6.1 Credit Policy
Firms routinely extend credit to their customers. This increases the sales of the firms and enables the customers to buy the goods even when they do not have any cash available. For instance a lumberyard may sell building material to a contractor on 60-day credit. When the contractor finishes his job and gets paid, he will also pay the bill from the lumberyard. The cost to the lumberyard for extending credit consists of two items: the cost of capital made available to the contractor, and second, the possibility of default. In some cases the seller is unable the recover the proceeds of a sale from a customer. The seller is prepared to take this risk.
Suppose the cost of capital to a firm is r. The firm has decided to make a credit sale to customer for an amount S. The cost of goods sold is C. The default risk of the firm is expressed in terms of a discrete probability distribution, Pi that the payment Si will be received after time i. The firm does not impose any penalties for late payments. Then we can write the NPV of this credit policy as
n PiSi
NPV = − C + (1 + r)i(6.1)
i=1
We can also compare the NPV of two different credit policies with the help of this expression.
6.2 Analysis of Credit Decisions
The corporations have to make the decision whether to grant credit to a customer, or to reject his application. This credit decision is usually based on the previous experience with this customer. If it is a new customer, the firm must make appropriate inquiries about the credit history of the new customer. There are well established credit reporting agencies that will rank the customers according to their creditworthiness. Based on their recommendations, the company must make its own credit-granting decision.
One way to analyze credit problems is to compute the NPV of a credit decision. Let us develop such a model. Suppose a company sells goods to a customer with value S, while the cost of goods sold is C. The probability that the customer will default is p, thus the probability that he will pay on time is 1 − p. If the customer pays on time, he will do so after time n. Since the billing cycles are generally in months, we assume that the time is n months. If the customer defaults, the firm may still be able to recover a fraction R of the
(
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original sales after time m months. The risk-adjusted cost of capital to the firm is r, per month. The NPV of this decision is
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This is a single period decision model. It ignores the possibility that the customer may come back for additional purchases. The NPV of the first encounter of the customer is N1.
To make a somewhat better model, we assume that the customer will buy an amount S every month. The firm will continue to give the customer credit until he actually defaults. The NPV of a two-period .
Basic Civil Engineering notes on Transportation Engineering, Modes of Transpo...
6.1 Credit PolicyFirms routinely extend credit to their customer.docx
1. 6.1 Credit Policy
Firms routinely extend credit to their customers. This increases
the sales of the firms and enables the customers to buy the
goods even when they do not have any cash available. For
instance a lumberyard may sell building material to a contractor
on 60-day credit. When the contractor finishes his job and gets
paid, he will also pay the bill from the lumberyard. The cost to
the lumberyard for extending credit consists of two items: the
cost of capital made available to the contractor, and second, the
possibility of default. In some cases the seller is unable the
recover the proceeds of a sale from a customer. The seller is
prepared to take this risk.
Suppose the cost of capital to a firm is r. The firm has decided
to make a credit sale to customer for an amount S. The cost of
goods sold is C. The default risk of the firm is expressed in
terms of a discrete probability distribution, Pi that the payment
Si will be received after time i. The firm does not impose any
penalties for late payments. Then we can write the NPV of this
credit policy as
n PiSi
i=1
We can also compare the NPV of two different credit policies
with the help of this expression.
6.2 Analysis of Credit Decisions
The corporations have to make the decision whether to grant
credit to a customer, or to reject his application. This credit
decision is usually based on the previous experience with this
customer. If it is a new customer, the firm must make
appropriate inquiries about the credit history of the new
2. customer. There are well established credit reporting agencies
that will rank the customers according to their creditworthiness.
Based on their recommendations, the company must make its
own credit-granting decision.
One way to analyze credit problems is to compute the NPV of a
credit decision. Let us develop such a model. Suppose a
company sells goods to a customer with value S, while the cost
of goods sold is C. The probability that the customer will
default is p, thus the probability that he will pay on time is 1 −
p. If the customer pays on time, he will do so after time n. Since
the billing cycles are generally in months, we assume that the
time is n months. If the customer defaults, the firm may still be
able to recover a fraction R of the
(
105
)
original sales after time m months. The risk-adjusted cost of
capital to the firm is r, per month. The NPV of this decision is
(
T
re
a
s
u
r
y
M
a
n
a
g
4. NPV(one-time) = − C + (1 + r)n + (1 + r)m = N1
This is a single period decision model. It ignores the possibility
that the customer may come back for additional purchases. The
NPV of the first encounter of the customer is N1.
To make a somewhat better model, we assume that the customer
will buy an amount S every month. The firm will continue to
give the customer credit until he actually defaults. The NPV of
a two-period model will be
NPV(two-time) =
(1 − p)S
+
(
(
1 −
p
)
S
−
C
+
(
1 +
5. r
)
n
+
pRS
m
(
1 +
r
)
) (
(1 + r)n+1
In the above expression, the tan part of the equation represents
the NPV of the first encounter, N1. The blue part of the
equation represents the probability that the customer paid his
bill for the first month. The lavender part of the eqution is the
value of the NPV of the second month. It assumes that the
customer will buy the amount of merchandise in the second
month. The soncd month’s NPV is similar to that for the first
month, except that the terms are multiplied by 1/(1 + r) because
of delay in the event by another month.
We can simplify the above equation as
6. 1−p
NPV(two-
The probabilities in the second and subsequent periods are
conditional probabilities. The probability that the customer pays
in the second month is contingent upon his prompt payment in
the first month, and thus the probability is (1 − p)2. This is the
probability that he will pay in the first month and the second
month.
We now extend the calculation to three periods as
1−p
NPV(three-
The third term, [(1 − p)/(1 + r)]2 in the above equation
represents that fact the third month’s shopping by the customer
is contingent upon the payment for the first two periods; and the
fact that the third month is delayed by two months.
Next, we can extend this analysis to an infinite-period model as
7. follows.
1−p
NPV(infinite-
+ … ∞ ]
For the summation of infinite series, use the equation
which gives
S = a + ax + ax2 + … ∞ =a
1 − x
NPV(infinite-times) =N1
1 − (1 − p)/(1 + r)
After some simplification, it becomes
8. NPV(infinite-
We can do the above summation with the help of Maple. We
type in
sum((1-p)^i/(1+r)^i,i=0..infinity);
–
1 + r p + r
i=0
Simplifying the expression further, we get
(1−p)S
pRS
(6.2)
If we assume that the customer is never going to default, then p
9. = 0. In that case, the above expression simplifies to
(6.3)
This represents the value of a well-established, long-term
customer with perfect credit record.
We use the following symbols in (6.2):
C = cost of goods sold each month
S = sales per month
r = cost of capital to the firm, per month
p = probability that the customer will default in a given month
n = normally the time taken by the customer to pay his bill, in
months
m = time taken by the firm to recover the money in case of
default, in months
R = the fraction of the total sale recovered in case of default
6.3 Valuation of a Credit Card Portfolio
Credit cards have become a permanent fixture on the national
scene. Some of the largest banks, such as Citibank, have
millions of cards in the hands of cardholders. The total credit
card debt for the entire American population is hundreds of
billions of dollars.
There is fierce competition among the card issuers. After
saturating the adult population, the banks are offering credit
cards to students and young adults. To gain customers, many of
10. the card issuers have dropped the annual fees, and they are
offering promotional rates as low as 0% for the first six months.
There is also consolidation in this industry. Many of the smaller
regional banks are out of this business. Most of them have
simply sold their credit card portfolios to national banks. The
Federal Reserve reported that the credit card delinquencies hit
4.86 percent in the first quarter in 2008, while revolving debt—
or the type used in credit purchases—hit
$957.2 billion in March, a 7.9 percent increase. [CNBC, 6/3/08]
We may estimate the value of a credit card portfolio as follows.
Suppose a bank has issued N cards altogether. The bank charges
an annual fee of F per card at the end of each year. Let us
assume that the cost of capital to the bank is r. The bank is able
to collect total fees NF per year from the cardholders. The value
of this perpetuity is
Annual feesV1 =
NF
r(6.4)
Let us assume that n of these cardholders pay off their entire
balance every month near the end of the grace period, and they
do not pay any interest at all. Such customers use the bank’s
capital for their personal use and are thus creating a loss for the
bank. Suppose the average balance on these accounts at the end
of each month is B, and the grace period is g days. The amount
of this monthly loss per card is thus
B−g/365
− B + (1 + r)g/365 = B[(1 + r)
11. − 1]
This monthly loss continues forever, and thus it becomes a
perpetual thing. The value of this perpetuity is calculated by
dividing the above quantity by the monthly cost of capital to the
bank, namely, r/12. Thus, we have the value of perpetuity
B[(1 + r)−g/365 − 1]
r/12=
12B[(1 + r)−g/365 − 1]
r
The total value of n such cards is thus
12nB[(1 + r)−g/365 − 1]
Free ridersV2 =
r(6.5)
The bank does not make a permanent investment in such
customers.
