## 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:

## NPV: Net Present Value

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

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### 1. Net Present Value (NPV) explained

THE net present value (NPV) of an investment is the present value of the series of cash flows generated by the investment minus the cost of the initial investment. Each cash inflow/outflow is discounted back to its present value and then summed together:

Where t is the time of the cash flow; r is the discount rate (see below for further explanation); Ct is the net cash flow (cash inflow minus cash outflow) at time t; and Co is the cost of the initial investment made at time zero.

NPV is used to analyse the profitability of an investment. As a general rule, assuming you have selected an appropriate discount rate, only those investments that yield a positive NPV should be considered for investment.

### 2. The discount rate

The rate used to discount future cash flows to their present value is an important variable in the net present value calculation. To some extent, the selection of the discount rate depends on the use to which the NPV calculation will be put.

2.1 Option 1 – cost of capital:

One option that is often used is to use a firm’s weighted average cost of capital.

There are two problems with using the cost of capital for the discount rate. Firstly, it may not be possible to know what the cost of capital will be in the future. One solution to this problem is to use a variable discount rate that increases over time to reflect the yield curve premium for long-term debt. A yield curve is the relation between the interest rate (or cost of borrowing) and the time to maturity and is usually upward sloping asymptotically. There are two common explanations for why the yield curve is upward sloping. Firstly, it might be that rising interest rates are expected in the future and investors who are willing to lock their money in now therefore need to receive a higher rate of interest. Secondly, even if interest rates are not expected to rise, longer maturities involve greater risks to an investor and so, to compensate for these inherent uncertainties about the future, a risk premium should be paid.

The second problem with using the cost of capital for the discount rate is that it does not take into account opportunity costs. A positive NPV calculation would tell us that the investment is profitable, but would not tell us whether the investment should be undertaken because there may be more profitable investment opportunities.

2.2 Option 2 – opportunity cost:

A second option is to use a discount rate that reflects the opportunity cost of capital. The opportunity cost of capital is the rate which the capital needed for the project could return if invested in an alternative venture. Obviously, where there is more than one alternative investment opportunity, the opportunity cost of capital is the expected rate of return of the most profitable alternative.

For example, assume that a firm has two investment options, investing in Project A (its existing line of business) or Project B (a new line of business). Based on past experience, the firm knows that it can obtain a 15% return from investing in Project A. This means that the opportunity cost of capital for investing in Project B is 15%. Thus, an NPV calculation for Project B will use a discount rate of 15%.

### 3. Common pitfalls

3.1 Negative future cash flows:

One potential problem with NPV is that if, for example, the future cash flows are negative (for example, a mining project might have large clean-up costs towards the end of a project) then a high discount rate is not cautious but too optimistic. A way to avoid this problem is to explicitly calculate the cost of financing any losses after the initial investment.