## Case Math

(Source: Flickr)

In this post we outline some mathematical concepts that may prove useful for solving consulting case questions.

## 1. Break Even Analysis:

Relevant when trying to decide whether to launch a new product or invest in a project with high fixed costs.

Customer lifetime value is a prediction of the entire future value that a company expects to derive from its relationship with a customer. It is a useful tool for a company that is trying to decide which customer segments to target and how much to spend on customer acquisition.

## 3. Net Present Value:

The NPV of an investment is the present value of the series of expected future cash flows generated by the investment minus the cost of the initial investment.

Where r = discount rate; CFt = expected cash flow in year t; CFn = expected cash flow in final year; g = long term cash flow growth rate.

## 4. Perpetuity:

A perpetuity is a constant stream of identical cash flows with no end.

## 5. Price elasticity of demand:

Price elasticity of demand is a measure of the responsiveness of quantity demanded to a change in price, and is relevant when formulating pricing strategy.

If demand is elastic (Ed > 1) then changes in price will have a relatively large effect on the quantity demanded, and total revenue will rise if prices are lowered.

If demand is inelastic (Ed < 1) then changes in price will have a relatively small effect on the quantity demanded, and total revenue will rise if prices are raised.

## 6. Product life cycle:

The product life cycle is relevant when calculating the expected lifetime revenue of a product.

## 7. Profit Margin:

Gross Profit Margin: Gross profit margin measures how much of every dollar of sales revenue remains after subtracting the cost of goods sold.

Net Profit Margin: Net profit margin measures how much out of every dollar of sales revenue a company actually keeps. Net profit margin is useful when comparing companies in similar industries. A higher net profit margin indicates a more profitable company that has better control over its costs compared to its competitors.

Contribution Margin: A cost accounting concept that allows a company to determine the profitability of individual products.

## 8. Return on Investment:

ROI is a performance measure that a company can use to evaluate the efficiency of an investment or to compare a number of different investments.

## 9. Rule of 70:

The Rule of 70 is a simple rule of thumb that can be used to figure out roughly how long it will take for an investment to double, given an expected growth rate.

The rule can be described by the following equation:

## Product Life Cycle Model

The Product Life Cycle Model can be used to analyse the maturity stage of products and industries

### 1. Background

THE idea of the Product Life Cycle was first developed in 1965 by Theodore Levitt in an article entitled “Exploit the Product Life Cycle” published in the Harvard Business Review on 1 November 1965.

### 2. Benefit of the Product Life Cycle model

For a business, having a growing and sustainable revenue stream from product sales is important for the stability and success of its operations. The Product Life Cycle model can be used by consultants and managers to analyse the maturity stage of products and industries. Understanding which stage a product is in provides information about expected future sales growth, and the kinds of strategies that should be implemented.

### 3. Product Life Cycle model

The “Product Life Cycle” is the name given to the stages through which a product passes over time. The classic Product Life Cycle has four stages:

3.1 Introduction

At the market introduction stage the size of the market, sales volumes and sales growth are small. A product will also normally be subject to little or no competition. The primary goal in the introduction stage is to establish a market and build consumer demand for the product.

There may be substantial costs incurred in getting a product to the market introduction stage. Substantial research and development costs may have been incurred, for example, thinking of the product idea, developing the technology, determining the product features and quality level, establishing sufficient manufacturing capacity, preparing the product branding, ensuring trade mark protection, etc. Marketing costs may be high in order to test the market, launch and promote the product, develop a market for the product, and set up distribution channels.

The market introduction stage is likely to be a period of low or negative profits. As such, it is important that products are carefully monitored to ensure that sales volumes start to grow. If a product fails to become profitable it may need to be abandoned.

Some of the considerations in the introduction stage include:

• Product development: research and development of the basic technology and product concept, determining the product features and quality level.
• Pricing: should penetration pricing or a skimming price strategy be used? A skimming price strategy might be appropriate where there are very few competitors.
• Distribution: distribution might be quite selective until consumer acceptance of the product can be achieved.
• Promotion: marketing efforts are aimed at early adopters, and seek to build product awareness and to educate potential consumers about the product.

