Question

Consider a homebuyer/investor who plans to buy a new house, the price of which is $ 500,000. Suppose the buyer does not have any initial savings for the down payment. To buy the house, he needs to borrow $ 500,000 from a bank in the form of a mortgage loan. The mortgage is a 30-year-fixed-rate mortgage. After buying the house, the buyer budget is $ 30,011 per year. That is, if the mortgage asks him to repay more than $ 30,011 per year (this happens when the interest rate is too high), then he cannot afford it and would choose not to buy this house. Derive the investment of this buyer as a function of interest rate r.

Answer 1

The homebuyer's investment can be derived by solving the fixed-rate mortgage payment formula with the annual payment set to $30,011 and the principal amount at $500,000, using numerical methods to find the maximum affordable interest rate.

Step-by-step explanation:

Given that the homebuyer/investor seeks to derive their investment as a function of interest rate r, we have to calculate the maximum amount they can repay annually and then solve for the interest rate that makes this possible. The homebuyer has a budget of $30,011 per year for mortgage repayments, and they need to borrow $500,000. To find the relationship between the annual payment and the interest rate, we can use the formula for the annual payment on a fixed-rate mortgage, which is:

PMT = P × [r(1+r)^n] / [(1+r)^n - 1]

Where PMT is the annual payment, P is the principal amount ($500,000), r is the annual interest rate, and n is the number of payments (30 years).

However, without a specific interest rate, we cannot provide an exact figure for the investment. Assuming r is the maximum interest rate the buyer can afford, the equation becomes:

30,011 = 500,000 × [r(1+r)^30] / [(1+r)^30 - 1]

This equation can then be solved for r to find the maximum interest rate the buyer can afford. The exact interest rate would depend on the provided terms of the mortgage and may require numerical methods or financial calculators to derive the precise interest rate value. To calculate the investment of the buyer as a function of interest rate r, we need to consider the annual mortgage payment for a 30-year-fixed-rate mortgage. The annual payment can be calculated using the formula for the present value of an annuity:

PV = PMT * (1 - (1 + r)^(-n)) / r

Where PV is the loan amount, PMT is the annual payment, r is the interest rate, and n is the number of periods (30 years in this case).

For the given budget of $30,011 per year, we can substitute PMT with this value:

500,000 = 30,011 * (1 - (1 + r)^(-30)) / r

Now, we solve this equation for r to find the investment as a function of interest rate.