Baumol’s Model gives us a framework for managing cash balances. It helps determine the optimal amount of cash to convert into marketable securities to minimize transaction costs and opportunity costs. I'll guide you through each of the parts using the information given.
Given:
- Cash use rate (D) = $200,000 per year
- Interest rate (i) = 2% or 0.02
- Transaction cost (T) = $20 per sale
a) What is the economic level of cash that the store should convert from marketable securities at a time?
The optimal cash balance (C*) can be calculated using the formula:
\[ C^* = \sqrt{\frac{2DT}{i}} \]
Substituting the values:
\[ C^* = \sqrt{\frac{2 \times 200,000 \times 20}{0.02}} \]
Calculating that:
\[ C^* = \sqrt{\frac{8,000,000}{0.02}} = \sqrt{400,000,000} \approx 20,000 \]
Thus, the economic level of cash that the store should convert from marketable securities at a time is $20,000.
b) How many times a year should the store sell securities?
The number of times (N) the store should sell securities per year can be determined by dividing the annual cash requirement by the optimal cash balance:
\[ N = \frac{D}{C^*} \]
Substituting the values:
\[ N = \frac{200,000}{20,000} = 10 \]
Hence, the store should sell securities 10 times a year.
c) What is its average cash balance?
The average cash balance (ACB) can be calculated as half of the optimal cash balance, because the cash balance decreases from \(C^*\) to $0 before needing to sell more securities:
\[ ACB = \frac{C^*}{2} \]
Substituting the value:
\[ ACB = \frac{20,000}{2} = 10,000 \]
Thus, the average cash balance is $10,000.
d) What is the annual opportunity cost of holding cash for the store?
The opportunity cost of holding cash can be calculated as the interest that could have been earned on the average cash balance:
\[ \text{Opportunity Cost} = ACB \times i \]
Substituting the values:
\[ \text{Opportunity Cost} = 10,000 \times 0.02 = 200 \]
Therefore, the annual opportunity cost of holding cash is $200.
e) What is the annual cost of selling securities?
The annual cost of selling securities is equal to the number of times securities are sold multiplied by the cost of each sale:
\[ \text{Cost} = N \times T \]
Substituting the values:
\[ \text{Cost} = 10 \times 20 = 200 \]
Thus, the annual cost of selling securities is $200.
Summary:
- a) Economic level of cash: $20,000
- b) Sales of securities per year: 10 times
- c) Average cash balance: $10,000
- d) Annual opportunity cost of holding cash: $200
- e) Annual cost of selling securities: $200
This analysis indicates that under the given conditions, the cash management strategy in place is efficient regarding the trade-offs between transaction costs and opportunity costs.
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