$2)$ Assume that the yield curve for zero$-$coupon bonds as a function of maturity T is

y$($T$)=0\mathrm{.}01+0\mathrm{.}002$ T

Consider a $10-$year, $4\%$ coupon bond $($paid annually$)$ with face value $=100.($Recall that a

coupon bond should be thought of as a portfolio of zero$-$coupon bonds. The equation

above tells you how to discount future cash flows for any maturity T$).$ Determine:

$2$A$)$ The present value of the bond

$2$B$)$ Its Macaulay duration

$2$C$)$ Its modified duration

$2$D$)$ Assume that $$overnight$($that is$,$ the bond has still $10$ years to maturity$),$ the yield

curve shifts to

y$($T$)=0\mathrm{.}02+\mathrm{.}002$ T

Determine the actual change in bond value, and an estimate of this change based on its

modified duration.