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# P2.T7.303. Liquidity and Leverage (Malz)

#### Low

##### New Member

Don't quite get the solution posted for 303.3

Let A = Assets and E = Equity with D = A - E.
ROE = ROA/Equity = [(5%*A) - (A - E)*4%]/E. Since ROE must be at least 15%, the minimum leverage is achieved when ROE = 15% such that:
[(5%*A) - (A - E)*4%]/E = 15%, and
[(5%*A) - (A - E)*4%] = 15%*E, and
5%*A - 4%*A + 4%*E = 15%*E, and
1%*A = 11%*E, such that:
A/E = 11%/1% = 11.0

Wouldn't LR = ROA/ROE rather than ROE = ROA/Equity?
where
• ROA = Inc / A and
• ROE = Inc / E
• Combining both E * ROE = A * ROA --> A/E = ROE/ROA - LR
any help appreciated. thanks

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Low

I have an interm typo, sorry that confused you, but I think everything else is okay, at the source Q&A here I have edited to:
Let A = Assets and E = Equity with D = A - E.
ROE = Income/Equity = [(5%*A) - (A - E)*4%]/E.
Since ROE must be at least 15%, the minimum leverage is achieved when ROE = 15% such that:
[(5%*A) - (A - E)*4%]/E = 15%, and
[(5%*A) - (A - E)*4%] = 15%*E, and
5%*A - 4%*A + 4%*E = 15%*E, and
1%*A = 11%*E, such that:
A/E = 11%/1% = 11.0

I don't see where Malz (who is simple here) gives the numerator a name, so I am safely going to call it: "Income" = ROA(%)*Assets($) - DebtCost(%)*Debt($);
in this example, Income = [(5%*A) - (A - E)*4%], where Debt = A-E.

I don't *think* we can quite use yours because the numerator in ROA is different than the numerator in ROE (I might mis-understand you, sorry if that is the case). But although these are all % just imagine the assets are $100, A =$100, then the "income before interest expense" (aka, EBITDA or Gross Operating Income) would be $5.0 million such that stylized income statement would look something like: Sales - COGS etc = Gross Operating Income (EBITDA) =$5.0 MM such that ROA = $5 EBITDA/$100 Assets = 5.0% ROA
- Cost of debt service $3.64 = 4.0% *$90.91 Debt
= (Net) Income of $1.36 =$5.0 - \$3.64, such that Income/Equity = ROE = 1.36/9.09 = 15.0% ROE

And leverage = A/E = 100/9.09 = 11.0. So I think that is an internally consistent implementation of the question, but notice that "return" numerators in ROA and ROA are very different (this is why ROE can be increased by the leverage of "cheaper" debt). Thanks,

Got it. Thanks!

#### hamu4ok

##### Active Member
As per my understanding, some of the important ideas behind the lengthly Malz' chapter on Liquidity & Leverage in one slide:

#### southeuro

##### Member
indeed. using the leverage equity return, the 303.3 lends itself to solution by itself <so to speak>

R(e) = L * R(a) - ( L - 1) * R(d)
15 = 5L + 4 - 4L --> 11 = L hence leverage 11%

##### Active Member
Subscriber
Hi @David Harper CFA FRM ,

Malz, Chapter 12: Liquidity and Leverage

Subtopic :
Explain the impact on a firm’s leverage and its balance sheet of the following transactions: purchasing long equity positions on margin,
entering into short sales, and trading in derivatives.

Can you explain below line I did n't understand how

"short positions generate leverage, they reduce risk if there are long positions with which they are positively correlated or other short positions with which they are negatively correlated"

Thanks a ton