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: $4$i$5$e $=6\mathrm{.}3\%$

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

The liquidity premium for holding a $3$ year bond purchased today is $0$l$3=2\mathrm{.}2\%$

The liquidity premium for holding a $4$ year bond purchased today is $0$l$4=2\mathrm{.}6\%$

The liquidity premium for holding a $5$ year bond purchased today is $0$l$5=3\mathrm{.}4\%$

The liquidity premium for holding a $6$ year bond purchased today is $0$l$6=3\mathrm{.}9\%$

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

$($a$)$ According to the liquidity premium theory of term structure, what is the YTM of a $4$ year bond

purchased today?

$($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?