Consider a $20-$year, $$180,000$ mortgage with a rate of $6\mathrm{.}3$ percent. Four years into the mortgage, rates have fallen to $5$ percent.

Suppose the transaction cost of obtaining a new mortgage is $$1,900.$

a$.$ Should the homeowner refinance at the lower rate?

Yes

No

b$.$ Quantify the effect of the homeowner's decision.

Note: Do not round intermediate calculations. Round your answer to $2$ decimal places.

Answer is complete but not entirely correct.