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Can I calculate Beta of a Portfolio like this?
Posted on 6/13/13 at 11:47 pm
Posted on 6/13/13 at 11:47 pm
From CAPM:
Return on Investment = Risk Free Return + Beta * Market Risk Premium
i.e, Ri = Rf + Bi * (Rm - Rf)
Reorganizing,
Ri - Rf = Bi * (Rm - Rf),
(or)
Bi = (Ri - Rf) / (Rm - Rf) <--- Can I calculate Beta using this formula? I know Ri, Rf and Rm. Only unknown is Bi(Beta).
This gives Beta of 1 asset. Similarly I calculate the Betas of other individual assets in the portfolio.
Finally using weighted averages of the betas of all the assets, I find the Beta of the entire Portfolio.
I don't want to use linear regression to calculate Beta because I only have 5 data points for both Ri and Rm (year end annual returns over 5 years) and there really is no correlation between them. My R^2 values are like 0.02 and 0.09.
So back to my original question: Can I calculate Beta using the CAPM formula like shown above?
Return on Investment = Risk Free Return + Beta * Market Risk Premium
i.e, Ri = Rf + Bi * (Rm - Rf)
Reorganizing,
Ri - Rf = Bi * (Rm - Rf),
(or)
Bi = (Ri - Rf) / (Rm - Rf) <--- Can I calculate Beta using this formula? I know Ri, Rf and Rm. Only unknown is Bi(Beta).
This gives Beta of 1 asset. Similarly I calculate the Betas of other individual assets in the portfolio.
Finally using weighted averages of the betas of all the assets, I find the Beta of the entire Portfolio.
I don't want to use linear regression to calculate Beta because I only have 5 data points for both Ri and Rm (year end annual returns over 5 years) and there really is no correlation between them. My R^2 values are like 0.02 and 0.09.
So back to my original question: Can I calculate Beta using the CAPM formula like shown above?
This post was edited on 6/13/13 at 11:54 pm
4
Posted on 6/13/13 at 11:54 pm to rickgrimes
You are forgetting epsilon. Kinda important IMHO.
Basically CAPM tries to make the epsilon term randomly distributed, but back when I studied this I didn't find that CAPM was all that valuable at doing so. The big problem with most of the models is that variation in epsilon is vastly greater than the variance in returns explained by beta.
Basically CAPM tries to make the epsilon term randomly distributed, but back when I studied this I didn't find that CAPM was all that valuable at doing so. The big problem with most of the models is that variation in epsilon is vastly greater than the variance in returns explained by beta.
Posted on 6/14/13 at 12:05 am to foshizzle
quote:
You are forgetting epsilon.
So from my limited knowledge of financial statistics, I understand epsilon represents the idiosyncratic risk and it tries to capture external factors that influence the return of the asset. How important is this for an academic exercise? Is there an easy way to estimate it?
Posted on 6/14/13 at 12:09 am to rickgrimes
It has been a VERY long time since I worked with any of this, so take this with a grain of salt.
Your analysis of a weighted average beta is basically ignoring the key premise of asset allocation, which is reducing the overall volatility of the portfolio by mixing in investments which are weakly correlated or at least imperfectly correlated.
I remember that there were some equations to deal with this and actually calculate portfolio beta, with an equation similar to yours above but then also a covariance term added for each possible combination of investments in your portfolio.
Sorry I can't be more specific, but this is off the top of my head on a cell phone. Then again, I always thought Markowitz portfolio theory and beta were kind of silly anyway.
Your analysis of a weighted average beta is basically ignoring the key premise of asset allocation, which is reducing the overall volatility of the portfolio by mixing in investments which are weakly correlated or at least imperfectly correlated.
I remember that there were some equations to deal with this and actually calculate portfolio beta, with an equation similar to yours above but then also a covariance term added for each possible combination of investments in your portfolio.
Sorry I can't be more specific, but this is off the top of my head on a cell phone. Then again, I always thought Markowitz portfolio theory and beta were kind of silly anyway.
Posted on 6/14/13 at 12:17 am to Bayou Tiger
quote:
I remember that there were some equations to deal with this and actually calculate portfolio beta, with an equation similar to yours above but then also a covariance term added for each possible combination of investments in your portfolio.
You can calculate Beta using the formula:
Bi = Co-variance(Ri,Rm)/Variance(Rm).
But in my case, the problem with using a statistical model to calculate Beta is that I only have 5 data points for both Return on Asset (Ri) and Return on Market/Index (Rm). My co-efficient of determination (R-squared) when I plot the 5 years of asset return (y-axis) vs 5 years of market return (x-axis) are really low. I see R^2 values of 0.02 and 0.09 for a couple of them. There really is no correlation between the asset return and the underlying market index return.
If I had daily or monthly returns over a few years, I'd feel more confident about fitting a line on that scatter data to find the slope (Beta). But I only have 5 data points for year-end returns for both the asset and the market.
This post was edited on 6/14/13 at 12:33 am
Posted on 6/14/13 at 12:21 am to rickgrimes
*Posted below before I had read your post immediately above (was typing at same time), so some redundancy.
Also, I don't think you can really do a single point beta calculation as the investment return above risk free rate divided by market return above risk free rate.