Next, let us consider those cardholders who carry an average
balance C on their cards at the end of each month, and pay
interest on them regularly. The annual rate of interest charged
by the bank on the outstanding balance on an account is R. The
number of such cards is, say m. The annual interest generated
by these accounts is mCR. This is another
12. perpetuity whose present value is given by mCR/r. The
bank has already made an investment mC to finance these
accounts. The NPV these cards is thus
Paying customersV3 = − mC +
mCR
r=
mC(R −r)
r(6.6)
These accounts create value for the bank because the bank
charges a much higher rate of interest R (perhaps 15%) than the
cost of capital for the bank r (around 6%).
We now look at the total annual expenses in maintaining this
credit card operation. This includes printing and mailing of the
bills, administrative expenses, and the losses due to default on
the loans by the cardholders. Assuming these costs average out
to be L at the end of each year, their present value this
perpetuity is
L
Administrative expenses V4 = − r
(6.7)
There is another important source of revenue for the bank.
Whenever a merchant sells something to a customer, who uses a
credit card for the purchase, the merchant must also pay a
percentage of the sale price to the bank. In practice, the bank
collects roughly 1% of the total credit card sales. Suppose the
total monthly sales on all credit cards are NS. The merchants
pay a fraction a of this amount to the bank at the end of every
month. The monthly income to the bank is thus aNS. This
13. equals 12aNS per year. The value of this perpetuity to the bank
is
Merchant feesV5 =
12aNS
r(6.8)
In the above expression S is the average monthly sale per card.
The total NPV of the credit-card portfolio is the sum of the five
components of the value given by equations (6.3-8). Thus
NF12nB[(1 + r)−g/365 − 1]
mC(R −r)
L12aNS
Or,
NPV = r +
r+r− r +r
NPV =
NF + 12nB[(1 + r)−g/365 − 1] + mC(R − r) − L + 12aNS
r(6.9)
If the bank wants to sell this portfolio to another bank, they
must ask for its NPV, plus the amount payable to the bank by
14. the cardholders at a given instant in time. Let us call the total
outstanding balance on all credit cards to be P. The selling price
of the portfolio is thus
Selling price = P +
NF + 12nB[(1 + r)−g/365 − 1] + mC(R − r) − L + 12aNS
r(6.10)
In the above expression we define the symbols as follows:
P = receivables, or the total amount due from the customers at a
given instant
N = n + m = total number of cards issued
n = number of card-holders who pay on time, and do not pay
any interest
B = average monthly balance on such free-rider cards
m = number of card-holders who pay interest every month
C = average monthly balance on these paying-customer cards
F = annual membership fee charged by the bank, at the end of
each year
R = annual rate of interest on the outstanding balance
r = annual cost of capital to the bank
L = annual administrative expenses for credit card portfolio,
including defaults
a = percentage paid by the merchant to the bank on each credit
sale
g = grace period in days
In May 2012, Capital One Financial Corporation purchased the
GM Card credit-card accounts from HSBC Bank Nevada, N.A.
Capital One paid a premium of $2.5 billion on the credit card
loans. According to TREFIS analysis, the stock of Capital One
is worth
15. $58 a share, on May 20, 2013, although in the market it is
trading at about $61 a share. Most of the value of the stock,
61.4%, lies in its credit card operations.
Source: TREFIS, May 20, 2013
6.4 Altman's Zeta Model
Edward Altman (1941-)
A loan officer at a lending institution is evaluating the
creditworthiness of a marginal borrower. A firm is considering
selling goods on credit to another firm that has shaky financial
condition. A security analyst is looking at risky bonds of a firm.
They must all look at the probability of default. In particular,
they are concerned about the possibility of bankruptcy by the
borrower. Edward Altman developed a well-known model for
predicting bankruptcy of firms. The model uses the information
about the financial ratios of the firm and discriminant analysis
to find the probability of bankruptcy.
In the original (1968) model, Altman studies publicly traded
manufacturing companies. He examined companies that went
bankrupt and compared them with those that remained solvent.
He focused on five ratios, shown in the following table, which
seemed to be the most significant factors in predicting
bankruptcy.
(
17. capital Total assets
Accumulated retained earnings Total assets
EBIT Total Assets
Market value of equity Book value of debt
Sales Total assets
Source: Edward I. Altman, Corporate Financial Distress and
Bankruptcy, John Wiley & Sons, (1993), p. 109.
Altman defined the Z-score of a firm as follows:
EBIT
Net working capital
Sales
Z = 3.3 Total Assets + 1.2
Total assets+ 1.0 Total assets
+ 0.6
Market value of equity Book value of debt+ 1.4
Accumulated retained earnings
Total assets(6.11)
where Z is an index of bankruptcy. The value of Z is interpreted
as follows:
sure
IfZ > 2.99, no threat of bankruptcy.
18. This is a statistical model with a fairly good track record. It is
employed by practitioners who have to decide on the possibility
of a firm going bankrupt.
A revised model (1984) looks at non-manufacturing, privately
owned companies. The Z- score in this case is defined as
Net working capital
Accumulated retained earnings
EBIT
Z = 6.56
Total assets+ 3.26
Total assets+ 1.05 Total assets
where Z < 1.23 indicates a bankruptcy prediction,
no bankruptcy.
+ 6.72
Book value of equity
Total liabilities(6.12)
Altman’s model is used by banks and other lenders in evaluating
the creditworthiness of borrowers.
Examples
19. 6.1. Pittston Company has annual sales of $2 million, while the
cost of goods sold is $1.2 million. All sales are cash sales. The
marketing manager at Pittston has come up with the plan of
giving credit to the customers. He believes that this will
increase the sales by 20% without increasing any of the fixed
costs. He thinks that 30% of the customers will pay within 30
days, 30% within 60 days, 38% within 90 days, and 2% of the
customers will default on the payments. The cost of capital to
Pittston is 15%. Should the company introduce the credit sales
system?
To simplify the problem, we consider the cash flows for a single
year. We are just comparing one policy against the other, and if
one policy is better for one year, it will be better for all
subsequent years. First, we find the NPV of the current (cash
only) policy. It comes out to be
NPV(cash) = − 1.2 + 2 = $0.8 million
By extending credit, the sales, and the corresponding cost of
goods sold, will increase by 20%. We can take care of it by
multiplying the entire calculation by a factor of 1.2. Next, the
$2 million in sales comes in three batches, 30%, 30%, and 38%.
This also accounts for the 2% loss by the defaulters. The cash
comes in after 30, 60, and 90 days. We have to discount the
cash by the proper discount factor. This is done as follows:
.3
5 million
20. By comparing the two results, we conclude that Pittston
Company should implement the credit policy. The 20% increase
in the sales, and the extra profits generated by it are sufficient
to offset the delay, and default, in payments. ♥
6.2. Taylor Company has the following credit policy at present:
2/10, net 30 days. At this time, Taylor receives 30% of the sales
within 10 days, and the rest within 30 days. The manager of the
firm believes that if the terms of sales were relaxed, the sales
would increase by 10%. He proposes that Taylor should
eliminate the discount, and allow all customers to pay within 60
days. The cost of capital for the company is 12% and the cost of
goods sold is 65% of the total sales. Should Taylor adopt the
new policy?
The term "2/10, net 30 days" means that those customers who
pay within 10 days of the sale are entitled to a 2% discount,
otherwise they must pay the full amount within 30 days. Many
businesses offer a discount for prompt payment for the sales. It
is a win-win situation: the customers get the merchandise at a
2% discount, and the company improves its cash flow. The
customers, who wait 30 days before they can make a payment,
must pay the full amount.
Let us assume that there are no defaults under either policy.
Suppose the total annual sales are $1 million. Let us find the
NPV of both the policies. For the current policy, we get
.3*.98
.7
NPV(old) = − .65 + 1.1210/365 + 1.1230/365 = $0.3366 million
In the above calculation, .65 stands for $.65 million in cost of
21. goods sold, .3 is 30% of the customers who pay within 10 days,
and they pay only 98% of the bill after taking the 2% discount.
The remaining 70% pay within 30 days.
For the new policy, there is a 10% increase in sales, but the
customers also delay the payments to 60 days. This gives us
Because the NPV of the second policy is higher, Taylor should
accept the new policy. The higher NPV is due to higher sales
and the elimination of the discount. But it is also reduced by the
delay in collecting money for the sales. ♥
6.3. Moosic Auto Parts is considering the credit application of
an established customer, Duryea Garage. The customer buys
$12,000 worth of merchandise annually and pays in cash.
