Hi guys. Having a look at some Qbank questions for this year and came across an answer which I think is wrong. Essentially the questions asks which is correct: a) All portfolios on the CML are unrelated except that they contain the risk free asset b) All portfolios on the CML are distinct from each other c) All portfolios on the CML are perfectly positively correlated I said B. They said C. Their justification for C is that the CML is a straight line thus everything on a straight line is perfectly positively correlated. But I disagree, because (as an example) the risk-free rate and the market portfolio both lie on the CML, yet their correlation with each other is 0, not 1. B seems right to me, because every portfolio on the CML has a different risk return profile thus one cuold argue they are distinct from each other. Am I missing something? Cheers

The word “distinct” in this context does not mean “has a different risk return profile”, but it means “perfectly uncorrelated” (correlation with each other = 0) I cannot get what you mean when you said “But I disagree, because (as an example) the risk-free rate and the market portfolio both lie on the CML, yet their correlation with each other is 0, not 1.” because “the risk-free rate” and “the market portfolio” are not the same concept, maybe you mean “the totally risk-free asset portfolio”? if so, they are perfectly positively correlated because they both lie on the CML. Maybe I should clarify the concept of correlation of portfolio in this context. Portfolios are presented by points defined by return (y-axis) and risk (x-axis), if we connect those points and we have a perfectly straight line (all the points lie on that straight line), we say those portfolios are perfectly correlated with correlation = 1 or -1, if no any line contains more than 2 points, we say those portfolios are perfectly uncorrelated (distinct in the context of the question) with correlation = 0, other cases we say they are imperfectly correlated (correlation can get any value but not 0,1,-1)

not sure why you’re bringing the RF rate into the equation here. The CML is the line where all possible risky assets in the whole world are, so if they are all on the line then they are perfectly positively correlated. There is only one CML. The tangency or “market” portfolio is the best portfolio as it holds all risky assets in market value weights. Let me know if that helps or not.

quangnguyen82 Wrote: ------------------------------------------------------- > The word “distinct” in this context does not mean > “has a different risk return profile”, but it > means “perfectly uncorrelated” (correlation with > each other = 0) > > I cannot get what you mean when you said “But I > disagree, because (as an example) the risk-free > rate and the market portfolio both lie on the CML, > yet their correlation with each other is 0, not > 1.” because “the risk-free rate” and “the market > portfolio” are not the same concept, maybe you > mean “the totally risk-free asset portfolio”? if > so, they are perfectly positively correlated > because they both lie on the CML. > > Maybe I should clarify the concept of correlation > of portfolio in this context. Portfolios are > presented by points defined by return (y-axis) and > risk (x-axis), if we connect those points and we > have a perfectly straight line (all the points lie > on that straight line), we say those portfolios > are perfectly correlated with correlation = 1 or > -1, if no any line contains more than 2 points, we > say those portfolios are perfectly uncorrelated > (distinct in the context of the question) with > correlation = 0, other cases we say they are > imperfectly correlated (correlation can get any > value but not 0,1,-1) Thanks quangnguyen82. It does help a lot. I have been pretty unsure of this ‘correlation’ concept too since I just understand it in terms of covariance and SDs. This does make sense that if they lie on a straight line, they are perfectly correlated. But what I don’t get either is that what do they mean by ‘A Risk free asset and the market portfolio have correlation of 0 i.e. risk free asset is uncorrelated with the market’? I am having trouble understanding the physical meaning of correlation. It would be great if you could explain it a little more.

