Suppose the current term structure of interest rates (0.5 year, 1 year, 1.5 year, 2 year maturities annualized with semi annual compounding) is (5.00%, 6.00%, 7.00%, 8.00%). A 2-year vanilla interest rate swap having a notional principal of $5,000,000 is being configured. This swap will make payments at the end of each half year pegged to the 0.5 year interest rate posted at the beginning of that half year. Determine the annualized swap rate to the nearest basis point.
This part I fully understand how it works. My answer is 3.95% (395 basis point)
The next part is: Suppose 1.5 years later, immediately after the swap exchange, the swap value is zero. Determine the implied 0.5 year interest rate (annualized, semi annual compounding) at that time to the nearest basis point.
Can someone explain how to derive for this part? I'm unsure what the meant by implied 0.5 year interest rate.
The value of the fixed-float swap is given by
$$V = \sum_{i=1}^40.5RZ_i-(Z_0-Z_4).$$
With semi-annual compounding the zero-coupon bond prices at semi-annual intervals are related to interest rates as
$$Z_i = \frac{1}{(1 + 0.5R_i)^i}.$$
For the given term structure the values are
$Z_0 = 1$, $Z_1 = 0.975610$, $Z_2 = 0.942596$, $Z_3 = 0.901943$, and $Z_4 = 0.854804$.
The par swap rate is the value of $R$ that makes $V=0$. Solving we get $R = 7.90 \%.$
Your answer for the swap rate is off by a factor of $2$ -- probably because you forgot the day-count factor.
For the second part, after the exchange there is one remaining payment. Let $f$ denote the future $0.5$-year rate in $1.5$ years. Set the forward value of the remaining single-payment swap to zero to find the implied short-term rate
$$V = \frac{0.5R}{1+0.5f} - \left(1 - \frac{1}{1+0.5f}\right)=0.$$
Substitute for $R$ and solve for $f$.