Moosic believes that the customer will buy $12,500 worth of
merchandise if he is given credit under the terms 2/10, net 60
days. The company is not sure for what percentage of the sales
will the customer elect to pay within 10 days. The cost of
capital for Moosic is 14%, and the variable cost factor is 70%.
Should Moosic extend credit to this customer?
To simplify the calculations, we assume that the customer
makes the purchases just once a year. Thus for the current
policy
NPV(cash) = − .7(12,000) + 12,000 = $3600
For the new policy, the customer buys $12,500 in merchandise.
Suppose the customer takes advantage of the discount for a
fraction p of the sales. Then
22. NPV(credit) = − .7(12,500) +
(.98)p(12,500) 1.1410/365+
(1 −p)(12,500) 1.1460/365
= 3483.64266 − 27.53907p
If p = 0, that is, the customer does not take advantage of the
discount and delays the payment for 60 days, then
12‚500
NPV(credit) = − .7(12,500) + 1.1460/365 = $3483.64
Moosic will lose in this arrangement, and the customer will
come out ahead. If p = 1, the customer pays promptly and takes
the discount for all sales, then
NPV(credit) = − .7(12,500) +
.98*12‚500
1.1410/365 = $3456.10.
This is still not profitable for the company. In fact, the
customer will always try to pay within 10 days. The company
should not implement the new policy.
The company loses primarily because of the 2% discount, and
the 10-day delay in collecting for the sales. The increase in
sales, $500 per year, is not enough to make this policy
worthwhile. ♥
23. Suppose the marketing manager comes with a higher sales
estimate for the customer. He believes that this customer will
now buy S dollars worth of auto parts annually because of the
2% discount for prompt payments. What is the minimum sales
for the customer that will make the new discount policy
profitable for Moosic?
The NPV with new sales figure S will be
(.98)S NPV(credit) = − .7S + 1.1410/365 = .276488 S
Equating it with the previous cash NPV, we get
.276488 S = 3600
Solving for S, we get S = 13,020
Therefore, the customer must buy at least $13,020 worth of
parts to make this new policy to be profitable for Moosic. ♥
6.4. Avoca Company has the following credit policy 2/10, net
30. Avoca also charges 1% per month interest on all accounts
after 30 days. The sales collection schedule of the company is
according to the following table:
Collection within
10 days
30 days
60 days
90 days
Percentage20%30%40%10%
To improve the collection rate, Avoca is thinking of imposing a
24. higher interest rate, 1.5% on all accounts paid after 30 days. It
will continue the 2/10, net-30 policy. Avoca believes that the
new policy will change the collection schedule as follows:
Collection within
10 days
30 days
60 days
90 days
Percentage20%50%20%10%
There will be no change in the total sales as a result of this new
policy. The cost of capital for Avoca is 15%. Should it try the
new policy?
In this problem, we consider only the net present value of the
change in the policy. We ignore such variables as the cost of
goods sold, and the value of the collection after ten days,
because they are the same in both cases.
Suppose the total sales are $1 million. For the current policy,
the present value of sales, in
$million, is
.2*.98
.3
.4(1.01)
.1(1.01)2
25. PV(current) = 1.1510/365 + 1.1530/365 + 1.1560/365 +
1.1590/365 = $.985203 million
For the proposed policy, the PV of sales is
.2*.98
.5
.2(1.015)
.1(1.015)2
PV(new) = 1.1510/365 + 1.1530/365 + 1.1560/365 +
1.1590/365 = $.987462 million
The second policy is only slightly better. The difference
is .987462 − .985203 =
$0.002259 million = $2259. Avoca should implement the new
policy. ♥
6.5. Wilkes Corporation is reviewing the credit application of a
customer. The company expects to sell $3000 worth of
merchandise every month and expects to receive the payment
for it within the 30-day grace period. The cost of goods sold is
$2000. There is a 15% probability that the customer may not be
able to pay his bill in a certain month. In that case, the company
will be able to collect 10% of the balance 4 months after the
sale. The risk-adjusted discount rate is 12%. Should Wilkes give
credit to this customer?
To calculate the NPV of this decision, we use the equation
27. The customer will add $3585 to the value of the firm. Because
the NPV is positive, Wilkes should give credit to the customer.
♥
When the customer becomes well established and has a perfect
credit record, then we may assume that p = 0. In that case we
may use (6.3)
(6.3)
This dramatically increases the value of the customer to the
firm. Thus, the firms keep a wary eye on potentially delinquent
customers. ♥
6.6. Throop Corporation is reviewing the credit application of a
customer. The customer is expected to buy $4000 worth of
merchandise every month and is expected to pay for it within
the 30-day grace period. The cost of goods sold is $2500. There
is a 10% probability that he may not be able to pay his bill in a
certain month. In that case the company will be able to collect
10% of the balance 6 months after the sale. The risk- adjusted
discount rate is 12%. Should Throop give credit to this
customer?
28. To calculate the NPV of this decision, we use the equation
pRS
(6.2)
Putting the numbers, C = 2500, p = .1, S = 4000, r = .01, n = 1,
R = .1, m = 6,
1.011 +
1.016
29. The NPV is positive, thus Throop should give credit to the
customer. ♥
6.7. Dupont National Bank issues credit cards with the
following terms. There is no interest charge if the entire bill is
paid within 25 days. Interest at the rate of 15% per annum is
added to the outstanding balance if the bill is paid after 25 days.
The new balance also includes any purchases made during the
month. This procedure continues until the entire bill is paid.
The cost of capital to the bank is 7%. The balance on a credit
card, on the average, is $1000 at the end of a month. Eighty
percent of the cardholders pay off their entire balance within 25
days, while the remaining carry the average balance.
Under proposed terms, Dupont wants to reduce the interest rate
to 10% on the unpaid balance. It expects that 50% of the
cardholders will now carry the balance, while the other 50%
will still pay in full within 25 days. Dupont expects the balance
of each bill will rise to $1100. Should Dupont introduce the new
terms?
Let us find the value of an average card for the bank. If a
cardholder is carrying a constant balance of $1000 every month,
and the interest rate is 15% per annum, then he is paying $150
per year in interest charges. From the point of view of the bank,
it is carrying a perpetual bond whose annual interest payment is
$150. The cost of capital to the bank is 7%. Recalling the value
of a perpetual bond as
B = C/r(6.3)
where B is the value of the bond, C is the annual interest
payment, and r is the discount rate, or the cost of capital. This
comes out to be 150/.07 = $2142.86. The bank has invested
30. $1000 in this card, namely the balance carried by the
cardholder.
For the bank, the NPV of this credit card balance is
therefore −1000 + 150/.07 =
$1142.86. At the moment, we shall ignore the credit risk of the
cardholder that the bank must also take into consideration.
Twenty percent of the cardholders are in this category.
Next, we analyze the burden on the bank due to a customer who
pays on time. This person is using $1000 of bank's money free
for 25 days, every month. The NPV for one month is thus
1000
− 1000 + 1.0725/365
To find its value as a perpetuity, we divide this by the monthly
discount rate, namely
.07/12. This comes out to be
Eighty percent of the customers are paying within the grace
period. Combining these two numbers, we can find the NPV of
the average card under the old terms as follows:
150/.07) = −$405.50
31. With the new policy, the interest rate is 10% charged on the
unpaid balances. The average balance rises to $1100. Thus, the
annual interest collected on each $1100 balance will be $110.
The value of a perpetual bond with $110 annual interest is
110/.07 =
$1571.43. The bank has invested $1100 in this bond. Thus the
NPV of this bond is –1100
+ 1571.43 = $471.43. We also include the factor .5 as 50%,
representing the percentage of customers who are paying their
bills every month, and the others who are not paying. The NPV
of an average card under the new terms is thus
+ 110/.07) = −$200.21
Suppose we want to use (6.9),
NF + 12nB[(1 + r)−g/365 − 1] + mC(R − r) − L + 12aNS
NPV =
r(6.9)
In (6.9), we let N = 1, n = .8, m = .2, F = 0, C = 1000, R = .15, r
= .07, L = 0, a = 0, B =
1000, g = 25, then
NPV(old) =
32. 12*.8*1000*[1.07−25/365 – 1] + .2*1000*(.15 − .07)
.07= –$405.50
For the new policy, we put N = 1, n = .5, m = .5, F = 0, C =
1100, R = .1, r = .07, L = 0,
a = 0, B = 1100, g = 25, then
12*.5*1100*[1.07−25/365 − 1] + .5*1100*(.1 − .07)
NPV(old) =
.07= −$200.21
This analysis reveals that the credit card operations for the bank
are unprofitable. Each card represents a negative value of about
$405 under the old policy, and negative $200 with the new
policy. This means that the bank has improved the operations,
but they are not profitable yet. ♥
6.8. Scranton National Bank has a portfolio of 20,000 credit
card accounts. The bank charges $25 annual fee on these cards.