3.2 Growth

If the public gains awareness of a product and consumers come to understand the benefits of the product and accept it then a company can expect a period of rapid sales growth, enter the “Growth Stage”. In the Growth Stage, a company will try to build brand loyalty and increase market share.

Profits are driven by increased sales volume (due to growth in market share as well as an increase in the size of the overall market). Profits might also be driven by cost reductions gained from economies of scale, and perhaps more favourable market prices. Competition in the Growth Stage remains low, although new competitors are expected to enter the market. When competitors enter the market a company might be subject to price competition and increase its marketing expenditure.

Some of the considerations in the Growth Stage include:

• Product improvement: product quality might be improved, additional features and support services added, and packaging updated.
• Pricing: if consumer demand is high the price might be maintained at a high level.
• Distribution: distribution channels might be added as consumer demand increases.
• Promotion: promotion is aimed at a broader audience. A company might spend a lot of resources on promotion during the Growth Stage to build brand loyalty.

3.3 Maturity

When a product reaches maturity, sales growth slows and sales volume eventually peaks and stabilises. This is the stage during which the market as a whole makes the most profit. A company’s primary objective at this point is to defend market share while maximising profit.

In this stage, prices tend to drop due to increased competition. A company’s fixed costs are low because it is has well established production and distribution. Since brand awareness is strong, marketing expenditure might be reduced, although increased marketing expenditure might be needed to retain market share and fight increasing competition. Expenditure on research and development is likely to be restricted to product modification and improvement, and perhaps research into improved production efficiency and product quality.

Some considerations for the mature product market include:

• Product differentiation: increased competition in the mature product market means that a company must find ways to differentiate its product from that of competitors. Strong branding is one way to do this.
• Pricing: prices may be reduced because of increased competition. Firms in the market should be careful not to start a price war.
• Distribution: distribution intensifies and incentives may be offered to encourage preference to be given over competing products.
• Promotion: promotion will focus on emphasising product differences and creating/maintaining a strong brand.

3.4 Decline

A product enters into decline when sales and profits start to fall. The market for that product shrinks which reduces the amount of profit available to the firms in the industry. A decline might occur because the market has become saturated, the product has become obsolete, or customer tastes have changed.

A company might try to stimulate growth by changing their pricing strategy, but ultimately the product will have to be re-designed, or replaced. High-cost and low market share firms will be forced to exit the industry.

As sales decline, a company has three strategy options:

• Hold: maintain production and add new features and find new uses for the product. Reduce the cost of manufacturing (e.g. move manufacturing to a low cost jurisdiction). Consider whether there are new markets in which the product might be sold.
• Harvest: continue to offer the product, reduce marketing expenditure, and sell possibly to a loyal niche segment of the market.
• Divest: Discontinue production, and liquidate the remaining inventory or sell the product to another firm.

Some considerations for a declining market include:

• Product consolidation: the number of products may be reduced, and surviving products rejuvenated.
• Price: prices may be lowered to liquidate inventory, or maintained for continued products.
• Distribution: distribution becomes more selective. Channels that are no longer profitable are phased out.
• Promotion: Expenditure on promotion is reduced for products subject to the Harvest and Divest strategies.

### 4. Criticisms

The Product Life Cycle is useful for monitoring sales results over time and comparing them to products with a similar life cycle. However, the Product Life Cycle model is by no means a perfect tool. Products often do not follow a defined life cycle, not all products go through each stage, and it is not always easy to tell which stage a product is in at any one time. Consequently, the life cycle concept is not well-suited for the forecasting of product sales.

The length of each stage will vary depending on the product and the marketing strategies employed. A Product Life Cycle may be as short as a few months for a fad or as long as a century or more for a product like petrol cars. In many markets the product life cycle is longer than the planning cycle of the organisations involved. Major products often hold their position for several decades or more, indeed, Coca-Cola was introduced in 1886 and is still the leading brand of cola.

The Product Life Cycle is only one of many considerations that a company must bear in mind. The product life cycle of many modern products is shrinking, while the operating life for many of these products is lengthening. For example, the operating life of durable goods like household appliances has increased substantially. As a result, a company that produces these products must take their market life and service life into account when planning.

Some critics have argued that the Product Life Cycle may become self-fulfilling. For example, if sales peak and then decline a manager may conclude that a product is on the decline and cut back on marketing, thus precipitating a further decline.