Grover Clark, an analyst at Gypum Securities, compiled the following information about two securities, Forest Cutters, Inc. (FCT) and Peerless Industries (PRL): FCT PRL Stock Price 50.00 120.00 Dividend per share 2.00 3.00 Expected Growth Rate 5% 6% Standard Deviation of Returns 25% 30% Correlation of Returns with the Stock Market’s Returns .90 .70 Grover also observes that the risk-free interest rate is currently equal to 3%. The consensus expectation is that the general stock market will produce an 8% return over the long run and the historical standard deviation in the stock market’s return has been, and is expected to remain at 20%. From this information, Grover would like to understand something about other characteristics of these two stocks and how they would behave if combined into a portfolio. The standard deviation of the return for a $100,000 portfolio consisting of $50,000 of FCT and $50,000 of PRL is closest to: a. 64.50% b. 24.85% c. 55.00% d. 27.50%

Not enough info - you know the corr. of FCT and PRL with the stock market but not with each other. With enough work, you can calculate bounds on the portfolio std. dev. but that’s all.

that’s what I was thinking. Stalla says the answer is B and they have R=.63 but they don’t show the calculation.

Actually I guess it’s not that hard to get bounds. The smallest the correlation can be would be cos(arccos(.7) + arccos(0.9)) = 0.318 and the biggest would cos(arccos(.7) - arccos(0.9)) = 0.941

JoeyDVivre Wrote: ------------------------------------------------------- > Actually I guess it’s not that hard to get > bounds. > > The smallest the correlation can be would be > cos(arccos(.7) + arccos(0.9)) = 0.318 and the > biggest would cos(arccos(.7) - arccos(0.9)) = > 0.941 I really don’t understand what that means. Can you explain further? How do I calcualte this?

:P, err that should be beyond the scope of Level 1… don’t worry about it whodey

xck2000 Wrote: ------------------------------------------------------- > :P, err that should be beyond the scope of Level > 1… don’t worry about it whodey Oh, I know it’s not part of level I, I was honestly just curious how he calculated this. The average of the 2 numbers he has = .629 which solves the problem.

Not yet at portfolio management, but from what I remember here is my take: you actually can calculate the correlation between and covariance of the two stocks. From the Stock prices, Dividends and expected growth rate plugged into the Gordon model get the estimated return on each stock: For FTC: 2*(1+5%)/50+5% = 9.2% For PRL: 3*(1+6%)/120+6%=8.65% Use Correlation of Returns with the Stock Market’s Returns to determine beta for each company: Beta = Covariance (stock, market)/Variation of the market = (Correlation (stock, market)*STDV stock *STDV Market)/ Variation market For FTC: beta FTC = (90%*25%*20%)/(20%^2)=1.125 For PRL: beta PRL = (70%*30%*20%)/(20%^2)=1.05 Plug beta into the CAPM model to determine expected return of each portfolio: E® = Rf+beta*(Rmarket-Rf) For FTC: E(R FTC) = 3%+1.125*5%=8.625% For PRL: E(R PRL) = 3%+1.05*5%=8.25% Hence, the Covariance of the two stocks: Cov (FTC, PRL) = (9.2-8.625)*(8.65-8.25)=0.23 The problem is that when plugged into the STDV of the portfolio formula, I get none of the answers. What’s wrong?

