A couple purchased a home and signed a mortgage contract for $900,000 to be paid with half-yearly payments over a 25-year period. The interest rate
applicable is j2 = 5.5% p. a. applicable for the first five years, with the condition
that the interest rate will be increased by 12% every 5 years for the remaining
term of the loan.
(a) Calculate the half-yearly payment required for each five-year interval​

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Step-by-step explanation:

The amortization formula can be used to find the payment value.

  A = P(r/2)/(1 -(1 +r/2)^(2n))

where P is the principal amount at the beginning of the loan period, r is the annual interest rate, n is the number of years remaining on the loan

The value of A in this scenario is the semi-annual payment.

Initially, the interest rate is 0.055 and the number of years remaining is 25.

After the first 5-year period, the interest rate goes up by 12%, so is multiplied by a factor of 1.12 to become 6.16% per year. The same growth factor is applied to the interest rate at the beginning of each 5-year period.

Of course, the number of years remaining is decreased by 5 years at the beginning of the next 5-year period.

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The Principal remaining at the end of each 5-year period is the starting principal for the next period. It is calculated from ...

  FV = P(1 +r/2)^10 -A((1 +r/2)^10 -1)/(r/2)

Where P is the starting principal, A is the loan payment as calculated above, and r is the annual interest rate.

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These formulas are built into spreadsheet functions, so the desired set of loan payments can be calculated easily by that technology. The result is attached. In the "# pmts" column is the value used to amortize the loan for the 5-year period. The semiannual payment is calculated as though the loan would be completely paid off in that number of payments, keeping the same interest rate for the duration. Of course, that payment series is interrupted and the loan recalculated at the beginning of the next 5 years.

The half-yearly payments for each 5-year interval are ...

  $33,336.83

  $35,176.68

  $36,869.23

  $38,266.75

  $39,158.08

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Additional comment

The values displayed in the spreadsheet are rounded to the values shown. The values used in calculation have 14 or more significant digits. This means the numbers here may vary from those provided by a lending institution. In the real world, the principal and interest values are rounded to cents with each payment, so actual results may vary by a few cents either way.

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step-by-step explanation: