1. **Problem statement:** We need to find the Z-spread of a risky corporate bond with the following details: - Coupon rate: 3% annually, paid semi-annually (so 1.5% every 6 months) - Maturity: 1.5 years (3 periods of 0.5 years each) - Risk-free semi-annual APR rates: 1% for 0.5 years, 2% for 1 year, 3% for 1.5 years - Market price: 99 - Z-spread: the constant spread added to each risk-free rate to discount the bond's cash flows to its market price 2. **Formula and explanation:** The bond price $P$ is the sum of discounted cash flows: $$ P = \sum_{i=1}^n \frac{C}{(1 + r_i + z)^t} + \frac{F}{(1 + r_n + z)^n} $$ where: - $C$ = coupon payment per period = $\frac{3\%}{2} = 1.5\%$ of face value (assume face value $F=100$) - $r_i$ = risk-free rate for period $i$ (semi-annual) - $z$ = Z-spread (unknown) - $t$ = period number (1, 2, 3) - $n=3$ (since 1.5 years with semi-annual coupons) We want to find $z$ such that the discounted cash flows equal the market price 99. 3. **Given data:** - $C = 1.5$ - $F = 100$ - $r_1 = 1\% = 0.01$ - $r_2 = 2\% = 0.02$ - $r_3 = 3\% = 0.03$ - $P = 99$ 4. **Set up the equation:** $$ 99 = \frac{1.5}{1 + 0.01 + z} + \frac{1.5}{(1 + 0.02 + z)^2} + \frac{101.5}{(1 + 0.03 + z)^3} $$ 5. **Solve for $z$ numerically:** We try values near 0.002 (20 basis points = 0.002) to check if the price matches 99. - At $z=0.002$: $$ \frac{1.5}{1.012} + \frac{1.5}{1.022^2} + \frac{101.5}{1.032^3} \approx 1.482 + 1.435 + 95.9 = 98.82 $$ Price is slightly less than 99, so try a slightly smaller $z$. - At $z=0.0015$: $$ \frac{1.5}{1.0115} + \frac{1.5}{1.0215^2} + \frac{101.5}{1.0315^3} \approx 1.483 + 1.438 + 96.1 = 98.99 $$ Closer to 99. - At $z=0.0013$: $$ \frac{1.5}{1.0113} + \frac{1.5}{1.0213^2} + \frac{101.5}{1.0313^3} \approx 1.484 + 1.439 + 96.2 = 99.12 $$ Price slightly above 99. By interpolation, $z \approx 0.0014$ or 14 basis points. 6. **Final answer:** The Z-spread is approximately **14 basis points** (0.0014) added to the risk-free rates to price the bond at 99. This means the bond's yield is about 14 basis points higher than the risk-free curve to reflect its credit risk.