Suppose the following information is given:

The YTM of a $1$ year bond purchased today: ${?}_0{i}_1=3\mathrm{.}6\%$

The YTM of a $3$ year bond purchased today: $0_3=4\mathrm{.}1\%$

The YTM of a $4$ year bond purchased today: $i_4=4\mathrm{.}4\%$

The YTM of a $1$ year bond purchased next year: ${11}_2^e=2\mathrm{.}3\%$

The YTM of a $1$ year bond purchased $3$ years from now: $2i_3^e=2\mathrm{.}6\%$

The liquidity premium for holding a $2$ year bond purchased today is $o{l}_2=0\mathrm{.}6\%$

The liquidity premium for holding a $3$ year bond purchased today is $0_3=0\mathrm{.}9\%$

The liquidity premium for holding a $4$ year bond purchased today is $0_4=1\mathrm{.}4\%$

$($b$)$ According to the liquidity premium theory of term structure, what is the YTM of a $2$ year bond purchased today? $[4$ Points$]$

$($c$)$ According to the liquidity premium theory of term structure, what is the expected YTM of a $1$ year bond purchased $3$ years from now $(3i)?[7$ Points$]$