Question #$4$: Liquidity Premium Theory of Term Structure $[16$ Points$]$

Suppose the following information is available to all

The YTM of a $1$ year bond purchased today: ${?}_0{i}_1=5\mathrm{.}2\%$

The YTM of a $6$ year bond purchased today: ${?}_0{i}_6=11\mathrm{.}4\%$

The YTM of a $7$ year bond purchased today: ${?}_0{i}_7=11\mathrm{.}6\%$

The YTM of a $1$ year bond purchased next year: ${?}_1{i}_2^e=5\mathrm{.}4\%$

The YTM of a $1$ year bond purchased $2$ years from now: ${?}_2{i}_3^e=5\mathrm{.}7\%$

The YTM of a $1$ year bond purchased $3$ years from now: ${?}_3{i}_4^e=6\mathrm{.}0\%$

The YTM of a $1$ year bond purchased $4$ years from now: $4i_5^e=6\mathrm{.}3\%$

The YTM of a $1$ year bond purchased $5$ years from now: $5i_6^e=6\mathrm{.}6\%$

The liquidity premium for holding a $3$ year bond purchased today is ${?}_0{1}_3=2\mathrm{.}2\%$

The liquidity premium for holding a $4$ year bond purchased today is $1_4=2\mathrm{.}6\%$

The liquidity premium for holding a $5$ year bond purchased today is $1_5=3\mathrm{.}4\%$

The liquidity premium for holding a $6$ year bond purchased today is $1_6=3\mathrm{.}9\%$

The liquidity premium for holding a $7$ year bond purchased today is ${?}_{0}17=5\mathrm{.}0\%$

$($a$)$ According to the liquidity premium theory of term structure, what is the YTM of a $4$ year bond purchased today? $[6$ Points$]$

$($b$)$ According to the liquidity premium theory of term structure, what is the expected YTM of a $1$ year bond purchased $6$ years from now? $[10$ Points$]$