cash conversion models

DETERMINING THE TARGET CASH

BALANCE

Target Cash Balance

The target cash balance involves a trade-off between the opportunity costs of holding too much cash and the trading cost of holding too little cash.

If a firm keeps its cash holdings too low, it will have to sell marketable securities more frequently than if the cash balance was higher.

The opportunity costs of holding cash rise and trading costs fall as the cash holdings rise.

Determining the Target Cash Balance

Cash Conversion Models balance the relevant

costs and benefits of holding cash versus

investing in marketable securities to determine

the economically optimum quantity of each.

The Baumol Model

The Miller-Orr Model

THE BAUMOL MODEL

Baumol Model

Assumptions of the model:

1. Net cash flow is the same everyday

2. Net cash flow is known with certainty

Baumol Model

Golden Peak Corp. began week 0 with a cash balance (C) of P1.2 million. Each week, outflows exceed inflows by P600,000. The cash balance drops to zero at the end of week 2.

Average cash balance = (Beginning balance + Ending balance)

over the 2-week period 2

= (P1,200,000 + P0)

2

= P600,000

At the end of Week 2, the company replenishes its cash by selling marketable securities.

Baumol Model

Baumol Model

If C were set higher, say P2.4 million, cash would last

4 weeks before the firm would have to sell

marketable securities, but the firms average cash

balance would increase to P1.2 million (from

P600,000).

If C were set at P600,000, cash would run out in 1

week, and the firm would have to replenish cash

more frequently, but the average cash balance would

fall from P600,000 to P300,000.

Baumol Model

To determine the optimal strategy, the firm needs to

know the following:

F = the fixed cost of selling securities to

raise cash (trading or transaction cost)

T = the total amount of cash needed over the

relevant planning period (for ex., one year)

K = the opportunity cost of holding cash (interest

rate)

Baumol Model

Opportunity costs (interest forgone):

= Average Cash Balance x Interest Rate

= (C/2) x K

Trading costs:

= No. of times the firm sells marketable securities x Fixed Cost of trading

= (T/C) x F

Baumol Model

Assuming Golden Peak Corporations opportunity cost is 10% and incurs $1,000 each time it sells its marketable securities. Weekly cash requirement is $600,000; therefore total cash needed in a year is $600,000 x 52 weeks or $31,200,000.

Baumol Model

Total Costs = Opportunity costs + Trading costs

= [(C/2) x K] + [(T/C) x F]

Using the numbers generated earlier, we have:

Notice that total costs start out at P246,500 and declines to about P82,000 before starting to

rise again.

So what is the optimal cash balance (ECONOMIC CONVERSION QUANTITY or ECQ)?

The Baumol Model

Opportunity Costs = Trading Costs

The optimal cash balance is found where the opportunity

costs equals the trading costs

Multiply both sides by C

F C

T K

C =

2

F T K C

= 2

2

K

F T C

= 2

2

K

TF C

2 * =

The Baumol Model

C* Size of cash balance

FT

KC

=C2

cost Total

FT

CTrading costs

The optimal cash balance is found where the opportunity

costs equals the trading costs

F K

T C =

2 *

Opportunity Costs K C

2

The Baumol Model

For Golden Peak Corp, we have T = P31.2,

F = P1,000, and K = 10%.

Therefore the optimum cash balance (ECQ) is:

_______________________

ECQ = (2 x P31,200,000 x P1,000)/.10

= P789,937

The Baumol Model

We can verify the answer by calculating the various costs at

this balance, as well as a little above and a little below:

The total cost at the ECQ level is P78,994, and it does appear to

increase as we move in either direction.

The Baumol Model

The Bulusan Corp. has cash outflows of P100

per day, seven days a week. The interest rate

is 5%, and the fixed cost of replenishing cash

balances is P10 per transaction. What is the

ECQ? What is the total cost? Assume a 365-

day year.

The Baumol Model

Total cash needed for the year is 365 x P100 = P36,500.

