Part $1$: Duration and Convexity Calculations

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 of $6\%.$ Also, assume that

the bond will make its next coupon payment in exactly $\frac{1}{2}$ years.

$1.)$ Compute the exact modified duration of the bond

$2.)$ Approximate the modified duration of the bond

$3.)$ Compute the exact convexity of the bond $($not the approximation$)$

$4.)$ Approximate the convexity of the bond

$5.)$ Suppose now that the discount rate increased to $10\%$

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