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% coupon rate 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 $F = 5000$ - Coupon rate = 12% annually, so semi-annual coupon rate = 6% - Coupon payment $C = 5000 \times 6\% = 300$ - Number of years = 6, so number of periods $N = 6 \times 2 = 12$ 4. **Case a: Cost of capital 10% annually (5% semi-annually):** - $r = 5\% = 0.05$ - Calculate present value of coupons: $$PV_{coupons} = 300 \times \frac{1 - (1+0.05)^{-12}}{0.05}$$ - Calculate present value of face value: $$PV_{face} = \frac{5000}{(1+0.05)^{12}}$$ - Calculate values: $$PV_{coupons} = 300 \times \frac{1 - (1.05)^{-12}}{0.05} = 300 \times 8.8633 = 2658.99$$ $$PV_{face} = \frac{5000}{1.7959} = 2785.01$$ - Total bond price: $$P = 2658.99 + 2785.01 = 5444.00$$ 5. **Case b: Cost of capital 14% annually (7% semi-annually):** - $r = 7\% = 0.07$ - Calculate present value of coupons: $$PV_{coupons} = 300 \times \frac{1 - (1+0.07)^{-12}}{0.07}$$ - Calculate present value of face value: $$PV_{face} = \frac{5000}{(1+0.07)^{12}}$$ - Calculate values: $$PV_{coupons} = 300 \times 7.0236 = 2107.08$$ $$PV_{face} = \frac{5000}{2.2522} = 2219.00$$ - Total bond price: $$P = 2107.08 + 2219.00 = 4326.08$$ 6. **Investment Decision:** - If cost of capital is 10%, bond price $5444$ is above face value $5000$, so bond is priced at a premium. - If cost of capital is 14%, bond price $4326.08$ is below face value, so bond is priced at a discount. - An investor should buy if the bond price is less than or equal to their required return value. --- 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 price 55. 2. **Dividend Discount Model 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 steady growth: $$P_0 = \sum_{t=1}^{n} \frac{D_t}{(1+k)^t} + \frac{P_n}{(1+k)^n}$$ where $P_n = \frac{D_{n+1}}{k - g_{steady}}$ 3. **Given Data:** - Current dividend $D_0 = 3$ - Growth rates: 30% for 2 years, 12% for next 4 years, then 6% forever - Required return $k = 13\% = 0.13$ 4. **Calculate dividends:** - $D_1 = 3 \times 1.30 = 3.9$ - $D_2 = 3.9 \times 1.30 = 5.07$ - $D_3 = 5.07 \times 1.12 = 5.6784$ - $D_4 = 5.6784 \times 1.12 = 6.3606$ - $D_5 = 6.3606 \times 1.12 = 7.1239$ - $D_6 = 7.1239 \times 1.12 = 7.9760$ 5. **Calculate price at year 6 (start of steady growth):** $$P_6 = \frac{D_7}{k - g_{steady}} = \frac{7.9760 \times 1.06}{0.13 - 0.06} = \frac{8.4546}{0.07} = 120.78$$ 6. **Calculate present value of dividends and $P_6$:** $$PV = \sum_{t=1}^6 \frac{D_t}{(1.13)^t} + \frac{P_6}{(1.13)^6}$$ Calculate each term: - $\frac{3.9}{1.13} = 3.45$ - $\frac{5.07}{1.13^2} = 3.97$ - $\frac{5.6784}{1.13^3} = 3.95$ - $\frac{6.3606}{1.13^4} = 3.96$ - $\frac{7.1239}{1.13^5} = 4.04$ - $\frac{7.9760}{1.13^6} = 3.99$ - $\frac{120.78}{1.13^6} = 60.39$ Sum all: $$PV = 3.45 + 3.97 + 3.95 + 3.96 + 4.04 + 3.99 + 60.39 = 83.75$$ 7. **Investment Decision:** - Intrinsic value is approximately 83.75 - Market price is 55 - Since intrinsic value > market price, the stock is undervalued and an investor should buy it.