$1.$ Suppose a discount bond, free of credit $($default$)$ risk, is issued today. It has a face value of $$10,000$ and will mature in two years. Assume the current interest is $10\%$ for the coming year and will also be $10\%$ for the following year.

$($a$)$ What is the present value of the discount bond given the interest rate? Show your calculation.

$($b$)$ If the market price of the bond was higher than its present value, say $$100$ above the value from part $($a$),$ would a rational investor be willing to buy the bond? Explain why or why not.

$($c$)$ If the market price of the bond was lower than its present value, say $$100$ below the value from part $($a$),$ explain how you could make a risk$-$free $($arbitrage$)$ profit.