At least the way I envision it, that is a single point along a scatter of data, with different investment returns on the y-axis versus market returns on the x-axis. Thus it is more of a linear best fit to solve the least squares fit for slope (beta) and intercept (alpha) over a given scattered data set to characterize the investment.
Each point on the graph represents a certain pair of data for one of your investments (market return, investment return) over a specific time interval. So the date range you use for this data set and also the time window over which each return is measured also play a part. Autocorrelation effects can also be interesting.
I am unaware of any industry standards for any of these evaluation parameters.
Also, I don't think you can really do a single point beta calculation as the investment return above risk free rate divided by market return above risk free rate.
At least the way I envision it, that is a single point along a scatter of data, with different investment returns on the y-axis versus market returns on the x-axis. Thus it is more of a linear best fit to solve the least squares fit for slope (beta) and intercept (alpha) over a given scattered data set to characterize the investment.
Each point on the graph represents a certain pair of data for one of your investments (market return, investment return) over a specific time interval. So the date range you use for this data set and also the time window over which each return is measured also play a part. Autocorrelation effects can also be interesting.
I am unaware of any industry standards for any of these evaluation parameters.
This post was edited on 6/14/13 at 12:56 am
Posted on 6/14/13 at 12:27 am to rickgrimes
quote:Well, I am not a finance guy and am quickly stepping out of my expertise.
If I had daily or monthly returns over a few years, I'd feel more confident about fitting a line on that scatter data to find the slope (Beta). But I only have 5 data points.
However, if I had to come up with a beta, my uneducated opinion was to compare the returns of a synthetic portfolio itself over time against the market rather than deal with 88 covariance terms (or whatever). Why bother with all of these terms when I could compare the back-tested covariance / beta directly?
I guess that doesn't really help you with the limited data issue you are facing, but as mentioned before I never really cared much for beta anyway.
This post was edited on 6/14/13 at 12:30 am
Posted on 6/14/13 at 12:29 am to Bayou Tiger
quote:
Bayou Tiger
Thanks for taking the time to reply.
quote:
Also, I don't think you can really do a single point beta calculation as the investment return above risk free rate divided by market return above risk free rate.
My idea was to calculate the Beta using the average of the 5 year return for both the Return on Asset and Return on Market. I thought that it was much better than a single point of reference.
Posted on 6/14/13 at 12:34 am to rickgrimes
Best of luck, since my limited expertise is out of ideas.
Is this for a class problem? If not, I would be way more concerned with the cash flows, trends, strategic placement, etc of the underlying investment (with associated upside/downside). But that is the simplistic view that I usually take.
Is this for a class problem? If not, I would be way more concerned with the cash flows, trends, strategic placement, etc of the underlying investment (with associated upside/downside). But that is the simplistic view that I usually take.
Posted on 6/14/13 at 12:39 am to rickgrimes
The way I remember maybe an elementary calculation and not what you are asking for. However, the calculation is to multiply each asset's beta by its portfolio weight and add up the results.
Posted on 6/14/13 at 12:42 am to Bayou Tiger
quote:
Best of luck, since my limited expertise is out of ideas.
Is this for a class problem? If not, I would be way more concerned with the cash flows, trends, strategic placement, etc of the underlying investment (with associated upside/downside). But that is the simplistic view that I usually take.
Thanks. This is not a class problem. It is something that I am working on on the side during my free time to whet my interest in finance. It is a purely academic venture. I am an engineer by day trying to meddle in finance at night.
Posted on 6/14/13 at 12:45 am to Notro
quote:
The way I remember maybe an elementary calculation and not what you are asking for. However, the calculation is to multiply each asset's beta by its portfolio weight and add up the results.
That is exactly what I am doing for the portfolio data. But my original question was more about the mechanics of calculating those indvidual betas given only 5 data points (for year end returns) for each of the asset returns and market returns.
This post was edited on 6/14/13 at 12:46 am
Posted on 6/14/13 at 12:52 am to rickgrimes
quote:With you being an engineer I don't know how you could have any confidence or significant digits in a beta pulled out of a statistically insignificant sample of highly variable data.
That is exactly what I am doing for the portfolio data. But my original question was more about the mechanics of calculating those indvidual betas given only 5 data points (for year end returns) for each of the asset returns and market returns.
My answer (as an engineer) would be to fall back on some expected norms or analogies of beta from similar types of investments. Then all sorts of arm waving and caveats would be attached to this assumption.
Posted on 6/14/13 at 12:52 am to rickgrimes
quote:
epsilon represents the idiosyncratic risk and it tries to capture external factors that influence the return of the asset.
Academically speaking, yes.
But in English epsilon represents everything not explained by beta. The idea behind basically every linear regression model is to try to reduce variation in epsilon to something that is purely randomly distributed and unexplainable.
I will argue that beta fails to make epsilon notably more random that this equation:
total return = epsilon
Notice the lack of beta in that equation. Most CAPM studies that include beta have an r-squared of roughly 2-3%. That isn't exactly an academic triumph.
That doesn't mean beta has no relevance at all, but it does mean beta by itself isn't a major factor in reducing the error in predicting returns.
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