There is a 25-day grace period on the accounts, and after that
the cardholders pay interest at the rate of 1.25% per month on
the unpaid balance. Half of the cardholders pay their balance in
full every month, and their average monthly bill is $1000. The
remaining cardholders carry an average balance of $1500
continuously. The operating expenses for the credit card
portfolio, including defaults, are
$150,000 annually. The merchants who accept his card do not
pay any fee to the bank. The cost of capital to the bank is 8%.
Citibank plans to buy Scranton's credit card portfolio. How
much should Citibank pay, excluding the receivables?
33. We use the following equation to find the selling price of the
portfolio,
NF + 12nB[(1 + r)−g/365 − 1] + mC(R − r) − L + 12aNS
Selling price = P +
r(6.10)
In the above expression, a = 0, P = 0, N = 20,000, F = $25, m =
10,000, n = 10,000, C =
$1500, B = $1000, R = .15, r = .08, g = 25 days, L = $150,000.
Putting these numbers, we find the selling price as follows.
NPV =
20‚000(25) + 12(10‚000)(1000)[1.08−25/365 − 1] +
10‚000(1000)(.15 − .08) − 150‚000
.08
= $5,238,847
Suppose the total outstanding balance for all credit cards is $20
million on the day the final deal is signed, then Citibank should
pay Scranton National Bank at least $25.239 million for the
credit card portfolio. ♥
6.9. Peckville National Bank has a portfolio of 30,000 credit
card accounts. The bank charges $25 annual fee on these cards.
There is a 25-day grace period on the accounts, and after that
the cardholders pay interest at the rate of 1% per month on the
unpaid balance. Half of the cardholders pay their balance in full
every month, and their monthly bill is $600, on the average. The
remaining cardholders carry an average balance of
$1200 continuously. The average monthly sale for all cards is
34. $800. The operating expenses for the credit card portfolio,
including defaults, are $120,000 annually. The cost of capital to
the bank is 7%. The outstanding balance of all credit cards is
$27 million.
The merchants pay 1% of the sales to the Bank. Citibank plans
to buy Peckville's credit card portfolio. How much should
Citibank pay, including the receivables?
We may start by using the expression
NF + 12nB[(1 + r)−g/365 − 1] + mC(R − r) − L + 12aNS
Selling price = P +
r(6.10)
In this formula, we have P = $27 million, N = 30,000, F = $25,
m = 15,000, n = 15,000, a = .01, R = .12, r = .07, g = 25 days, B
= $600, S = $800, L = $120,000. Substituting these values, we
get
Selling price = 27,000,000 +
30‚000(25) + 12(15‚000)(600)[1.07–25/365 − 1] +
.07
= 82,866,703
Thus selling price = $82.867 million. ♥
6.10. Archbald Bank is analyzing its credit card portfolio. It
classifies its cardholders into two types: 5000 "free riders", and
10,000 "paying customers." The free riders charge
$300 worth of merchandise every month, on the average, and
35. pay off the full balance after 25 days. The paying customers
charge $100 a month, on the average, but they continuously
carry a balance of $400 of debt. The cost of capital to the bank
is 9%, and it charges 15% interest on the unpaid balance. The
participating merchants pay 1% of the sales, charged on a credit
card, to the bank at the end of each month. Find the value of
this credit-card operation to the bank.
First, we look at the merchant fees. The total monthly sales =
5000*300 + 10,000*100 =
$2,500,000. This produces a revenue of $25,000 at the end of
each month for the bank. To find the value of this income
stream, we discount it at the monthly discount rate of 9/12 =
.75%. This comes out to be
∞ 25000
25000
i=1
Second, we consider the cost of having the free riders. When a
person charges $300 and pays for it after 25 days, the PV of this
transaction to the bank is
300
PV = − 300 + 1.0925/365 = − 1.7655586
If this person keeps on doing this, month after month, the PV of
this to the bank becomes
∞ 1.7655586
36. 1.7655586
i=1
1.0075i= −
.0075= − 235.4078133
The PV to the bank, for all 5,000 such cardholders, is
PV = − 235.4078133*5,000 = − $1,177,039(2)
Third, we evaluate those people who carry a balance of $400
every month. They pay interest at the rate of 400*.15/12 = $5
per month. There are 10,000 such cardholders and their total
contribution to the bank is $50,000 a month. The value of this
income stream to the bank is
∞ 50‚000
i=1
50‚000
.0075 = $6,666,667(3)
The net present value of the credit card operation to the bank is
thus the sum of the three parts of the operation outlined above.
NPV = 3,333,333 − 1,177,039 + 6,666,667 = $8,822,961
This comes out to be around $8.823 million. However, because
of the administration costs, defaults by cardholders, and
fraudulent use of the cards, the actual value is much less.
37. 1. Interpreting Bond Yields. Suppose you buy a 7 percent
coupon, 20-year bond today when it’s first issued. If interest
rates suddenly rise to 15 percent, what happens to the value of
your bond? Why?
2. Bond Yields. The Timberlake-Jackson Wardrobe Co. has 10
percent coupon bonds on the market with nine years left to
maturity. The bonds make annual payments. If the bond
currently sells for $1,145.70, what is its YTM?
3. Coupon Rates. Osborne Corporation has bonds on the market
with 10.5 years to maturity, an YTM of 9.4 percent, and a
current price of $945. The bonds make semiannual payments.
What must the coupon rate be on the bonds?
4. Stock Values. The next dividend payment by Mosby, Inc.
will be $2.45 per share. The dividends are anticipated to
maintain a 5.5 percent growth rate, forever. If the stock
currently sells for $48.50 per share, what is the required return?
5. Stock Values. Ziggs Corporation will pay a $3.85 per share
dividend next year. The company pledges to increase its
dividend by 4.75 percent per year, indefinitely. If you require a
12 percent return on your investment, how much will you pay
for the company’s stock today?
6. Growth Rates. The stock price of Jenkins Co. is $53.
Investors require a 12 percent rate of return on similar stocks.
If the company plans to pay a dividend of $3.15 next year, what
growth rate is expected for the company’s stock price?
PLEASE REPLY IN HIGLITE TO EACH QUESTION ON THIS
SHEET IN SENTENCE FORM, FOLLOWING EACH
PROBLEM AS: 2003 WORD .doc.
Please also SEPARATELY include the Excel Sheet showing
how each problem was solved.
AGAIN, PLEASE USE THIS SHEET TO ADD THE WRITTEN
SOLUTIONS TO.
5.1 Cash Management
38. The management of cash, or treasury management, is perhaps
the most important aspect of working capital management of a
firm. There should always be an adequate amount of cash
available to the corporation. If there is an unexpected shortage
of cash, the company must also have proper means to raise the
needed cash. This requires careful planning and cash budgeting.
Cash is a necessary resource in business, but too much of it is
also wasteful. Usually corporations keep cash in a checking
account, or several accounts, and the excess cash in marketable
securities through a brokerage firm.
These days it is possible to keep both the checking account and
the brokerage account at a single institution. For instance, a
corporation can have checking and brokerage accounts at PNC
Bank. It is also possible to have both these accounts at a
brokerage firm, such as Merrill Lynch. There are some
restrictions, however, on the checking accounts maintained at a
broker. At one time, the banks were not allowed to sell stocks,
and the brokers could not give check-writing privileges to
customers. However, the current trend is to blur the distinction
between banks and brokers. The policy of the Federal Reserve is
to move in that direction.
Large corporations, such as Walmart, Ford, or Microsoft, have
billions of dollars in cash. They have full-time staff who track
the cash flows and cash balances constantly. Even smaller
companies have to watch their cash accounts carefully.
5.2 Baumol Model (1952)
Perhaps the earliest quantitative analysis of the cash
management of a firm was done by William Baumol in 1952.
We study his approach more in a historical context, rather than
a practical tool to manage cash and marketable securities at a
firm.
39. (
82
)
William Baumol (1922- )
We assume that the corporation maintains two accounts: the
checking account for daily expenditure of cash, and the
brokerage account to keep the marketable securities.
