# Basis of Calculation for HPA Refund for Borrower-Paid Single-Premium Plans

In the sections below, the following symbols are used:

Symbol | Definition |
---|---|

S | Starting month of calculation July 2014 |

T_{earn} |
Earning period in months plus 1 |

n | Loan term in months |

F | Factor for monthly earnings |

P | Premium amount |

p | Monthly payment |

a | Original principal balance |

r | Annual interest rate / 12 |

a_{78} |
Loan balance at 78% LTV |

n_{78} |
Number of payments remaining at 78% LTV |

M | Number of (current) months |

EP_{m} |
Earnings for month |

### Single Premium Certificates—Borrower Paid

**To determine the amount of the refund due for certificates issued on or after July 1, 2014, subtract the earned premium as calculated below from the single premium paid:**

Value | Calculation |
---|---|

p = | a × r × (1+r)^{n} / ((1+r)^{n}-1) |

a_{78} = |
78 / 100 × original loan amount / original LTV percentage |

n_{78} = |
1n (p / (p - a_{78} × r)) / 1n (1+r) |

F = | 1 / (T_{earn} + 12) |

For Earnings for Month (EP_{m}), the calculation rule is as follows:

Month_{m} |
Earnings in month m EP_{m} |

1 | F × P |

2 – 12 | 2 × F × P |

13 | 1.5 × F × P |

14 – T_{earn} |
F × P |

T_{earn} + 1 |
p-sum of all previous EP_{m} |

### For certificates with effective dates prior to July 1, 2014:

The calculation of F (factor for monthly earnings) will depend on s (starting month of calculation) = month in force at July 1, 2014.

Starting month s | Factor F |
---|---|

S = 1 (this is listed as a comparison) | F = 1 / (T_{earn} - s + 1 + 12) = 1/( T_{earn} +12) as described above |

S = 2 – 13 | F = 1/( T_{earn} - s + 1 + 12 + 2 - s) = 1 / ( T_{earn} - 2s +15) |

S = 14 – T_{earn} - 1 |
F = 1/( T_{earn} – s + 1 + 0.5) = 1/( T_{earn} – s + 1.5) |

S > = T_{earn} |
F = 1 |

### Annual Certificates

Annual premium certificates will be earned over 13 months, with the following distribution, where P is the premium amount paid at the last anniversary:

Month m | Earnings Per Month EP_{m} |
---|---|

1 | 0.5 × P / 12 = P / 24 |

2 | P / 12 |

3 | P / 12 |

4 | P / 12 |

5 | P / 12 |

6 | P / 12 |

7 | P / 12 |

8 | P / 12 |

9 | P / 12 |

10 | P / 12 |

11 | P / 12 |

12 | P / 12 |

13 | 0.5 × P / 12 = P / 24 |

For questions about Arch MI’s calculation method, send an email to the Servicing team at servicing@archmi.com.