How CAPM Connects to Mean-Variance Portfolio Optimization

# How are modern portfolio theory (MPT) and CAPM related?


# How are modern portfolio theory (MPT) and CAPM related?












#### 1. Question

- In what sense Capital Asset Pricing Model(CAPM) is related with Modern Portfolio Theory(MPT)?

- Why do we need to check whether the current price of assets is overvalued or undervalued using CAPM when we already have historical price movements of assets, that are all the information needed to come up with the Capital Allocation Line? (We can calculate the expected return, variance, and covariance of an individual asset with historical price movements, and those 3 things are all we need to make a CAL with the highest sharpe ratio)

- Where in steps shown below do I need to use CAPM?

#### 2. My understandings of MPT (Any corrections are welcome):

- Out there in the world, we have thousands of risky assets such as stocks, natural resources and bonds, and one risk-free asset, which is T-bills in usual cases.

- With the assumption that return of all assets follow the normal distribution, we can use 3 information( expected return, variance, covariance with all other assets) to come up with the mean-variance frontier, a group of portfolios with the least risk at a given level of return. The portfolios are comprised of all risky assets. These 3 kinds of information are from the historical price movements of assets.

- There is only one best risky asset portfolio that all the investors are holding, and that is the tangency portfolio. This tangency portfolio is on the mean-variance frontier of risky assets and when it is mixed with risk-free asset, it has the higher sharpe ratio than any other combination of other risky asset portfolio on the efficient frontier and risk-free asset.

- The combination of the tangency portfolio and a risk-free asset can be done with several different weights in each. Since it is a linear combination of tangency portfolio and risk-free asset, this combination can be shown as a line and it is called Capital Allocation Line(CAL).

- Based on an investor's risk aversion, investors choose how much weight of their wealth to invest in the risk-free asset, and the rest in tangency portfolio.

## Answer by MGL (score 8, accepted)

https://quant.stackexchange.com/a/43567

CAPM states that the expected return of any given asset should equal $ER_i=R_f+β_i (R_m-R_f)$, with α being the error term of the previous equation. Now, as α has an expected value of zero, then only way to achieve higher expected returns is taking on more β (given that $E[(R_m-R_f )]>0$). Every individual stock has some idiosyncratic risk in addition to its market β (true always when correlated less than perfectly with the market). Thus, we can get the best return/risk ratio by buying the market portfolio, as buying anything else, we could not get more expected return for the same β, but would only get some additional idiosyncratic risk. Now, if you use historical data to estimate expected returns, you imply nonzero expected α-s for all assets. This is not coherent with the CAPM framework, so using this methodology within MPT, it has nothing to do with CAPM. In effect by using MPT this way, you are generating a momentum based investment strategy, as you assume that assets that have had good returns historically will continue to have good returns in the future. Here is a paper in which a strategy is analyzed that utilizes short-term past historical returns as the expected returns for mean-variance optimization. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2606884

Edit: My initial answer was fairly ambiguous in terms of notation. To clarify my notion of the connection between CAPM, Jensen's Alpha and security characteristic line (SCL) in this context, as discussed in comments below:

We can define the SCL as $R_i = \alpha_i + \beta_i * R_m + \epsilon_i$ (with $R_i$ and $R_m$ being the realized security and market returns in excess of the risk-free rate and $\beta_i$ being the OLS regression beta with $R_i$ being the dependent variable and $R_m$ being the dependent variable).

We can define Jensen's alpha as $\alpha_i = R_i - \beta_i * R_m$ (with the variables defined as above). From here it can be seen that Jensen's alpha equation is just another form of the SCL (with $\alpha_i$ and $R_i$ switching sides, and the equation multiplied by $-1$).

When SCL and Jensen's alpha equations use realized returns, CAPM uses expected returns and can be formulated followingly: $E(R_i) = \beta_i * E(R_m) + \epsilon_i$ (notation similar to the previous equations, but with $E(R_i)$ being the expected excess return of the security and $E(R_m)$ being the expected return of the market portfolio), where $\epsilon_i$ is an error term , and $E(\epsilon_i) = 0$.

Now, when previous realized returns are used as proxies for the expected returns (i.e. $E(R_i) = R_i$ and $E(R_m) = R_m$), when plugged in to the CAPM, we find that it must be the case that $\alpha_i ≡ \epsilon_i$ for all $i$. As the (realized) $\alpha_i$ is a deterministic term and does not necessarily equal zero, we find that it can't be that $\alpha_i ≡ E(\epsilon_i) = 0$ for all $i$. Thus, using realized security returns as proxies for expected returns is not compatible with the CAPM.

Edit2: I figure I did not still actually answer your question very well. Sharpe’s development of the CAPM was originally spurred by the problem his graduate school supervisor Markowitz had with mean-variance optimization. As computers were slow and expensive, it was not feasible to do the calculations for a large number of securities.

Sharpe then first came up with the single-index model (SIM), which is basically what I previously referred to as the security characteristic line (SCL). The reasoning here was that the returns of different securities were related only through common relationships with some basic underlying factor. This being the case, instead of calculating all the pairwise covariances and the resulting portfolio volatilities the volatility of a (well diversified) portfolio (where all idiosyncratic risk is diversified away) could be approximated via securities’ weigthed covariances with the underlying factor (i.e. the market index). This decreased the computing power cost of the operation dramatically.

The SIM was thus used to decompose (“analyst’s”) estimates of expected returns on different securities for a more efficient calculation of the efficient frontier. CAPM followed soon after, when Sharpe concluded that (if alpha’s could not be predicted) the market portfolio itself is the tangency portfolio.

Now, as computing power is cheap today, and you can easily calculate the covariance matrix as well as the portfolio volatilies of a large number of different combinations, the SIM is no longer needed for the analysis.