In another paper in 1956, James Tobin (1918-2002), extended
this model. In the Baumolmodel, also called Baumol-Tobin
model, we consider two costs associated with managing cash:
the holding cost, and the ordering cost. The first cost is due to
the fact that the cash kept in a checking account, readily
available for any use, is not earning a
rate of return consonant with other operations of the firm. The
economists call this the opportunity cost, because the
opportunity to invest this cash in the business is lost. To
minimize this loss, the firm invests the surplus cash in
marketable securities, such as Treasury bills, and keeps them in
a brokerage account.
(
T
re
a
41. Cash inflow
Brokerage Account
x
Checking Account
Cash outflow
Fig. 5.1: Cash flow in the Baumol model of cash management.
The company maintains a checking account and a brokerage
account. The company deposits all incoming cash in a brokerage
account, which is invested in high-grade bonds and Treasury
securities. The company transfers $x at regular intervals from
the brokerage account to the checking account. It writes checks
to pay all the bills.
The second cost is that of converting marketable securities into
cash. This includes the transactions cost of selling these
securities, the cost of sending the order to the broker, and the
cost of transferring the money from the broker to the local bank
where the checking account is maintained.
Fig. 5.2: An example of a firm that pays out $10,000 uniformly
every week in bills. It starts with $10,000 in the checking
account, and when the balance drops to zero, it replenishes the
cash by another deposit of
$10,000. The average amount in the checking account is thus
$5000.
42. Let us find an optimal way to manage cash. Suppose a company
needs a total amount of cash C in a whole year to pay all its
bills by check. This could be the amount paid to the workers,
suppliers, utilities, rent, and so on. Rather than keeping the
entire amount in a checking account, the company invests most
of the money in marketable securities. When the cash in the
checking account is depleted, it sells an amount x of these
securities and
puts the money in the checking account from which it writes
checks to pay the bills. When the money is exhausted, it
replenishes the checking account by selling another x dollars’
worth of marketable securities. The number of times this
process is repeated in a year is C/x.
The maximum amount of money in the checking account is x,
and the minimum zero. Assuming that the money is used
uniformly, then the average amount of money in the checking
account is x/2. The cost of keeping this amount in the checking
account for a year depends on the return generated by the next
available investment opportunity of these funds, namely,
marketable securities. Suppose this rate is r per annum. Then
the carrying cost of cash, that is, the cost of maintaining cash in
the checking account per year is rx/2.
Next, we look at the ordering cost. This equals the commission
charged by the broker to sell the securities, plus the costs
related to the order of the sale and transfer of the money,
perhaps by wire, to the checking account. Suppose this cost is b
every time this procedure is repeated. The total number of
transfers per year is C/x, and so the total ordering cost per year
is bC/x.
The total of carrying and ordering cost is rx/2 + bC/x per year.
43. To minimize this cost, we have to differentiate the cost function
with respect to x, which is an independent variable.
Let the total cost of cash management per year be T, where
T = rx/2 + bC/x
dTrbC
Then
dx = 2 − x2
At the optimal point, the total cost T is minimized, and its
derivative is zero. Thus
Solving for x, we find
rbC
2 − x2 = 0
2bC
Optimal amount of transfer,x =
r(5.1)
Equation (5.1) implies that to minimize the total cost of
managing cash and marketable securities, the optimal amount of
transfer from brokerage account to the checking account is x =
2bC/r . The equation also implies the following:
44. (1) The optimal transfer x, is directly proportional to the total
spending per year, C
(2) The optimal transfer x, is directly proportional to the
transfer cost, b
(3) The optimal transfer x, is inversely proportional to the
interest rate, r
We can also calculate the following costs.
Total ordering cost, per year = (transaction cost per
transfer)(number of transfers per
year) = bC/x = bC
r
2bC =
rbC
2
Total interest forgone, per year = (average balance in the
checking account)(rate of
interest) = xr/2 =
2bC
rr/2 =
rbC
2
Note that the two costs are equal. Add them to find the total
cost as
45. Total cost of cash management system, per year =
rbC
2 +
rbC
2 =2rbC(5.2)
Example
5.1. Alabama Corporation has to pay $32 million in bills
annually. It has managed to stretch them out uniformly
throughout the year. Alabama has a checking account at a bank
in Scranton, and keeps its excess cash in the form of high grade
bonds in a brokerage account in Philadelphia. The checking
account pays no interest, and the average transaction cost in the
brokerage account is $125. The average interest on the bond
portfolio is 8.5%. Explain how Alabama should optimize its
cash system.
Put b = 125, C = 32,000,000, r = .085, in the equation
2bC
x =r(5.1)
2(125)(32‚000‚000)
x =.085= $306,786
Alabama should keep at most $306,786 in the checking account,
and keep replenishing it when the money runs out. Alabama has
to do it 32,000,000/306,786 = 104.31 times a year. This is
equivalent to one transaction every 365/104.31 = 3.5 days,
twice a week. ♥
5.2. Alaska Corporation spends $25 million a year to pay its
bills. The cost of ordering the sale of securities is $100 per
46. order. The securities are earning 6% per annum. How often
should Alaska sell the securities, and in what amount, in order
to keep its checking account running at the optimal level?
Putting b = 100, C = 25,000,000, and r = .06, in (5.1), we find
2*100*25‚000‚000
x =.06= $288,675
The number of orders per year is 25,000,000/288,675 = 86.6025.
This is equivalent to an order every 365/86.602 = 4.21 days, on
the average.
The ordering cost per year is 86.6025(100) = $8660.25. The
carrying cost is (288,675/2).06 = $8660.25. The ordering cost is
equal to the carrying cost at the optimal point. ♥
The Baumol-Tobin model (5.1) is derived under the following
simplifying assumptions:
1. The company knows its cash expenditures in advance. There
is no uncertainty in these cash payments. These expenses occur
uniformly with time.
2. The cash outflow from the company remains constant with
time, that is, it is not increasing or decreasing.
Alternatively, we can use NPV of cash management to find the
optimal solution. The company has to convert marketable
securities into cash, C/x times per year. The time interval
between two conversions is x/C years. At each order point, the
cash required is x, and the transaction cost is b. The carrying
cost for the first cycle is given by
(av
47. of first cycle, in years) That is,(x/2)(r)(x/C) = rx2/(2C).
The total cost for the first cycle = transaction cost + carrying
cost = b + rx2/2C.
Assuming that this cost is incurred at the end of the first cycle,
the present value of the cost of the first cycle is thus
PV cost of one cycle =
b + rx2/(2C)
(1 + r)x/C
where r is the rate of return available on the marketable
securities. The company will continue to use this procedure as
long as possible, provided the cash flows remain constant. The
present value of the total cost for infinite many cycles is
∞ b + rx2/(2C)
i=1
(1 + r)ix/C
Carrying out the summation and simplifying, we get
2bC + rx2
PV of infinite cycles = 2[(1 + r)x/C − 1]
Differentiate the above expression with respect to x, and set it
equal to zero.
2rxC[1 − (1 + r)x/C] + (2bC + rx2) ln(1 + r) (1 + r)x/C
48. 2[C(1 + r)x/C − 1]2= 0
Or, 2rxC[1 − (1 + r)x/C] + (2bC + rx2) ln(1 + r) (1
+ r)x/C = 0
Put b = 100, C = 25,000,000, and r = .06, in the above equation,
which gives 3,000,000x(1.06x/25,000,000 – 1) −
1.06x/25,000,000 ln(1.06) (5,000,000,000 + .06x2) = 0
Solving for x, we get x = $288,772. This is quite close to the
result obtained by using (5.1), namely, $288,675. Theoretically,
the second method is superior to the previous one because it
looks at the time value of the cash flows, but practically, the
difference is very small.
5.3 Miller-Orr Model (1966)
Another method used for estimating the optimal amount of cash
for a firm was developed by Merton Miller and Daniel Orr.
Miller-Orr model assumes that the cash inflows and outflows
are completely random. It further assumes that the mean cash
flow is zero, and variance of the cash flows is known.
Merton Miller (1923-2000)
From a practical point of view the company maintains a
checking account where all incoming cash and checks are
deposited daily. The company also writes checks on this
account to pay all bills as they come due. The firm monitors two
items on a daily basis:
the net cash flow in the account, and the cash balance in the
49. account. The cash balance should not be too high, because that
is wasteful, and it should not be too low, otherwise the checks
written by the company may start to bounce.
Cash inflow
Checking Account
Cash outflow
Brokerage Account
Fig. 5.3: Cash flow in the Miller-Orr cash management system.
All incoming cash is deposited in a checking account and the
company pays the bills out of this account. When there is too
much cash in the checking account, the companys transfers $2x
from the checking to a brokerage account. When there is not
enough cash in the checking account, an amount $x is
transferred from the brokerage account to the checking account.
