Question $1$:

A straight, option$-$free, bond with a coupon rate of $8$ percent sells at a yield to maturity of $9$ percent. The bond matures in $10$ years.

a$.$ Compute the bond's modified duration. $($Your financial calculator will most efficiently help solve for this.$)$ How can this value be used?

b$.$ Compute the Macaulay duration of the bond and explain what it means.

c$.$ Suppose the yield to maturity was originally $7$ percent. Re$-$compute the Macaulay duration. What does this tell you about the relationship between duration and the yield to maturity?

Question $2$:

Lifeco Insurance Company needs to have $\backslash (\backslash$$ $100\backslash )$ million $5$ years from today to satisfy life insurance claims. Lifeco purchases $8$ percent coupon bonds selling at par with a maturity of $6\mathrm{.}2$ years. Assume immediately after the purchase, an interest rate shock causes bond yields to jump up to $\backslash (10\backslash \%\backslash ).$

If immediately after the price shock, Lifeco Insurance Company sold the $8$ percent coupon bonds and purchased bonds selling at par that offer a $10$ percent coupon. What will the maturity of these bonds be$?($Round to $2$ decimal places$)$