Consider a bond with the following terms:

$10$ years to maturity

$$ $$1,000$ face value

$$ Coupons are paid $2$ times per year

$$ Annual coupon rate is $5\%$

For problems $1-5,$ assume a constant discount rate across maturities has increased from $6\%$ to $10\%.$ Also, assume that the bond will make its next coupon payment in exactly $1/2$ years.

$1.$ Find the exact price of the bond under the new discount rate.

$2.$ Approximate the new bond price using modified duration.

$3.$ Repeat the approximation using modified duration and convexity

$4.$ How far off was the MoD approximation from the exact answer?

$5.$ Explain why the modified duration approximation yielded a different answer than the actual approximation.