The money manager at the firm first decides the minimum
amount of cash that the checking account must have. Let us say,
this amount is L.
When the balance in the checking account drops to L, the
manager sells x amount of marketable securities and puts the
50. cash in the checking account. The balance now becomes L + x,
which is supposed to be the optimal amount of cash in the
account. When the cash in the account rises to a level equal to L
+ 3x, the manager buys securities worth 2x, so that the cash
balance drops down once again to the optimal level L + x. In
this way the cash in the checking account remains between the
limits L + x and L + 3x.
The amount x depends upon the following factors:
1. The transactions cost, b. This is the cost of converting excess
cash into securities, or converting securities back into cash.
This includes the brokerage commissions, and the value of the
time of the person managing the money. If the cost per
transaction is high, one should move large amounts of cash at
each transaction.
2. The daily variance of the cash flows, σ2. The greater is the
variance of the cash flows, the greater should be the amount
transferred each time. If the cash flows are very predictable, or
known with certainty, then there is no need for the movement of
large blocks of money.
3. The daily interest rate, r. One can easily see that if the
interest rates are high, one should keep as little money in the
checking account as possible. This means that for high interest
rates, x should be small.
Cash, $
L + 3x
51. L + 4/3 x L + x
L
Time
Fig. 5.4. The cash balances in the Miller-Orr model for cash
management.
The mathematical derivation of the optimal transfer amount x is
somewhat complicated, but the result is that
(
) (
)x =
(5.2)
Further, the model specifies the follows values: The minimum
amount of cash = L
The optimal amount of cash = L + x
52. The maximum amount of cash = L + 3x The average amount of
cash = L + (4/3)x
To use equation (5.2) in practice, one has to develop estimates
for the three parameters, b,
again. They expect to spend
$25 million in cash payments annually. Suppose the standard
$6 million, on an annual basis. The cost of each transaction is
still $100, and the rate of return on the marketable securities is
6%. Putting these values in (5.2), we find
(
2
1
/
3
4*.06
So this is what Alaska Corporation should do. They should start
with a cushion of, say,
$50,000, and add $355,689 to it. The starting balance is then
$405,689. Then they should keep putting collections in this
account, and also write checks out of it. If the collections are
running at a faster pace, the balance in the account will keep on
53. rising. When it reaches 50,000 + 3*355,689 = $1,117,067, they
should buy 2*355,689 = $711,378 worth of securities and the
bring the balance down to 1,117,067 − 711,378 = $405,689.
This is the optimal amount of money in the checking account.
On the other hand, if the disbursements are going ahead faster
than the collections, the balance in the account will drop
gradually. When it reaches $50,000, they should sell
$355,689 worth of securities and replenish the cash balance,
bringing it to its optimal level of $405,689 once again.
Example
5.3. Arizona Company uses the Miller-Orr model to manage
cash. The ending balance in their checking account, including
checks and deposits, for 10 consecutive business days is:
Day
Balance
Day
Balance
1$15,625622,725
212,225719,000
313,825817,775
417,375912,125
521,9001010,225
The cost of each transaction is $100, whereas the return on
securities is 5%. They would like to maintain a minimum
balance of $5,000. How should they manage their cash?
First we have to find the variance of the cash flows. We may do
so by augmenting the above table as following.
We calculate the net cash flow each day by subtracting the first
day's balance from the second day's balance, and so on. Then we
add these daily net cash flows, and divide the total by 9. The
54. average net cash flow per day is therefore −5400/9 = −600. We
can also check the result by calculating the difference in the
balance on the first and the tenth day, and dividing by 9. This
comes out to be (10,225 − 15,625)/9 = −600, as before.
Day
Balance
Net Cash
Flow
Difference
(Difference)2
1
$15,625
2
12,225
12,225 – 15,625 =
–3,400
–3,400 + 600
7,840,000
3
13,825
13,825 – 12,225 =
1,600
1,600 + 600
4,840,000
4
17,375
17,375 – 13,825 =
3,550
3,550 + 600
17,222,500
56. Total
–5,400
95,547,500
Next we find the difference between the individual daily net
cash flows and the average. This is set up in the next column.
Then we find the square of all these differences and place them
in the next column marked (Difference)2. Then we add the
numbers in this column and divide the result by 8, because we
have lost another degree of freedom. The final result,
95,547,500/8 gives us the variance of the net cash flows. The
resulting number is σ2 = 11,943,437.5, on a daily basis.
The interest rate is the daily interest rate, because we are
dealing with daily cash flows. That is, r = .05/365. We also
know that b = 100. Putting these numbers in (5.2), we find
4*.05/365
They should start out with the optimal balance of 5,000 +
18,700 = $23,700. If the account balance drops to $5,000, they
should sell $18,700 worth of securities and put the money in the
checking account. If the account balance rises to 5,000 +
57. 3*18,700 =
$61,100, they should take 2*18,700 = $37,400 out of it and buy
securities from this money. This brings the level back to the
optimal point at $23,700. The average balance in this account is
5,000 + (4/3)*(18,700) = $19,933. ♥
5.4. Arkansas Company's checking account balance on 12
successive business days is given in the table below. It uses the
Miller-Orr model for cash management. Arkansas requires a
minimum balance of $3,000 in its checking account. The return
on the securities is 8%, and the cost of each transaction is $150.
How should it set up its cash system?
Day
Balance
Day
Balance
Day
Balance
Day
Balance
1$15,6254$17,3757$19,00010$10,225
212,225521,900817,7751114,000
313,825622,725912,1251215,000
To do the problem with the help of Maple, we type in the
following. The lines starting with the symbol # are comment
lines. They are not part of the instructions for the computer, but
are merely an aid to understand the program.
# n is the number of data items n:=12;# a is an array to store the
data, with size n a:=array(1..n):# put the data in
placea[1]:=15625.: a[2]:=12225.: a[3]:=13825.:
a[4]:=17375.:a[5]:=21900.: a[6]:=22725.: a[7]:=19000.:
a[8]:=17775.:a[9]:=12125.: a[10]:=10225.: a[11]:=14000.:
a[12]:=15000.:print (a);#ncf is an array to store the net cash
58. flows, with size n-1 ncf:=array(1..n-1);# The next statement
fills out the net cash flows for i to n-1 do ncf[i] := -a[i]+a[i+1]
od;# avncf is the average net cash flow avncf:=(a[n]-a[1])/(n-
1);# diffsq is an array to store (difference)^2, with size n-1
diffsq:=array(1..n-1);# fill in the data for (difference)^2for i to
n-1 do diffsq[i]:=(ncf[i]-avncf)^2 od;# var is the variance =
(sigma)^2var:=sum(diffsq[j],j=1..n-1)/(n-2);
# x is the Miller-Orr order quantity x:=(3*b*var/4/r)^(1./3);
subs(b=150,r=.08/365,x);
There are several do statements in the above program. They
give instructions to repeat a certain operation a given number of
times. Each do statement must end with od, which is do spelled
backwards. One must follow the syntax carefully.
The final result of the above calculations is x = $18,020. The
company should start with a cash balance of $21,020, replenish
cash when the balance drops to $3000, and buy
$36,040 worth of securities when the balance reaches $57,060.
♥
5.5. California Company maintains a checking account and a
brokerage account to manage its cash. It writes $45,000 in
checks every week on the average, and the standard deviation of
net cash flows is $15,000 per week. California keeps the excess
cash in the brokerage account that pays 5.25% in interest. The
cost of transferring money between the accounts is $200 per
transaction. California maintains a minimum of $30,000 in the
checking account. Using Miller-Orr model, explain how it
should manage its cash in an optimal manner. In particular:
A. What is the minimum balance in the checking account that
triggers a transfer of money from the brokerage to checking
account? How much money is transferred?
59. B. What the maximum balance in the checking account that
requires a transfer of money from the checking to the brokerage
account, and how much is this amount?
C. What is the interest forgone each year?
A. The minimum balance in the checking account is $30,000.
Use the formula
(
3
b
σ
)2 1/3
(
(5.2)
and put the numerical values for b = 200, σ2 = 15,0002 =
225,000,000, r = .0525/52. Note that we have weekly cash
flows, and thus we must use the weekly rate of interest. This
gives us x = (3*200*225,000,000/4/.0525*52)1/3 = $32,214.
Thus the amount of money transferred is $32,214. ♥
B. Since California would like to keep a minimum of $30,000 in
the checking account, they should start out by keeping 30,000 +
32,214 = $62,214 in this account, which is at the optimal level.
