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 ...
- 6,924
14
votes
Accepted
Estimate Beta of CAPM from Implied Volatility?
Yes it is a better way.
Just take a look to figure 3, from Buss and Vilkov (2012, RFS):
- 8,062
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)...
- 6,924
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 ...
- 15.3k
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-...
- 6,924
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 ...
- 10.9k
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 ...
- 15.3k
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 ...
- 10.9k
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.
...
- 5,891
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 ...
- 6,924
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 ...
- 702
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 ...
- 6,924
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 ...
- 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, ...
- 6,445
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 ...
- 21k
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, ...
- 15.3k
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 ...
- 6,445
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 ...
- 2,906
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 ...
- 2,638
5
votes
Fundamental CAPM questions
Because you have CAPM therefore the following holds:
$$r_i = r_f + \beta_i (r_M - r_f) + \epsilon_i$$
where $r_i$ is the expected return of stock $i$, $r_f$ is the risk free return and $r_M$ is ...
- 307
5
votes
Accepted
Why are investors risk-averse?
Below you find some observations...
In CAPM, we assume people are risk-averse and people get compensated
for the systematic risk they suffer.
The assumption that most people are risk-averse ...
- 363
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 ...
- 13.6k
5
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 ...
- 702
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^...
- 6,924
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 ...
- 6,924
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, $...
- 6,924
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 ...
- 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 ...
- 1,396
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 ...
- 6,445
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 ...
- 8,062
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