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Part 1
The Fast Reader Company supplies bulletin board services to numerous hotel chains nationwide. The owner of the firm is investigating the desirability of employing a billing firm to do her billing and collections. Because the billing firm specialises in these​ services, collection float will be reduced by 11 days. Average daily collections are $ 4 comma 900​, and the owner can earn 7 % annually​ (expressed as an APR with monthly​ compounding) on her investments. If the billing firm charges ​$400 per​ month, should the owner employ the billing​ firm?Assuming the billing firm can reduce the collection float​ immediately, as soon as Fast Reader starts the service they will receive ​$

To determine whether the owner of Fast Reader should employ the billing firm, we need to compare the financial benefits of reducing the collection float to the cost of hiring the billing firm. Here's a step-by-step approach to solving this problem:

### Step 1: Calculate the financial benefit of reducing collection float
The reduction in the collection float is 11 days, meaning the owner will get paid 11 days earlier. The average daily collections are $4,900. By reducing the float by 11 days, the owner would receive money 11 days earlier, which can then be invested.

Total collections affected by the reduced float:
\[
\text{Total collections} = 11 \times 4900 = 53,900
\]

### Step 2: Calculate the interest earned on the $53,900 over 11 days
The owner earns 7% annually, compounded monthly. To find out how much interest can be earned on $53,900 for 11 days, we can use the formula for compound interest:

\[
A = P \times \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- $P = 53,900$ (the total collections),
- $r = 0.07$ (the annual interest rate),
- $n = 12$ (since it's compounded monthly),
- $t = \frac{11}{365}$ (since we are looking at 11 days out of the year).

We can simplify this to determine the interest gained from receiving this money 11 days earlier.

### Step 3: Calculate the monthly cost of hiring the billing firm
The cost of employing the billing firm is $400 per month.

### Step 4: Compare the benefit and the cost
We need to compare the interest earned from the reduced float with the cost of hiring the billing firm. If the interest earned is greater than $400 per month, then it is financially beneficial for the owner to employ the billing firm.

Let me compute the interest earned.The interest earned from receiving the collections 11 days earlier is approximately $113.50.

### Step 5: Conclusion
- The interest earned due to the reduction in collection float is **$113.50**.
- The cost of hiring the billing firm is **$400** per month.

Since the interest earned is less than the cost of hiring the billing firm, it is **not financially beneficial** for the owner to employ the billing firm.

### Do you want more details or have any questions?

Here are 5 related questions you could explore:
1. How would the result change if the average daily collections were higher?
2. What if the billing firm's monthly fee were lower?
3. How would an increase in the annual interest rate affect the decision?
4. What if the billing firm reduced the collection float by a greater number of days?
5. How does monthly compounding compare to daily compounding in this scenario?

**Tip:** Always compare the costs and benefits when deciding to outsource financial services.

The Fast Reader Company supplies bulletin board services to numerous hotel chains nationwide. The owner of the firm is investigating the desirability of employing a billing firm to do her billing and collections. Because the billing firm specializes in these services, collection float will be reduced by 11 days. Average daily collections are $4,900, and the owner can earn 7% annually (expressed as an APR with monthly compounding) on her investments. If the billing firm charges $400 per month, should the owner employ the billing firm?

Explore a detailed cost-benefit analysis for the Fast Reader Company on whether to employ a billing firm. Understand the impact of reducing collection float by 11 days on $4,900 average daily collections, with a 7% APR compounded monthly. Suitable for undergraduate finance or business students.

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