Price $of$ $$100$ face value zero$-$coupon bond $=\frac{\text{\$}100}{(1+i{)}^n}$

$P_{CB}=[\frac{\text{C}\text{o}\text{u}\text{p}\text{o}\text{n}\text{p}\text{a}\text{y}\text{m}\text{e}\text{n}\text{t}}{(1+i{)}^1}+\frac{\text{C}\text{o}\text{u}\text{p}\text{o}\text{n}\text{p}\text{a}\text{y}\text{m}\text{e}\text{n}\text{t}}{(1+i{)}^2}+$cdots$+\frac{\text{C}\text{o}\text{u}\text{p}\text{o}\text{n}\text{p}\text{a}\text{y}\text{m}\text{e}\text{n}\text{t}}{(1+i{)}^n}]+\frac{\text{F}\text{a}\text{c}\text{e}\text{v}\text{a}\text{l}\text{u}\text{e}}{(1+i{)}^n}$

Suppose you have a one$-$year bond with a face value of $$100.$ The interest rate for this specific bond is $8\%.$

a$.$ Suppose this is a zero$-$coupon bond. What is the fair price $($present value$)$ of owning this bond?

b$.$ Suppose instead that there is a $$3$ coupon awarded after one year of owning this bond, as well as the face value yield. The interest rate remains at $8\%.$ What is the new fair price of this bond?

c$.$ Suppose instead that the original bond $($zero$-$coupon$)$ is now a $2-$year bond with face value $=$$$100,$ and the interest rate is at $5\%.$ What is the fair price of this bond?

d$.$ Lastly, suppose that the chance that this company defaults on this bond is $10\%.$ What is the new fair price of this riskier bond $($use all the assumptions from part $($c$))?$