Ignore Joey boy’s explanation. He is just trying to show off his knowledge on the geometric interpretation of correlation. The joke, of course, is on him, because this problem is perfectly solvable in the CAPM framework and not “Not enough info - you know the corr. of FCT and PRL with the stock market but not with each other” as he claims. Let me show you how. Let X=FTC return ; Y=PRL return ; M= Market return; Rf = risk-free rate; B = Beta Then, Corr(X,M) = Cov(X,M) / sd(X)sd(M) -----------------(*) Here’s the key step: Under CAPM framework, we can write Y = Rf + B(Y,M)[M-Rf] Therefore, M = (Y - Rf)/B(Y,M) + Rf M = Y/B(Y,M) - Rf(1/B(Y,M)-1) --------------------(1) Now, you need to know a few correlation identities here, namely: A) Cov(X, constant) = 0 B) Cov (aX, Y) = aCov(X,Y) C) Cov (X, Y+Z) = Cov(X,Y) + Cov(X,Z) Now subst (1) into Cov(X,M), you have, Cov(X,M) = Cov(X, {Y/B(Y,M) - Rf(1/B(Y,M)-1)}) From C, you have, Cov(X,M) = Cov(X,Y/B(Y,M)) + Cov (X, Rf(1/B(Y,M)-1)) Since B(Y,M) and Rf(1/B(Y,M)-1) are constant, from A and B, we have Cov(X,M) = Cov(X,Y)/B(Y,M) + 0 = Cov(X,Y)/{Cov(Y,M)/Var(M)} = Var(M) * COv(X,Y)/Cov(Y,M) Subst this into (*), you then have, Corr(X,M) = {Var(M) * Cov(X,Y)} / {Cov(Y,M)*sd(X)*sd(M)} Corr(X,M) = Cov(X,Y) / {[Cov(Y,M)/sd(M)*sd(Y)]*sd(Y)*sd(X)} Corr(X,M) = {Cov(X,Y) / sd(Y)*sd(X)} / Corr(Y,M) Therefore, Corr(X,Y) = Corr(X,M) * Corr(Y,M) Thus the schweser answer.

Sigh. Sure. Think hard…

propanol Wrote: ------------------------------------------------------- > > Therefore, Corr(X,Y) = Corr(X,M) * Corr(Y,M) > > Thus the schweser answer. That’s a formula I MUST remember withut the calculus:) Thanks a lot.

Thanks a lot, that makes sense but I would have never got that on my own.

Except it’s not right (see if you can figure out why not). Edit: Please do not remember that formula…

OK Joey now I’m curious because I can’t seem to identify an error with propanol’s explanation, in addition to the fact that the solution Corr(FCT, PRL) = Corr(FCT,Market)*Corr(PRL,Market) yields the correlation value that evidently produces the correct answer 0.630 = 0.90 * 0.70 is this a fluke?

also I guess you could just answer this question with reason… For example if Rho = 1 the portfolio standard deviation would have to be 27.50 since the assets are weighted 50/50. The fact that they are not perfectly correlated with each other means their combined standard deviation must be lower then 27.50 when considering the attributes of diversification… the only logical choice is (b) 24.85

- Just because Schweser says it’s the answer doesn’t mean anything 2) There is nothing in the question to suggest that we should bring CAPM into the picture even if it would help (It won’t except in some kind of asymptotic way) 3) CAPM doesn’t say "Here’s the key step: Under CAPM framework, we can write Y = Rf + B(Y,M)[M-Rf] ". CAPM says the E(Y) = Rf + B(Y,M)[M-Rf] so right away we have a problem. We are trying to get the correlation of a r.v. Y with another r.v. and suddenly we have lost it to an expectation. In any event, all the following stuff about covariances and stuff is made unnecessary by the observation that correlations are not affected by linear transformations so if the key step was true, we could just bag all that stuff. If Y = Rf + B(Y,M)[M-Rf] and X = Rf + B(X,M)[M-Rf] then X and Y are just linear transformations of M and their correlation is 1. Fortunately, propanol makes it more interesting by adding another error after missing those facts. 4) Even if we kept in the normal errors stuff, you can’t say something like Y = a + b X + error so X = (Y - a)/b + error because the regression lines are different so all that stuff about flipping the regression estimator over is BS (think about the least squares fit that “allows” the substitution Cov(X,M) = Cov(X,Y)/B(Y,M) + 0 = Cov(X,Y)/{Cov(Y,M)/Var(M)} That least squares estimation is for regressing Y on M. We regress M on Y and we don’t get 1/B(Y,M)). I don’t know why propanol is so aggressive about this stuff. Sigh.

agree the aggressive stuff is unnecessary… moving on… any thoughts on my second post regarding the reason approach?

Char-Lee, I think you’re right. Adding the two stock into a portfolio would indeed result in a portfolio with a SD lower than that of the stock with the highest SD (diversification benefit). Having said that, strike out 2 of the answers. Working back the 27.5% answer I get a correlation of 1, which obviously is not the case, so the only solution is 24.85%.

I agree with that too - given the choices, that is certainly the best choice.

I emailed Stalla about this question, here’s the answer I got: The correlation of 0.63 is calculated as the product of 0.90 x 0.70, the correlations for Cutters and Peerless, respectively. Cheers. Ben PIerce, CFA Stalla Academic Support Team Based on the tread above, is this incorrect?