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I follow this paper and estimated two different asset pricing models via systems of deep neural networks. Both models have the exact same input: firm-specific features for 10'000 (unique) US stocks and a large set of macroeconomic variables over 20 years (training period = 1970-1990), but differ slightly in architecture. More specific: I have estimated two different stochastic discount factors and consequently two different tangency portfolios $F_{t}$ that account for time-variation. From there on I have estimated via another neural network the risk loadings as output $y = R^e_{t,i}F_t$ to estimate $\beta_{t,i}=\mathbb{E}_t[R^e_{t,i}F_t]$ which is proportional to the true $\beta_{t,i}$.

Some out-of-sample differences in results for the period 1995-2019 (after ensembling over 10 runs):

  • Sharpe Ratio of $F_t$ for model 1 is 2.8, model 2 only 0.9
  • Total cumulative return for model 1 is about 8 (800%), for model two 14.
  • Maximum drawdown for model 1 is 8% over the whole period, for model 2 it is 59%
  • Monthly avarge turnover for model 1 is 0.6, for model 2 it is 0.07
  • Model 2 loads more heavily on small- and micro-cap stocks, but does not, at no point in time, load up on short positions in stocks. Model 1 does and, so I assume, can achieve this higher Sharpe ratio.

In short, model 2 is more index-like with high exposure to the MKT factor, and as mentioned, to the SMB factor. Model 1 is nearly MKT and SMB neutral. Model 2 can be very well explained by FF5, whereas model 1 cannot: enter image description here To investigate the SML, I normalized all obtained $\beta_{t,i}$ cross-sectionally by making sure that $\beta_t^{F_t}=1$, since I do not estimate it directly and built equally-weighted $\beta$-decile portfolios for both models. Then, I did a plain-vanilla regression and obtained this:

enter image description here

Given no-arbitrage, the intercept should be at 0, since I operate in the excess-return space. Both models do a quite satisfying job. But I am still puzzled: model 1 indicates an equity risk premium of 4.65% p.a. which is, given all the results from research, a reasonable result. Model 2 has an even better fit, but indicates a whopping premium of 12.1% p.a.

My current interpretation is that model 2 indicates a higher premium since it loads more heavily on small-caps, which offer (on paper) a attractive risk-return profile.

Q: Do you have any suggestions, or interpretations on why both models obtain such a good fit but differ that much in slope for the exact same input data?

EDIT 1 (in response to Chris' questions)

  • Holding period on your two strategies? Monthly rebalancing for both models.
  • A Sharpe of 2.8 is quite high (like, top decile+) for a long-only buy and hold mid to low frequency equity strategy. Even .9 is on the high side for a broad-based US index. Model 1 is not long-only, it also takes short positions in stocks. Model 2 is long-only. The original authors obtained also a similar SR of 2.65 for a slightly different period.
  • Using 70-90 as a training set and 95-present as an out of sample set seems prime for issues related to change in regime. I can't think of a cut-off (ie, 1995) more aligned with a general shift in underlying business models, specifically the emergence of tech as a primary driver in US equities. I agree here, I didn't think of that. I choose 20 years of trianing in order to have a large set available for training but not a too large one, since training on this already takes 23 hours per ensemble (10 runs per ensemble). Then I took 5 years (1990-1995) for validation. And the rest is testing. This way I have a very long out-of-sample period which I deem to be important to test such models, especially to analyze their behavior during the dot-com bubble and GFC 2008. Moreover I want to compare my results to the original paper with similar periods.
  • Regarding your follow-up question, why are you 'investigating SML'? How are you normalizing your betas across assets? How are you constructing your subsequent portfolio(s)? You'll need to provide more detail. I'm new to this area of empirical asset pricing research and, correct me if I'm wrong, as far as I understand is that classical models fail to capture equity risk premia correctly and that many models return flat SML which, compared to historical (positive) risk premia, is something like a paradox, right? So, besides nice Sharpe ratios, a model must be able to explain differences in returns via differences in risk exposures to the systematic factor $F_t$. Well, I follow this paper and estimate via neural networks a portfolio weight $\omega_{t,i}$ for the implicit tangency portfolio $F_{t+1}=\sum_{i=1}^{N_t}\omega_{t,i}R_{t+1,i}$, where $\omega_{t,i}$ is the output of the NN-system. The betas I estiamte not via linear-regression but with another NN and only as a proportion. This way, I (and the original authors) can avoid estimating a covariance matrix for thousands of stocks every month. This means, for every stock I get a approximate beta that should be proportional to its true beta. Since I get this for all stocks, I can compute the weighted average beta for $F_{t}$ given the weights above. This $\beta^F_{t}$ will not be 1, so I have to scale all single stock betas in that cross-section with 1/$\beta^F_t$, such that finally $\beta^F_t=1$. Then I sort stocks cross-sectionally into deciles based on their beta and weigh them equally in these portfolios. Every month, to be evaluated in the subsequent month. Finally, for the SML i compute the mean excess returns and mean betas for all these porftolios over the testing period and perform this regresion above.
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There's a lot to pick apart here, but to start:

(1) Holding period on your two strategies? A Sharpe of 2.8 is quite high (like, top decile+) for a long-only buy and hold mid to low frequency equity strategy. Even .9 is on the high side for a broad-based US index.

(2) Using 70-90 as a training set and 95-present as an out of sample set seems prime for issues related to change in regime. I can't think of a cut-off (ie, 1995) more aligned with a general shift in underlying business models, specifically the emergence of tech as a primary driver in US equities.

(3) Regarding your follow-up question, why are you 'investigating SML'? How are you normalizing your betas across assets? How are you constructing your subsequent portfolio(s)? You'll need to provide more detail.

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  • $\begingroup$ Thanks for the questions, I made an update. Hope it helps. $\endgroup$ – budfox3 Oct 11 at 12:50

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