1. **Problem Statement:** Calculate the value of a bond issued by City Development Corporation Ltd. with a face value of 5000, 6 years maturity, 12% annual coupon payable semi-annually, for two cost of capital rates: 10% and 14%. Then decide if an investor should buy it. 2. **Bond Valuation Formula:** The price of a bond is the present value of its coupon payments plus the present value of the face value at maturity: $$P = \sum_{t=1}^{N} \frac{C}{(1+r)^t} + \frac{F}{(1+r)^N}$$ where: - $C$ = coupon payment per period - $r$ = cost of capital per period - $N$ = total number of periods - $F$ = face value 3. **Given Data:** - Face value $F = 5000$ - Annual coupon rate = 12%, so annual coupon = $5000 \times 0.12 = 600$ - Coupons paid semi-annually, so coupon per period $C = 600/2 = 300$ - Number of years = 6, so total periods $N = 6 \times 2 = 12$ 4. **Case a: Cost of capital 10% annually** - Semi-annual cost of capital $r = 10\%/2 = 5\% = 0.05$ - Calculate present value of coupons: $$PV_{coupons} = 300 \times \frac{1 - (1+0.05)^{-12}}{0.05}$$ Calculate: $$1+0.05=1.05$$ $$1.05^{-12} = \frac{1}{1.05^{12}} \approx 0.5568$$ So: $$PV_{coupons} = 300 \times \frac{1 - 0.5568}{0.05} = 300 \times \frac{0.4432}{0.05} = 300 \times 8.864 = 2659.2$$ - Present value of face value: $$PV_{face} = \frac{5000}{1.05^{12}} = 5000 \times 0.5568 = 2784$$ - Total bond price: $$P = 2659.2 + 2784 = 5443.2$$ 5. **Case b: Cost of capital 14% annually** - Semi-annual cost of capital $r = 14\%/2 = 7\% = 0.07$ - Present value of coupons: $$PV_{coupons} = 300 \times \frac{1 - (1+0.07)^{-12}}{0.07}$$ Calculate: $$1+0.07=1.07$$ $$1.07^{-12} = \frac{1}{1.07^{12}} \approx 0.4269$$ So: $$PV_{coupons} = 300 \times \frac{1 - 0.4269}{0.07} = 300 \times \frac{0.5731}{0.07} = 300 \times 8.187 = 2456.1$$ - Present value of face value: $$PV_{face} = \frac{5000}{1.07^{12}} = 5000 \times 0.4269 = 2134.5$$ - Total bond price: $$P = 2456.1 + 2134.5 = 4590.6$$ 6. **Should an investor buy the bond?** - If the bond price is below face value (5000), it sells at a discount. - At 10% cost of capital, price is 5443.2 > 5000, so bond sells at a premium. - At 14% cost of capital, price is 4590.6 < 5000, so bond sells at a discount. - An investor should buy if the bond price is less than or equal to their required return valuation. --- 1. **Problem Statement:** Calculate the intrinsic value of Prime Motors Ltd. stock with dividend growth rates changing over time and decide if an investor should buy it at 55. 2. **Dividend Discount Model (DDM) with multiple growth rates:** Value is sum of present values of dividends during high growth periods plus the present value of the stock price at the start of perpetual growth. 3. **Given Data:** - Current dividend $D_0 = 3$ - Growth rates: - 30% for 2 years - 12% for next 4 years - 6% forever after - Required return $k = 13\% = 0.13$ 4. **Calculate dividends for first 6 years:** - Year 1: $D_1 = 3 \times 1.30 = 3.9$ - Year 2: $D_2 = 3.9 \times 1.30 = 5.07$ - Year 3: $D_3 = 5.07 \times 1.12 = 5.6784$ - Year 4: $D_4 = 5.6784 \times 1.12 = 6.361$ - Year 5: $D_5 = 6.361 \times 1.12 = 7.124$ - Year 6: $D_6 = 7.124 \times 1.12 = 7.979$ 5. **Calculate stock price at year 6 (start of perpetual growth):** - Dividend at year 7: $$D_7 = D_6 \times 1.06 = 7.979 \times 1.06 = 8.458$$ - Price at year 6 using Gordon Growth Model: $$P_6 = \frac{D_7}{k - g} = \frac{8.458}{0.13 - 0.06} = \frac{8.458}{0.07} = 120.83$$ 6. **Calculate present value of dividends and $P_6$ at year 0:** - Present value of dividends: $$PV = \sum_{t=1}^6 \frac{D_t}{(1+k)^t} + \frac{P_6}{(1+k)^6}$$ Calculate each term: - $\frac{3.9}{1.13} = 3.45$ - $\frac{5.07}{1.13^2} = \frac{5.07}{1.2769} = 3.97$ - $\frac{5.6784}{1.13^3} = \frac{5.6784}{1.4429} = 3.94$ - $\frac{6.361}{1.13^4} = \frac{6.361}{1.629} = 3.90$ - $\frac{7.124}{1.13^5} = \frac{7.124}{1.841} = 3.87$ - $\frac{7.979}{1.13^6} = \frac{7.979}{2.08} = 3.84$ - $\frac{120.83}{1.13^6} = \frac{120.83}{2.08} = 58.10$ Sum all: $$PV = 3.45 + 3.97 + 3.94 + 3.90 + 3.87 + 3.84 + 58.10 = 80.07$$ 7. **Should an investor buy the stock?** - Intrinsic value is 80.07 - Market price is 55 - Since intrinsic value > market price, the stock is undervalued and a good buy. **Final answers:** - Bond value at 10% cost of capital: 5443.2 (buy if price ≤ this) - Bond value at 14% cost of capital: 4590.6 (buy if price ≤ this) - Stock intrinsic value: 80.07 (buy since market price 55 < 80.07)