Theory suggests that, for any given level of risk, there will be one portfolio that offers the highest expected return. The continuum of these portfolios forms the well-known Efficient Frontier. The Efficient Frontier slopes upward to the right, as added risk brings added reward; but there is a limit to the reward an investor can expect, so the slope is not constant. There will, then, be one portfolio along the Efficient Frontier that offers the highest ratio of expected return to risk.
Is this optimal portfolio a constant? One would think that, as interest rates increase, raising the prevailing riskless rate of return, the optimal portfolio would have to move further and further to the right along the Efficient Frontier. In other words, investors will have to accept higher and higher risks in order to achieve a return above the riskless interest rate.
If this is true, then the share of high beta stocks in the market should increase as interest rates rise and decrease as interest rates fall. The share of low beta stocks, on the other hand, should vary inversely with the direction of interest rates.