60. They should put the rest of the cash in the brokerage account.
When the amount in the checking account drops to $30,000,
then they should replenish it with
$32,214 additional cash from the brokerage account. The
maximum amount of money in the checking account should be
30,000 + 3*32,214 = $126,642. At that point California should
transfer $64,428 from the checking to brokerage account. ♥
C. The company keeps on the average 30,000 + (4/3)*32,214 =
$72,952 in the checking account. The annual interest foregone
is 72,952*.0525 = $3,830. ♥
5.6. Colorado Company uses Miller & Orr model for its cash
management by maintaining a checking account and a brokerage
account. It writes $65,000 in checks on the average per week,
and the standard deviation of its net cash flows is $25,000 per
week. Colorado requires a minimum of $40,000 in the checking
account. Colorado keeps the excess cash in the brokerage
account that pays 4.75% in interest. The cost of transferring
money between the accounts is $150 per transaction. Explain
how it should manage its cash in an optimal manner. In
particular:
A. What is the minimum balance in the checking account that
triggers a transfer of money from the brokerage to checking
account? How much money is transferred? Minimum balance =
$40,000. We use the formula,
(
3
b
σ
)2 1/3
61. (
(5.2)
and put the numerical values for b = 150, σ2 = 25,0002 =
625,000,000, r = .0475/52. This gives us x =
(3*150*625,000,000/4/.0475*52)1/3 = $42,538. Colorado
should transfer
$42,538 from the brokerage account to checking account. ♥
B. Find the maximum balance in the checking account that
requires a transfer of money from the checking to the brokerage
account, and the amount of this transfer.
The maximum amount in the checking account is 40,000 +
3*42,538 = $167,614. At that time they should transfer
2*42,538 = $85,076 from the checking account to the brokerage
account. ♥
C. What is the interest forgone per year?
The average cash in the checking account is 40,000 +
4*42,538/3 = $96,717. The interest on this amount is
.0475*96,717 = $4,594. ♥
5.4 Speeding Up Collections
The business firms like to get hold of cash from their customers
as soon as possible. Traditionally, the customers pay their bills
by mailing a check. This delays the actual payment because of
62. slow mail, depositing the check, and then clearance of the check
before the funds become available to the payee.
To speed up the collections, the firms with lots of customers,
such as the utility companies, or credit card companies, have
devised several schemes. The two important ones are electronic
collections, and lock-box arrangements.
(a) Electronic Collections
It is possible to transfer funds electronically from bank to bank
by using a system known as the federal wire. This enables the
payer to send the money securely, and precisely at a given time.
In order to collect bills when they are due, some corporations
and banks will enter into an agreement with the buyer to
transfer the money directly from the checking account of the
buyer to their own account. For example, when an insurance
company sells a policy, it may allow the buyer of the policy the
option of monthly payments, whereas the money will be
transferred directly from the account of the policyholder to the
account of the insurance company. The main advantage of this
method is that the insurance company will get the installments
on time, and the policyholder does not have to worry about
writing checks and mailing them.
(b) Lock-box Arrangement
Did you ever notice that your credit card bill, or the telephone
bill, has a post office box as the return address? To speed up the
collections, many corporations, such as Citibank, or Discover
Card, or Sears, who have accounts all over USA, will set up
lock-box arrangements. For example, Sears may have return
addresses with post office numbers in Boston, Atlanta, Chicago,
Houston and San Francisco. Customers in the neighboring states
will send their bills to the nearest post office address. Once the
checks from the customers reach the post office, they are
immediately deposited in a local bank. That bank, in turn, will
63. credit the national account of Sears on a daily basis. This can
reduce the collection period by two to four days.
Sears had annual revenue of $51.78 billion in 12 months ending
January 21, 2008. This comes out to be about $4.315 billion a
month. Suppose Sears is able to reduce the collection period by
4 days each month by using a lock-box arrangement, and its cost
of capital is 12%, then this arrangement is saving them
4315*.12*4/365 = $5.675 million every month. This adds up to
$68 million every year.
One of the optimization problems in cash management is to
properly plan the location and the number of the lock-boxes.
5.5 Treasury Bills
Part of the efficient cash management system of a company is to
invest the free cash in interest-bearing securities, which are
very liquid, and also very safe. The best securities for this
purpose are short-maturity Treasury securities. They are also
known as Treasury bills.
The United States government, through the Department of
Treasury, sells bonds with various times to maturity. Because
the US government, through its power to tax people, has always
been able to pay the interest and principal back to the investors,
such investments are known as risk-free securities. These
securities have various times to maturity ranging from a few
days up to 30 years.
The Treasury Department auctions these securities every week.
These securities are sold at a discount from their face value. In
other words, you can buy a $1000 T-bill for perhaps $990.
When this T-bill matures, you can cash it in for $1,000. Thus
the difference, $10, is the interest earned on the $990
64. investment.
After they have been issued by the Federal Government, the
Treasury bills are then traded in the capital markets. The market
value of these securities changes daily due to the fluctuations in
the interest rates. The market value also drifts slowly towards
the face value of the bonds with the passage of time.
The Wall Street Journal provides two discounts for these
securities. The asked discount gives the purchase price, and the
bid discount the selling price of the T-bill. The discount is
quoted as a percentage of the face amount, but it is annualized
with a 360-day year. The relationship between the dollar
discount and the percentage discount is thus
We can express it as
dFn D = 360
65. The market price of the bond isB = F − D,
(
) (
(5.3)
Once we know the market price of a T-bill, we can also
calculate its bond equivalent yield, which is defined as
Using (5.3), we get
66. (
(
) (
365d
Or,BEY = 360 − nd(5.4)
Another way to look at these securities is to consider them as
zero-coupon bonds. Their present value and the future value are
related by the expression
F = B(1 + r)T(5.5)
Here F is the final value, or face value of the bond, B is its
present value, r is the implied rate of interest on the bond, and
T is the time to maturity in years.
The US Treasury also issues bonds with maturity longer than
one year. These securities carry a coupon and their interest is
paid semiannually. The bonds with maturity less than 5 years
are called notes, while the securities with maturity longer than 5
years are known as bonds.
67. At one time, prices for long-term bonds were quoted in the
newspaper in 32nds of dollars. For example, on May 25, 1994,
the notes maturing in October 1999 with 6% coupon had bid
price listed as 96:16 and asked price 96:18. This means that an
investor can sell such bonds for 9616/32 percent of their face
value and another investor can buy them for 9618/32 percent of
the face amount. For example, one has to pay $96,562.50 to buy
a bond with $100,000 face amount. An investor who is holding
a similar bond can sell it for $96,500. The difference between
these numbers, $62.50, is the profit of the dealer in such bonds.
These days, all prices are quoted in decimals. For instance, on
December 30, 2007, the 3.875% Treasury note, maturing on
February 15, 2013, was selling for 95.08% of its face value. It
paid interest semiannually. Its current yield was 4.076% and the
yield to maturity was 4.957%.
Examples
5.7. A Treasury bill with face value $100,000 will mature in 73
days. Glenn Corporation has bought the bill at a discount of
6.08%. How much did it pay for the T-bill?
Using (5.3), we get
B = 100,000(1 – 0.0608*73/360) = $98,767.11 ♥
5.8. For the T-bill in the previous problem, what is its yield to
maturity considering it to be a zero coupon bond and annual
compounding?
Using the relation (5.5), we have
100,000 = 98,767.11(1 + r)73/365
68. which gives the zero-coupon yield,
r = 6.40% ♥
The zero-coupon yield is higher than quoted discount of 6.08%
because of two reasons. First, the yield is compounded for a full
year, and not just for 73 days, and second, the year is now
counted as being equal to 365 days, and not 360 days. There is
also an inherent inconsistency in this calculation because the
year is being considered to be equal
to 360 days in the first part and then equal to 365 days in the
second part. Anyway, that is the common practice.
5.9. For the T-bill in the previous problem, what is its bond
equivalent yield? We find the bond equivalent yield by using
(5.4). This comes out to be
365*.0608
Bond equivalent yield = 360 – 73*.0608 = 6.24% ♥
5.10. For a T-bill that matures after 135 days, the bid discount
is 3.51% and the asked discount 3.49%. Calculate the buying
and selling price of a bill with face value $100,000. By using
the average of the bid and asked discount, find the zero-coupon
yield and the bond equivalent yield.
Using (5.3), we get
Asked price = buying price = 100,000[1 − .0349(135/360)] =
$98,691.25 ♥
Bid price = selling price = 100,000[1 − .0351(135/360)] =
69. $98,683.75 ♥
Using the mean value of the bid-asked spread = ½(3.49% +
3.51%) = 3.50%, we find
B = 100,000[1 − .035(135/360)] = $98,687.50
For zero-
– 1 = 3.64% ♥
365*.035
For bond equivalent yield, r = 360 – 135*.035 = 3.60% ♥
5.6 Other short-term investments
Besides the Treasury securities, the corporations also invest in
the following:
(a) Repurchase Agreements ("Repos")
Suppose Akron Corporation has Treasury securities with face
amount $1 million. Their market value is, say $960,000.
