1. **Problem Statement:** (a)(i) Compute the forward rate premium of the Tanzania Shilling (Tsh) with respect to the Kenyan Shilling (Ksh) using Interest Rate Parity (IRP). (a)(ii) Compute the six-month forward exchange rate given the spot rate and interest rates. Demonstrate IRP applicability for a Kenyan investor with 1,000,000 Ksh. (b) Compute the value of a European call option on Kakuzi Ltd shares using the Black-Scholes model. 2. **Formulas and Rules:** - Interest Rate Parity forward premium formula: $$\text{Forward Premium} = \frac{(1 + i_{domestic} \times t)}{(1 + i_{foreign} \times t)} - 1$$ where $i$ is the annual interest rate and $t$ is the fraction of the year (0.5 for 6 months). - Forward exchange rate: $$F = S \times \frac{(1 + i_{domestic} \times t)}{(1 + i_{foreign} \times t)}$$ where $S$ is the spot rate. - Black-Scholes call option price: $$C = S_0 N(d_1) - K e^{-rT} N(d_2)$$ where $$d_1 = \frac{\ln(\frac{S_0}{K}) + (r + \frac{\sigma^2}{2})T}{\sigma \sqrt{T}}$$ $$d_2 = d_1 - \sigma \sqrt{T}$$ $N(\cdot)$ is the cumulative standard normal distribution, $S_0$ is current stock price, $K$ strike price, $r$ risk-free rate, $\sigma$ volatility, $T$ time to maturity in years. 3. **Calculations:** (a)(i) Forward rate premium: - $i_{Tsh} = 0.2025$ per annum - $i_{Ksh} = 0.1775$ per annum - $t = 0.5$ $$\text{Forward Premium} = \frac{1 + 0.2025 \times 0.5}{1 + 0.1775 \times 0.5} - 1 = \frac{1 + 0.10125}{1 + 0.08875} - 1 = \frac{1.10125}{1.08875} - 1 = 1.0113 - 1 = 0.0113$$ So, the forward premium is approximately 1.13%. (a)(ii) Forward exchange rate: - Spot rate $S = 0.12$ $$F = 0.12 \times \frac{1.10125}{1.08875} = 0.12 \times 1.0113 = 0.12136$$ So, the six-month forward exchange rate is approximately 0.12136 Ksh per Tsh. (a)(iii) Applicability of IRP for Kenyan investor with 1,000,000 Ksh: - Invest in Kenya at $i_{Ksh}$: $$\text{Amount after 6 months} = 1,000,000 \times (1 + 0.1775 \times 0.5) = 1,000,000 \times 1.08875 = 1,088,750$$ - Convert to Tsh at spot rate: $$1,000,000 \times \frac{1}{0.12} = 8,333,333.33 \text{ Tsh}$$ - Invest in Tanzania at $i_{Tsh}$: $$8,333,333.33 \times (1 + 0.2025 \times 0.5) = 8,333,333.33 \times 1.10125 = 9,177,083.33 \text{ Tsh}$$ - Convert back to Ksh at forward rate: $$\frac{9,177,083.33}{0.12136} = 75,635,135.14 \text{ Ksh}$$ This large number suggests a miscalculation in currency conversion; correct approach is: - Convert initial Ksh to Tsh: $$1,000,000 \times \frac{1}{0.12} = 8,333,333.33 \text{ Tsh}$$ - Invest in Tanzania: $$8,333,333.33 \times 1.10125 = 9,177,083.33 \text{ Tsh}$$ - Convert back to Ksh at forward rate: $$9,177,083.33 \times 0.12136 = 1,113,750 \text{ Ksh}$$ Compare with Kenyan investment: - Kenyan investment after 6 months: 1,088,750 Ksh - Investing in Tanzania and converting back yields 1,113,750 Ksh Difference due to forward premium confirms IRP applicability. (b) Black-Scholes option pricing: - $S_0 = 4.50$ - $K = 3.50$ - $r = 0.10$ - $\sigma = 0.40$ - $T = 0.25$ (3 months) Calculate $d_1$: $$d_1 = \frac{\ln(\frac{4.50}{3.50}) + (0.10 + \frac{0.40^2}{2}) \times 0.25}{0.40 \times \sqrt{0.25}} = \frac{\ln(1.2857) + (0.10 + 0.08) \times 0.25}{0.40 \times 0.5} = \frac{0.2513 + 0.045}{0.20} = \frac{0.2963}{0.20} = 1.4815$$ Calculate $d_2$: $$d_2 = 1.4815 - 0.40 \times 0.5 = 1.4815 - 0.20 = 1.2815$$ Using standard normal distribution values: - $N(d_1) \approx 0.9307$ - $N(d_2) \approx 0.8997$ Calculate call price: $$C = 4.50 \times 0.9307 - 3.50 \times e^{-0.10 \times 0.25} \times 0.8997 = 4.188 - 3.50 \times 0.9753 \times 0.8997 = 4.188 - 3.07 = 1.118$$ So, the current value of the call option is approximately 1.12. **Final answers:** - Forward premium: 1.13% - Six-month forward rate: 0.12136 Ksh/Tsh - Kenyan investor gains by investing in Tanzania and converting back, confirming IRP. - Call option value: 1.12