Estimating Hansen–Jagannathan Distance from Asset Returns

# Hansen and Jagannathan distance


# Hansen and Jagannathan distance












Hansen and Jagannathan distance, or HJ-distance for time-series regression of excess test assets return on excess factor return reads:

$HJ = \sqrt{\alpha'(E[RR']^{-1})\alpha}$

However, I am little bit confused whether the alpha here is the intercept term or the residual? Second, how do I calculate the middle term because it is involved with the expectation ?

More detail: For the equation above, you can refer to the document (equation 4, page 18) https://studunifrankfurtde-my.sharepoint.com/:b:/g/personal/s1358868_stud_uni-frankfurt_de/ESpAu6KjycFAkfB65J9z-v8B_Yb--bptY4baTrOoRtoeJQ?e=cKz1pJ

Besides, you can read equation 12 on page 7 of this paper: https://studunifrankfurtde-my.sharepoint.com/personal/s1358868_stud_uni-frankfurt_de/_layouts/15/onedrive.aspx?id=%2Fpersonal%2Fs1358868_stud_uni-frankfurt_de%2FDocuments%2FTailieu%2FMaster Thesis%2FMain Focus%2FEvaluatingtheSpecificationErrors.pdf&parent=%2Fpersonal%2Fs1358868_stud_uni-frankfurt_de%2FDocuments%2FTailieu%2FMaster Thesis%2FMain Focus&ga=1

@phdstudent: When you mean the covariance matrix $\Sigma = E[RR']$. Is this correct for the explicit form:

$E[RR'] = \frac{(R-\bar{R})*(R-\bar{R})'}{T-1}$ where $\bar{R}$ is average return

Btw, should $R$ in this case is return or excess return ?

## Answer by phdstudent (score 6, accepted)

https://quant.stackexchange.com/a/39562

It would be easier to answer if you tell us where that equation came from (there are many ways of deriving the HJ distance) - in any case the numerator of your equation should be the expected return on the efficient portfolio and the denominator the expected variance/covariance.

Let me give you a the same equation using simpler notation (and derive it!). From the law of one price:

\begin{equation} 1 = E [R_{i,t+1} m^\star_{t+1}] \end{equation}

Therefore:

\begin{equation} 1 = E(R_{i,t+1}) E(m^\star_{t+1}) + Corr(R_{i,t+1}, m^\star_{t+1}) Std(R_{i,t+1}) Std(m^\star_{t+1}) \end{equation}

Rewrite the equation above using $R_{f,t+1} = 1/E(m^\star_{t+1})$ to get:

\begin{equation} \frac{E(R_{i,t+1}) - R_{f,t+1}}{Std(R_{i,t+1}) } \leq \frac{Std(m^\star_{t+1})}{E(m^\star_{t+1})} \end{equation}

The left hand side equation is the equivalent to your equation (the maximum attainable sharpe ratio). And the equation on the right gives you the bound so: The max Sharpe ratio in the economy is then bounded by the minimum variance SDF volatility over mean!

How do we use these?

- Take $N$ assets. Compute excess returns.

- Estimate variance covariance matrix of returns $\Sigma = E[R R']$ and average payoffs $E(R_{t+1})$. Usually the first one we estimate by taking a large sample and computing covariance matrix and the latter just by averaging returns.

- Plot the above locus and compare with your candidate SDF;

The locus should deliver something like this:

Edit: After some clarifications above.

I now see where your derivation comes from (see equation 10 from Hodrick and Zhang), which is not about the bound itself but the distance. Basically your equation comes form solving the following problem:

\begin{equation} \min_m E[(y_{t+1}-m_{t+1})^2] + 2\lambda (E[m_{t+1}R_{t+1}]-1) \end{equation}

where the first term is the JG distance and the second term the constraint.

Take f.o.c. with respect to $m$ for every asset and you get: \begin{equation} E[(y_{t+1}-m^\star_{t+1})^2] = [1-E(R_{t+1}y_{t+1})]'[E(R_{t+1}R'_{t+1}][1-E(R_{t+1}y_{t+1})] \end{equation}

So indeed your $\alpha$ are the time-series intercepts of the time-series regression and the denominator is just a matrix with expected values of products of returns for all assets (just like a variance-covariance matrix when means are close to zero).