For this problem, round all of your answers to three decimal places.

An annuity purchased today to yield $6\mathrm{.}5\%$ annually has the following set of payments:

$$ $$45002$ years from today;

$$ $$22003$ years from today;

$$ $$40506$ years from today.

$($a$)$ Find the modified duration of this annuity: $(0\mathrm{.}065,1)$ years

$($b$)$ Find the modified convexity of this annuity: $(0\mathrm{.}065,1)$ years

$($c$)$ Use the modified duration and convexity to estimate the relative change in price if the interest rate increases by $50$ basis points:

Relative price change $$

Recall: An increase from an interest rate $6\mathrm{.}5\%$ by $50$ basis points means that the interest rate changes from $6\mathrm{.}5\%$ to $7\%.$ The relative price change if the interest changes from $$ to $0$ is given by

$(0)()()(,1)(0)+(,1)(0)22$