# Risk and Leverage by stariya

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```									                                                         Risk and Leverage – Saul W. Adelman

Risk
Sources of Risk:
1. Sales – Uncertainty of sales level
2. Operating Leverage – amount of fixed versus variable costs in operations
3. Financial Leverage – amount of debt in the capital structure
4. Systematic Risk – risk associated with the general economy

I.     Risk concerning future sales (ability to predict)
a. Calculate expected value X
b. Calculate variance sqrt(var) = standard deviation (sd)

Coefficient of variation = sd/x = measure of risk
The higher the number the more risk; the higher the number the more dispersion and less
confidence in prediction.

II.    Operating Leverage
a. Based on break-even analysis
b. Break-even point is where total costs = total revenue; i.e. profit = 0

SP (Q) – [VC(Q) + FC] = 0
Where: SP = Sales Price; VC = Variable Costs; FC = Fixed Costs

SP (Q) – VC(Q) – FC = 0
Q(SP- VC) – F = 0
Q = F/ (SP-VC)
Q = break even quantity
Q(SP) = Dollar amount of Sales at break even

\$

Units

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Risk and Leverage – Saul W. Adelman

Example:
Suppose you were advising a firm as to the addition of a new product and there are two
alternatives for manufacture. Sales Price = \$12.00
A. Machine cost \$60,000; \$3 a unit variable cost
B. Machine cost \$100,000; \$1.70 a unit variable cost
Which alternative would you choose?

A. Q = F/(SP-VC)                                     B. Q = F/(SP-VC)
Q = \$60,000/(\$12-\$3)                                 Q = \$100,000/(\$12-\$1.70)
Q = 6,666 Units                                      Q = 9,708 Units

Therefore, risk is introduced by the structure of fixed versus variable costs.
This is operating leverage

BE B

Profit B

Profit A
\$                               Loss B

BE A
Loss A

Units

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Risk and Leverage – Saul W. Adelman

Operating Leverage – Measurement
SP = 4
VC = 3
FC = 20,000
BE = FC / (SP-VC) = 20,000 / (4-3) = 20,000 units

At production units        40,000
Total Revenue             160,000
Variable Cost             120,000
Fixed Cost                 20,000
EBIT                       20,000

Assume a 10% increase in sales (+10%)
What does a 10% increase in sales do to operating profit?
40,000 x 1.10 = 44,000 new units

At production units        44,000
Total Revenue             176,000
Variable Cost             132,000
Fixed Cost                 20,000
EBIT                       24,000

(24 – 20)/20 = 4/20 = 20% increase in operating income with a 10% increase in sales.

%∆ EBIT / %∆ Sales = .20 / .10 = 2 = Degree of operating leverage at 40,000 units of production

Starting at 40,000 units any change %∆ in sales will result in a 2 x %∆ in operating income.

The further you move away from BE point the lower the operating leverage will be. Operating leverage
changes for every level of production.

Operating Leverage
= %∆ EBIT / %∆ Sales = DOL = Degree of Operating Leverage

DOL = [S – VC ] / [S- VC – F] = [Q (P-V) ] / [Q (P-V)-F]

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Risk and Leverage – Saul W. Adelman

Example: Sales = 9,000 units, VC= \$3/unit, FC = \$28,000, P=\$10
DOL = [9,000 (10-3)] / [9,000 (10-3) – 28,000] = 63,000/(63,000 – 28,000) = 1.8

So %∆ EBIT /%∆ Sales = 1.8 at 9,000 units; so, if %∆ Sales = +100%, %∆ EBIT increases 180% (1.8 x
100%)

If sales go to 110,000 from 90,000 (dollars not units)
Sales                      90,000         110,000
Variable Cost              27,000          33,000
Fixed Cost                 28,000          28,000
EBIT                       35,000          49,000

%∆ Sales = (110 - 90) /90 = 22.22%; so %∆ EBIT = 22.22% x 1.8 = .4
35,000 x (1+.4) = 49,000 = EBIT at 110,000 sales (see box above)

IMPLICATIONS: The greater the operating leverage the more a firm’s EBIT will vary given a %∆ Sales.
Operating leverage arises from the tradeoff of fixed versus variable costs in the operating structure.

∆ Sales
- COGS                              Operating Leverage
- Total Variable Costs
- Fixed Costs
= EBIT                                                                    Combined Leverage
- Interest
= EBT                                Financial Leverage
- Tax
∆ EAT
- Preferred Stock Div
EPS

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Risk and Leverage – Saul W. Adelman

III.    Financial Leverage

Financial leverage arises from the use of fixed payments – Debt – versus variable payments - %∆ EAT for
stockholders.

Financing with Bonds                   Financing with Stock
10,000 old shares                      10,000 old shares
No new stock                           5,000 new shares
\$30,000 Int. expense                   0 Int. Expense; but 15,000 total shares
EBIT                     100,000                                100,000
Less: I                  30,000                                 0
EBT                      70,000                                 100,000
Less: T (50%)            35,000                                 50,000
EAT                      35,000                                 50,000
EPS                      \$3.50                                  \$3.33

But if EBIT is low:
EBIT                     10,000                                 10,000
Less: I                  30,000                                 0
EBT                      -20,000                                10,000
Less: T (50%)            -10,000                                5,000
EAT                      -10,000                                5,000
EPS                      \$-1.00                                 \$0.33

The more leverage (debt) the more you can make, but the more you can lose.

