Patacon Wrote: 2) Beta is NOT a measure > of an assets correlation with the market > benchmark, that would be the R^2, in a single variate regresssion, R^2 = beta^2 ain’t they the same measure?
If both managers achieved the same return, then based on CAPM the manager with lower beta must have generated higher alpha (since market risk premium is the same for both). This by itself makes the IR for manager with lower beta higher provided the active risk is same. I cannot think of a way to judge the latter proposition.
Well, I don’t think the statment R^2=beta^2 is true. You can get the R^2 of an asset by dividing its beta by its std but not by squaring it. The equations are similar, however. R^2=cov(A,M)/[std(A)*std(M)] beta=cov(A,M)/std(M) Honestly, I don’t think R^2 is in the CBOK for L3 so I wouldn’t worry about it. But here’s a little more… THE FOLLOWING EXPLANATION IS NOT CFA L3 EXAM RELEVANT: So, you plot a regression of market returns onto asset returns and get a scatter-plot, right? Imagine the cloud of data points. Graphically, the cloud might be very diffuse or within a narrow or wide band. That dispersion (variance) of data points is captured in your ANOVA. When you divide the variance that is explained by the regression (implying a relationship between variables) by the overall variance, you get your R^2. The higher the R^2, the more of the variance is explained by the regression, the more confidence you have that the movement of the independent variable predicts movement in the dependent. You may remember R^2=SSR/SST from level 2, or a stat class. But the R^2 only tells you how clustered the data is and if a relationship is likely to exist. To estimate how _much_ your dependent variable will move for a given movement in the independent, the equation for the regression line is calculated (usually) by the least squares method, minimizing the sum of squared errors. Think of it as a line through your cloud of data points that minimizes the dispersion of the datapoints from that line. The slope of that line is your beta and is a measure of magnitude of movement in the dependent variable wrt the independent. The problem is, you can calculate a least-squares line through a diffuse cloud of data points and get a slope (beta). But the dispersion will be so high, as reflected in a low R^2, does the beta mean anything? That’s where the F-statistic comes in handy.
ws Wrote: ------------------------------------------------------- > ^Thanks man!! Have a question. You 1) alpha is > indeed dependent on beta. Using your equation > Alpha = Ra - Rf - beta(Rm-Rf). I can see that. > Is this in CAPM context, or in general? Can you > please explain a little? When I said Alpha is not > determined by beta, I was thinking in absolut > terms. > > Thanks If I understand your question, and I may not, this calculation of alpha follows the same regression principals as CAPM, yes. It’s also the way the CFAI text defines the ex-post alpha, aka Jensen’s alpha. At least according to my notecard; I would need to verify in the text. It’s also what I use in practice. In general, there are other loosey-goosey measures of alpha that in my opinion, are either disingenuous or just not helpful. I have encountered peers who try to tell me that alpha, or active return, is simply realized return minus benchmark return. More of an absolute difference. I think maybe you were hitting on something along those lines? There are also people who posit that with multi-factor models, all components of return can be modeled by different factors and alpha explained away completely! It’s all almost enough to convert me into an indexer! (not really)
^Got it! I am clear now. Thx
rand0m Wrote: ------------------------------------------------------- > > sig(R-Rb) = sqrt(cov(R,Rb)) = sqrt(beta * var(bm)) Actually this is incorrect. Sig(R-Rb) = SQRT[SigR^2 + SigRb^2 - 2*Cov(R,Rb)]
i believe that standard deviation represents both systematic risk (Beta) and unsystematic risk. If the portfolio is not diversified then systematic risk would represent a high percentage of standard deviation. Since beta is not equal to one in your example it implies that there is some systematic risk and having a low beta reduces the standard deviation. A lower standard deviation would lead to a higher IR
Whats BETA ?