hi chunty, In calculating correlation coefficient, we usually use 2 kinds of data , time series data and cross-sectional data. In time series data, two variables of 1 entity get different values across a time horizon (i.e. the data of interest rate and inflation rate of the US between 1900 and 2000). In cross-sectional data, two variables get different values across different entities at a point of time (i.e. the data of interest rate and inflation rate of the US, Japan, England, Canada. French in 2008) In the CML model, suppose that we currently only have the market portfolio (with 100% risky assets), now we want to reduce the risk of the portfolio, we decide to add a risk-free asset to our portfolio, but, as a principle, first we must calculate the correlation between the risk-free asset and our portfolio, if that correlation is less than 1, we accept the risk-free asset. How do we calculate the correlation? by calculating the covariance between the risk free asset and our portfolio, and by using the time series data which have this form: year…market portfolio return (%)…risk-free asset return(%) 1990…12…5 1991…15…5 … and so on. because the risk-free asset return is constant over the time horizon, its variance is 0, so the covariance (and the correlation) between the risk-free asset and the market portfolio is 0, which less than 1, so we accept this risk-free asset. Maybe this is what you’re thinking about. Because the correlation between the risk-free asset and the market portfolio is 0, we calculate the expected portfolio return and standard deviation of portfolio (the new portfolio including the risk-free asset and the market portfolio) as follow: E(Rp)= RFR + Wm(Rm - RFR) STDEVp= Wm STDEVm (STDEVp: standard deviation of portfolio) with Wm is the weight of the market portfolio in the new portfolio. We can easily calculate E(Rp) and STDEVp by changing the value of Wm from 0% (in this case, we have the 100% risk-free asset portfolio which we are talking about) to 100% (100% risky asset portfolio, or 100% market portfolio), then we will have the data like this (I supposed some value of E(Rp) and STDEVp): Wm(%)…E(Rp)(%)…STDEVp 0…12…0 1%…12.002…0.0011 … 100%… Each pair in the data represents for a portfolio which lies on the CML, and to calculate the correlation between these portfolios, we use those series data of E(Rp) and STDEVp (it’s a kind of cross-sectional data). Because the value of E(Rp) and STDEVp is not constant, the correlation calculated is not zero as you thought (exactly it equals 1).

quangnguyen82 Wrote: ------------------------------------------------------- > > > > Each pair in the data represents for a portfolio > which lies on the CML, and to calculate the > correlation between these portfolios, we use those > series data of E(Rp) and STDEVp (it’s a kind of > cross-sectional data). Because the value of E(Rp) > and STDEVp is not constant, the correlation > calculated is not zero as you thought (exactly it > equals 1). Wow, its hit me now… Thanks, it makes a lot of sense…

Still confused. Let me put my question another way. The CML is the line connecting the risk-free asset with the portfolio of all risky assets (The market portfolio). The answer to that Qbank question suggests any portfolio on the CML (thus with varying weights between the risk-free asset and the market portfolio) are perfectly positively correlated with each other. Consider two portfolios: A - 1% in risk-free, 99% in market portfolio B - 99% in risk-free, 1% in market portfolio Now these two portfolios both lie on the CML. The Qbank answer implies that both portfolios have a correlation coefficient of 1. How can these two portfolios be perfectly positively correlated? For example, if the market drops 10%, it is going to smash portfolio A by a lot more than portfolio B. Does that make sense?

All portfolio on CML are bearing the same risk i.e. market risk, and weight doesn’t account for the risk. Now why I say that weight doesn’t account because when a risk free asset asset is mixed with a risky asset, the only risk portfolio bears is risky asset risk which is essentially market risk in this in case. When all assets are bearing the same risk, they should ideally have the same risk-return characteristic which implies that they will be moving together in either direction that means they are positively correlated. That’s my understanding of the CML. Now to your point, your portfolio A on the CML will be very close to the market portfolio but portfolio B will be very far towards center. portfolio B will less returns than A because it’s far down and bearing less risk, so yes, it will smash the portfolio A a lot more than portfolio B.

I understand that portfolios with above mentioned proportions (1/99 and 99/1) correlate perfectly and it makes a lot of sense. Unfortunetaly, I can’t get why portfolio with zero risky asset (0,100) should correlate with risky asset (100/0). Or we just assume that there never is such portfolio with only risk free asset and without risky asset?

hi chunty, Actually those portfolios perfectly positively correlated in all case, if you want to test it, feel free to use this file I’ve created http://www.mediafire.com/?1hqemao0uc1re7m