U$.$S$.$ Treasuries represent a significant holding in many pension portfolios. You decide to analyze the yield curve for U$.$S$.$ Treasury notes.

a$.$ Using the data in the table below, calculate the $5-$year spot and forward rates assuming annual compounding. Show your calculations.

U$.$S$.$ Treasury Note Yield Curve Data

$\backslash$table$[[$Years to Maturity,$\backslash$table$[[$Par Coupon$],[$Yield to Maturity$]],\backslash$table$[[$Calculated$],[$Spot Rates$]],\backslash$table$[[$Calculated$],[$Forward Rates$]]],[1,5\mathrm{.}00,5\mathrm{.}00,5\mathrm{.}00],[2,5\mathrm{.}20,5\mathrm{.}21,5\mathrm{.}42],[3,6\mathrm{.}00,6\mathrm{.}05,7\mathrm{.}75],[4,7\mathrm{.}00,7\mathrm{.}16,10\mathrm{.}56],[5,7\mathrm{.}00,?,?]]$

b$.$ Define and describe each of the following three concepts:

i$.$ Short rate

ii$.$ Spot rate

iii. Forward rate

Explain how these concepts are related.

c$.$ You are considering the purchase of a zero$-$coupon U$.$S$.$ Treasury note with $4$ years to maturity. On the basis of the above yield$-$curve analysis, calculate both the expected yield to maturity and the price for the security. Show your calculations.