the question - compare EX-POST IR if beta if two PMs if one portfolio beta is lower than the other…mathematically, since portfolio return is in the numerator and return when calc using CAPM, the higher the beta the higher the return…hence high beta high IR… Another way of thinking - you are not compensated for non-systematic risk…thus the portfolio with the higher beta exposure, if it had a higher return, then it is better on a risk adjusted basis. If returns are same, then portfolio with lower beta is better… I’m confused, I contradicted myself…
annadru Wrote: ------------------------------------------------------- > i believe that standard deviation represents both > systematic risk (Beta) and unsystematic risk. If > the portfolio is not diversified then systematic > risk would represent a high percentage of standard > deviation. Since beta is not equal to one in your > example it implies that there is some systematic > risk and having a low beta reduces the standard > deviation. A lower standard deviation would lead > to a higher IR Hmmmm…I thought that a port is not diverisifed (not as diversified) if beta(systematic risk) makes up a smaller portion of the std. relatively.
Just a couple of points on recent comments: 1) the numerator in the IR equation is not portfolio return but alpha, or risk-adjusted excess return. 2) a higher beta implies the _expected_ return of the manager is higher. In other words, his threshold for adding value is higher, the bar is higher. You only add value, thus alpha, by achieving higher than expected returns. Lower beta, lower threshold. Beta is the market risk exposure of the asset/portfolio. If I have higher exposure to the market risk, my returns will be expected to be higher for positive market returns (but actually lower for negative market returns). 3) By throwing in some active risk (tracking error), I hope to get better returns than the market bm. But how do you know if my excess return is worth the additional risk? Well, if you divide my excess/active return (alpha) by my active risk (std of alpha, also known as omega for those interested), you can judge the excess/active return per unit of active risk I can provide. This is the information ratio. Kind of like how the Sharpe ratio measures how much total return you’re getting per unit of total risk. Higher the better. For CFA testing purposes, just keep in mind: IR = alpha/std(alpha) alpha = Rp-Rf-beta(Rm-Rf)
Patacon Wrote: ------------------------------------------------------- > Just a couple of points on recent comments: > 1) the numerator in the IR equation is not > portfolio return but alpha, or risk-adjusted > excess return. > For CFA testing purposes, just keep in mind: > IR = alpha/std(alpha) > alpha = Rp-Rf-beta(Rm-Rf) Actually, information ratio does not use risk adjusted alpha, it uses a simple excess return calculation (Rp-Rb) in the numerator, divided by tracking error.
Not so, my friend. Review eq. 43-22 of the CFAI text, vol 6 pg 59. IRa=(Ra-Rb)/std(a-b) [ignoring the bar and hat notations of R and std] Yes, it would seem like a simple excess return calc but now review eq. 43-17 on pg. 56. CFAI text explicitly defines Ra as the return of the asset, given its beta, aka risk-adj return. Tricky, tricky, but important. Furthermore, when addressing IR, Reading 44, pg. 95 under eq. 44-11 refers to the excess return as alpha, which was defined, ex post, as risk-adjusted. And of course, IR in both readings are found under the heading, “Risk-Adjusted Performance”. Hope this helps.
Patacon Wrote: ------------------------------------------------------- > Not so, my friend. Review eq. 43-22 of the CFAI > text, vol 6 pg 59. > > IRa=(Ra-Rb)/std(a-b) > > Yes, it would seem like a simple excess return > calc but now review eq. 43-17 on pg. 56. > CFAI text explicitly defines Ra as the return of > the asset, given its beta, aka risk-adj return. > Tricky, tricky, but important. > > Furthermore, when addressing IR, Reading 44, pg. > 95 under eq. 44-11 refers to the excess return as > alpha, which was defined, ex post, as > risk-adjusted. > > And of course, IR in both readings are found under > the heading, “Risk-Adjusted Performance”. > > Hope this helps. I believe you’re over-reading all this. Check out the CFAI exam of last year. They ask you to calculate IR simply as excess return/ St Dev of excess return (It’s all relative to a benchmark off-course and not to the expected return of the portfolio based on its beta).
