# 5.4 The Crossover Point

The EBIT “crossover” point represents the unique level of EBIT for both plans, at which the EPS and ROE respectively will simultaneously be the same for both plans. In other words, at a certain EBIT (“the crossover point”), the EPS for both plans will be the same, as will the ROE be the same for each. (In fact, the crossover point will be the same regardless of the amount of leverage employed; we will see this later.)

We shall – arbitrarily – choose to use the ROE (rather than the EPS) formula in order to calculate the EBIT crossover point. (We know this choice does not matter as the choices – EPS and ROE – are congruent diagrammatically and mathematically identical; this too we will see later.) Using the prior example, we will set the unknown EBIT to equate to the unique ROE of each alternative, as follows, and then solve for the EBIT. In other words, we will set the ROE formula for each plan to be equal to one another and then solve for EBIT!

ROE Case 1 = ROE Case 2

ROE = NI ÷ Eq. = [(EBIT – Interest Exp.) (1- T)] ÷ Eq.

Again, by definition, the EBIT value where the ROE (or EPS) of each alternative is the same is the crossover point! The formula for ROE, in both cases, is simple accounting (as above in general and as solved below):

ROE1 = (EBIT – 0) (1 – .40) ÷ \$200,000 =

ROE2 = (EBIT – 12,000) (1 – .40) ÷ \$100,000 = EBIT (.6) ÷ \$200,000 = (EBIT – \$12,000) (.6) ÷ \$100,000

To simplify, let’s solve for EBIT:

(.6) (EBIT) / 200 = [(.6) (EBIT – 12)] / 100

(.6) (EBIT) / [(.6) (EBIT – 12)] = 2

EBIT = 2 (EBIT – 12)

EBIT – 2 (EBIT) = -24

EBIT = 24

When EBIT = \$24,000:

ROE                = NI/Equity

ROE              = 14.4/200 = 7.2%

ROE2               = 7.2/100 = 7.2%

Here is a tabular summary:

 No Leverage 50/50 Leverage EBIT 24 24 Interest Exp. 00 (12) Taxes (9.6) (4.8) Net Income 14.4 7.2 ROE 14.4/200 = .072 7.2/100 = .072 EPS 14.4/10 = \$1.44 7.2/5 = \$1.44

Note: EPS = NI  NOSO

The exact value of the crossover point matters in terms of capital planning. If the firm projects an EBIT level to the right – or left – of the crossover point, the capital funding decision will be affected accordingly. To the right of the point, leverage is favorable.

We note that we have used just two cases above (i.e., no-leverage, and leverage set at 50%).  In fact, we could use multiple cases, addressing increasing levels of leverage and risk. We could still calculate a crossover point – and it would be the same locus point regardless of the degree of leverage (see below)!

Finally, we ask that if debt under certain conditions maximizes EPS and ROE, how much debt should the company use? Although we have only illustrated 50/50 leverage, it is an easy step to show that more debt provides even greater earnings per share leverage – at greater risk!

Note 1:

Had we used, say, 25% leverage, the crossover point would still be the same. See if you can explain this solution and how it may relate to the choice of degree of leverage. (The solution below compares 0% and 25% leverage, but would work for, and come out with the same solution as, any other leverage ratio.)

[(EBIT) (0.6) ÷ 200] = [(EBIT – 6) (0.6) ÷ 150]

EBIT = \$24

Note 2:

We could have solved for the crossover point using EPS rather than ROE. Here we go.

Again, let’s assume that EPS1 = EPS2.

• And EPS = [(EBIT – I) (1 – T)] ÷ NOSO
• (“NOSO” = number of shares outstanding)
• Only the denominator is different in this formulation
• This time, we’ll substitute “x” for EBIT. Next….
• [(x – 0) (1 – 0.40)] ÷ 10,000 = [(x – 12) (1 – 0.40)] ÷ 5,000
• Skipping along….
• X = 2 (x -12)
• X = EBIT = 24