Question $1$

$($a$)$ What happens to the price of a $1$ year bond with a face value $1,000$ and a coupon of $8\%$ when the interest rate decreases from $8\%$ to $6\%.$

$($b$)$ Calculate the present value of the following:

$($i$)$ A perpetuity with a face value of $100$ and a coupon of $6\%$ where the interest rate is $10\%$ and the $1$st payment is made at the end of year $1.$

$3$

$($ii$)$ A perpetuity with a face value of $100$ and a coupon of $6\%$ where the interest rate is $10\%$ and the $1$st payment is made at the end of year $10.$

$3$

$($iii$)$ A $9$ year annuity with a face value of $100$ and a coupon of $6\%$ where the interest rate is $10\%$ and the $1$ st payment is made at the end of year $1.$

$4$

$($c$)$ Derive the formula for an annuity using the perpetuity formula.

$($d$)$ Derive the formula for a growing perpetuity.

$6$