Assume that you were recently discharged from the US Navy after six years of active$-$duty service and have been hired by a defense contractor who is quite interested in your military$-$acquired skills, knowledge, and experience. To help induce you to sign with their company, the contractor is paying you an up$-$front signing bonus of $$50,000.$ You are now considering buying a new house for $$250,000.$ Because of your honorable service, you are eligible for a Veteran Administration $($VA$)100\%$ loan$-$to$-$value $($LTV$)$ home loan. VA mortgage loans do not incur PMI for LTV ratios of greater than $80\%$ as conventional mortgage loans do$.$ This is because the US government insures VA mortgage loans. Let us also say you have a certified $10\%$ VA disability that waives the $1\%$ VA origination fee. You are considering borrowing the entire $$250,000$ at a nominal annual interest rate of $4\mathrm{.}75\%.$ We$$ll call this loan alternative Loan A$.$ Alternatively, you could use the $$50,000$ signing bonus as a down payment on the house and only borrow $80\%$ of the purchase price, or $$200,000,$ at a rate of $4\mathrm{.}25\%.$ We$$ll call this loan alternative Loan B$.$ Both loan alternatives are fixed$-$rate, fully amortizing $30-$year loans with monthly payments.

Assume that your new employer has told you that in $5$ years the project you will be working on will be transferred to the Air Force Research Laboratory at Wright$-$Patterson AFB near Dayton, Ohio. The project is being transferred for further technological development and overall operational support. You will follow the project to Wright$-$Patterson AFB and work at the air force laboratory as a representative and technical expert for your company. This will result in you being at Wright$-$Patterson AFB for several years working on this project; hence, you decide to sell your new house at your current location $($the house you are buying in Question $1).$

a$.$ What are the principal balances for the two loan alternatives at the end of Year $5($use the FV method$)$

b$.$ What is the difference between the Year $5$ principal balances of the two loan alternatives?

c$.$ What is the incremental cost of borrowing the extra $$50,000$ with Loan A if you sell the house at the end of Year $5?$