Essay: Compound interest

Answer for Q1.

COMPOUND INTEREST: The initial point for understanding the time value of money is to develop an appreciation for compound interest. While it is not clear that Albert Einstein actually held compound interest in such high regard, it is clear that understanding the forces of compound interest is a powerful tool. Very simply, money can be invested to earn money. In this context, consider that when you spend a dollar on a soft drink, you are actually foregoing 10� per year for the rest of your life (assuming a 10% interest rate). And the annual dime of savings builds to much more because of interest that is earned on the interest! This is the almost magical power of compound interest.

Compound interest calculations can be used to compute the amount to which an investment will grow in the future. That is why it’s also called future value. If you invest $1 for one year, at 10% interest per year, how much will you have at the end of the year? The answer, of course, is $1.10. This is calculated by multiplying the $1 by 10% ($1 X 10% = $0.10) and adding the $0.10 to the original dollar. And, if the resulting $1.10 is invested for another year at 10%, how much will you have? The answer is $1.21. That is, $1.10 X 110%. This process will continue, year after year.

The annual interest each year is larger than the year before because of “compounding.” Compounding simply means that your investment is growing with accumulated interest, and you are earning interest on previously accrued income of interest that becomes part of your total investment pool. This formula expresses the basic mathematics of compound interest:

(1+i)n

Where “i” is the interest rate per period and “n” is the number of periods

So, how much would $1 grow to in 25 years at 10% interest? The answer can be determined by taking 1.10 to the 25th power [(1.10)25], and the answer is $10.83. Future value tables provide predetermined values for a variety of such computations (such a table is found at this FUTURE VALUE OF $1 link). To experiment with the future value table, determine how much $1 would grow to in 10 periods at 5% per period. The answer to this question is $1.63, and can be found by reference to the value in the “5% column/10-period row.” If the original investment was $5,000 (instead of $1), the investment would grow to $8,144.45 ($5,000 X 1.62889).

The “period” might be years, quarters, months, etc. It all depends on how frequently interest is to be compounded. For instance, a 12% annual interest rate, with monthly compounding for two years, would require you to refer to the 1% column (12% annual rate equates to a monthly rate of 1%) and 24-period row (two years equates to 24 months). If the same investment involved annual compounding, then you would refer to the 12% column and 2-period row. The frequency of compounding makes a difference in the amount accumulated — for the given example, monthly compounding returns 1.26973, while annual compounding returns only 1.25440!

PRESENT VALUE: Future value calculations provide useful tools for financial planning. But, many decisions and accounting measurements will be based on a reciprocal concept known as present value. Present value (also known as discounting) determines the current worth of cash to be received in the future. For instance, how much would you be willing to take today, in lieu of $1 in one year. If the interest rate is 10%, presumably you would accept the sum that would grow to $1 in one year if it were invested at 10%. This happens to be $0.90909. In other words, invest 90.9� for a year at 10%, and it will grow to $1 ($0.90909 X 1.1 = $1). Thus, present value calculations are simply the reciprocal of future value calculations:

1/(1+i)n

Where “i” is the interest rate per period and “n” is the number of periods

The PRESENT VALUE OF $1 TABLEreveals predetermined values for calculating the present value of $1, based on alternative assumptions about interest rates and time periods. To illustrate, a $25,000 lump sum amount to be received at the end of 10 years, at 8% annual interest, with semiannual compounding, would have a present value of $11,410 (recall the earlier discussion, and use the 4% column/20-period row — $25,000 X 0.45639).

Answer for Q2:

Breakeven analysis is a great management tool, and one that is significant in planning, decision-making, and expense control. Breakeven analysis can be invaluable in determining whether to buy or lease, expand into a new area, build a new plant, and many other such considerations. Breakeven analysis can also show the impact on your business of changing your price structure. As the price goes down (and so your gross margin goes down), breakeven shoots up – usually very rapidly. Breakeven analysis will not force a decision, of course, but it will provide you with additional insights into the effects of important business decisions on your bottom line.

Breakeven refers to the level of sales necessary to cover all of the fixed and variable costs.

Fixed costs are those costs or expenses that are expected to remain fairly constant over a reasonable period of time. These costs are relatively unaffected by changes in output or sales up to the point where the level of operation reaches the capacity of the existing facilities. At that point, major variety of changes would have to be taken, such as the expansion of existing plant and equipment or the construction of new facilities. Such actions would increase the fixed costs.

However, under normal operating conditions, the fixed costs (also referred to as indirect costs, overhead, or burden) will remain constant. Some examples of fixed costs include rent or mortgage payments, interest on loans, executive and office salaries, and general office expenses.

Variable costs are those costs or expenses that vary or change directly with output. These costs are associated with production and/or selling and are frequently identified as “costs of goods sold.” As compared with the fixed costs, which continue whether the firm is doing business or not, variable costs do not exist if the firm is not doing business. Thus, by definition, variable costs are zero when no output is being produced. At that time, fixed costs are the only costs that will be incurred. Examples of variable costs include cost of goods sold, factory labour, and sales commissions.

However, most overhead costs for most businesses are fixed over a volume of productionand there fore fixed costs. However, some costs have an element of variable and fixed cost elements called semi-fixed or semi-variable costs. These costs have to be separated using statistical regression analysis. That is the costing system has to produce for each product what is the unit variable cost, selling price of each unit, fixed cost for a period. Maximum sales possible, which is estimated for a future period.

Then one can determine the production point where the profit is zero. For some products the break-even point will be at higher level and for some products the break even point will be at a lower level of production. As well, the margin of safety that the excess profit that can be earned after the break even point also varies. There fore, to maximize profit earned from each product is to reduce variable cost and reduce overhead and increase sales by cost effective promotions and advertising and improving the quality of the products compared to its competitors. There fore break-even analysis gives a tool for a manger to analyze the mix of products that maximize profit for a period and have cost control systems so that it can minimize waste and improve productivity of labor force and streaming production methods and operations.

In effect break even analysis enable business managers to make effective decisions based on sound rational basis and based on cost information and other limiting factors. As well, it gives the manager how a manger can improve profitability of the business as a whole in a dynamic and uncertain market place my monitoring cost and improving the efficiency of the organization on a continuous basis.

Answer for Q3:

Even Period:-

Example:

The data below shows the sales for a certain product of a company:

Calculate the a suitable moving average to isolate the trend, then draw the time series and the trend on separate graphs to isolate the trend.

Comment on your graphs.

Answer:-

Odd period:

The following data gives the total profits of a company:

_calculate the 5 year moving average for these data.

_plot the moving average and the trend on different graphs.

Comment on what you see.

Answer:-

References:

� http://www.buzgate.org/8.0/nh/ft_beven.html

� http://www.cliffsnotes.com/study_guide/Production-Costs-and-Firm-Profits.topicArticleId-9789,articleId-9759.html