Worked Example: Mortgages, Loans, and Repayments



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Suppose you enter into a loan agreement to borrow $90,000 to help finance the purchase of your new home. The agreement specifies a term of 20 years with monthly repayments determined at a fixed rate of 9% p.a. (compounding monthly).

(a) What is the amount of your monthly repayments?

To compute the amount of the monthly repayment, set the present value of the series of repayments (an annuity) equal to the amount borrowed.

|Summary Information: Original Loan |

|Original Loan Amount (PV) |90,000 |

|Monthly Repayment (a) |? |

|Monthly Interest Rate (i) |0.075% (the annual rate divided by 12) |

|Number of Repayments (n) |240 (20 years of monthly repayments) |

This information allows us to apply the formula for the present value of an annuity and compute the monthly repayment:

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in which case the monthly repayment, a, is $809.75.

(b) Now suppose that exactly five years have passed (you made the 60th repayment yesterday). A rival lender offers to refinance your loan at a fixed rate of 8% p.a. (compounding monthly). Costs associated with this refinancing amount to $1,500. Should you refinance?

To determine whether it makes sense to refinance, simply compute the monthly repayment required under the rival lender's terms and compare this with the monthly repayment you are making under the existing loan. The amount you would have to borrow from the rival lender is the principal outstanding on the original loan plus any refinancing costs. The principal outstanding at the end of five years is determined by computing the present value of all remaining repayments:

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With refinancing costs of $1,500 the specifics of the new agreement are:

|Summary Information: After Refinancing |

|Original Loan Amount (PV) |1,500 + 79,836 = 81,336 |

|Monthly Repayment (a) |? |

|Monthly Interest Rate (i) |0.067% (the annual rate divided by 12) |

|Number of Repayments (n) |180 (15 years of monthly repayments) |

To determine the new repayment amount we again apply the annuity formula to find a monthly payment scheme that has a present value equal to the amount of the loan:

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in which case the monthly repayment, a, is $779.17. Therefore you should refinance since this would save you 809.75 – 779.17 = $30.58 on each monthly repayment.

(c) Now suppose that exactly nine years have passed (you made the 108th repayment yesterday). You are considering making an extra payment of $10,000 off your loan. If you plan to keep the term of the loan the same, by how much will your monthly repayments reduce?

First determine the outstanding principal after nine years:

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The lump sum payment will reduce this principal to 68,127.17 – 10,000 = $58,127.17. In this case the unknown is the new repayment amount required (a). Solving for this in the standard way yields:

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in which case the monthly repayment, a, is $664.80. The reduction in the monthly repayment in this case is 779.17 – 664.80 = $114.37.

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