Bootstrapping is the process of carving out spot rates from the market prices of a set of coupon-paying bonds. The spot rates are determined in a sequential manner.

Step 1: Find the 6-month spot rate ($Z_{0.5}$)

For the 6-month T-bond:

$`$101.5` = \frac{100 + 5/2}{\left(1 + \frac{Z_{0.5}}{2}\right)^1}$$

$`$101.5` = \frac{102.5}{1 + \frac{Z_{0.5}}{2}}$$

$`$1` + \frac{Z_{0.5}}{2} = \frac{102.5}{101.5} = 1.0099$$

$Z_{0.5} = 0.0099 \times 2 = 1.98\%$

Thus, the six-month spot rate is 1.98% with semiannual compounding.

Step 2: Find the 1-year spot rate ($z_1$)

For the 1-year T-bond:

$`$102.6` = \frac{2.625}{\left(1 + \frac{z_{0.5}}{2}\right)^1} + \frac{100 + 2.625}{\left(1 + \frac{z_1}{2}\right)^2}$$

$`$102.6` = \frac{2.625}{1.0099} + \frac{102.625}{\left(1 + \frac{z_1}{2}\right)^2}$$

$`$102.6` = 2.60 + \frac{102.625}{\left(1 + \frac{z_1}{2}\right)^2}$$

$\left(1 + \frac{z_1}{2}\right)^2 = \frac{102.625}{100} = 1.02625$

$\left(1 + \frac{z_1}{2}\right) = \sqrt{1.02625} \approx 1.0131$

$\frac{z_1}{2} = 0.0131$

$z_1 = 2 \times 0.0131 = 2.61\%$

Therefore, the one-year spot rate is 2.61% with semiannual compounding.