When you pay a mortgage/loan, the interest rates may not always be same, interest rates may vary in the due course of the loan depending on the market rates and a given interest rate may be applicable for only some period of time as per the contract, in this case what's the best thing to do when the interest rate changes (assume rates increase)? For answering this question we will look into a classic example from Investment Science by Luenberger (Problem 3.8).

Problem 3.8

Variable rate mortgage: The smith family just took out a variable-rate mortgage on their new home. The mortgage value is $100,000, the term is 30 years and initially the interest rate is 8%. The interest rate is guaranteed for 5 years, after which time the rate will be adjusted according to prevailing rates. The new rate can be applied to their loan by either changing the payment amount or by changing the length of the mortgage.

(a) What is the original yearly mortgage payment? (Assume payments are yearly)

(b) What will be the mortgage balance after 5 years?

(c) If the interest rate rate on the mortgage changes to 9% after 5 years, what will be the new yearly payment that will keep the termination time the same?

(d) Under the interest change in (c), what will be the new term if payments remain the same?

We know that the present value (P) of an annuity that pays an amount A every individual period for 'm' periods for 'n' years at an yearly interest rate 'r' is given by

Given

(a) P= $100,000 n = 30, m = 1, n * m = 30, r = 8%, r/m = 8% = 0.08

Therefore

A =

(b) The mortgage balance after 5 years is given by (refer this article)

Solving, we get, Mortgage balance after 5 years =

Now we have a new set of values after 5 years, which is

P =

(c) Here we need to keep the termination time the same (which is 30 - 5 = 25), therefore we need to find the new payment which will complete the loan at this termination time, or in other words,

A =

The new yearly payment that will keep the termination time the same is $9653.4.

(d) Here we need to keep the same old monthly payments as we got in (a), but need to find the new term (termination time) at which the loan will complete after the interest rate change (which increased to 9% from 8%, therefore if your payments are same at 9% interest rate, the new term is going to increase).

n =

In other words, it will take (37.56 + 5 = 42.56 years) to pay out the loan when you keep the payments same after 5 years at 9% interest rate.

The learning exercise: Which is beneficial, option (c) or (d)?

Problem 3.8

Variable rate mortgage: The smith family just took out a variable-rate mortgage on their new home. The mortgage value is $100,000, the term is 30 years and initially the interest rate is 8%. The interest rate is guaranteed for 5 years, after which time the rate will be adjusted according to prevailing rates. The new rate can be applied to their loan by either changing the payment amount or by changing the length of the mortgage.

(a) What is the original yearly mortgage payment? (Assume payments are yearly)

(b) What will be the mortgage balance after 5 years?

(c) If the interest rate rate on the mortgage changes to 9% after 5 years, what will be the new yearly payment that will keep the termination time the same?

(d) Under the interest change in (c), what will be the new term if payments remain the same?

We know that the present value (P) of an annuity that pays an amount A every individual period for 'm' periods for 'n' years at an yearly interest rate 'r' is given by

Given

(a) P= $100,000 n = 30, m = 1, n * m = 30, r = 8%, r/m = 8% = 0.08

Therefore

A =

**$8882.74**(The original yearly mortgage payment)

(b) The mortgage balance after 5 years is given by (refer this article)

**$94,821.12**

Now we have a new set of values after 5 years, which is

P =

**$94,821.12**(This becomes the new outstanding principal), m = 1, r = 9%, r/m = 9% = 0.09

(c) Here we need to keep the termination time the same (which is 30 - 5 = 25), therefore we need to find the new payment which will complete the loan at this termination time, or in other words,

**$9653.4**

(d) Here we need to keep the same old monthly payments as we got in (a), but need to find the new term

n =

**37.56 years**

In other words, it will take (37.56 + 5 = 42.56 years) to pay out the loan when you keep the payments same after 5 years at 9% interest rate.

The learning exercise: Which is beneficial, option (c) or (d)?

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