Quantity of risk

[afterworkguinness]

Hello,
I don't understand the solution to the below practice question from the Amenc chapter 4 notes:

4. Assume the riskfree rate is 4%, the overall market volatility is 20% and the volatility of our
portfolio (P) is 25%. If the price of risk is 6% and the quantity of risk is 0.8, what is the (i)
correlation between the portfolio and the market and (ii) the portfolio’s expected return?
a) 0.80 and 7.5%
b) 0.80 and 8.8%
c) 0.64 and 7.5%
d) 0.64 and 8.8%

Solution given:

4. D. 0.64 and 8.8%
As beta = cov()/var() = correlation(M,P)*volatility(P)/volatility(M),
correlation(M,P) = beta*volatility(M)/volatility(P) = 0.8 * 20%/25% = 0.64
E(portfolio return) = 4% + 6%*0.8 = 8.8%
i.e., price of risk = excess market return = E(market return) - riskfree rate

I would have thought the answer to be b: 0.8 and 8.8% because the correlation between the portfolio and the market is given by beta and the question provides the value for beta: "and the quantity of risk is 0.8".

 

[David Harper CFA FRM]

Hi afterwork,

It's tempting to equate beta with correlation (and often the press mistakenly does this) but, as in the answer given, beta is given by covariance(M,P)/variance(M); this is a really fundamental formula that is more general than equity beta: beta (instrument, portfolio) = covariance(instrument, portfolio)/variance(portfolio). By canceling the volatility(M), we end up with another version of beta: beta(i, M) = correlation(i,M)*volatility(i)/volatility(M), or put another way, "beta is correlation scaled (multiplied) by cross-volatility"

beta and correlation are both "standardized covariance" but whereas correlation standardized by dividing by product of both volatilties, beta squares only one of the volatilities into a variance (in this way, beta is "with respect to [market | portfolio]")

In the above, beta is 0.8 such that correlation is also 0.8 if and only if the volatilties are equal (if cross volatility = 1.0). I hope that explains, thanks,

 

[afterworkguinness]

Thanks for your fast reply (on a Saturday none the less). I always understood beta as the correlation between the portfolio and the market, thanks for clearing that up for me.

 

[David Harper CFA FRM]

Sure thing (I'm working on content all day anyway!). Another perspective that might reinforce this is to observe something sort of interesting:

• There is only one correlation between two variables, in this case, correlation(M,P)= correlation(P,M) = 0.64 above
• But beta(M,P) <> beta(P,M) unless cross vol = 1.0. In the above, while beta(P,M) = 0.80, beta(M,P) = 0.64*25%/20% = 0.5120 ... this would be an unnatural expression but would refer to "beta of the market with respect to the portfolio." When we say asset/portfolio, we implicitly mean "with respect to the market" but it can be with respect to other portfolios. So, in my opinion, the order of the elements inside beta(.) is significant! FWIW

 

[0791]

Sure thing (I'm working on content all day anyway!). Another perspective that might reinforce this is to observe something sort of interesting:

• There is only one correlation between two variables, in this case, correlation(M,P)= correlation(P,M) = 0.64 above
• But beta(M,P) <> beta(P,M) unless cross vol = 1.0. In the above, while beta(P,M) = 0.80, beta(M,P) = 0.64*25%/20% = 0.5120 ... this would be an unnatural expression but would refer to "beta of the market with respect to the portfolio." When we say asset/portfolio, we implicitly mean "with respect to the market" but it can be with respect to other portfolios. So, in my opinion, the order of the elements inside beta(.) is significant! FWIW

Hi David,

I'd like to ask one question,
So based on what you were saying, "quantity of risk is 0.8" means beta of portfolio with respect to the portfolio, and how so?
I just started preparing for the FRM exam. So forgive me for asking dumb questions like this.
And I really appreciate anyone that could answer this.

Thank you.

 

[David Harper CFA FRM]

Hi 0791

Sure (welcome to the beginning!). The question includes "quantity of risk" only because the associated CAPM-reading references that term; "quantity of risk" is beta but, IMO, is just a term of art that connotes beta in the CAPM context. The original question is sort of typical in using a CAPM beta, where the beta is "with respect to the overall market."

But beta generalizes. Firstly, it generalizes to beta we typically read about in regard to hedge funds: beta with respect to some common factor (e.g., interest rates). Secondly, it generalizes all the way to "beta with respect to anything."

Maybe the best is to imagine a univariate regression: Y regressed against X. Beta is the slope. Now we can imagine several varieties of regression, including

• In a CAPM model (~ market index), Y is the asset or portfolio regressed against the market index. Equity beta is the slope
• We can regress change in futures prices against change in spot prices. This beta is the minimum (optimal) variance hedge ratio (Hull Chapter 3)
• We can regress a position within a portfolio (Y) against the portfolio that includes the position (X). The slope is a beta that is a direct function of marginal VaR (Part 2, Topic 8)
• Ultimately, we can regress any Y against any X

In such a regression, beta = covariance(Y,X)/variance(X) = correlation(Y,X)*volatility(Y)/volatility(X).

Note if we switch Y & X in a regression, the the difference between beta and correlation appears. If we switch Y & X, correlation (X,Y) = correlation (Y,X) ... correlation is unchanged. However, since beta(Y,X)= correlation(Y,X)*volatility(Y)/volatility(X), beta changes if we invert because beta(X,Y) =correlation(Y,X)*volatility(X)/volatility(Y). In this way, CAPM beta is sensitivity "with respect to the market," or with respect to whatever occupies the Y axis" but we could awkwardly invert the regression and look at beta of the market with respect to position, merely to illustrate that it's just a relationship between two variables. I hope that helps,