a$.$ Find the convexity of a $6-$year, zero$-$coupon bond with redemption value of $$1,000,$ assuming an effective annual rate of interest of $10\%.$

b$.$ Calculate the price of this zero$-$coupon bond if interest rates drop to $9\%$ under the three methods below:

i$.$ Exact Price

ii$.$ Estimate using modified duration

iii. Estimate using modified duration and convexity