 # PM: beta and standard deviation

Source: Reading 60 - Schweser pg. 175 (top of page)

Question: Can standalone risk be represented by both beta and standard deviation for a well-diversified portfolio? My initial thoughts: We know that beta is represented by the independent variables in the SML - and since beta includes the market portfolio, it can be said that beta reflects systematic risk. Standalone risk, in contrast, has inclusion of both systematic and unsystematic risk - however, this is only true for a concetrated portfolio. For a well-diversified portfolio, standalone risk is only systematic risk because unsystematic risk has been eliminated. So, this means - for a well-diversified portfolio, standalone risk (represented by standard deviation) equals systematic risk - BUT systematic risk is also represented by beta, which leads us to conclude that standalone risk can be both represented by standard deviation and beta. Is this right?

Standard deviation of a security is its total risk. Beta is its risk relative to the market.

I’m not sure to understand what ‘standalone’ mean in this case. If this means something like "financial product risk’, then I would say that beta represents only the systematic part of the whole standalone risk.

We know that:

beta = systematic risk

standalone risk = standard deviation = systematic risk + unsystematic risk

If we diversify away the ‘unsystematic risk’ in the above equation we are only left with systematic risk which leads us to conclude:

beta = standalone risk = standard deviation

What am I missing.

What you call standalone risk is total risk = systematic + unsystematic.

If you form a large portfolio of risky assets, the total risk of the portfolio = systematic risk.

Beta of portfolio does not equal standard deviation of portfolio:

Beta_p = r_p, market * (stdDev_p/stdDev_m). If the correlation between the portfolio and the market is +1.0, then:

Beta_p = stdDev_p/stdDev_m

If your portfolio and the market have the same stdDev, then beta_p = 1.0, and its expected return is same as the market’s.

If your portfolio is more risky than the market, its beta > 1.0, and you can easily calculate its expected return.

So, even though we can diversify away the ‘unsystematic risk’ in standalone risk leaving us with:

standalone risk = systematic risk

this does NOT mean standalone risk is now measured by beta - it STILL is measured by standard deviation.

Given that’s the case, we can safely state the following:

Systematic risk is always measured by beta

Standalone risk (even when unsystematic risk is diversified away) is always measured by standard deviation

Is this right.

another way of stating this.

total risk = systematic risk + unsystematic risk

= market risk (beta risk) + specific risk

adding a large # of stocks will cause diverisifcation to the extent of being able to eliminate only SPECIFIC RISK. (or unsystematic risk). Market risk is still left behind, even after all the diversification that you do.

yes - systematic risk is measured by beta

total risk is measured by std deviation.

Thanks for this discussion - it is so clarifying

Just want to say I think the Beta for P is not stdDev_p/stdDev_m. It should be the correlation between the portfolio and the market. Like if actually want Beta in real life we would regress the profolio’s return with the market to find the R^2.

So it’s COV(Portfolio, Market)/Variance(Market). Correct me if I am wrong.

Kys916, above I said:

• Beta_p = r_p, market * (stdDev_p/stdDev_m). If the correlation between the portfolio and the market is +1.0, then:

• Beta_p = stdDev_p/stdDev_m

That’s the same as saying Beta_p = COV(Portfolio, Market)/Variance(Market)

Remember that COV(P,M) = Corr(P,M) * stdDev_p * stdDev_m.

Sorry I missed the then part.