You are in charge of the bond trading and forward loan department of a large investment bank.

You have the following YTMs for five default$-$free pure discount bonds as displayed on your

computer terminal: YEAR TO MATURITY $1,2,3,5,10$ YTMi $6\%,6\mathrm{.}5\%,7\%,6\mathrm{.}5\%,8\%.$

where YTMi denotes the yield to maturity of a default$-$free pure discount bond maturing at time

i$.$

$(1)$ A new summer intern from Harvard tells you that $3$ year treasury notes with annual

coupons of $$100$ and face value of $$1,000$ are trading for $$1,000.$ Would you ask the

intern to recheck the price of this coupon bond? If so$,$ why?

$4$

$(2)$ What is the annualized forward interest rate between the end of year $3$ and the end of

year $5?$ In other words, what is the geometric average forward interest rate for years $4$

and $5?$

$(3)$ If the expected one year short rate for year $2($i$.$e$.,$ from year $1$ to year $2)$ is $5\%,$ then what

is the liquidity premium for year $2?$