___________________

ECQ = (2 x P36,500) x 10)/.05 = P3,821

Average cash balance = P3821/2 = P1,911

Opportunity cost = P1,911 x .05 = P96

Because Bulusan needs P100 a day, the P3,821 balance will last 38.21 days (P3,821/P100). The firm needs to replenish the account 365/38.21 or 9.6 times per year. So the trading cost is:

Trading cost = 9.6 x P10 = P96

Total cost = Opportunity cost + Trading cost

= P 96 + P96

= P192

The Baumol Model

Advantage of the Baumol Model:

Simple to use

Limitations of the Baumol model:

assumes steady, certain cash flows

assumes a constant disbursement rate

ignores cash receipts during the period

does not allow for safety cash reserves


assumes that cash inflows and outflows fluctuate randomly from day to day

management sets the lower limit (or safety stock), L, depending on how much risk of a cash shortfall the firm is willing to tolerate.

as with the Baumol model, the optimal cash balance (Z) depends on trading costs and opportunity costs.

the only extra data needed is 2, the variance of the net cash flow per period. The period can be a day, a week, a year, for example, as long as the interest rate and the variance are based on the same length of time.


The firm allows its cash balance to wander randomly between

upper and lower control limits.

$

Time

H

Z

L

When the cash balance reaches the upper control limit H cash

is invested elsewhere to get us to the target cash balance Z.

When the cash balance

reaches the lower

control limit, L,

investments are sold

to raise cash to get

us up to the target

cash balance.


Given L, which is set by the firm, the Miller-Orr

model solves for Z and H

LK

FZ = 3

2*

4

3 LZH 23** =

where s2 is the variance of net daily cash flows.

The average cash balance in the Miller-Orr model

is

3

4balancecash Average

* LZ =


Example:

F = P10

Minimum cash balance L = P100

Interest rate = 1% per month

Standard deviation of the monthly net cash

flows = P200


The variance of the monthly net cash flows is 2 = P2002 =P40,000

Z = L + (3/4 x F x 2/K)1/3

= P100 + (3/4 x P10 x P40000/.01)1/3

= P411

The upper limit, H = ( 3 x Z ) (2 x L)

= (3 x P411) (2 x P100)

= P1,033

Average cash balance = [(4 x Z) L]/3

= [( 4 x P411) P100]/3

= P515

Implications of the Miller-Orr Model

To use the Miller-Orr model, the manager must

do four things:

1. Set the lower control limit for the cash balance.

2. Estimate the standard deviation of daily cash flows.

3. Determine the interest rate.

4. Estimate the trading costs of buying and selling

securities.

Exercises

Baumol Model

Your firm utilizes P165,000 a week to pay bills. The standard deviation of these cash flows is P20,000. The fixed cost of transferring funds is P48 a transfer. The applicable interest rate is 6%. The firm has established a lower cash balance limit of P100,000. Answer these five questions using the Baumol model:

What is the optimal cash balance?

What is the optimal average cash balance?

What is the opportunity cost of holding cash?

What is the trading cost of holding cash?

What is the total cost of holding cash?

Baumol Model

Your firm utilizes P165,000 a week to pay bills. The standard deviation of these cash flows is P20,000. The fixed cost of transferring funds is P48 a transfer. The interest rate is 6% per annum. The firm has established a lower cash balance limit of P100,000. Answer the following using the Baumol model:

What is the optimal cash balance? (P117,167)

What is the optimal average cash balance? (P58,584)

What is the opportunity cost of holding cash? (P3,515)

What is the trading cost of holding cash? (P3,515)

What is the total cost of holding cash? (P7,030)

Baumol Model

What is the optimal initial cash balance?

167,117$

55.166,117$

06.

000,680,823$

06.

48$52000,165$2

R

)FT2(*C

=

=

=

=

=

Miller-Orr model

Your firm utilizes P130,000 a week to pay bills. The standard deviation of these

cash flows is P15,000. The fixed cost of transferring funds is P51 a transfer. Your

firm has established a lower cash balance limit of P80,000. The weekly interest

rate is .067%. Use the Miller-Orr model to answer these three questions.


What is the optimum upper limit?

What is the average cash balance?

Miller-Orr model


417,103$

26.417,103$

26.417,23$000,80$

00067.

000,15$51$

4

3000,80$

4

3*

33333.2

3/12

=

=

=

=

=

RFLC

s

Miller-Orr model

What is the optimum upper limit?

251,150$

000,160$251,310$

)000,80$2()417,103$3(

)L2(*)C3(*U

=

=

=

=

Miller-Orr model

What is the average cash balance?

223,111$

67.222,111$

3

000,80$)417,103$4(

3

L-C*)(4 balancecash Average

=

=

=

=

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