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I asked this question on Personal Finance and Money but since I don't know where to place it I placed it here also.

On the Coursera course Portfolio and Risk Management, on Week 2, I am having trouble finding the link between the following 2 formulas from the following 2 slides:

From the slides above my conclusion would be the following equation:

$$ \beta_i = \frac{cov(R_i,R_M)}{var(R_M)} = \frac{\sigma_i}{\sigma_M} $$

So I have the following doubts:

  1. Is this right?
  2. How do you get from one expression to the other?
  3. In Formula 2, the expression for $\beta$ includes a relation between the asset and the market. But in Formula 1 the relation $\frac{\sigma_i}{\sigma_M}$ suggest there is no relation between the market and the asset.
    Why is this?
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    $\begingroup$ You are mixing up two different concepts. The red line shown on the left (the CML) is a different line than the one discussed on the right (the CAPM equation). In particular the slope of the red line is not Beta and has nothing to do with Beta. $\endgroup$
    – nbbo2
    Jun 18, 2021 at 2:15
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    $\begingroup$ Afaik, formula 1 is not looking at single assets, but on portfolios, hence $\sigma_i$ is the vola of a portfolio, not of an asset. The background is that assuming all investors are maximizing utility, everyone will hold the same two assets (i.e. the risk free asset and the market portfolio), just a different combination of it. All possible combinations are the CML. $\endgroup$
    – simzoor
    Jun 18, 2021 at 6:15
  • $\begingroup$ In all brevety, the market portfolio with weights $w_M$ is the portfolio that is tangent to the CML - See your first diagram. The CAPM now tells you what kind of excess return you may expect, in equilibrium, from holding an asset as a function of that asset's covariance with the market portfolio, i.e. $cov(R_i,R_M)= w_i^T \Sigma w_M$, with $w_i$ a vector that is all zeros expect for the $i$th entry, where it is one. $\endgroup$ Jun 18, 2021 at 6:15

2 Answers 2

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For short:

  • 1 ) Is this right? Yes
  • 2 ) How do you get from one expression to the other? You did it already
  • 3 ) In Formula 2, the expression for β includes a relation between the asset and the market. But in Formula 1 the relation suggest there is no relation between the market and the asset. Why? It suggests a direct relation actually.

Long answer:

1 ) First start with what the capital market line is. It describes the capital allocation (line) that is constructed from a risk free rate (usually described to be T-bills) and the market portfolio (the blue dot called market in the figure on the left hand side). The market portfolio in turn is the optimal tangency portfolio on the efficient frontier (green line). In the aggregate, lending and borrowing cancels and the value of the aggregate risky portfolio equals the entire wealth of the economy (the market portfolio). In other words, the market value (price per share times number of shares) of each stock divided by sum of market values of all stocks equals the proportion held of each stock.

If you denote A as the investor's risk aversion, you can write the proportion $y$ allocated to the optimal portfolio $M$ as

$$y = \frac{(E[R_M] - R_F)}{A\sigma_M}$$

Since net borrowing is zero, the average position in the risky portfolio is $100%$, hence $y=1$, we can solve for $$(E[R_M] - R_F) = \hat{A}\sigma_M$$ where $\hat{A}$ is the average degree of risk aversion.

Ignoring the ridiculous outcome that this is equal for everyone, as the CAPM implies that all individuals arrive at the same portfolio, you will see that it must be related to individual stocks somehow, otherwise you would not end up with identical portfolios.

2 ) and 3 ) How does it relate?

  • The risk premium of the market portfolio $$(E[R_M] - R_F) = \hat{A}\sigma_M$$ and
  • The risk premium on individual securities is $$(E[R_i] - R_F) = \beta_i(E[R_M] - R_F)$$

If this holds for any individual asset, it must hold for any market portfolio. Hence, $$(E[R_M] - R_F) = \beta_M(E[R_M] - R_F)$$ This is a tautology because $\beta_M = 1$ since $$\beta_M = \frac{Cov(R_M,R_M)}{\sigma^2_M} = \frac{\sigma^2_M}{\sigma^2_M} $$

Now, you can clearly see that $$\frac{\sigma^2_i}{\sigma^2_M} = \frac{\sigma_i}{\sigma_M}$$ which is your result (if variance or standard deviation is used makes little difference). The expression $$\frac{(E[R_M] - R_F)}{\sigma_M} = \frac{\Delta y}{\Delta x}$$ used in the course is simply the slope as explained in the video.

This is generally called the expected return-beta relationship in the literature. In general, if the covariance between $asset_i$ and the rest of the market is negative, then the asset makes a negative contribution to portfolio risk (or positive if positive). The contribution of one stock to portfolio variance is expressed as the sum of all covariance terms in the variance covariance matrix (I omitted the weights, each column and row corresponds to weight $w_1, ... , w_i, ..., w_n$). $$\begin{bmatrix}Cov(r_1,r_1) & Cov(r_1,r_2) & ... & Cov(r_1, r_i) & ... & Cov(r_1,r_n)\\Cov(r_2,r_1) & Cov(r_2,r_2) & ... & Cov(r_2, r_i) & ... & Cov(r_2,r_n)\\. & . & ... & . & ... & .\\. & . & ... & . & ... & .\\. & . & ... & . & ... & .\\Cov(r_i,r_1) & Cov(r_i,r_2) & ... & Cov(r_i, r_i) & ... & Cov(r_i,r_n)\\. & . & ... & . & ... & .\\. & . & ... & . & ... & .\\. & . & ... & . & ... & .\\Cov(r_n,r_1) & Cov(r_n,r_2) & ... & Cov(r_n, r_i) & ... & Cov(r_n,r_n) \end{bmatrix}$$

