Chapter 19, Problem 6QP

Using Weighted Average Delay [LO1] A mail-order firm processes 4,900 checks per month. Of these, 60 percent are for $39 and 40 percent are for $71. The $39 checks are delayed two days on average; the $71 checks are delayed three days on average. Assume 30 days in a month.

a. What is the average daily collection float? How do you interpret your answer?

b. What is the weighted average delay? Use the result to calculate the average daily float.

c. How much should the firm be willing to pay to eliminate the float?

d. If the interest rate is 7 percent per year, calculate the daily cost of the float.

e. How much should the firm be willing to pay to reduce the weighted average float to 1.5 days?

Summary Introduction

To compute: The average daily collection floats.

Introduction:

The float is the difference between the bank cash and the book cash denoting the net effects of checks in the clearing process.

Answer to Problem 6QP

The average daily float is $21,560.

Explanation of Solution

Given information:

The mail-order company processes checks of $4,900 per month. From these checks, 60% are for $39 and 40% are from $71. There is a delay of two days on an average for $39 checks and a delay of three days for $71 checks.

Explanation:

Formula to compute the average daily float:

$\text{Average daily float}=\left[\frac{\begin{array}{l}\text{Sum of percentage of checks}\left(\text{Number of checks}\right)\\ \left(\text{Amount of check}\right)\\ \left(\text{Number of days for the clearance of the check}\right)\end{array}}{\text{Number of days in a month}}\right]$

Compute the average daily float:

$\begin{array}{c}\text{Average daily float}=\left[\frac{\begin{array}{l}\text{Sum of percentage of checks}\left(\text{Number of checks}\right)\\ \left(\text{Amount of check}\right)\\ \left(\text{Number of days for the clearance of the check}\right)\end{array}}{\text{Number of days in a month}}\right]\\ =\left[\frac{0.60\left(4,900\right)\left(\$39\right)\left(2\right)+0.40\left(4,900\right)\left(\$71\right)\left(3\right)}{30}\right]\\ =\$21,560\end{array}$

Hence, the average daily float is $21,560.

Summary Introduction

To compute: The weighted average delay.

Introduction:

The float is the difference between the bank cash and the book cash denoting the net effects of checks in the clearing process.

Explanation of Solution

Given information:

The mail-order company processes checks of $4,900 per month. From these checks, 60% are for $39 and 40% are from $71. There is a delay of two days on an average for $39 checks and a delay of three days for $71 checks.

Answer: The weighted average delay is 2.55 days.

Formula to compute the total collections:

$\text{Total collections}=\left[\begin{array}{l}\text{Sum of percentage of checks}\left(\text{Number of checks}\right)\\ \left(\text{Amount of check}\right)\end{array}\right]$

Compute the total collections:

$\begin{array}{c}\text{Total collections}=\left[\begin{array}{l}\text{Sum of percentage of checks}\left(\text{Number of checks}\right)\\ \left(\text{Amount of check}\right)\end{array}\right]\\ =0.60\left(4,900\right)\left(\$39\right)+0.40\left(4,900\right)\left(\$71\right)\\ =\$253,820\end{array}$

Hence, the total collection is $253,820.

Formula to compute the weighted average delay:

$\text{Weighted average delay}=\left[\left(\begin{array}{l}\text{Number of days for }\\ \text{clearance of check}\end{array}\right)\left[\frac{\begin{array}{l}\text{Sum of percentage of checks}\\ \left(\text{Number of checks}\right)\\ \left(\text{Amount of check}\right)\end{array}}{\text{Total collections}}\right]\right]$

Compute the weighted average delay:

$\begin{array}{c}\text{Weighted average delay}=\left[\left(\begin{array}{l}\text{Number of days for }\\ \text{clearance of check}\end{array}\right)\left[\frac{\begin{array}{l}\text{Sum of percentage of checks}\\ \left(\text{Number of checks}\right)\\ \left(\text{Amount of check}\right)\end{array}}{\text{Total collections}}\right]\right]\\ =2\left[\frac{0.60\left(4,900\right)\left(\$39\right)}{\$253,820}\right]+3\left[\frac{0.40\left(4,900\right)\left(\$71\right)}{\$253,820}\right]\\ =2\left(0.451737451\right)+3\left(0.548262548\right)\\ =2.55\text{ days}\end{array}$

Hence, the weighted average delay is 2.55 days.

