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I have one portfolio with high beta stocks, and one with low beta stocks. Is it better to have higher expected return with high volatility, or medium expected return with medium volatility? (All from a asset allocation, efficient frontier, risk/reward prospective.)

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From a theoretical point of view (you mentioned beta, so assume we're in a CAPM world), you should hold the market portfolio (let's assume S&P500 index) and be long (or short) the risk-free asset to decrease (or increase) your return and risk. That is, if you'd like higher returns than the S&P500 offers and are willing to accept the risk, trade the S&P500 index on margin. Again, theoretically, you want to hold a portfolio along the line tangent to the efficient frontier.

In the real world, maybe you can't lever up and trade on margin (maybe borrowing from the bank to buy an S&P500 index fund), so then you'd want to look at an equity only portfolio with a beta greater than one.

If you're looking at portfolio with a beta below one, then it seems that you should just hold the market and risk-free debt (maybe a money market savings account or low-yield corporate debt index fund, depending on your liquidity needs), which should outperform your low beta portfolio for a given level of risk.

Assuming that you've minimized risk for a given return (i.e., you're on the efficient frontier), then ultimately your risk aversion will determine your trade off between risk and return (i.e., picking the right beta). I agree with Chris that the Sharpe and Treynor ratios are good ways of quantifying the risk-return trade-offs in each of these portfolios.

Although not viewed as rational in the strict sense, given that you likely need this capital for retirement or buying a house, you may want to also look at value-at-risk (VaR) or expected shortfall (ES) to quantify how much you could lose in some hypothesized worst-case scenario. Of course, there are some problems with these measures, but they're illustrative.

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Most portfolio managers look at the Sharpe ratio, or occasionally the Treynor ratio. In general, you want to maximize one of the these metrics, though there could be other issues that you haven't currently considered, like turnover or transaction costs associated with obtaining the portfolio.

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The empirical evidence shows that low to medium beta portfolios beat high beta portfolios on a risk adjusted basis. Search SSRN for "betting against beta." This flies completely in the face of CAPM but frankly CAPM is crap.

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Do you have a specific link? – Ralph Winters Mar 31 '11 at 1:12
@ralph winters - ssrn.com/abstract=1723048 – Joshua Chance Mar 31 '11 at 16:33
Interesting but sounds like it is might be a very controlled situation. Possibly you could put together a case for the opposite? But +1 for the reference. – Ralph Winters Apr 1 '11 at 14:48
@Ralph Winters - I searched through my links but can't seem to find research done by Eric Falkenstein (i think) that shows the same thing, with a different methodology. His blog is falkenblog.blogspot.com. If I remember correctly his theory is that low beta stocks get neglected and undervalued relative to high beta "newsworthy" stocks. A possible risk based reason might be that although one could buy the low beta basket and lever up you are exposed to potential tail risk, while using high beta stocks to get "implied leverage" removes this risk and any extra return from taking the risk. – Joshua Chance Apr 2 '11 at 8:32

You still want to perform portfolio optimization. Put everything into one bucket, run 'global' portfolio optimization, build the portfolio. Even if you prefer Sharpe ratios, you should do that on the overall portfolio - not just on individual ones. Be careful of sharpe ratios for low risk, low return assets. Dividing one small number by another small number can sometimes give you a giant number and you can't always lever that up as you please (you hit the friction of overleverage => higher borrowing rate)

Now if you must pick only one over the other (I'm assuming you've questioned why this limitation applies to you), then you should run a monte carlo simulation or perform some back testing of both portfolios. In a temporal way, the world is similar (same management, similar human attributes etc) but not exact (tech bust won't be exactly played out next time). Therefore I prefer Monte Carlo (using historical distributions from which to draw simulation outcomes) over straight up back testing ("In 2001, this portfolio would have looked like ...")

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