1. **Problem Statement:** Calculate the initial balance sheet liability and the liability one year after issuing 10 million Naira in 7% annual pay, 4-year bonds when the market rate is 7.25%. 2. **Formula and Explanation:** The bond price (initial liability) is the present value of future cash flows discounted at the market rate. The cash flows are: - Annual coupon payment: $10,000,000 \times 7\% = 700,000$ - Principal repayment at maturity: $10,000,000$ The present value of coupons (annuity) is: $$PV_{coupons} = C \times \frac{1 - (1 + r)^{-n}}{r}$$ The present value of principal (lump sum) is: $$PV_{principal} = F \times (1 + r)^{-n}$$ Where: - $C = 700,000$ - $F = 10,000,000$ - $r = 7.25\% = 0.0725$ - $n = 4$ 3. **Calculate Present Value of Coupons:** $$PV_{coupons} = 700,000 \times \frac{1 - (1 + 0.0725)^{-4}}{0.0725}$$ Calculate $(1 + 0.0725)^{-4}$: $$1.0725^{4} = 1.0725 \times 1.0725 \times 1.0725 \times 1.0725 \approx 1.3310$$ So, $$(1 + 0.0725)^{-4} = \frac{1}{1.3310} \approx 0.7512$$ Then, $$PV_{coupons} = 700,000 \times \frac{1 - 0.7512}{0.0725} = 700,000 \times \frac{0.2488}{0.0725} \approx 700,000 \times 3.4324 = 2,402,680$$ 4. **Calculate Present Value of Principal:** $$PV_{principal} = 10,000,000 \times 0.7512 = 7,512,000$$ 5. **Initial Liability (Bond Price):** $$PV = PV_{coupons} + PV_{principal} = 2,402,680 + 7,512,000 = 9,914,680$$ 6. **Liability One Year Later:** After one year, the bond has 3 years remaining. Calculate new present value of coupons (3 payments): $$PV_{coupons,1yr} = 700,000 \times \frac{1 - (1 + 0.0725)^{-3}}{0.0725}$$ Calculate $(1 + 0.0725)^{-3}$: $$1.0725^{3} = 1.0725 \times 1.0725 \times 1.0725 \approx 1.2315$$ So, $$(1 + 0.0725)^{-3} = \frac{1}{1.2315} \approx 0.8117$$ Then, $$PV_{coupons,1yr} = 700,000 \times \frac{1 - 0.8117}{0.0725} = 700,000 \times \frac{0.1883}{0.0725} \approx 700,000 \times 2.5986 = 1,819,020$$ Calculate present value of principal (3 years): $$PV_{principal,1yr} = 10,000,000 \times 0.8117 = 8,117,000$$ 7. **Liability One Year After Issue:** $$PV_{1yr} = 1,819,020 + 8,117,000 = 9,936,020$$ **Final answers:** - Initial liability: approximately $9,914,680$ - Liability one year later: approximately $9,936,020$