$2-20.$ Suppose UTM issues bonds with a one$-$year maturity and $$1000$ face value on January $1,2019.$

These bonds pay quarterly coupons of $$50$ on April $1,2019$; July $1,2019$; October $1,2019$;

and January $1,2020.$

$($a$)$ Suppose we're computing the present value of this bond using two different scenarios. In

the first scenario, we assume an APR of $1\mathrm{.}3\%,$ and in the second scenario we assume an

APR of $2\%.$ Without doing any computations, which scenario results in a lower present

value?

$($b$)$ Confirm your guess in part $($a$)$ by computing the present value in each scenario, assuming

the given APR compounds monthly.

$($c$)$ Assume instead that the compounding periods occur quarterly, and coincide with the

coupon payments. Modify your calculation in part $($a$)$ to compute the present value of

the $1\mathrm{.}3\%$ APR bond.

$($d$)$ Again assume quarterly compounding periods and an APR of $1\mathrm{.}3\%.$ This time use the

present value of an annuity formula to confirm the answer you got in part $($c$).$