Summary Introduction

To compute: The amount the company will be willing to pay to avoid the float.

Introduction:

The float is the difference between the bank cash and the book cash denoting the net effects of checks in the clearing process.

Answer to Problem 6QP

The company will be willing to pay to avoid the float is $21,560.

Explanation of Solution

Given information:

The mail-order company processes checks of $4,900 per month. From these checks, 60% are for $39 and 40% are from $71. There is a delay of two days on an average for $39 checks and a delay of three days for $71 checks.

Formula to compute the average daily float:

$\text{Average daily float}=\text{Weighted average delay}\left(\frac{\text{Total collections}}{\text{Number of days in a month}}\right)$

Compute the average daily float:

$\begin{array}{c}\text{Average daily float}=\text{Weighted average delay}\left(\frac{\text{Total collections}}{\text{Number of days in a month}}\right)\\ =2.55\left(\frac{\$253,820}{30\text{ days}}\right)\\ =\$21,560\end{array}$

Hence, the average daily float is $21,560.

Summary Introduction

To compute: The daily float cost, if the rate of interest is 7% per year.

Introduction:

The float is the difference between the bank cash and the book cash denoting the net effects of checks in the clearing process.

Answer to Problem 6QP

The daily float cost is $4.00.

Explanation of Solution

Given information:

The mail-order company processes checks of $4,900 per month. From these checks, 60% are for $39 and 40% are from $71. There is a delay of two days on an average for $39 checks and a delay of three days for $71 checks.

Formula to compute the average daily rate of interest:

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

Compute the average daily rate of interest:

$\begin{array}{c}\left(1+\text{Rate of interest}\right)={\left(1+R\right)}^{365}\\ 1.07={\left(1+R\right)}^{365}\\ {\left(1.07\right)}^{\frac{1}{365}}=1+R\end{array}$

$\begin{array}{c}\text{1}+R\text{ =1}\text{.0001854}\\ R=1.0001854-1\\ R=0.0001854\times 100\\ R=0.01854\%\text{ per day}\end{array}$

Hence, the average daily rate is 0.01854% per day.

Formula to compute the daily float cost:

$\text{Daily float cost}=\text{Average daily float}\left(\text{Average daily rate}\right)$

Compute the daily float cost:

$\begin{array}{c}\text{Daily float cost}=\text{Average daily float}\left(\text{Average daily rate}\right)\\ =\$21,560\left(0.0001854\right)\\ =\$4.00\end{array}$

Hence, the daily float cost is $4.00.

Summary Introduction

To compute: The amount that the company will be willing to pay to decrease the weighted average float to 1.5 days.

Introduction:

The float is the difference between the bank cash and the book cash denoting the net effects of checks in the clearing process.

Answer to Problem 6QP

The amount that the company will be willing to pay to decrease the weighted average float to 1.5 days is $8,869.

Explanation of Solution

Given information:

The mail-order company processes checks of $4,900 per month. From these checks, 60% are for $39 and 40% are from $71. There is a delay of two days on an average for $39 checks and a delay of three days for $71 checks.

Formula to compute the new average daily float:

$\text{New average daily float}=\left[\left(\begin{array}{l}\text{New weighted }\\ \text{average float}\end{array}\right)\left(\frac{\text{Total collections}}{\text{Number of days in a month}}\right)\right]$

Compute the new average daily float:

$\begin{array}{c}\text{New average daily float}=\left[\left(\begin{array}{l}\text{New weighted }\\ \text{average float}\end{array}\right)\left(\frac{\text{Total collections}}{\text{Number of days in a month}}\right)\right]\\ =1.5\left(\frac{\$253,820}{30}\right)\\ =\$12,691\end{array}$

Hence, the new average daily float is $12,691.

Formula to compute the maximum payment amount:

$\text{Maximum amount}=\text{Average daily float}-\text{New average daily float}$

Compute the maximum payment amount:

$\begin{array}{c}\text{Maximum amount}=\text{Average daily float}-\text{New average daily float}\\ =\$21,560-\$12,691\\ =\$8,869\end{array}$

Hence, the maximum amount the firm will be willing to pay is $8,869.