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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

( 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 e m e n t ) ( 6. A cco u n ts Rec ei v a bl e M a n a g e m e n t )

(1−p)S

pRS

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) − C

(1 − p)S

+

pRS 

+ 

( ( 1 − p ) S − C + ( 1 + r ) n + pRS m ( 1 + r ) ) (  )1 + r

(1 + r)n+1

(1 + r)m+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

1−p

 1 − p

NPV(two-times) = N1 + 1 + r N1 = N11 + 1 + r

 

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

1 − p2

NPV(three-times) = N1[1 + 1 + r + 1 + r]

 

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 follows.

1−p

1 − p2

NPV(infinite-times) = N1[1 + 1 + r + 1 + r

+ … ∞ ]

 

For the summation of infinite series, use the equation

which gives

S = a + ax + ax2 + … ∞ = a

1 − x

NPV(infinite-times) = N 1

1 − (1 − p)/(1 + r)

After some simplification, it becomes

1 + r

NPV(infinite-times) = N1p + r

 

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 – pi

1 + r

1 + r p + r

 =

i=0

Simplifying the expression further, we get

1 + r

(1−p)S

pRS

1 + r

Or, NPV = N1p + r=− C + (1 + r)n + (1 + r)m 

(6.2)

 

p + r

If we assume that the customer is never going to default, then p = 0. In that case, the above expression simplifies to

 S 1 + r

  

(6.3)

NPV = − C + (1 + r)nr 

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 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 fees V1 =

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)

− 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 riders V2 =

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

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 customers V3 = − 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 equals 12aNS per year. The value of this perpetuity to the bank is

Merchant fees V5 =

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

NF 12nB[(1 + r)−g/365 − 1]

mC (R −r)

L 12aNS

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 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

$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.

( B an k rupt fi r ms N o n -ba n kru p t fir m s  0.06 0 1 0.414  0.626 0.355  0.318 0.154 0.401 2.477 1.50 1.90 )Average ratio one year before bankruptcy of Net working 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:

If Z < 1.81, bankruptcy is imminent If 1.81 Z 2.99, not sure

If Z > 2.99, no threat of bankruptcy.

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,

1.23 Z 2.90 indicates a gray area, and Z > 2.90 indicates 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

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

.3

.38 

NPV(credit) = 1.2[− 1.2 + 21.1530/365 + 1.1560/365 + 1.1590/365] = $0.8565 million

 

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 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

 1 

NPV(new) = 1.1− .65 + 1.1260/365= $0.3647 million

 

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

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. ♥

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
Percentage 20% 30% 40% 10%

To improve the collection rate, Avoca is thinking of imposing a 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
Percentage 20% 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

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

 (1−p)S

pRS

1 + r

 m 

(6.2)

NPV = − C + (1 + r)n + (1 + r)

p + r

Putting the numbers, C = 2000, p = .15, S = 3000, r = .01, n = 1, R = .1, m = 4, we get

 .85*3000

.15*.1*30001 + .01 

 

= $3585

NPV = − 2000 +

1.011 +

1.014

.15 + .01

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)

 S 1 + r

  

(6.3)

NPV = − C + (1 + r)nr 

 30001.01

This gives us NPV = − 2000 + 1.01 .01 = $98,000

  

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?

To calculate the NPV of this decision, we use the equation

 (1−p)S

pRS

1 + r

 m 

(6.2)

NPV = − C + (1 + r)n + (1 + r)

p + r

Putting the numbers, C = 2500, p = .1, S = 4000, r = .01, n = 1, R = .1, m = 6,

 .9*4000

.1*.1*40001 + .01 

 

= $10,119

NPV = − 2500 +

1.011 +

1.016

.1 + .01

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 $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

 1000  1 

− 1000 +

25/365 

 1.07

.07/12

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:

 1000  1 

NPV(old) = .8− 1000 + 1.0725/365.07/12+ .2(−1000 + 150/.07) = −$405.50

  

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

 1100  1 

NPV(new) = .5− 1100 + 1.0725/365.07/12+ .5(− 1100 + 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) =

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?

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 $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] + 15‚000(1200)(.12 .07) 120‚000 + 12(.01)(30‚000)(800)

.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 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

PV = 1.0075i = .0075 = $3,333,333 (1)

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

1.7655586

PV = − 

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

PV = 1.0075i =

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.