1. **Problem 1:** Calculate the issue price of a bond with face value P100,000, coupon 9% per year paid semi-annually, maturity 10 years, and market YTM 8% per annum compounded semi-annually. 2. **Formula:** The price of a bond is the present value of its coupon payments plus the present value of the face value at maturity: $$\text{Price} = \sum_{t=1}^{N} \frac{C}{(1+r)^t} + \frac{F}{(1+r)^N}$$ where: - $C$ = coupon payment per period - $r$ = market interest rate per period - $N$ = total number of periods - $F$ = face value 3. **Step 1:** Calculate coupon payment per period: Annual coupon = 9% of 100,000 = 9,000 Since coupons are semi-annual, coupon per period $C = \frac{9,000}{2} = 4,500$ 4. **Step 2:** Calculate market interest rate per period: Annual YTM = 8%, semi-annual rate $r = \frac{8\%}{2} = 4\% = 0.04$ 5. **Step 3:** Calculate total number of periods: $N = 10 \times 2 = 20$ 6. **Step 4:** Calculate present value of coupons (annuity): $$PV_{coupons} = C \times \frac{1 - (1+r)^{-N}}{r} = 4,500 \times \frac{1 - (1+0.04)^{-20}}{0.04}$$ Calculate: $(1+0.04)^{-20} = (1.04)^{-20} \approx 0.45639$ So, $$PV_{coupons} = 4,500 \times \frac{1 - 0.45639}{0.04} = 4,500 \times \frac{0.54361}{0.04} = 4,500 \times 13.5903 = 61,156.35$$ 7. **Step 5:** Calculate present value of face value: $$PV_{face} = \frac{100,000}{(1+0.04)^{20}} = 100,000 \times 0.45639 = 45,639$$ 8. **Step 6:** Calculate bond price: $$\text{Price} = PV_{coupons} + PV_{face} = 61,156.35 + 45,639 = 106,795.35$$ --- 9. **Problem 2:** Find the price Mara should pay for a corporate bond with face P10,000, annual coupon 7%, 6 years maturity, market yield 5% annual. 10. **Step 1:** Coupon payment per year: $C = 7\% \times 10,000 = 700$ 11. **Step 2:** Market interest rate per period $r = 5\% = 0.05$ 12. **Step 3:** Number of periods $N = 6$ 13. **Step 4:** Present value of coupons: $$PV_{coupons} = 700 \times \frac{1 - (1+0.05)^{-6}}{0.05}$$ Calculate: $(1+0.05)^{-6} = (1.05)^{-6} \approx 0.74622$ So, $$PV_{coupons} = 700 \times \frac{1 - 0.74622}{0.05} = 700 \times \frac{0.25378}{0.05} = 700 \times 5.0756 = 3,552.92$$ 14. **Step 5:** Present value of face value: $$PV_{face} = \frac{10,000}{(1.05)^6} = 10,000 \times 0.74622 = 7,462.20$$ 15. **Step 6:** Bond price: $$\text{Price} = 3,552.92 + 7,462.20 = 11,015.12$$ --- 16. **Problem 3:** Find the yield to maturity (YTM) of a bond with face P1,000, annual coupon 5%, 8 years maturity, price P950. 17. **Step 1:** Coupon payment per year: $C = 5\% \times 1,000 = 50$ 18. **Step 2:** We want to find $r$ such that: $$950 = \sum_{t=1}^8 \frac{50}{(1+r)^t} + \frac{1,000}{(1+r)^8}$$ 19. **Step 3:** This equation cannot be solved algebraically easily, so we use trial and error or financial calculator. Try $r=6\% = 0.06$: Calculate present value of coupons: $$PV_{coupons} = 50 \times \frac{1 - (1+0.06)^{-8}}{0.06} = 50 \times \frac{1 - 0.62741}{0.06} = 50 \times 6.2098 = 310.49$$ Present value of face value: $$PV_{face} = \frac{1,000}{(1.06)^8} = 1,000 \times 0.62741 = 627.41$$ Sum: $$310.49 + 627.41 = 937.90 < 950$$ Try $r=5.5\% = 0.055$: $$PV_{coupons} = 50 \times \frac{1 - (1.055)^{-8}}{0.055} = 50 \times 6.3835 = 319.18$$ $$PV_{face} = 1,000 \times (1.055)^{-8} = 1,000 \times 0.6470 = 647.00$$ Sum: $$319.18 + 647.00 = 966.18 > 950$$ Since 5.5% price is higher than 950 and 6% price is lower, YTM is between 5.5% and 6%. 20. **Step 4:** Linear interpolation: $$r \approx 5.5\% + \frac{966.18 - 950}{966.18 - 937.90} \times (6\% - 5.5\%) = 5.5\% + \frac{16.18}{28.28} \times 0.5\% = 5.5\% + 0.286\% = 5.79\%$$ **Final answers:** 1. Issue price = P106,795.35 2. Price Mara should pay = P11,015.12 3. Yield to maturity = 5.79% per annum