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Note I have added much more detail to this question I have decided to add the detail without altering the original text since a number of those of you offering assistance asked for clarification. Please scroll down to the bolded section below for a more complete problem description.

With relatively few assumptions (stationary excess returns), and arbitrarily small (but positive) predictive power for an algorithmic trading signal, I can construct a series of strategies with excess returns $S_n$ with arbitrarily large Sharpe Ratio, i.e., $$\frac{E[S_n]}{\sigma[S_n]}\rightarrow\infty$$ and at the same time, $Prob\{|S_n|>0\}\rightarrow 0$ (i.e., the probability of trading goes to zero).

This would seem to contradict Hansen-Jagannathan bounds, which show that for all priced assets in the economy with excess returns $R_e$, and pricing kernel $M$, we would necessarily have $$0=E[M R_e]$$ and this gives us the bound: $$\frac{|E[R_e]|}{\sigma[R_e]} \le \frac{\sigma[M]}{E[M]}$$ i.e., the Sharpe ratio of assets in our economy is bounded by the inverse of the Sharpe of the pricing kernel, otherwise known as a good deal bound.

In other words, near arbitrages would exist, although the probability of getting any return at all is negligible.

What are the consequences of this?

Would this then be a pricing anomaly?

New Material

Pricing Anomaly for Stationary Returns with non-zero ACF?

We consider a dynamic (e.g., momentum or mean-reversion) strategy on a stationary time-series of excess returns, $R_t$ which has mean zero and fixed autocorrelation function (ACF) and is driven by Gaussian innovations (i.e., $R_t\sim \mathscr{N}(0,\sigma_R^2)$ and $E[R_tR_s]=C(t-s)$ with $C(k)\ne 0$ for some $k\geq 1$). We have here assumed that at least one of the elements of the ACF is non-zero.

We have a signal $X_t$ which is unknown but also has no prevision. So for instance at time $t$ we could have it be an exponentially weighted moving average, $X_t=\sum_1^\infty \lambda^k R_{t-k}$ or let it be the forecast from an ARIMA model among other things. Irrespective, the signal $X$ and the returns $R$ are assumed to be jointly normal. (i.e., $X_t\sim \mathscr{N}(0,\sigma_X^2)$ and $E[X_t R_t]=\rho$)

For those who are concerned with notion of prevision: we will assume that at time $t$, $X_t$ is fully known whereas $R_t$ is not seen. Consequently, the moments of the conditional returns of the strategy $E_t[(X_t R_t)^k]=X_t^k E[R_t^k]$ are merely moments of the conditionally normal distribution $R_t$. We are interested in looking at the unconditional one-period moments of the strategy, e.g., $E[(X_t R_t)^k]$ which can be characterised by Isserlis' or Wick's theorem, and have properties like positive skewness and excess kurtosis, as is seen commonly in CTA returns. Moreover the distribution of $X_tR_t$ is fully known.

As I mentioned, we are interested in the unconditional properties of the strategy $X_tR_t$ which are easy to characterise due to the joint normality of $X$ and $R$ in terms of the correlation $\rho$:

$$ SR = \frac{\rho}{\sqrt{\rho^2+1}}$$ and Skewness is $$ \gamma_3=\frac{2\rho(3+\rho^{2})}{(1+\rho^{2})^{\frac{3}{2}}}, $$ and kurtosis is $$ \gamma_4 = \frac{3(3+14\rho^2+3\rho^4)}{(1+\rho^2)^2} $$

An immediate consequence is that in order to maximize the Sharpe ratio, we should not be concerned with prediction (i.e., minimizing the MSE), but rather should consider maximizing the correlation between our signal and returns. Consequently, OLS is not the optimal framework, (i.e., Yule-Walker equations aren't quite right) and rather orthogonal or total least squares (TLS) is more appropriate. TLS is derived from PCA.

Moreover it is possible to determine standard errors on Sharpe ratios for dynamic strategies which are a refinement of those of Lo (Lo, Andrew W. The statistics of Sharpe ratios. Financial analysts journal 58.4 (2002): 36-52), although for all $|\rho|<0.5$ they are reasonably approximated using adjusted stderr calculations of Mertens (E. Mertens, Comments on variance of the IID estimator in Lo, Research Note, 2002, http://www.elmarmertens.com/research/discussion\#TOC-Correct-variance-for-estimated-Sharpe-Ratios).

Nonlinear Transforms Our study takes us tononlinear transforms of the signal $X$, which means that we will scale each trades by $f(X)$ rather than $X$ itself. In our paper, we consider optimal transformations (as in those which maximize the Sharpe ratio among all smooth transforms, etc). Among smooth nonlinear transformations $f()$ of our signal $X$, it can be shown that the optimal (Sharpe ratio maximizing) smooth transformation is given by $$f^*(x)= \frac{E[R|X=x]}{E[R^2|X=x]}$$ and in the case of jointly normal signal and returns, this reduces to $$f^*(x)= \frac{\beta X}{\beta^2 X^2 +\sigma^2_{R|X}}$$ where $R=\beta X+\eta$ with $\eta\sim \mathcal{N}(0,\sigma_{R|X}^2)$.

The optimal Sharpe ratio obtained by a smooth transformation can be found as $$SR^*=\frac{A}{(1-A)^{1/2}}$$ where $$A=E_x\bigl[\frac{E[R|X=x]^2}{E[R^2|X=x]}\bigr].$$ In the case of jointly Gaussian signal and returns, this can be found in closed form (FWIW).

