an entity is generally calculated for a designated time period, usually one year. We will assume here that this time period is one year, even though it can conceptually be any duration.

In respect of an entity, such a net profit is usually based on the following formula:

Where, in respect of one year, "Sales" are all the sales of the entity, "Cost Of Goods Sold" is the cost incurred or associated with those sales, "Expenses" are all other expenses, and of course "Net Profit" is the resultant calculation or balance remaining.

"Cost Of Goods Sold" is sometimes referred to as the "Cost of Sales." The two expressions are usually used interchangeably, meaning the same thing.

		$	$
SALES

	Gross Sales		900,000

LESS: COST OF GOODS SOLD

	Opening Stock	100,000
	Add: Purchases	700,000
		–––––––
		800,000
	Less: Closing Stock	200,000
		–––––––
	Cost Of Goods Sold		600,000
			–––––––
	Gross Profit		300,000

LESS : EXPENSES

	All Other Expenses		200,000
			–––––––
NET PROFIT

	Net Profit for the year		100,000
			=======

In practice, the above formula is usually expanded and adapted to meet the nature of the business as well as the legal structure of the entity concerned.

The above "net profit" figure is the usual accounting meaning of "net profit," and it may differ from the fiscal authorities' definition.

Generally Accepted Accounting Principles (GAAP). As can be seen from the above example, the amounts used are represented by the amounts as actually incurred, a practice which is patently logical and completely consistent with the historical cost accounting principle.

In addition, the calculation must also comply with the matching principle, which requires that all sales, costs, and expenses be related solely to the period or year in respect of which the net profit calculation is made. This necessitates year end adjustments.

There are, of course, many other accounting principles which need to be complied with. All these principles are referred to as the generally accepted accounting principles (GAAP), and they all govern how net profit should be calculated. Departures from GAAP are usually frowned upon by the accounting profession, and would result in a qualified audit statement if the net profit calculation were subject to auditing requirements. However, in practice, not all departures from GAAP necessarily lead to a qualified audit statement.

Legislative Requirements. In addition to the generally accepted accounting principles, there are also certain legislative requirements that may need to be adhered to in calculating the net profit. And of course, things can get quite tricky if these requirements conflict with GAAP, as some do.

Complications. To calculate its yearly net profit, a firm needs good record keeping systems and procedures in place, and working properly. This is not as easy as it may sound. Though there are numerous firms that have computer based accounting systems, many of these systems simply do not work properly. In due course, when reality cannot finally be further covered up, these dud computer systems eventually get discarded at huge cost to the firm, by which time the persons responsible, together with all the money they got for the systems, would have long moved on elsewhere, to another firm, to repeat the process.

The problems of bad computer accounting systems are more acute in large organisations, especially in those where a computer guy is made responsible for operating the system, and where the accountant responsible for the financial data simply has to accept the output generated from the system and not create internal office waves by questioning the quality of that output. In this environment, approximate results get accepted repeatedly at all levels as the modus operandi, until such time as the whole process really hits the fan. Such an environment also nurtures and perpetuates a corporate culture of "second best" which, sooner or later, results in the firm coming to grief, either temporarily or terminally.

Unfortunately, it is beyond the scope of this elementary explanation to elaborate further.

12. Working Capital. The working capital of a business entity is generally calculated at any point in time, and is usually based on the following formula:

Where "Current Assets" are the assets that are expected to be converted to cash or cash equivalent within the ensuing 12 months, "Current Liabilities" are the liabilities that fall due for payment within the ensuing 12 months, and of course "Working Capital" is the resultant calculation or balance remaining.

Examples of current assets are cash on hand, bank accounts, accounts receivable (debtors), and inventories (stocks); and examples of current liabilities are salaries and wages, accounts payable (creditors), and rents payable. Short term finance from a bank is sometimes utilised in order to meet temporary shortfalls of working capital. Prudent banks usually secure this finance by some sort of mortgage over valuable assets.

Working capital constitutes the financial wherewithal or liquidity allowing a business enterprise to function and to carry on business on a day to day basis. Failure to have sufficient working capital could mean that an entity may not be able to pay its debts as and when they fall due. Such an entity could therefore become technically insolvent and, if actually unable to pay its debts when demanded to do so by a creditor, it could then be liable to bankruptcy proceedings provided certain legal conditions are satisfied, in accordance with the applicable bankruptcy laws. Bankruptcy proceedings, though, are a costly matter for all parties concerned, and not all application for bankruptcy are granted by the Courts. The clear winners are usually all the lawyers involved.

Complications. Unfortunately, it is beyond the scope of this elementary explanation to elaborate further.

13. Gross Profit Margin. The gross profit margin in respect of a business entity is normally calculated for a designated time period, normally one year, and is usually based on the following formula:

Where, in respect of one year, "Gross Sales" are all the sales of the entity, and "Gross Profit" is the balance remaining after having deducted the "Cost Of Goods Sold" from "Gross Sales," as illustrated above in respect of the calculation of Net Profit.

The gross profit margin is really a ratio, but its history of use has always denoted it as a margin, mainly to convey what's left over, i.e. the margin left over to defray all other expenses.

The gross profit margin, or ratio, is a very important tool in business management. Most businesses aim it to be about 33%, but the figure usually varies widely, not only between business sectors, but also obviously between firms. Slow moving items of stock usually have a higher gross profit margin than slower moving ones, for obvious reasons. They need to recover the extra costs of sitting around, such as interest costs, storage costs, and so on. These issues involve inventory management, a specialist are of management which is as much an art as a science. Like most things in life, some people are good at it, while some are not.

Complications. Unfortunately, it is beyond the scope of this elementary explanation to elaborate further.

Warning and Disclaimer on the above Financial Mathematics. Kindly consult your own financial mathematics adviser. Cosepp Corporation Pty Limited does not accept any responsibility of any kind whatsoever in relation to the above financial mathematics. Readers who rely on the above financial mathematics do so at their own risk. Refer also to the warning and disclaimer below and to the adjacent Table of Contents for further details on Cosepp Corporation Pty Limited's non acceptance of any responsibility of any kind whatsoever.

Warning. Readers or other who deal in the above stocks, and in those mentioned below or elsewhere in this publication, do so at their own risk. Refer also to the disclaimer below and to the adjacent Table of Contents for further details on Cosepp Corporation Pty Limited's non acceptance of any responsibility of any kind whatsoever. Any of the above stocks, those below, or any other, may be sold or purchased at any time, in any quantities and in any manner whatsoever.

Disclaimer. Cosepp Corporation Pty Limited does not provide advice on shares to the general public nor to clients in any way whatsoever. Cosepp Corporation Pty Limited, its officers, its employees, and any other person or entity mentioned herein accept or bear no responsibility for this publication's contents, accuracy, completeness, reliability, timeliness, or other. This publication is only about general discussion, and is not intended to constitute any form of recommendation in any way whatsoever, including any possible recommendation which may so be construed on what to buy or not buy or sell or not sell or hold or other, either similar or distant or directly or indirectly, or other. A person reading this publication should not rely on the information provided, but instead should discuss the issues raised with his or her own adviser who, necessarily having details of his or her individual particulars, would have due regard to his or her specific circumstances, requirements, and objectives.

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Internal Rate of Return: 7.59%
Calculation Date: 19/11/99
Date of Transaction	Value
Date of Transaction	$
27/02/95	-9234.99
30/07/96	-345.00
30/07/97	-500.00
30/07/98	+3456.00
30/07/99	+2345.15
19/11/99	+8000.00
Total	+3721.16