# Understanding what is 'special' about the security market line

I am trying to get my head around the CAPM model and all the intricacies of portfolio management. I have written some code to help me visualise what happens to the risk-return characteristics of my portfolio as I vary the weightings amongst three stocks (classic bullet shape).

What I don't understand is what is special about the security market line.

$$\bar{r_i}-r_{f} = \beta_{i}(\bar{r_{M}}-r_{f})$$

In short, I already know how to calculate $$\bar{r_i}$$ (by supposing each stock as a random variable with returns following the normal distribution). Sooooo, great, the security market line gives me a new way of calculating $$\bar{r_i}$$ with respect to how it covaries with the market portfolio, why is that special? or any more revealing than simply calculating $$\bar{r_i}$$ as mentioned above?

I hope this makes sence.

• What is surprising is that $\bar{r}_i$ for stock i depends only on one fact about the stock, its Beta $\beta_i$. It does not directly depend on $\sigma_i$ or any other thing about stock i (the nature of the business, who is CEO,...). So this new thing Beta is really at the heart of the CAPM, it is the one thing that drives expected return. (Of course in later theories they found ways to bring in other pricing factors, but in 1964 this was seen as an amazing fact: we can determine lg term expected return in equilibrium on a stock from 1 easily observable measurement about the stock). Apr 11, 2020 at 23:57
• Perfect---that is what I was looking for.
– Andy
Apr 16, 2020 at 12:02

The most special thing about the securites market line (or rather "capital market line" in this context), ML, is that it connects all portfolios with the highest ratio of return to standard deviation in the return-standard deviation space. See the diagram where the ML is denoted "Market Line" The most efficient non-leveraged fully-invested portfolio, "Market Portfolio", MP, lies where the SML touches the set of portfolios (inside the Portfolio Efficient Frontier), to which it forms the tangent.

Another special thing of the ML is that it shows how an investor (with known utility function) should leverage or dilute the MP with funds on the "Risk free rate" (diagram). The investor's most prefered exposure (Optimal Portfolio) to the MP lies where the SML forms the tangent to the investor's indifference curve.

CAPM is the graphical representation of the security market line. What is special about it is that it can tell you where you expect the return for an asset to be given its level of systematic risk - beta. CAPM in a way measured the price the market would expect for that level of volitility in an asset. For instance a stock is supposed to be a good investment if it has an abnormal return- that is an actual return much higher that the CAPM expected return. That way the asset is actually producing wealth above what the market says it should. CAPM is a good comparison of stock returns and their value when stocks have no dividends and hence the Gordon growth model and others are not helpful in estimating price /worth of the said asset.

Regarding portfolio weighting, to weight a portfolio requires weights and a return like you mentioned. CAPM is good as the higher the expected return of one asset, the higher the maximum portfolio return can be. It gives an estimate of what the market would say a good weighting would be for a number of stocks.

I hope this clarifies some points for you

• Ok that makes some sense. What CAPM tells us is that looking at how the expected return relates to its own risk (standard deviation) is irrelevant...What matters in determining the expected return is how it covaries with the market. You could have a stock with high std but low expected return which only makes sense because that particular stock has a low beta...In short, beta is a new (more revealing) measure of risk.
– Andy
Apr 12, 2020 at 11:02

In CAPM (security market line), the returns are endogenous. It is the result of the equilibrium of the entire market according to the mean-variance criterion, and is not determined by the so-called risk (variance, beta, or covariance). The beta value is calculated from the equilibrium return, using beta value to explain the expected return is a circular argument. In addition, calculating the beta value and then solving the expected return by CAPM is not as easy as taking the expectation of return directly.

In the framework of CAPM equilibrium, risky securities are priced as a whole, and the security returns and the market return are endogenous. Examining the return of individual security in isolation leads to the wrong causal inference that beta determines the expected return.

For the closed-form solution of CAPM formula, see CAPM: Absolute Pricing, or Relative Pricing? or Arbitrage Opportunity, Impossible Frontier, and Logical Circularity in CAPM Equilibrium

So you have your bullet-shaped “efficient frontier”. There is no portfolio offering a better return for lower vol, assuming - of course - that your return assumptions and covariance matrix have perfect foresight?

The textbooks habitually hint this; but this is not so. The hint is based on the supplementary assumption of zero cash, zero leverage, and no shorting. Break these assumptions: and you can do better. This is why people fuss about the “securities market line”.

So imagine interest rates lie on a spectrum at zero risk, the y-axis of your bullet chart. As rates change, the maximum slope of that line to any point on your efficient frontier will change.

The point at which this steepest slope shaves your efficient frontier is called the “tangency portfolio”.

So draw that line. Any portfolio mixing tangency and cash will have a better risk-reward than a safer portfolio of your assets (moving down your bullet). And a portfolio levering up tangency will also have better risk-reward than one swapping safe assets for riskier ones (moving up your bullet).

This why people care about the SML. Scrap the nil cash/short/leverage assumption, and there becomes only one efficient point on your efficient frontier.

In my personal opinion, all of CAPM is highly dubious. But I hope above conveys an intuitive answer to your question.

Very best, DEM