Paying Off That Dream House When Jacqueline and Keith Sommers were \"house hunti

Question

Paying Off That Dream House When Jacqueline and Keith Sommers were "house hunting" five years ago, the mortgage rates were pretty high. The fixed rate on a 30-year mort- gage was 7.25%, while the 15-year fixed rate was at 6.25%. After walking through many homes, they finally reached a consensus and decided to buy a $300,000, two-story house in an up-and-coming suburban neighborhood in the Midwest. To avoid prepaid mortgage insurance (PMI), the couple had to borrow from family members and come up with a 20% down pay- ment and the additional required closing costs. Since Jacqueline and Keith had already accumulated significant credit card debt and were still paying off their college loans, they decided to opt for lower monthly payments by taking on a 30-year mortgage, despite its higher interest rate. Currently, due to worsening economic conditions, mortgage rates have come down significantly and a refinancing frenzy is underway. Jacqueline and Keith have seen 15-year fixed rates (with no closing costs) advertised at 2.75%, and 30-year rates at 3.75%. Jacqueline and Keith realize that refinancing is quite a hassle due to all the paperwork involved, but with rates being down to 30-year lows they don't want to let this opportunity pass them by. About two years ago, rates were down to similar levels, but they procrastinated and missed the boat. This time, however, the couple called their mortgage officer at the Uptown Bank and locked in the 2.75%, 15-year rate. Nothing was going to stop them from reducing the costs of paying off their dream house this time!

Explanation / Answer

We are given the following information:
House cost = $300,000 ; Loan Amount = 80% * 300,000 = 240,000 ; Loan Tenure = 30 years 0r 360 months and Loan interest rate = 7.25% p.a. or monthly (7.25%/12) = 0.60%

Monthly mortgage payment formula = Loan Amount * [r * (1+r)t ] / [(1+r)t - 1]; where r is the applicable monthly rate and t is the tenure in months.

Monthly mortgage payment = 240000 * [0.60% * (1+0.60%)360]/[(1+0.60%)360 - 1] = 1637.22

Now the amount they have paid towards mortgage over 5 years = 1637.22 * 60 = $98233.38

Residual loan balance formula after k months = Loan Amount * [(1+r)t - (1+r)k] / [(1+r)t - 1]

Hence loan balance after 60 months = 240000 * [(1+0.60%)360 - (1+0.60%)60] / [(1+0.60%)360 - 1] = 226509.07

Hence they have only paid loan principal of (240000 - 226509.07) = 13490.93 in 5 years and balance amount (98233.38 - 13490.93) = 84742.45 has been applied towards interest

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