Suppose Akron needs $960,000 right away, but they also expect
to receive $960,000 after two weeks.
One possibility is to sell these securities on the open market and
get the needed cash. Another possibility is that Akron may sell
these securities for their market value to another corporation,
Toledo Company, with the agreement that Akron will
repurchase these securities after 14 days for $962,000. This
enables Akron to effectively borrow
70. $960,000 for 14 days by paying $2000 in interest costs. The
effective annual interest rate comes out to be
2000*365
r = 960‚000*14 = 5.431%
The advantage to Akron for this arrangement is that its interest
cost is fixed at $2000. It does not have to worry about the
interest-rate fluctuations in the bond market. Toledo corporation
has the advantage of investing at a fixed rate in a risk-free
environment. The interest rate in this case will be slightly
higher than that offered by the Treasury bills.
In the above example, Toledo Corporation is entering into a
reverse-repo agreement, whereby it can invest its spare cash for
a short time. The advantage of a repo or a reverse- repo
arrangement is that a company can borrow or lend money,
without taking any risks, for a fixed period of time, at a fixed
rate.
(b) Certificates of Deposit (CDs)
Certificates of deposit, CD's, are issued by banks at relatively
higher rate of interest, but the investors must put the money for
a fixed period of time. For individual investors, these CD's are
insured by FDIC up to $100,000. The corporations may also buy
"jumbo" CD's, with a minimum of $100,000, and usually in
denomination of $1 million each. The corporations are able to
lock in a fixed rate of return for their idle cash for a set period
of time. However, these CD’s are not federally insured and they
do carry a certain risk.
(c) Money Market Funds
Many mutual fund companies also offer money-market funds.
These funds simply take the money from individual investors
and corporations, and buy jumbo CD's with the pool of money.
71. The managers at the mutual fund company, such as Fidelity
Money MarketFund, select the CD's that are issued by
financially secure banks. They also buy the CD's with staggered
maturity dates. Another feature of the money market funds is
that their share value is fixed daily at $1 per share. The interest
on the account is generally credited once a month.
For a corporation, it is an excellent cash-management tool. The
spare cash earns a fairly high rate of interest, it is kept in a very
safe investment, and it is totally liquid. The fund also offers
check-writing privileges, meaning that the cash is available
immediately.
5.7 Choice of Marketable Securities
The corporations investing in marketable securities look at three
main characteristics of these investments: (a) Maturity, (b)
Credit risk, and (c) Income taxes.
If a company needs cash immediately, it should preferably keep
it in a money-market account. If a company will need the cash
after, say, six months, it is better off buying a
CD that will mature after six months. The cash needs must be
matched with maturity date of the investments.
The marketable securities are issued by other commercial
entities. The risk of the securities depends on the financial
wherewithal of the issuing corporations. The firm that is
purchasing marketable securities must look at the credit quality
of the instruments that it is buying. Of course, lower grade
investments have a higher degree of risk, but they also provide
higher rate of return.
Some companies buy the preferred stock of other companies for
72. investment purposes. This is because the dividends on this type
of stock quite secure, and most of this dividend income is tax
exempt. Consider the following example.
Suppose a company is in the 32% tax bracket. It buys a
preferred stock with a dividend yield of 5%. Suppose 70% of
the dividend income is tax exempt. What is the pre-tax rate of
return on another investment that will provide the same after-
tax return?
Suppose the required return is x. Suppose the firm invests
$100,000 and it gets 100,000x in dividends. It pays 32% in
taxes, which gives 100,000x(1 − .32) = $68,000x after taxes.
Suppose the company invests $100,000 in 5% preferred stock.
The dividend is $5000. 70% of this amount is tax free, or only
30% is taxable. The tax is thus .32(.3)(5000) =
$480. After paying taxes, the net amount is 5000 − 480 = $4520.
Equating the two possibilities, we get
68,000 x = 4520
Or, x = 4520/68,000 = 6.647%
You may simplify the calculation as
x(1 − .32) = .05 − .05(1 − .7)(.32)
This gives x = 6.647%. Therefor, another investment whose
return is fully taxable, must provide 6.647% return to compete
with the 5% return, of which 70% is tax free.
6.1
Collection and NPV from the credit policy of 2/10, net 30 and
73. 1% per month interest on all accounts after 30 days.
OLD POLICY
Collection within
10 days
30 days
60 days
90 days
Percentage
10%
30%
40%
20%
Discount/interest
-2%
0%
1%
1%
Collection
10%*(1-2%)
30%
40%+(40%*1%)
20%+(20%*1%*2)
0.098
0.3
0.404
0.204
Discount Rate
12%
pa
76. Yes, it should try the new policy
For calculating the NPV first the collections have to be
determined after taking into account the discounts given on
payment within 10 days and interest charged on payment made
after 30 days.
After that, the discount rate considered by the company is
12%p.a and days in a year are assumed to be 365. So using this
rate the present value of the collection is calculated and a sum
total of the amount gives the Net Present Value. The same
method applies for both, the old policy as well as the new credit
policy.
Since the NPV of the new credit policy is higher the new credit
policy should be implemented.
6.2
Cash Sale
Annual Sale
5
COGS
-3.2
78. 60 days
90 days
Bad Debt
Total
6.25*20%
6.25*40%
6.25*37%
6.25*3%
1.25
2.5
2.3125
0.1875
6.25
Discount Rate
12%
pa
-4
PV
1.25/(1+12%)^30/365
2.5/(1+12%)^60/365
2.3125/(1+12%)^90/365
0
1.238
2.454
2.249
0.000
5.941
COGS
79. -4.000
NPV
1.941
After calculating the present value of cash flows from credit
sale at a discounted rate of 12% with days in a year taken at 365
days is calculated.
Since the NPV of credit sale is higher than that of cash sale,
Credit sale should be encouraged.
The minimum increase in sale to justify credit sale should be
such that the NPV of Credit sale is at least equal to NPV of
Cash Sale
NPV
1.800
COGS
4.000
PV of Sale
5.800
82. The following equation to find the selling price of the portfolio,
NF + 12nB[(1 + r)−g/365 − 1] + mC(R − r) − L + 12aNS
Selling price = P +
r(6.10)
In the above expression, a = 0, P = 0, N = 10,000, F = $25, m =
10,000, n = 10,000, C =
$1200, B = $800, R = .15, r = .08, g = 25 days, L = $100,000.
Putting these numbers, we find the selling price as follows.
NPV =10‚000(25) + 12(10‚000)(800)[1.08−25/365 − 1] +
10‚000(1200)(.15 − .08) − 100‚000
.08
=3.971
Since the amount offered by Mellon Bank at $5 mn is higher
than the NPV of $3.971 of credit card portfolio, First National
Bank of Jermyn should accept the offer.
SOLUTIONS
5.1. Campbell Corporation uses Baumol model to manage cash.
The cost of transferring money from a money-market fund,
which pays 6% interest on balances, to a checking account is
$32 per transaction. Campbell needs $13 million annually to pay
its bills. Find the annual cost of interest forgone.
$3533 ♥
83. Solution
:
Annual requirement of cash (A) = $ 13 million
Transaction cost (T) = $ 32 per transaction
Opportunity cost of holding cash (R) = 6%
According to Baumol model to manage cash
Total cost = Transaction cost + Opportunity cost
Let C* be the optimum cash balance that minimizes the cost of
holding cash
Transaction cost = A x T
C*
Opportunity cost = C* x R
2
Total cost = A x T + C* x R
C* 2
Differentiating with respect to C* we get
0 = - A x T + 1 x R
C*2 2
84. - A x T = 1 x R
C*2 2
C* = √(2AT) / R
Therefore
C* = √(2*13,000,000* 32) / 0.06
C* = √13866666666.67
C* = $ 117756.81
The annual cost of interest foregone is the opportunity cost
Opportunity cost = C* x R
2
=($ 117756.81/2) *0.06
= $ 3533
The annual cost of interest foregone is $ 3533.
141
85. 5.2. Genentech Corporation, by analyzing its weekly balances in
its checking account, has determined that the variance of cash
flows is $3,000,000. Further, the cost of transferring money
from the checking account to a money market account is $65 per
transfer. The interest on the checking account is 1%, while that
on the market account is 6%. Genentech wants to keep $5,000
as a minimum balance in the checking account. Find the annual
cost of interest forgone.
$606 ♥