A few things that I’ve noticed in the thread. First: R^2 is not equal to Beta^2, although the two can be related to each other. a) Pearson R = Beta * SD(y)/SD(x) b) R^2 =(Pearson R)^2 (for a bivariate regression (1 indep var)) Beta can be any real number; pearson R can only go from -1 to +1. Second: Beta is the slope of the best fit line relating the benchmark performance (in this case) to the performance of the portfolio. You can use beta as a measure of market risk or “benchmark risk,” because it tells you how much your portfolio return is expected to increase for every 1% return on the benchmark. R^2 tells you how “closely bound” the real world results are to that line. If things are pretty disperse in that “cloud” on the scatterplot, R^2 may be low, but the slope of the best fit line might still be high. You can have (high beta, low R^2), (low beta, high R^2), or any other combination. Other things equal, higher betas tend to have higher R^2, but in practice this relationship is fairly weak. If you have a low R^2, there is potentially more opportunity or to find alpha, but you still need to come up with a way to forecast it first. Third: I’ll have to go back and check, but I don’t think that Ra is a risk adjusted return. Ra does not get lower as the risk increases, which is what a risk adjusted return would be. Information Ratios are designed to be risk adjusted measures that tell you how effectively the manager uses deviations from the benchmark to create excess returns over the benchmark. Fourth: Two sources of alpha are security selection and market timing. If you are trying to measure alpha from security selection (people generally believe this is more “doable” than market timing), then you need to say : Alpha from Selection = [Port Return] - [Rfr] - [(port. beta)*(Rmkt - Rfr)] I’m a little less sure about this, but Alpha from market timing is about figuring out how changes in Beta have contributed to portfolio returns over time, so, I think: : Alpha from Mkt Timing = (portfolio beta - 1) * (Rmkt - Rfr) Which would mean : Total Return = [RfR] + [Rmkt - Rfr] + [Alpha from Selection] + [Alpha from Timing] It’s a little more complex, I’d assume, if your benchmark is something other than just S&P, russel, or something like that. I’M NOT SURE POINT #4 IS CFA KOSHER, or even 100% correct, but this is how I understood it.
One more comment regarding IR, the CFAI material indicates that the IR is a generalized form of Sharpe Ratio. = (RP- RB)/Sigma(RP-RB) Sharpe ratio is actually the special case where the benchmark is the Risk-Free rate in this special case : Sigm(RP- RF) = Sigm(RP) ( RF is constant and is not correlated with any asset return).
True… When beta is lower, E® = Rf + B (Rm- Rf)… will be lower … and hence alpha will be higher which translates into higher IR
Excuse me…I agree that alpha will be higher if beta is lower, but the tracking error, that is the std. of alpha is uncertain, isn’t it?
Its true that tracking error will change but it will be very marginal… for eg… Standard deviation of three numbers …(2,3,4) and (3,4,5) is the same ie 1.41… so if all the numbers change due to the fall in beta the standard deviation will be more or less similar … so I think Active retuen dominates the ratio…
Patacon Wrote: ------------------------------------------------------- > Not so, my friend. Review eq. 43-22 of the CFAI > text, vol 6 pg 59. > > IRa=(Ra-Rb)/std(a-b) > > Yes, it would seem like a simple excess return > calc but now review eq. 43-17 on pg. 56. > CFAI text explicitly defines Ra as the return of > the asset, given its beta, aka risk-adj return. > Tricky, tricky, but important. > > Furthermore, when addressing IR, Reading 44, pg. > 95 under eq. 44-11 refers to the excess return as > alpha, which was defined, ex post, as > risk-adjusted. > > And of course, IR in both readings are found under > the heading, “Risk-Adjusted Performance”. > > Hope this helps. Hmmm… I see your point. It seems that alpha is derived by adjusting for market risk, but it isn’t adjusted for total risk. So if you see an alpha shown on the exam, you know that that number is independent of market risk, but you don’t know how much ideosyncratic risk you are taking to get that alpha. Tricky tricky. I hope they don’t try to mess with my mind on that stuff.