# How to Calculate a Crossover Rate

by Gerald Hanks; Updated September 26, 2017Since most businesses do not have unlimited resources, managers must choose how to allocate their limited resources for their various projects. Project A may deliver slow and steady returns with minimal risks, while Project B may deliver faster profits but at a higher risk. The **crossover rate** helps these managers evaluate the profits that each project will bring relative to its risk factors. Managers can then present the data to investors and show them the relative value of each potential project.

## Net Present Value

A key factor in calculating the crossover rate is the **net present value**, or **NPV**. Managers find the NPV by calculating the **present value** (PV) of the total revenues and costs of a project. Since the future revenues must be adjusted for its discount rate, the value of each year of future revenue must be discounted. The formula for NPV looks like this:

NPV = (SUM(C_{t}/(1+r)^{t}))-C_{0}

Where C_{t} = cash inflow at time period t

t = number of time periods

r = discount rate

C_{0} = initial cash outflow

*Example*: Golf-Hotel-Igloo.com wants to invest in a new resort management software system. The Bravo-Charlie system (B) will cost $200,000. This system will help the site bring in $50,000 in the first year, $75,000 in the second year and $100,000 in the third year. The discount rate is 4 percent.

NPV(B) = [(50,000/1.04) + (75,000/(1.04)^{2}) + (100,000/(1.04)^{3})] - 200,000 = $6,318.27

Another system, the Yankee-Zulu system (Y), will cost $250,000. This system will bring in $50,000 in the first year, $100,000 in the second year, and $150,000 in the third year. The discount rate is 4 percent.

NPV(Y) = [(50,000/1.04) + (100,000/(1.04)^{2}) + (150,000/(1.04)^{3})] - 250,000 = $23,882.00

## Internal Rate of Return

Another factor used to calculate the crossover rate is the internal rate of return, or IRR. **The IRR measures the rate of return an investment gives based on the initial cash outflow and the subsequent cash inflows**. The IRR can be found by using the NPV formula, setting the NPV to zero and solving for the discount rate.

For the Bravo-Charlie software package:

[(50,000/(1+IRR(B)) + (75,000/(1+IRR(B))^{2}) + (100,000/(1+IRR(B))^{3})] - 200,000 = 0 => IRR (B)= 5.4853%

For the Yankee-Zulu software package:

[(50,000/(1+IRR(Y)) + (100,000/(1+IRR(Y))^{2}) + (150,000/(1+IRR(Y))^{3})] - 250,000 = 0 => IRR(Y) = 8.2083%

## Crossover Rate Calculation

The crossover rate (CR) is the discount rate at which **both projects deliver the same net present value**. The crossover rate formula is the same as that for the IRR, but each factor is replaced by the difference between the projects. In this example, we use the Bravo-Charlie package and the Yankee-Zulu (Y) package.

C_{0}(Y-B) = 250,000 - 200,000 = 50,000

C_{1}(Y-B) = 50,000 - 50,000 = 0

C_{2}(Y-B) = 100,000 - 75,000 = 25,000

C_{3}(Y-B) = 150,000 - 100,000 = 50,000

[(0/(1+CR) + (25,000/(1+CR)^{2}) + (50,000/(1+CR)^{3})] - 50,000 = 0 => CR = 16.5374%

Both projects would deliver the same net present value at a discount rate of 16.5374 percent.

#### References

- The Free Dictionary: Crossover Rate
- Calculator Soup: Net Present Value (NPV) Calculator
- McGraw Hill Education: Calculating the Crossover Rate
- Calkoo: Internal Rate of Return (IRR) Calculator
- University of West Georgia: The Basics of Capital Budgeting - Evaluating Cash Flows
- Springer International Publishing: Derivation of Crossover Rate