Barclay Brothers Company is a large manufacturer of adult parlor games. Its marketing vice president, Rudy Barclay, must make the decision whether to introduce a new game called ­Strategy into the competitive market. Naturally, the company is concerned with costs, potential demand, and profit it can expect to make if it markets Strategy.

Rudy identifies the following relevant costs:

The selling price per unit is set at $10.

The break-even point is the number of games sold at which total revenues are equal to total costs. It can be expressed as follows:¹

So in Barclay’s case,

Any demand for the new game that exceeds 6,000 units will result in a profit, whereas a demand less than 6,000 units will cause a loss. For example, if it turns out that demand is 11,000 games of Strategy, Barclay’s profit would be $30,000:

If demand is exactly 6,000 games (the break-even point), you should be able to compute for yourself that profit equals $0.

Rudy Barclay now has one useful piece of information that will help him make the decision about introducing the new product. If demand is less than 6,000 units, a loss will be incurred. But actual demand is not known. Rudy decides to turn to the use of a probability distribution to estimate demand.

Actual demand for the new game can be at any level—0 units, 1 unit, 2 units, 3 units, up to many thousands of units. Rudy needs to establish the probability of various levels of demand in order to proceed.

In many business situations, the normal probability distribution is used to estimate the demand for a new product. It is appropriate when sales are symmetric around the mean expected demand and follow a bell-shaped distribution. Figure M3.1 illustrates a typical normal curve that we discussed at length in Chapter 2. Each curve has a unique shape that depends on two factors: the mean of the distribution $\left(\mu \right)$ and the standard deviation of the distribution $\left(\sigma \right).$

For Rudy Barclay to use the normal distribution in decision making, he must be able to specify values for $\mu$ and $\sigma$. This isn’t always easy for a manager to do directly, but if he or she has some idea of the spread, an analyst can determine the appropriate values. In the Barclay example, Rudy might think that the most likely sales figure is 8,000 but that demand might go as low as 5,000 or as high as 11,000. Sales could conceivably go even beyond those limits; say, there is a 15% chance of being below 5,000 and another 15% chance of being above 11,000.

Because this is a symmetric distribution, Rudy decides that a normal curve is appropriate. In Chapter 2, we demonstrated how to take the data in a normal curve such as Figure M3.2 and compute the value of the standard deviation. The formula for calculating the number of standard deviations that any value of demand is away from the mean is

where Z is the number of standard deviations above or below the mean, $\mu .$ A standard Z-normal table is provided in Appendix A.

We see that the area under the curve to the left of 11,000 units demanded is 85% of the total area, or 0.85. From Appendix A, the Z value for 0.85 is approximately 1.04. This means that a demand of 11,000 units is 1.04 standard deviations to the right of the mean, $\mu$.

With $\mu =8,000, Z=1.04,$ and a demand of 11,000, we can easily compute $\sigma$:

or

or

or

At last, we can state that Barclay’s demand appears to be normally distributed, with a mean of 8,000 games and a standard deviation of 2,885 games. This allows us to answer some questions of great financial interest to management, such as what the probability is of breaking even. Recalling that the break-even point is 6,000 games of Strategy, we must find the number of standard deviations from 6,000 to the mean:

This is represented in Figure M3.3. Because Appendix A is set up to handle only positive Z values, we can find the Z value for $\mathrm{+}0.69,$ which is 0.7549 or 75.49% of the area under the curve. The area under the curve for $\mathrm{-}0.69$ is just 1 minus the area computed for $\mathrm{+}0.69,$ or $1-\mathrm{0.7549.}$ Thus, 24.51% of the area under the curve is to the left of the break-even point of 6,000 units. Hence,

The fact that there is a 75% chance of making a profit is useful management information for Rudy to consider.

Before leaving the topic of break-even analysis, we should point out two caveats:

1. We have assumed that demand is normally distributed. If we find that this is not reasonable, other distributions may be applied. These distributions are beyond the scope of this book.

2. We have assumed that demand is the only random variable. If one of the other variables (price, variable cost, or fixed costs) were a random variable, a similar procedure could be followed. If two or more variables are both random, the mathematics becomes very complex. This is also beyond our level of treatment. Simulation (see Chapter 13) could be used to help with this type of situation.

In addition to knowing the probability of suffering a loss with Strategy, Barclay is concerned about the expected monetary value (EMV) of producing the new game. He knows, of course, that the option of not developing Strategy has an EMV of $0. That is, if the game is not produced and marketed, his profit will be $0. If, however, the EMV of producing the game is greater than $0, he will recommend the more profitable strategy.

To compute the EMV for this strategy, Barclay uses the expected demand, $\mu$, in the following linear profit function:

Rudy has two choices at this point. He can recommend that the firm proceed with the new game; if so, he estimates there is a 75% chance of at least breaking even and an EMV of $12,000. Or he might prefer to do further market research before making a decision. This brings up the subject of the expected value of perfect information.