The following table shows the current and expected one$-$year interest rates in following four years.

$($a$)$ Estimate the YTM on bonds of $2,3,4$ and $5$ years to maturity and graphically illustrate the yield

curve under the expectations hypothesis. $[8$ Points$]$

$($b$)$ Calculate the price of a discount bond that is purchased in year $3$ that promises to pay $$1000$ in year $5.$

Hint: You must first calculate the YTM of a $2$ year bond that is purchased in year $3,315.$ Solve for:

$3i_5^e=\frac{3_4^e+4i_5^e}{2}[8$ Points$]$