FRM 2008 Practice PI question 27 - Asset vs. Equity volatility

[fullofquestions]

Question
Firm A has equity volatility of .3 and debt to firm value (debt to capitalization) of .4. Firm B has the same
debt to firm value but its asset volatility is .3. Which statement about firms A and B is true?
a. The capital of Firm A is less than the leverage of Firm B.
b. The volatility of Firm A’s operations is greater than the volatility of Firm B’s operations.
c. The equity of Firm B is less risky than the equity of Firm A.
d. The equity of Firm A is less risky than the equity of Firm B.
Answer: d
a. Incorrect. The two firms have the same capital ratio, so they have the same capital if they
have the same assets. As no mention was made of asset size, it could go either way.
b. Incorrect. Firm A actually has the lower asset volatility and the opposite of the sentence is true.
c. Incorrect. We know that asset volatility is smaller than equity volatility holding constant leverage,
so A has the lower asset volatility. This answer implies its asset volatility is higher.
a. Correct. See c for the explanation.

I cannot see why asset volatility is smaller than equity volatility. From the wording, the equity volatility of A is the same as the equity volatility of B. I think the question takes equity as a subset of assets. But beyond that I cannot confirm the assertions the answer provides. Any thoughts?

 

[fullofquestions]

I've searched the forum for this question using various portions of the question and some key words such as 'equity volatility' and 'asset volatility'. So far no luck. This one should be an easy one to tackle. I am missing the first part of the question, meaning, I am not getting why "We know that asset volatility is smaller than equity volatility." Once that is clear then I'm all set. Does anyone have any insight on this?
thanks

 

[David Harper CFA FRM]

Hi FoQ,

I think there are a few ways this is treated (e.g., in the KMV model, equity vol is related to asset vol by leverage) but i think it's always a function of financial leverage; i.e., since asset = debt + equity, for any given asset volatility, an increase in debt is "amplifying" equity volatility

here is how i think about this, although I'd imagine there are better ways: with multiples:

assume firm value = multiple of EBITDA or EBIT
assume equity value = multiple of EBT or net earnings
the earnings base of equity must be smaller, roughy by the debt service
e.g., if EBIT = 100, then -10 implies asset value goes form 100*multiple on firm value to 90*multiple

then if debt service is 50
EBT = 100 - 50 = 50
and -10 implies equity value goes from 50*multiple to 40* multiple

so if the debt service is 50 on EBIT base of 100, a 10% reduction in EBIT implies a -10% for asset value but a -20% on equity value

this is related to unlevered (asset) beta versus levered (equity) beta. See bottom of http://en.wikipedia.org/wiki/Beta_(finance)
where asset beta is a function of equity beta * equity value/firm value (can assume debt beta ~ 0 )

(also, we might connect to Hull's point about the volatitilty smile in equity options, although it's not a fundamental equation-type relationship, where he says the leverage could be an explanation for high implied volatility as leverage decreases)

David