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ch.-5.docx5.1 Cash Management
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.
( 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 Baumol model , 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 s u r y M a n a g e m e n t ) ( 5 . C a s h M a n a g e m e n t )
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.
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. 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
dT r bC
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
r bC
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:
(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
r r/2 =
rbC
2
Note that the two costs are equal. Add them to find the total cost as
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 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
(average amount of cash) (interest rate per year) (time 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)
PV cost of infinite cycles =
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
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 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
2x
x
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 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
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
3bσ2 1/3
( ) ( )x =
4r
(5.2)
Further, the model specifies the follows values: The minimum amount of cash = L
The optimal amount of cash = L + x
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,
2, and r. Consider the problem (5.2) of Alaska Corporation again. They expect to spend
$25 million in cash payments annually. Suppose the standard deviation of cash flows is
$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 )3*100*6‚000‚000
x =
4*.06
= $355,689
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 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 $15,625 6 22,725 |
|||
|
2 12,225 7 19,000 |
|||
|
3 13,825 8 17,775 |
|||
|
4 17,375 9 12,125 |
|||
|
5 21,900 10 10,225 |
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 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 |
|
5 |
21,900 |
21,900 – 17,375 = |
4,525 |
4,525 + 600 |
26,265,625 |
|
6 |
22,725 |
22,725 – 21,900 = |
825 |
825 + 600 |
2,030,625 |
|
7 |
19,000 |
19,000 – 22,725 = |
–3,725 |
–3,725 + 600 |
9,765,625 |
|
8 |
17,775 |
17,775 – 19,000 = |
–1,225 |
–1,225 + 600 |
390,625 |
|
9 |
12,125 |
12,125 – 17,775 = |
–5,650 |
–5,650 + 600 |
25,502,500 |
|
10 |
10,225 |
10,225 – 12,125 = |
–1,900 |
–1,900 + 600 |
1,690,000 |
|
|
|
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
3*100*11‚943‚437.51/3
that x =
4*.05/365
= $18,700
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 + 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 $15,625 4 $17,375 7 $19,000 10 $10,225 |
|||||||
|
2 12,225 5 21,900 8 17,775 11 14,000 |
|||||||
|
3 13,825 6 22,725 9 12,125 12 15,000 |
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 place
a[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 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)^2
for i to n-1 do diffsq[i]:=(ncf[i]-avncf)^2 od;
# var is the variance = (sigma)^2
var:=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?
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
( )x =
4r
(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. 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
( )x =
4r
(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 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 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 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
$ discount from the face
% discount as
face value of
Time to maturity in days
amount of the bond
= quoted in the paperthe bond in $
360 days
We can express it as
dFn D = 360
The market price of the bond is B = F − D,
dFn
dn
( ) ( )B = F − 360 = F1 −
360
(5.3)
Once we know the market price of a T-bill, we can also calculate its bond equivalent yield, which is defined as
$discount
365 days
Bond equivalent yield = $market valuedays to maturity
Using (5.3), we get
dFn
1 365
( )Bond equivalent yield = 360
( ) ( )F1 − dn n
360
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.
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
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)] = $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
F1/T
For zero-coupon yield, r = B
– 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
$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. The managers at the mutual fund company, such as Fidelity Money Market Fund , 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 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.
