a$)(1$ point$)$ First, consider a $10-$year bond with a coupon rate of $7\%$ and annual coupon payments. Draw a graph showing the relationship between the price and the interest on this bond. The price should be on the y$-$axis and the interest rate on the x$-$axis. To compute the various prices, consider interest rates between $2\%$ and $12\%($use $0\mathrm{.}5\%$ increments$).$ So your x$-$axis should go from $2\%,$ then $2\mathrm{.}5\%...$ until $11\mathrm{.}5\%,$ and then $12\%.$

Is the relationship linear $($i$.$e$.$ is the slope constant$)?$ Start at $7\%.$ If interest rates go up or down by $0\mathrm{.}5\%$ is the price changing by the same amount? What type of relationship do we observe between prices and interest rates $($liner$,$ concave, convex or something else$)?$

b$)(1$ point$)$ Now consider the same bond with a $10-$year maturity, a face value of $$1,000,$ a coupon rate of $7\%($coupon is paid annually$)$ and assume that the yield to maturity on the bond is $7\%.$ Compute the duration of this bond.

c$)(1$ point$)$ Next, we are going to analyze the effect of time to maturity on the duration of the bond. Compute the duration of a bond with a face value of $$1,000,$ a coupon rate of $7\%($coupon is paid annually$),$ and a yield to maturity of $7\%$ for maturities of $2$ to $18$ years in $1-$year increments $($so here we are going to vary the time to maturity and see how duration changes if N$=2,3...$ etc.$).$ What happens to duration as maturity increases?

d$)(1$ point$)$ Next, we are going to analyze the effect of the yield to maturity on the duration of the bond. Compute the duration of a bond with a face value of $$1,000,$ a coupon rate of $7\%($coupon is paid annually$)$ and a maturity of $10$ years as the interest rate $($or yield to maturity$)$ on the bond changes from $2\%$ to $12\%($consider increments of $1\%-$ so you need to compute the duration for various yields to maturity $2\%,3\%,...,12\%).$ What happens to duration as the interest rate increases?

e$)(1$ point$)$ Finally, we are going to analyze the effect of the coupon payment on the duration of the bond. Compute the duration of a bond with a face value of $$1,000,$ a maturity of $10$ years and a yield to maturity of $7\%.$ Compute the duration for coupon rates ranging from $2\%$ to $12\%($in increments of $1\%).$ What happens to duration as the coupon rate increases?