$1.(12$ points, $4$ points each part$)$ Quiet and Smooth Electric Motor Company $($QSEMC$)$ recently issued bonds that mature $20$ years from now, and pay $8\mathrm{.}85\%$ coupon interest. Each bond has $$1000$ face value, and half the annual coupon is paid each six months. Recent Federal Reserve announcements have led to reduced interest rates in the economy, and the yield to maturity on these bonds as of now is $8\mathrm{.}13\%.$

$($a$)$ What is today$$s market value of each QSEMC bond?

$($b$)$ Explain why these bonds sell at a price that differs from their $$1000$ face value.

$($c $)$ The yield to maturity on U$.$S$.$ Treasury bonds with $20$ years to maturity is barely over $4\%.$ Why is QSEMC$$s yield to maturity so much higher?