1. **State the problem:** You have existing bonds with a coupon rate of 5.75% paid semi-annually, 10 years to maturity, and a price of 1076. You want to issue new 10-year bonds at par (price = 1000). The question is: what coupon rate should the new bonds have? 2. **Identify known values:** - Existing bond coupon rate: 5.75% annually - Coupons paid semi-annually, so coupon payment per period = $\frac{5.75\%}{2} \times 1000 = 28.75$ - Price of existing bond = 1076 - Maturity = 10 years = 20 semi-annual periods - New bond price = 1000 (par) 3. **Calculate the yield to maturity (YTM) of the existing bond:** The YTM is the semi-annual discount rate $r$ that satisfies: $$ 1076 = \sum_{t=1}^{20} \frac{28.75}{(1+r)^t} + \frac{1000}{(1+r)^{20}} $$ 4. **Approximate YTM:** Since price > par, YTM < coupon rate. Coupon rate per period = 2.875%. Try $r = 2.5\% = 0.025$: Calculate present value of coupons: $$28.75 \times \frac{1 - (1+0.025)^{-20}}{0.025} = 28.75 \times 14.877 = 427.4$$ Present value of principal: $$\frac{1000}{(1.025)^{20}} = 1000 / 1.6386 = 610.3$$ Total = 427.4 + 610.3 = 1037.7 < 1076$$ Try $r = 2.2\% = 0.022$: Coupons PV: $$28.75 \times \frac{1 - (1.022)^{-20}}{0.022} = 28.75 \times 15.46 = 444.0$$ Principal PV: $$\frac{1000}{(1.022)^{20}} = 1000 / 1.538 = 650.0$$ Total = 444.0 + 650.0 = 1094 > 1076$$ By interpolation: $$r \approx 0.022 + \frac{1094 - 1076}{1094 - 1037.7} \times (0.025 - 0.022) = 0.022 + \frac{18}{56.3} \times 0.003 \approx 0.02296$$ So semi-annual YTM $r \approx 2.296\%$. 5. **Calculate annual YTM:** $$\text{Annual YTM} = 2 \times 2.296\% = 4.59\%$$ 6. **Find coupon rate for new bonds at par:** For bonds issued at par, coupon rate = YTM. So, new coupon rate = 4.59% annually. **Final answer:** You need to set a coupon rate of **4.59%**.