There is a $1-$year zero coupon bond that has a $$1000$ face value. We know the $1-$year risk$-$free interest rate is $4\%.$ The $1-$year zero$-$coupon bond with the following risky cash flows:

$$1,000$ with a probability of $90\%$

Payoffs $=500$ with a probability of $10\%$

The discount rate of the above risky bond is $8\%.$ Answer the following questions.

$1.$ What is the yield to maturity of the bond?

$2.$ Suppose there is a CDS completely hedging away the default risk of the risky bond. What is the price of the CDS$?$

$3.$ Suppose the bond's default probability is underestimated. What is the consequence to the CDS issuer?