To compute the 1.5-year spot rate, we will first derive discount factors from the given swap rates and then calculate the spot rate from the derived discount factors.

Step 1: Deriving discount factors from swap rates:
Consider an interest rate swap. If we assume that the notional amount is exchanged, the fixed leg of the swap would resemble a fixed coupon-paying bond, with fixed leg payments acting like semiannual, fixed coupons, and the notional amount acting like the principal payment. We can, therefore, write an equation for each “bond” that equates the present value of its cash flows to its price of par.

$$(100 + \frac{1}{2}) \times d(0.5) = 100 \Rightarrow d(0.5) = 0.9950$$ $$\frac{1.6}{2} \times d(0.5) + (100 + \frac{1.6}{2}) \times d(1.0) = 100 \Rightarrow d(1.0) = 0.9842$$ $$\frac{1.88}{2} \times d(0.5) + \frac{1.88}{2} \times d(1.0) + (100 + \frac{1.88}{2}) \times d(1.5) = 100 \Rightarrow d(1.5) = 0.9723$$

Step 2: Deriving the spot rate from discount factors:
Spot rates, $z(t)$ and discount factors, $d(t)$ are related as shown in the following formula, assuming semi-annual coupons:

$$z(t) = 2 \left[ \left( \frac{1}{d(t)} \right)^{\frac{1}{2t}} - 1 \right]$$ $$\Rightarrow z(1.5) = 2 \left[ \left( \frac{1}{0.9723} \right)^{\frac{1}{2 \times 1.5}} - 1 \right] = 1.88\%$$