Advise the client on how to use the hedging techniques (call option, money market hedge and forward contract) to get the BWP 500 000 at the lowest cost. Show comparison between unhedged and hedged scenarios.

The client will buy BWP 500,000 at the future spot rate. We calculate the expected cost based on the forecasted outcomes.

$$\begin{align*} \text{Cost for } S_{30} = 1.25 &= \text{BWP } 500,000 \times 1.25 = \text{ZAR } 625,000 \\ \text{Cost for } S_{30} = 1.32 &= \text{BWP } 500,000 \times 1.32 = \text{ZAR } 660,000 \\ \text{Cost for } S_{30} = 1.08 &= \text{BWP } 500,000 \times 1.08 = \text{ZAR } 540,000 \end{align*}$$

Expected cost of unhedged scenario:

$$\begin{align*} E[\text{Cost}_{\text{Unhedged}}] &= (\text{ZAR } 625,000 \times 0.30) + (\text{ZAR } 660,000 \times 0.40) + (\text{ZAR } 540,000 \times 0.30) \\ &= \text{ZAR } 187,500 + \text{ZAR } 264,000 + \text{ZAR } 162,000 \\ &= \boxed{\text{ZAR } 613,500} \end{align*}$$

The client locks in the exchange rate today for a future transaction using the 30-day forward rate.

$$\begin{align*} \text{Cost}_{\text{Forward}} &= \text{BWP } 500,000 \times F_{30} \\ &= \text{BWP } 500,000 \times \text{ZAR } 1.16/\text{BWP} \\ &= \boxed{\text{ZAR } 580,000} \end{align*}$$

The strategy for paying foreign currency (BWP) in the future is:

1. Calculate the present value of BWP 500,000 using the Botswana deposit rate.
2. Convert this BWP amount to ZAR today using the spot rate.
3. Borrow this ZAR amount in South Africa for 30 days using the South African borrowing rate.
4. The total ZAR to be repaid in 30 days is the cost.

First, calculate the 30-day interest rates:

$$\begin{align*} r_{BWP,dep,30} &= 0.06 \times \frac{30}{365} \approx 0.0049315 \\ r_{ZAR,bor,30} &= 0.105 \times \frac{30}{365} \approx 0.0086301 \end{align*}$$

Amount of BWP to deposit today ($PV_{BWP}$):

$$\begin{align*} PV_{BWP} &= \frac{\text{BWP } 500,000}{1 + r_{BWP,dep,30}} \\ &= \frac{\text{BWP } 500,000}{1 + 0.0049315} \\ &= \frac{\text{BWP } 500,000}{1.0049315} \approx \text{BWP } 497,540.82 \end{align*}$$

Convert $PV_{BWP}$ to ZAR today using the spot rate:

$$\begin{align*} ZAR_{\text{today}} &= \text{BWP } 497,540.82 \times \text{ZAR } 1.15/\text{BWP} \\ &\approx \text{ZAR } 572,171.94 \end{align*}$$

Borrow this ZAR amount for 30 days at the South African borrowing rate:

$$\begin{align*} \text{Cost}_{\text{MMH}} &= ZAR_{\text{today}} \times (1 + r_{ZAR,bor,30}) \\ &= \text{ZAR } 572,171.94 \times (1 + 0.0086301) \\ &= \text{ZAR } 572,171.94 \times 1.0086301 \\ &\approx \boxed{\text{ZAR } 577,110.04} \end{align*}$$

The client buys a call option on BWP.
Exercise price $X = \text{ZAR } 1.16 / \text{BWP}$
Premium $P = \text{ZAR } 0.025 / \text{BWP}$
Contract size = BWP 500,000

First, calculate the future value of the premium, which is paid upfront.

$$\begin{align*} \text{Total Premium Today} &= \text{BWP } 500,000 \times \text{ZAR } 0.025/\text{BWP} = \text{ZAR } 12,500 \\ \text{FV of Premium} &= \text{ZAR } 12,500 \times (1 + r_{ZAR,bor,30}) \\ &= \text{ZAR } 12,500 \times (1 + 0.0086301) \\ &= \text{ZAR } 12,500 \times 1.0086301 \approx \text{ZAR } 12,607.88 \end{align*}$$

Now, calculate the cost for each outcome in 30 days:

• Outcome 1: $S_{30} = 1.25$ Since $S_{30} (1.25) > X (1.16)$, the option will be exercised.

  $$\begin{align*} \text{Cost}_1 &= (\text{BWP } 500,000 \times X) + \text{FV of Premium} \\ &= (\text{BWP } 500,000 \times \text{ZAR } 1.16/\text{BWP}) + \text{ZAR } 12,607.88 \\ &= \text{ZAR } 580,000 + \text{ZAR } 12,607.88 = \text{ZAR } 592,607.88 \end{align*}$$

• Outcome 2: $S_{30} = 1.32$ Since $S_{30} (1.32) > X (1.16)$, the option will be exercised.

  $$\begin{align*} \text{Cost}_2 &= (\text{BWP } 500,000 \times X) + \text{FV of Premium} \\ &= (\text{BWP } 500,000 \times \text{ZAR } 1.16/\text{BWP}) + \text{ZAR } 12,607.88 \\ &= \text{ZAR } 580,000 + \text{ZAR } 12,607.88 = \text{ZAR } 592,607.88 \end{align*}$$

• Outcome 3: $S_{30} = 1.08$ Since $S_{30} (1.08) < X (1.16)$, the option will not be exercised. The client will buy BWP at the spot rate.

  $$\begin{align*} \text{Cost}_3 &= (\text{BWP } 500,000 \times S_{30}) + \text{FV of Premium} \\ &= (\text{BWP } 500,000 \times \text{ZAR } 1.08/\text{BWP}) + \text{ZAR } 12,607.88 \\ &= \text{ZAR } 540,000 + \text{ZAR } 12,607.88 = \text{ZAR } 552,607.88 \end{align*}$$

Expected cost of the call option:

$$\begin{align*} E[\text{Cost}_{\text{Call Option}}] &= (\text{ZAR } 592,607.88 \times 0.30) + (\text{ZAR } 592,607.88 \times 0.40) + (\text{ZAR } 552,607.88 \times 0.30) \\ &= \text{ZAR } 177,782.36 + \text{ZAR } 237,043.15 + \text{ZAR } 165,782.36 \\ &= \boxed{\text{ZAR } 580,607.87} \end{align*}$$

Here is a summary of the expected costs for each scenario:

Hedging Technique	Expected Cost (ZAR)
Unhedged Scenario	613,500.00
Forward Contract	580,000.00
Money Market Hedge	577,110.04
Call Option	580,607.87

Based on the expected costs, the Money Market Hedge offers the lowest cost for the client to acquire BWP 500,000 in 30 days. This method also provides certainty regarding the future ZAR cost, similar to a forward contract, but at a more favorable rate in this specific case.

Recommendation:
The client should implement a Money Market Hedge. This involves:

1. Calculating the present value of BWP 500,000 using Botswana's 30-day deposit rate.
2. Converting this BWP amount to ZAR today at the spot rate.
3. Borrowing the equivalent ZAR amount in South Africa for 30 days at the South African borrowing rate. This strategy will result in the lowest expected ZAR cost of ZAR 577,110.04 to obtain the required BWP 500,000 in 30 days.