DFL = Degree of Financial Leverage
%∆ EPS / %∆ EBIT = EBIT / [EBIT – I – PD (1 / (1-t)) ] where PD = Preferred Stock Dividends (Note: this
calculation PD (1/(1-t) calculates the amount of money required to pay taxes and pay the preferred
dividend.) For example if PD = \$1,000 and t = 40%, you need \$1,000/0.6 = \$1,666.67 before taxes. So,
\$1,666.67 x (1-0.4) = \$1,000 after tax PD payment. Remember preferred dividend payments are not tax
deductible.

Refers to the employment of funds for which the firm pays a fixed cost. Increased debt or financial
leverage means increased financial risk. The greater the I is in the above formula the greater the effect.

DFL measures the changes in EPS caused by changes in EBIT.

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Risk and Leverage – Saul W. Adelman

So :
%∆ EBIT / %∆ Sales = operating leverage = Q(P-v) / [Q(P-V) – F]

%∆ EPS / %∆ EBIT = financial leverage = EBIT / [EBIT – I – PD(1/(1-t))]

%∆ EPS / %∆ Sales = combined leverage = %∆ EBIT / %∆ Sales x %∆ EPS / %∆ EBIT

Combined leverage = DCL = DOL X DFL = Q(P-V) / [Q(P-V) – F - PD(1/(1-t))]

Comprehensive Example: 5,000 shares of stock; \$20,000 interest expense, Preferred Div = 12,000

Sales (units)                          20,000              30,000              + 50%
Sales Revenue                        \$100,000            \$150,000
Less: Variable Op. Costs               40,000              60,000          DOL = 1.2
Less: Fixed Costs                      10,000              10,000
EBIT                                   50,000              80,000               +60%

Less: Interest                          20,000              20,000                           DCL = 6.0
Net Profits Before Tax                  30,000              60,000
Less: Taxes (40%)                       12,000              24,000          DFL = 5.0
Net Profits After Tax                   18,000              36,000
Less: Preferred Div                     12,000              12,000
Earnings available for CS                6,000              24,000
EPS                                       1.20                4.80            +300%

DOL at 20,000 units = Q(P-v) / [Q(P-V) – F]
20,000 (5-2) / 20,000 (5-2) – 10,000 = 60,000 / 50,000 = 1.2

DFL at 20,000 units = EBIT / [EBIT – I – PD(1/(1-t))]
50,000 / 50,000 – 20,000 – 12,000/.6 = 50,000 / 10,000 = 5.0

DCL at 20,000 units = Q(P-V) / [Q(P-V) – F – I - PD(1/(1-t))] = DOL x DFL =
DOL x DFL = 1.2 x 5 = 6.0
[20,000 (5-2)] / [20,000 (5-2) – 10,000 – 20,000 – 12,000/.6] = 60,000 / 10,000 = 6.0

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Risk and Leverage – Saul W. Adelman

Operating Leverage:
%∆ Sales = 30,000 – 20,000 / 20,000 = 50%
50% x 1.2 = +60% = %∆ EBIT so ending EBIT = (0.60 + 1) x 50,000 = \$80,000 (at 30,000 units)

Financial Leverage:
%∆ EBIT = 80,000 – 50,000 / 50,000 = +60%
+60% x 5.0 = 3.0 = %∆ EPS so ending EPS = (3.0 + 1) x 1.20 = 4.80 (at 30,000 units)

Combined Leverage:
%∆ Sales = 30,000 – 20,000 / 20,000 = 50%
+50% x 6.0 = 3.0 so ending EPS = (3 +1) x 1.20 = 4.80 (at 30,000 units)

IV.     Systematic Risk
Systematic risk stems from the general economic conditions. This risk you can generally not
control nor can you directly impact it. However, there are some financial instruments that can
hedge against certain items. These methods are beyond this course.

Non-systematic risk is the risk association with an individual company. This is somewhat
controllable by the way the company operates and how successful it is with respect to its sales.

EBIT – EPS Break Even Analysis
Example: Bonds versus Preferred Stock Financing
A company is considering two financial structures A and B.
A: Add more debt to cause \$4,500 more in interest expense.
B: Add preferred stock to cause \$900 more in preferred dividends and add 2,000 common shares.
Analysis is done at two levels of EBIT which are \$30,000 and \$50,000.

Structure A           Structure A           Structure B           Structure B
EBIT                              30,000                50,000                30,000                50,000
Less: Interest                    12,000                12,000                 7,500                 7,500
Net Profits B/T                   18,000                38,000                22,500                42,500
Less: Taxes (40%)                  7,200                15,200                 9,000                17,000
Net Prof A/T                      10,800                22,800                13,500                25,500
Less: Pref. Div                    1,800                 1,800                 2,700                 2,700
Earnings                           9,000                21,000                10,800                22,800
EPS                                1.125                 2.625                 1.080                 2.280
Common Shares                      8,000                 8,000                10,000                10,000
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Risk and Leverage – Saul W. Adelman

At what point should the firm be indifferent? Solve for the EBIT level where EPS(A) = EPS(B).

EPS (A) = EPS (B)
[(EBIT – 12,000) (1-.4) – 1,800] / 8,000 = [(EBIT – 7,500) (1-.4) – 2,700] / 10,000; Solve for EBIT
EBIT = 27,000; at EBIT = 27,000 EPS = \$0.90

EPS

EBIT

Structure A has greater financial leverage.
If EBIT is expected to be less than \$27,000, Structure B is preferred.
If EBIT is expected to be more than \$27,000, Structure A is preferred.

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