Chapter 27, Problem 6QP

Using Weighted Average Delay A mail-order firm processes 5,450 checks per month. Of these, 70 percent are for $55 and 30 percent are for $80. The $55 checks are delayed two days on average; the $80 checks arc delayed three days on average.

1. a. What is the average daily collection float? How do you interpret your answer?
2. b. What is the weighted average delay? Use the result to calculate the average daily float
3. c. How much should the firm be willing to pay to eliminate the float?
4. d. If the interest rate is 7 percent per year, calculate the daily cost of the float.
5. e. How much should the firm be willing to pay to reduce the weighted average float by 1.5 days?

Summary Introduction

To compute: The average daily collections float.

Purchasing Power:

Purchasing power can be referred to the amount of goods and services that can be purchased by one unit of money. It decreases with increase in inflation or decreases with the decrease in inflation.

Float:

Float is defined as the difference between the balance shown in ledger of the company and the balance that is available at the bank. Available balance refers to the balance that is shown by the banks of that particular company.

Explanation of Solution

Solution:

Formula to calculate average collection,

$\text{Average}\text{}\text{collection}=\frac{\text{Value of check of }\$55+\text{Value of }\$80\text{ check}}{\text{Number}\text{of}\text{days}\text{in}\text{a}\text{month}}$

Substitute $419,650 for value of check of $55, $610,400 for value of $80 check and 30 for number of days in a month.

$\begin{array}{c}\text{Average}\text{}\text{collection}=\frac{\$419,650+\$610,400}{30}\\ =\frac{\$1,030,050}{30}\\ =\$34,335\end{array}$

The average collection is $34,335.

Working notes:

Given,

Total checks per month are 5,450.

Percentage is 70%.

Delay days are 2 days.

Computation of number of $55 check,

$\begin{array}{c}\text{Number}\text{of}\$55\text{check}=\text{Total}\text{checks}\times \text{Percentage}\\ =5,450\times 70\%\\ =3,815\end{array}$

Total number of $55 checks is 3,815.

Computation of value of check,

$\begin{array}{c}\text{Value}\text{of}\text{check}=\text{Number}\text{of}\text{checks}\times \$55\text{checks}\times \text{Delay}\text{days}\\ =3,815\times \$55\times 2\\ =\$419,650\end{array}$

Value of check of $55 is $419,650.

Given,

Total checks per month are 5,450.

Percentage is 30%.

Delay days are 2 days.

Computation of number of $80 check,

$\begin{array}{c}\text{Number}\text{of}\$80\text{check}=\text{Total}\text{checks}\times \text{Percentage}\\ =5,450\times 30\%\\ =1,635\end{array}$

Total number of $80 checks is 1,635.

Computation of value of check,

$\begin{array}{c}\text{Value}\text{of}\text{check}=\text{Number}\text{of}\text{checks}\times \$80\text{checks}\times \text{Delay}\text{days}\\ =3,815\times \$80\times 2\\ =\$610,400\end{array}$

Value of $80 check is $610,400.

Conclusion

Hence, the average collection is $34,335.

Summary Introduction

To compute: The weighted average delay and the average daily float.

Explanation of Solution

Solution:

Formula to calculate weighted average delay,

$\text{Weighted}\text{}\text{average}\text{}\text{delay}=\frac{\text{Value}\text{of}70\%\text{checks}}{\text{Total}\text{collection}}+\frac{\text{Value}\text{of}30\%\text{check}}{\text{Total}\text{collection}}$

Substitute $419,650 for value of 70% of checks, $340,625 for total collection, $610,400 for value of 30% check.

$\begin{array}{c}\text{Weighted}\text{}\text{average}\text{}\text{delay}=\frac{\$419,650}{\$340,625}+\frac{\$610,400}{\$340,625}\\ =1.232+1.792\\ =3.024\end{array}$

Weighted average delay is 3.024.

Formula to calculate average daily float,

$\text{Average}\text{daily}\text{float}=\text{Average}\text{daily}\text{receipt}\times \text{Weighted}\text{average}\text{delay}$

Substitute $340,625 for average daily receipt and 3.024 for weighted average delay,

$\begin{array}{c}\text{Average}\text{daily}\text{float}=\frac{\$340,625}{30}\times 3.024\\ =34,335\end{array}$

Average daily float is $34,335.

Working note:

Total number of $55 checks is 3,815

Checks are $55.

Total number of $80 checks is 1,635.

Checks are $80.

Computation of total collection,

$\begin{array}{c}\text{Total}\text{collection}=\left[\begin{array}{c}\left(\text{Total number of }\$55\text{ checks}\times \text{Checks}\right)\\ +\left(\text{Total number of }\$80\text{ checks}\times \text{Checks}\right)\end{array}\right]\\ =\left(3,815\times \$55\right)+\left(1,635\times \$80\right)\\ =\$209,825+\$130,800\\ =\$340,625\end{array}$

Hence, the total collection is $340,625.

Conclusion

Hence, the weighted average delay is 3.024 and average daily float is $34,335.

Summary Introduction

To identify: The firm willingness to pay to eliminate the company’s float.

Answer to Problem 6QP

Solution:

• Interest rate should be paid by the company to eliminate the float.
• Present value of the payment of future interest is to be computed.

Explanation of Solution

• Company’s willingness is to pay the present value on the interest income on the payment that is to be received today.

Conclusion

The firm should pay the interest rate on the present value.

Summary Introduction

To compute: The Company’s daily float.

Explanation of Solution

Solution:

Given,

Annual interest rate is 7%.

Formula to calculate rate,

$\text{Annual}\text{interest}\text{rate}={\left(1+\text{Rate}\right)}^{365}$

Substitute 7% for annual interest rate,

$\begin{array}{c}1+7\%={\left(1+\text{Rate}\right)}^{365}\\ {\left(1.07\right)}^{\frac{1}{365}}=\left(1+\text{Rate}\right)\\ 1.0001854=1+\text{Rate}\\ \text{Rate}=0.01854\%\text{per}\text{day}\end{array}$

Rate is 0.01854% per day.

Average daily float is $34,335.

Formula to calculate cost of float,

$\text{Cost}\text{of}\text{float}=\text{Average}\text{daily}\text{float}\times \text{Rate}\text{per}\text{day}$

Substitute $34,334.998 for average daily float and 0.01854% for rate per day.

$\begin{array}{c}\text{Cost}\text{of}\text{float}=\$34,335\times 0.01854\%\\ =\$6.3657\end{array}$

Cost of float is $6.3657.

Conclusion

Hence, the daily cost of float is $6.3657.

Summary Introduction

To compute: The amount the company is willing to pay to reduce the weighted average float.

Explanation of Solution

Solution:

Given,

Company wants to reduce to 1.5 days.

Formula to calculate Net average daily float,

$\text{Net}\text{}\text{average}\text{}\text{daily}\text{float}=\text{Reduced}\text{days}\times \left(\frac{\text{Total}\text{collection}}{30\text{days}}\right)$

Substitute 1.5 as reduced days and $340,625 for total collection.

$\begin{array}{c}\text{Net}\text{}\text{average}\text{}\text{daily}\text{float}=1.5\times \left(\frac{\$340,625}{30}\right)\\ =1.5\times \$11,354.1667\\ =\$17,031.2501\end{array}$

Net average daily float is $17,031.2501.

Conclusion

Hence, the amount the company is willing to pay to reduce the weighted average float is $17,031.2501.