Problem 42 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Josh purchased a condominium 5 yr ago for $$\$ 180,000$$. He made a down payment of \(20 \%\) and financed the balance with a 30 -yr conventional mortgage to be amortized through monthly payments with an interest rate of \(7 \% /\) year compounded monthly on the unpaid balance. The condominium is now appraised at $$\$ 250,000$$. Josh plans to start his own business and wishes to tap into the equity that he has in the condominium. If Josh can secure a new mortgage to refinance his condominium based on a loan of \(80 \%\) of the appraised value, how much cash can Josh muster for his business? (Disregard taxes.)

Step-by-Step Solution

Answer

By refinancing his condominium, Josh can obtain approximately \(62,378.07\) in cash for his business.

Step 1: Calculate the original mortgage amount

We need to find out the amount Josh financed for the condominium after making a down payment of 20%. Let's calculate the amount of the down payment as follows: Down payment = Original purchase price * Down payment percentage Down payment = \(180,000 * 0.2 = \)36,000 Now, we'll subtract the down payment from the original purchase price to find the amount that Josh financed with a mortgage: Original mortgage amount = Original purchase price - Down payment Original mortgage amount = \(180,000 - \)36,000 = $144,000

Step 2: Determine the current mortgage balance after 5 years

To calculate the current mortgage balance after 5 years, we need to find the monthly mortgage payment and the amount repaid after the first 5 years. The interest rate is 7% per year compounded monthly, so we'll first calculate the monthly interest rate: \(Monthly\ interest\ rate = (1 + Annual\ interest\ rate)^{1/12} - 1\) \(Monthly\ interest\ rate = (1 + 0.07)^{1/12} - 1 ≈ 0.00565\) Now, we can calculate the monthly mortgage payment using the following formula: \(Monthly\ payment = P*\frac{r*(1+r)^n}{(1+r)^n-1}\) where: P = original mortgage amount ($144,000) r = monthly interest rate (0.00565) n = number of payments (30 years * 12 months = 360 payments) \(Monthly\ payment ≈ 144000*\frac{0.00565*(1+0.00565)^{360}}{(1+0.00565)^{360}-1} ≈ \$958.63\) Now, we'll calculate the amount repaid after 5 years (60 payments). Using the remaining balance formula: \(Remaining\ balance = P*\frac{(1+r)^n-(1+r)^p}{(1+r)^n-1}\) where: P = original mortgage amount ($144,000) r = monthly interest rate (0.00565) n = total number of payments (360 payments) p = number of payments made (5 years * 12 months = 60 payments) \(Remaining\ balance ≈ 144000*\frac{(1+0.00565)^{360}-(1+0.00565)^{60}}{(1+0.00565)^{360}-1} ≈ \$137,621.93\)

Step 3: Calculate the maximum loan amount based on the appraised value

Now we need to find out the maximum loan amount Josh can get based on the new appraisal value. The new loan amount is 80% of the appraised value: New loan amount = Appraised value * 80% New loan amount = \(250,000 * 0.8 = \)200,000

Step 4: Calculate the cash Josh can get for his business

Finally, we can find out how much cash Josh can get for his business by subtracting the remaining balance from the new loan amount: Cash for business = New loan amount - Remaining balance Cash for business = \(200,000 - \)137,621.93 ≈ $62,378.07 So, Josh can obtain approximately $62,378.07 in cash by refinancing his condominium based on the 80% loan-to-value ratio.

Key Concepts

Amortization Schedule

An amortization schedule is a detailed table that shows how each payment Josh makes splits between paying off the principal and interest of his mortgage. As Josh makes his monthly payments, the interest portion of each payment decreases over time while the principal portion increases. Over the duration of the loan, this steady transition from interest-heavy to principal-heavy payments outlines the amortization process.

This schedule is essential for borrowers like Josh since it helps them to understand how much of their regular payment is actually reducing the loan's principal. Initially, more of his payment goes towards paying the interest. This means in the early years, the mortgage balance decreases slowly. However, as Josh continues with his payments, the reduction in the principal amount will accelerate. Understanding an amortization schedule can be helpful for planning purposes, whether deciding to refinance or to assess future equity growth in the property.

Equity Financing

Equity financing is when Josh taps into the value of his condominium to raise funds, in this case, for starting a business. As he has been paying off his mortgage, the equity in his home has increased. Equity is the difference between the home's current appraised value and the remaining balance on the outstanding mortgage.

In Josh's situation, he seeks to use this built-up equity as a means of financing. This typically involves refinancing the home to take out a new loan that reflects the present appraised value. By doing this, he can obtain a lump sum of money, which represents the equity in his property, to use as he sees fit. It's a popular method of financing large expenses or investments, providing Josh an opportunity to leverage the financial gain embedded in his property's value without having to sell it.

Loan-to-Value Ratio

The loan-to-value (LTV) ratio is a financial term that measures the loan amount compared to the appraised value of the property. When Josh wishes to refinance his condominium, this ratio becomes crucial. In Josh's case, the lender allows him to borrow up to 80% of the home's current appraised value as the new loan amount.

The formula to determine the LTV is: \[ LTV = \frac{Loan hinspace Amount}{Appraised hinspace Property hinspace Value} \]
This ratio is significant because it influences the terms of the mortgage, such as the interest rate and eligibility for refinancing. A lower LTV ratio often means the borrower has more equity, signalling less risk to the lender, potentially offering better loan terms. For Josh, maintaining an LTV of 80% or less was crucial in securing the refinancing deal to extract equity for his venture.