Skip to main content
16 votes
Accepted

Philosophical Question about Factor Models

This is actually a rather involved question, and different interpretations exist. A narrow, linear algebra based interpretation is that the stochastic discount factor lies in the linear span of the ...
Matthew Gunn's user avatar
  • 6,954
14 votes
Accepted

Calculating alpha and its meaning

Alphas from a time-series regression are error terms in the cross-sectional, linear relationship between expected returns and factor betas. If a factor model were correct those error terms (the alphas)...
Matthew Gunn's user avatar
  • 6,954
13 votes
Accepted

Is market price of risk always negative?

Summary Yes, the market price derived from the moments of the SDF is negative. Assets pay high returns (are risky) if they negatively correlate with the SDF (and thus positively correlate with ...
Kevin's user avatar
  • 16k
12 votes
Accepted

Fama-French factor model: why mimicking portfolios?

The innovation of Fama and French's Three Factor Model wasn't in finding book to market ratios forecast returns but in reconciling that empirical regularity with the standard framework of macro-...
Matthew Gunn's user avatar
  • 6,954
10 votes
Accepted

Excess, Residual and Active Return

Active return: $R-R_m$ i.e. your security (or portfolio) compared to the market portfolio. Used to judge performance before the CAPM was invented Excess return: $R-R_f$ the security compared to the ...
nbbo2's user avatar
  • 11.4k
9 votes

Definitions of Beta

I slightly disagree with Alex’s comment. The CAPM does not read as \begin{align*} r_{i,t} = r_{f,t}+ \beta_{i} (r_{m,t}-r_{f,t}) + \varepsilon_{i,t}. \end{align*} There is an important difference ...
Kevin's user avatar
  • 16k
8 votes
Accepted

Difference between CAPM and single index model

As you know the equation that describes them is the same. The single index model is an empirical description of stock returns. You do some regressions using data and you come up with Alphas, Betas ...
nbbo2's user avatar
  • 11.4k
8 votes
Accepted

How much capital to allocate between two trading strategies given average daily P&L and their Sharpe Ratios?

To be consistent with the average daily returns that you specified, your first strategy would need to have a daily standard deviation of 31,749 USD and the second a standard deviation of 7,937 USD. ...
Chris Taylor's user avatar
  • 5,931
8 votes
Accepted

How high can Beta be in CAPM?

Infinity is rather non-sensical. A better question perhaps is whether you can put some theoretical bounds on an asset's market beta. An asset's volatility bounds its market beta Let $R_i$ be the ...
Matthew Gunn's user avatar
  • 6,954
8 votes

Is CAPM a cross sectional or time series model?

To answer your question directly: CAPM is a cross-sectional model, and is NOT a time series model. CAPM aims at explaining variance of single asset's return by overall market return of the same ...
Will Gu's user avatar
  • 712
8 votes
Accepted

CAPM model as a regression

If you really believed the CAPM's prediction that $\alpha=0$, then imposing $\alpha=0$ in your estimation would indeed lead to your 2nd formula. The problems? The CAPM doesn't work so imposing a ...
Matthew Gunn's user avatar
  • 6,954
8 votes
Accepted

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

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 ...
MGL's user avatar
  • 516
8 votes
Accepted

Inverse Covariance Matrix Transformation from CAPM

This is the result of the Sherman-Morrison inversion for the sum of an invertible matrix and an outer product. You will find this (and many other helpful methods) in the Matrix Cookbook. Specifically, ...
Kermittfrog's user avatar
  • 6,663
7 votes
Accepted

Mathematical Derivation of Residual Risk

Note that $\beta$ is the coefficient of the portfolio regressed on the benchmark. That is \begin{align*} r_P = \alpha+\beta r_B + \varepsilon, \end{align*} where $\varepsilon$ is the residual. The ...
Gordon's user avatar
  • 21.1k
7 votes

What mechanisms does the market use to brining an asset back to the market line, as defined by CAPM?

Supply and Demand That's a great question and your question has a simple answer. @Daneel showed you the ``maths'' behind it. It boils down to: if an asset has too high returns than the CAPM predicts, ...
Kevin's user avatar
  • 16k
7 votes
Accepted

utility function and CAPM in portfolio theory

Please have a look at this image, which I have copied from here: Here, the point M is the tangency portfolio of the capital market line. As you can see, the investor A (left hand side) can attain ...
Kermittfrog's user avatar
  • 6,663
6 votes

How to use factor models for prediction?

