1.  Zahara Ltd. wants to maintain a minimum cash balance of $500,000. The estimated variance of the daily cash flows is $1,200,000 per day and the cost for each transaction of buying and selling marketable securities is fixed at $10,000. The marketable securities yield 3% per year. Zahara Ltd. wants to use the Miller-Orr model to determine its desired cash balance. Required: Calculate target cash level and the upper limit for Zahara Ltd.’s cash balance. 


To determine Zahara Ltd.'s desired cash balance using the Miller-Orr model, we need to calculate two main quantities: the target cash balance and the upper limit. The Miller-Orr model is used to manage cash balances by setting limits on how much cash a firm holds.

### Given:
- Minimum cash balance ($L$) = $500,000
- Variance of daily cash flows ($\sigma^2$) = $1,200,000
- Fixed transaction cost ($C$) = $10,000
- Daily interest rate ($i$) = 3% per year = $ \frac{3\%}{365} $ per day

### 1. **Calculating the Target Cash Balance** ($Z$):

The formula for the target cash balance in the Miller-Orr model is:

\[
Z = L + \frac{1}{3} \left(\frac{3C\sigma^2}{4i}\right)^{\frac{1}{3}}
\]

First, let's calculate the daily interest rate ($i$):

\[
i = \frac{3\%}{365} = \frac{0.03}{365} \approx 0.00008219 \text{ per day}
\]

Next, calculate the square root term:

\[
\left(\frac{3C\sigma^2}{4i}\right)^{\frac{1}{3}} = \left(\frac{3 \times 10,000 \times 1,200,000}{4 \times 0.00008219}\right)^{\frac{1}{3}}
\]

Now, compute this step by step:

\[
\frac{3 \times 10,000 \times 1,200,000}{4 \times 0.00008219} = \frac{36,000,000,000}{0.00032876} \approx 1.0956 \times 10^{14}
\]

Then, take the cube root of this value:

\[
\left(1.0956 \times 10^{14}\right)^{\frac{1}{3}} \approx 4.78 \times 10^4 = 47,800
\]

Finally, the target cash balance ($Z$) is:

\[
Z = 500,000 + \frac{1}{3} \times 47,800 \approx 500,000 + 15,933.33 = 515,933.33 \text{ (approximately $515,933)}
\]

### 2. **Calculating the Upper Limit** ($H$):

The upper limit ($H$) is calculated as:

\[
H = 3Z - 2L
\]

Substitute the values:

\[
H = 3 \times 515,933 - 2 \times 500,000
\]

\[
H = 1,547,799 - 1,000,000 = 547,799 \text{ (approximately $547,799)}
\]

### **Summary:**

- **Target Cash Balance**: \$515,933 (approximately)
- **Upper Limit**: \$547,799 (approximately)

These values indicate that when Zahara Ltd.'s cash balance reaches $547,799, they should purchase marketable securities to bring the balance down to $515,933. Similarly, if the cash balance drops to $500,000, they should sell marketable securities to bring the balance back to $515,933.

Would you like more details or have any questions? 

Here are 5 related questions to explore further:
1. How does the variance of daily cash flows affect the target cash balance in the Miller-Orr model?
2. What is the significance of the daily interest rate in the Miller-Orr model?
3. How would a change in transaction costs affect the upper limit and target cash balance?
4. Can the Miller-Orr model be applied if the interest rate is variable instead of fixed?
5. What are the potential risks of holding too much or too little cash in the context of the Miller-Orr model?

**Tip:** Understanding the implications of the upper limit can help you effectively manage the trade-offs between holding costs and transaction costs when managing cash balances.

Learn how to calculate Zahara Ltd.'s target cash balance and upper limit using the Miller-Orr model. Understand the impact of variance in daily cash flows, fixed transaction costs, and the daily interest rate of 3% per year. Discover how these factors determine optimal cash management strategies. Suitable for advanced finance professionals and students.

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