So much for the background (which is covered in an upcoming paper, Firoozye, N and Koshiyama, A, Optimal Dynamic Strategies on Guassian Returns, in preparation).

Do let me know if you are interested in the paper, since I can post a link here once we've tidied it up a little more.

If we instead restrict our attention to nonlinear transforms which are simple indicator functions, we let $$f_\epsilon(x)=1_{x>\epsilon}-1_{x<-\epsilon}$$ (i.e., we trade based on a threshold and are long or short one unit of the underlying) for $\epsilon>0$ then the Sharpe ratio for strategy $S_\epsilon=f_\epsilon(X)R$ can be calculated as $$SR[f(X)R]=\frac{\sqrt{2} \rho M(\lambda)}{\sqrt{1+\rho^2 M(\lambda)(\lambda -2M(\lambda))}}$$ where $\lambda =\epsilon/\sigma_X$ and $M(\lambda)$ is the inverse Mills ratio or $$M(\lambda)=\frac{\phi(\lambda)}{1-\Phi(\lambda)}$$ for $\phi$ and $\Phi$ being the Gaussian PDF and CDF respectively. While not obvious, we can show numerically that this Sharpe ratio becomes unbounded for $\epsilon$ large enough. Note I have not proved that $SR\rightarrow \infty$ but it appears that for $\lambda$ large enough it always becomes unbounded.

Consequently, with relatively few assumptions (stationary excess returns), and arbitrarily small (but positive) predictive power for an algorithmic trading signal, we can construct a series of strategies with excess returns $S_\epsilon$, with arbitrarily large Sharpe Ratio, i.e., $$\frac{E[S_\epsilon]}{\sigma[S_\epsilon]}\rightarrow\infty$$ but at the same time, we note that $Prob\{|S_\epsilon|>0\}\rightarrow 0$ (i.e., the probability of trading goes to zero).

This would seem to contradict Hansen-Jagannathan bounds, which show that for all priced assets in the economy with excess returns $R_e$, and pricing kernel $M$, we would necessarily have $$0=E[M R_e]$$ and this gives us the bound via CSB: $$\frac{|E[R_e]|}{\sigma[R_e]} \le \frac{\sigma[M]}{E[M]}$$ i.e., the Sharpe ratio of assets in our economy is bounded by the inverse of the Sharpe of the pricing kernel, otherwise known as a good deal bound.

In other words, in our economy, where excess returns are stationary with a fixed (nonzero) ACF, near arbitrages would exist, although the probability of getting any return at all is negligible.

  • What are the consequences of this?

  • Would this then be a pricing anomaly?

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    $\begingroup$ Are the market prices for any payoff $X$ in your world given by the inner product with the stochastic discount factor (SDF) i.e. $p = \operatorname{E}[MX]$? Does the SDF have finite 1st and 2nd moments? The Hansen-Jagannathan basically comes out of correlations being in the set $[-1, 1]$ (which itself comes from the Cauchy-Schwartz inequality). It sounds that you've got two different notions of the stochastic discount factor (SDF)? You have prices that AREN'T given by the SDF? $\endgroup$ – Matthew Gunn Nov 9 '17 at 23:13
  • $\begingroup$ Do you have a reference with more details on the setup you describe? $\endgroup$ – Daneel Olivaw Nov 10 '17 at 0:35
  • $\begingroup$ A reference. The bound holds when the correlation between the pricing kernel and the excess return is strictly positive. $\endgroup$ – Daneel Olivaw Nov 10 '17 at 0:39
  • $\begingroup$ Without analysing the derivation of your limiting behaviour it is difficult to make a judgement. $\endgroup$ – Daneel Olivaw Nov 10 '17 at 0:43
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    $\begingroup$ 1)It would be nice to show how you arrive at the point where $$\frac{E\left[S_n\right]}{\sigma\left(S_n\right)} \to \infty$$ 2) The statement of the Hansen-Jagannathan bound is a statement about returns in equilibrium, ruling out the existence of arbitrage. What you construct seems to be an arbitrage or something like that. Or at least some strategy that can be taken only by some and not resulting in equilibrium $\endgroup$ – fni Nov 10 '17 at 11:12
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The interpretation of the Hansen-Jagannathan bounds is that it's a bound on stochastic discount factors, not a bound on returns. If $\frac{\operatorname{E}[R_e]}{\sigma(R_e)}$ is unbounded above, then so is $\frac{\sigma(M)}{\operatorname{E}[M]}$.

Quick derivation of the Hansen Jagannathan bound

By the Cauchy-Schwartz inequality: $$ |\operatorname{Cov}(R_e, M) | \leq \sigma(R_e) \sigma(M)$$ If the pricing function for a payoff is linear (i.e. $p(\alpha X + \beta Y) = \alpha p(X) + \beta p(Y)$), then the pricing function can be written as the inner product with a stochastic discount factor $M$. Apply that the inner product of zero cost return $R_e$ with stochastic discount factor $M$ properly gives the price 0 (i.e. $ \operatorname{E}[R_e M] = 0$) $$ |\operatorname{E}[R_e]\operatorname{E}[M] | \leq \sigma(R_e) \sigma(M)$$ If no arbitrage opportunities exist then $M$ is strictly positive and $\operatorname{E}[M] > 0$. Furthermore if $\sigma(R_e) > 0$ then:

$$ \frac{|\operatorname{E}[R_e] |}{\sigma(R_e)} \leq \frac{\sigma(M)}{\operatorname{E}[M]}$$

The assumptions that got here are fairly minimal.

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