# Crossover Rate

Discount rate for equal NPVs

Discount rate for equal NPVs

Crossover Rate is the rate of return (alternatively called weighted average cost of capital) at which the Net Present Values (NPV) of two projects are equal. It represents the rate of return at which the net present value profile of one project intersects the net present value profile of another project.

In capital budgeting analysis exercises, crossover rate is used to show when one investment project becomes superior to another as a result of a change in the rate of return (cost of capital).

Consider the following example. An investor is faced with 2 investment projects: Project ABC and Project XYZ. Project ABC has an initial investment of A_{0} and will generate a steady stream of income over a 3-year period. The cash flows from Project ABC at the end of the first, second and third years are denoted by C_{1}, C_{2}, and C_{3, }respectively. Thus the Net Present Value of Project ABC is as follows:

Project XYZ has an initial investment of X_{0} and will generate a steady stream of income over a 2-year period. The cash flows from Project XYZ at the end of the first and second years are denoted by Z_{1 }and Z_{2}, respectively. Thus the Net Present Value of Project XYZ is as follows:

Crossover rate is defined as that rate of return at which the NPV_{ABC} = NPV_{XYZ}. Since C_{1}, C_{2}, C_{3}, A_{0}, Z_{1}, Z_{2}, and X_{0 }are all known, we can solve for the Crossover Rate.

- Determine the cash flow streams of the two projects.
- Find the difference in the initial investments between the two projects.
- Find the difference in the cash flow in each period between the two projects. (After a project ends, it generates zero cash flow in all subsequent periods).
- Develop an IRR equation by equating the net present value equation of the resulting differential cash flows to zero.
- Solve the equation for r.

Let us continue with the example above. An investor is faced with 2 investment projects, Project ABC and Project XYZ. The first project, Project ABC costs $5 million and will result in a cash flow of $6 million at the end of the first year, $4 million at the end of the second year and $2 million at the end of the third year. The second project, Project XYZ costs $8 million but will generate cash flow of $25 million at the end of the second year.

For simplicity let 1+r = v. Therefore the above equation reduces to the following:

We can solve the above equation using the cubic formula. Because it is a cubic equation, it will have 3 roots that satisfy it. For the above equation v= -3.85, 1.76 and 0.098. Therefore 1+r = -3.85, 1.76 or 0.098. Since the rate of return can never be negative, we can ignore the first and the third root. Therefore 1+r = 1.76 or r=0.76 or 76%.

We can arrive at the above solution using software such as Microsoft Excel and plot the NPV profiles of the two different projects and observe their point of intersection.

r |
NPV_{ABC} |
NPV_{XYZ} |

0.05 | 6.070078825 | 14.67573696 |

0.1 | 5.26296018 | 12.66115702 |

0.15 | 4.556998438 | 10.90359168 |

0.2 | 3.935185185 | 9.361111111 |

0.25 | 3.384 | 8 |

0.3 | 2.892580792 | 6.792899408 |

0.35 | 2.452116039 | 5.717421125 |

0.4 | 2.055393586 | 4.755102041 |

0.45 | 1.69646152 | 3.890606421 |

0.5 | 1.37037037 | 3.111111111 |

0.55 | 1.07297506 | 2.405827263 |

0.6 | 0.80078125 | 1.765625 |

0.65 | 0.550825055 | 1.182736455 |

0.7 | 0.320578058 | 0.650519031 |

0.75 | 0.10787172 | 0.163265306 |

0.8 | -0.089163237 | -0.283950617 |

0.85 | -0.272145776 | -0.695398101 |

0.9 | -0.442484327 | -1.074792244 |

0.95 | -0.601409329 | -1.425378041 |

From the above graph, we can see that the two NPV profiles intersect with each other at around r=0.76 or 76%, which is the same answer that we got when we solved the cubic equation. Thus, if the rate of return or cost of capital is less than 0.76, Project XYZ is superior to Project ABC. However, if the rate of return is more than 0.76, Project ABC is superior to Project XYZ.

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