Inverted Risk/Return Curves

Over 50 years of academic financial thinking is based on a kind of financial gravity:  the notion that for a relatively diverse investment portfolio, higher risk translates into higher return given a sufficiently long time horizon.  Stated simply: “Risk equals reward.”  Stated less tersely, “Return for an optimized portfolio is proportional to portfolio risk.”

As I assimilated the CAPM doctrine in grad school, part of my brain rejected some CAPM concepts even as it embraced others.  I remember seeing a graph of asset diversification that showed that randomly selected portfolios exhibited better risk/reward profiles up to 30 assets, at which point further improvement was minuscule and only asymptotically approached an “optimal” risk/reward asymptote.  That resonated.

Conversely, strict CAPM thinking implied that a well-diversified portfolio of high-beta stocks will outperform a marketed-weighted portfolio of stocks over the long-term, albeit in a zero-alpha fashion.  That concept met with cognitive dissonance.

Now, dear reader, as a reward for staying with this post this far, I will reward you with some hard-won insights.  After much risk/reward curve fitting on compute-intensive analyses, I found that the best-fit expected-return metric for assets was proportional to the square root of beta.  In my analyses I defined an asset’s beta as 36-month, monthly returns relative to the benchmark index.  Mostly, for US assets, my benchmark “index” was VTI total-return data.

Little did I know, at the time, that a brilliant financial maverick had been doing the heavy academic lifting around similar financial ideas.  His name is Bob Haugen. I only learned of the work of this kindred spirit upon his passing.

My academic number crunching on data since 1980 suggested a positive, but decreasing incremental total return vs. increasing volatility (or for increasing beta).  Bob Haugen suggested a negative incremental total return for high-volatility assets above an inflection-point of volatility.

Mr. Haugen’s lifetime of  published research dwarfs my to-date analyses. There is some consolation in the fact that I followed the data to conclusions that had more in common with Mr. Haugen’s than with the Academic Consensus.

An objective analysis of the investment approach of three investing greats will show that they have more in common with Mr. Haugen than Mr. E.M. Hypothesis (aka Mr. Efficient Markets, [Hypothesis] , not to be confused with “Mr. Market”).  Those great investors are 1) Benjamin Graham, 2) Warren Buffet, 3) Peter Lynch.

CAPM suggests that, with either optimal “risk-free”or leveraged investments a capital asset line exists — tantamount to a linear risk-reward relationship. This line is set according to an unique tangent point to the efficient frontier curve of expected volatility to expected return.

My research at Sigma1 suggests a modified curve with a tangent point portfolio comprised, generally, of a greater proportion of low volatility assets than CAPM would indicate.  In other words, my back-testing at Sigma1 Financial suggests that a different mix, favoring lower-volatility assets is optimal.  The Sigma1 CAL (capital allocation line) is different and based on a different asset mix.  Nonetheless, the slope (first derivative) of the Sigma1 efficient frontier is always upward sloping.

Mr. Haugen’s research indicates that, in theory, the efficient frontier curve past a critical point begins sloping downward with as portfolio volatility increases. (Arguably the curve past the critical point ceases to be “efficient”, but from a parametric point it can be calculated for academic or theoretical purposes.)  An inverted risk/return curve can exist, just as an inverted Treasury yield curve can exist.

Academia routinely deletes the dominated bottom of the the parabola-like portion of the the complete “efficient frontier” curve (resembling a parabola of the form x = A + B*y^2) for allocation of two assets (commonly stocks (e.g. SPY) and bonds (e.g. AGG)).

Maybe a more thorough explanation is called for.   In the two-asset model the complete “parabola” is a parametric equation where x = Vol(t*A, (1-t)*B) and y = ER( t*A, (1-t)*B.  [Vol == Volatility or standard-deviation, ER = Expected Return)].   The bottom part of the “parabola” is excluded because it has no potential utility to any rational investor.  In the multi-weight model, x=minVol (W), y=maxER(W), and W is subject to the condition that the sum of weights in vector W = 1.  In the multi-weight, multi-asset model the underside is automatically excluded.  However there is no guarantee that there is no point where dy/dx is negative.  In fact, Bob Haugen’s research suggests that negative slopes (dy/dx) are possible, even likely, for many collections of assets.

Time prevents me from following this financial rabbit hole to its end.  However I will point out the increasing popularity and short-run success of low-volatility ETFs such as SPLV, USMV, and EEMV.  I am invested in them, and so far am pleased with their high returns AND lower volatilities.

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NOTE: The part about W is oversimplified for flow of reading.  The bulkier explanation is y is stepped from y = ER(W) for minVol(W) to max expected-return of all the assets (Wmax_ER_asset = 1, y = max_ER_asset_return), and each x = minVol(W) s.t. y = ER(W) and sum_of_weights(W) = 1.   Clear as mud, right?  That’s why I wrote it the other way first.