### TV10, Break-even number of periods

a. Find the break-even numbers of periods (1 period = 1 quarter = 3 months) for an initial investment of \$30000 and a cost of \$1500 per period with an expected revenue of \$3200 per period.
b. Redo the above taking into account that operating funds cost a nominal 8% per year, but the outstanding balance is computed monthly and the interest due for the previous month is added to the outstanding balance.

Break-even analysis without considering the effects of time on money is usually fairly simple if the positive and negative aspects of the variable effects are bases on averages per unit. The break-even quantity is found by dividing the fixed quantity by the contribution which is the net amount of the average positive - negative varying amounts. In the a. part above the contribution is \$3200 - \$1500 = \$1700 per period. The number of periods to break-even is 30000 / 1700 = 17.65. In practical terms one would use 18 as the break-even time.

Because this calculation is so simple it is often used to judge whether it is feasible to take some action. It is well to remember that this is the minimum and if cost of money effects are included in the break-even analysis it will require more units than without. The b. part of the question suggests a simple set of conditions that include time value of money in this type of problem. The answer to the b. part should be greater than 17.65. How much greater will depend on the interest rate.

If we think of the \$30000 initial investment as P, the Present Value, and the contribution \$1700 as A, the period amount then 17.65 represents P/A. It is well to remember that the maximum value of i for P/A = 17.65 occurs when n is infinite, or the number of periods is also at a maximum. The value of i for this condition is 1/17.65 = 0.0567, or 5.67%. This means that if the discount or interest rate used i the problem conditions exceed 5.67% per quarter then the break-even for the b. part will never happen.

If you consider what has been demonstrated above it is obvious that the simple break-even calculation can be dangerous because it may suggest that break-even may happen, when in fact it may not. Therefore many who use the simple break-even calculation for decision making tend to only consider it valid if the numbers show that it will happen quickly.

It is also relatively simple to decide the limit of acceptable break even times by knowing the value of 1 / discount rate, or 1 / i. E.g., if the discount rate is 12% per year applied on the monthly balance then the effective rate is .12 / 12 = .01 and the maximum number of months for a break-even is 1 / .01 = 100. 100 months is a 8 years 4 months. A relatively short time for some kinds of investments such as real-estate, or heavy equipment.

Recall that Income Tax affects the discount rate, and one must carefully examine each situation to decide whether before or after tax minimum attractive rates of return are used as the discount rate. It is usually safer to err on the side of high rather low rate.

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