The diagional entries are the covariance of one security's return with itself, which is simply the variance of that security. E.g. $$Cov(r_i, r_i) = \sigma^2_i$$

The contribution of one $asset_i$ to total portfolio variance is the sum of all the covariance terms in the column corresponding to $asset_i$, where each covariance is multiplied by the weight from it's row and column. The rate of return for the market portfolio is $$R_M = \sum_{i=1}^n w_iR_i$$

It follows that the covariance of the rate of return of $asset_i$ with the market portfolio is $$Cov(R_i,R_M) = Cov(R_i, \sum_{i=1}^n w_iR_i) $$ and the contribution of holding $asset_i$ to the risk premium of the market portfolio is $$w_i[E(R_i)-R_F)]$$ In other words, the reward to risk ratio of $asset_i$ can be written as $$\frac{asset_i\ 's \ contribution\ to\ risk\ premium}{asset_i\ 's \ contribution\ to\ variance} = \frac{E(R_i)-R_F}{Cov(R_i,R_M)} $$

The market portfolio $M$ is the tangency (efficient frontier) portfolio with reward to risk ratio $$\frac{market\ risk \ premium}{market\ variance} = \frac{E(R_M)-R_F}{\sigma^2_M} $$ This is often called market price of risk. However, there is some ambiguity as this term is frequently used for the reward to volatility ratio as well $$\frac{E(R_M)-R_F}{\sigma_M} $$

In equilibrium, all investments should offer the same reward to risk ratio. There is a common misconception here. Well managed firms will produce high returns (as measured by return on plant and equipment or human capital). However, here, the notion of investment (in securities) return is used. Security prices should already reflect the information (depending on what efficient market hypothesis you prescribe to, public, or all information - also insider's) about the firm's prospects. Therefore, the stock price is already bid up for this exceptionally well lead firms, and returns for stockholder's will not be excessive.

This implies that the reward to risk ratio of $asset_i$ and the market portfolio should be equal: $$\frac{E(R_i)-R_F}{Cov(R_i,R_M)} = \frac{E(R_M)-R_F}{\sigma_M^2}$$

Rearranged, this yields $$E[R_i]-R_F = \frac{Cov(R_i,R_M)}{\sigma^2_M}[E(R_M)-R_F]$$ where $\frac{Cov(R_i,R_M)}{\sigma^2_M}$ corresponds to the contribution of $asset_i$ to the variance of the market portfolio as a fraction of total variance. This ratio is actually $\beta$ which allows us to rewrite the previous formula as . $$E[R_i] = R_F + \beta_i[E(R_M)-R_F]$$

If you are interested in a more comprehensive explanation, you can have a look at chapter 9 of Investments by Zvi Bodie et. al (P.290-P.298).

Having written that, I would recommend not to waste too much time with these ideas. They are intellectually interesting - they fit the conventional (neoclassical) mainstream economics, but are terrible trading strategies. This is not just my claim but something many experts like Graham Giller assert.

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  1. Is this right?

No. $\beta_{i,M} = \frac{\sigma_i \rho_{i,M}}{\sigma_M}$ (note the correlation $\rho_{i,M}$ that is missing from your formula.)

The first formula (CML) is only for portfolios on the CML, namely those portfolios mean-variance investors would actually hold (when there is a risk-free asset): portfolios that put some proportion in the risk-free asset, and some proportion into the market portfolio.

The second formula (CAPM) holds for any portfolio, and in particular also for portfolios that consist of only one asset. As such, it gives you the expected return for that asset (and therefore, if you know the expected payoff, its price: thus Capital Asset Pricing Model).

  1. How do you get from one expression to the other?

The first expression only holds for some portfolios. The second expression holds for all portfolios, and it also holds for portfolios on the CML. But those all have correlation 1 with the market portfolio - and therefore first and second formula coincide for those special portfolios.

  1. In Formula 2, the expression for 𝛽 includes a relation between the asset and the market. But in Formula 1 the relation suggest there is no relation between the market and the asset. Why is this?

In the CAPM formula, $\beta_{i,M}$ is the only thing (on the RHS) that depends on the specific portfolio or asset $i$. That's the point: the beta of an asset completely determines the expected return of that asset. Nothing else.

Formula 1, the CML formula, is not about specific assets. It is only about optimal portfolios one would hold (namely a mix of risk-free and the market portfolio). For those, the expected return is a linear function of its risk (standard deviation), which is in turn a linear function of its beta (because the correlation with the market is constant 1).

TLDR: For all portfolios (including individual assets), formula 2 (CAPM) holds: the expected return of a portfolio (or asset) is a function of the beta (of that portfolio with respect to the market)

For some portfolios (the optimal ones), the expected return is a function of the beta, also, but correlation to the market portfolio is constant = 1, and thus you can simplify and say that the expected return is a function of the risk (standard deviation). That's the CML.

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