ECON 3077: Management of Financial Institutions
Homework # 04
Direction: To receive full credit, do not leave any part of a question unanswered. Illegible writing will be marked incorrect, so please write clearly. Show your work.
Example:
Using a spreadsheet program or a calculator, solve Tracy’s problem of how often to go to the ATM when the nominal interest rate on her bank account is 10 percent, she spends $30 each day, it costs her $0.50 each time she uses the ATM, and she thinks that there is a 15 percent chance that she will lose her cash or have it stolen. Under these conditions, how often does Tracy go to the ATM, and how much cash does she take out each time?
Answer:
Cost = cost of going to ATM + opportunity cost + expected cost of loss or theft
= (365 × ATM cost)/T + ($30 × T × i)/2 + ($30 × T × probability of loss or theft)/2
= (365 × $0.50)/T + ($30 × T × 0.10)/2 + ($30 × T ×0.15)/2
= ($182.50/T) + ($3.75 × T).
See the spreadsheet results shown below:
T Total annual cost
of withdrawing
when spending =
$30/day
Amount withdrawn
when spending =
$30/day
1 186.25 30
2 98.75 60
3 72.08 90
4 60.63 120
5 55.25 150
6 52.92 180
7 52.32 210
8 52.81 240
9 54.03 270
10 55.75 300
11 57.84 330
12 60.21 360
13 62.79 390
14 65.54 420
15 68.42 450
Min 52.32
Result: To minimize her costs, she will go to the ATM every seven days and take out $210 each time.
2
1. Someone who visits the ATM once every 8 days, and who spends $25 per day. What is his average cash
balance?
2. Someone who visits the ATM once every 7 days, and who has an average cash balance of $70. How much does
this person spend per day?
3. If the nominal interest rate is 5 percent, someone who has a 15 percent probability of having his cash lost or
stolen and who spends $10 each day, and who has the total cost of holding cash = (365/T) + T. What is the cost
of going to the ATM of this person?
4. If the cost of going to the ATM is $1 and the nominal interest rate is 5 percent, someone who spends $10 each
day and has the total cost of holding cash = (365/T) + T. Find the probability of having his cash lost or stolen.
5. Suppose someone’s cost of going to the ATM is $1.50, there is a 12 percent probability of having his cash lost
or stolen, and he spends $5 each day. Suppose his total cost of holding cash = (547.50/T) + (0.375 × T). Find
the nominal interest rate.
6. If the nominal interest rate is 3 percent and the cost of going to the ATM is $1.50, someone who has a 12
percent probability of having his cash lost or stolen and has the total cost of holding cash = (547.50/T) + (0.375
× T). How much does he spend per day?
7. If the cost of going to the ATM is $1 and the nominal interest rate is 5 percent, someone who has a 15 percent
probability of having his cash lost or stolen and spends $10 each day. How often will he go to the ATM (in
days)?
8. If the cost of going to the ATM is $2 and the nominal interest rate is 1 percent, someone who has a 9 percent
probability of having his cash lost or stolen and spends $15 each day. How often will he go to the ATM (in
days)?
9. Suppose the nominal interest rate is 3 percent, the cost of going to the ATM is $1.50, you have a 12 percent
probability of having your cash lost or stolen, and you spend $5 each day.
a. What is your total cost of holding cash as a function of the number of days between trips to the ATM?
b. How often will you go to the ATM to minimize your costs?
10. Suppose you have a 20 percent probability of having your cash lost or stolen, and you spend $25 each day. Your
total cost of holding cash is (182.50/T) + (3.75 × T).
a. What is your cost of going to the ATM?
b. What is the nominal interest rate?
c. How often will you go to the ATM to minimize your costs?