You kinda mentioned in your questions, but the predictive model is essentially a lagged version of the "factor model". Part of the problem comes from the subscripts, the model itself doesn't really ...
Will Gu's user avatar
  • 712
6 votes
Accepted

CAPM - Expected vs. actual returns

Based on your comments on other answers, i would like to provide you a summary on the difference of the CAPM-Alpha and Jensen's-Alpha. CAPM The CAPM is an economic model for asset pricing. It states ...
skoestlmeier's user avatar
  • 2,916
6 votes
Accepted

CAPM yields very poor fit (low R-squared). Is that normal?

Question 1 There may be at least two reasons for this (aside from possible programming errors or poor data): The model is actually a very poor approximation of reality, as Matthew Gunn indicates in ...
Richard Hardy's user avatar
5 votes
Accepted

how can we know the residual return will be uncorrelated with the market return

Let us ignore the riskless rate for simplicity of the presentation. If you have (historical or simulated) return series $r_i$ for the portfolio and $r_i^M$ for the market, then the beta is the OLS ...
Richi Wa's user avatar
  • 13.7k
5 votes
Accepted

What are the assumptions in the first-stage of Fama-MacBeth (1973)?

The CAPM is an economic theory that expected returns in excess of the risk free rate should be linear in the regression beta on the market. $$ \operatorname{E}[R_i - R^f] = \beta_i \operatorname{E}[R^...
Matthew Gunn's user avatar
  • 6,954
5 votes
Accepted

Finding a minimum variance portfolio when using a regulariser?

You're not going to get an analytic formula except in special cases of function $\rho(x)$. And you're probably going to want $\rho$ convex. If $\rho$ is convex, the problem is a convex optimization ...
Matthew Gunn's user avatar
  • 6,954
5 votes

In the "betting against beta" paper, what exactly is the "BAB factor"?

An excess return is the payoff of a zero cost portfolio. For example: $R_i - R_f$ is an excess return. $c \left( R_i - R_f \right) $ is an excess return for any $c \in \mathbb{R}$,. More generally, $...
Matthew Gunn's user avatar
  • 6,954
5 votes
Accepted

Calculating beta to market

In a word, yes. That's a correct and valid view to take but, as you'll always find in finance, it really depends on context and the question that you're trying to answer. This is the case in markets ...
RobAbMo's user avatar
  • 131
5 votes

CAPM and Beta: problem with regression (Beta is too low yet statistically significant?)

You are right to be sceptical of the beta of an international portfolio when it is calculated using daily returns. Beta estimates are often low for international portfolios because stock market ...
Tim Wilding's user avatar
  • 1,406
5 votes

Covariance Between Two Frontier Portfolios

Let $\Sigma$ denote the covariance matrix of our asset universe, $\mu$ is the vector of expected returns. Further, $\mathbb{1}$ is a vector of ones. Let's identify the vector of the minimum variance ...
Kermittfrog's user avatar
  • 6,663
5 votes

Incorporating idiosyncratic risk as a pricing factor Fama-MacBeth style

By definition idiosyncratic volatility needs to be computed against a candidate asset pricing model. See for example this paper. So my suggestion is: Run your favorite asset pricing model (e.g. the ...
phdstudent's user avatar
  • 8,381
5 votes

Question about adding new investment A to portfolio B

The proposition is intuitive but the proof of this is not so straight forward in my opinion. The paper Benhamou & Guez (2021), Computation of the marginal contribution of Sharpe ratio and other ...
oronimbus's user avatar
  • 1,896
5 votes

What is the mathematical difference between Mean-Variance Optimization and CAPM?

The CAPM is an asset pricing model, while mean-variance optimization is a type of optimization. These are objects from two different categories. When you characterize the CAPM as an optimization ...
Richard Hardy's user avatar
4 votes

Correlation -1 and standard deviation

$\sigma_p=\sqrt{\omega_a^2 \sigma_a^2+(1-\omega_a)^2 \sigma_b^2+2 \omega_a (1-\omega_a) \rho_{ab} \sigma_a \sigma_b}$ with $\rho_{ab}=-1$ the term under the square root simplifies to $(\omega_a \...
Chris Degnen's user avatar

Only top scored, non community-wiki answers of a minimum length are eligible