There's a framework that is often used as a model for thinking about these things. I actually am not sure there's a name for the whole shebang, but it is the combination of Markowitz "modern portfolio theory" (MPT), the CAPM (capital asset pricing model), and Tobin's Separation Theorem.* I'm name-dropping these for identification but I don't actually understand them. However, when they come together, you see diagrams like this:
In this framework, yes. Assume that we have a portfolio of stocks, bonds, and the risk-free asset. Assume that our goal is to optimize risk-adjusted return--maximize the Sharpe ratio. All else being equal, if the only change that happens is that the risk-free rate increases, then the optimum allocation shifts toward more equities, and vice versa.
The
reason why this happens is that if you assume you have a "risk budget," a certain amount of risk you are willing to take, and if 100% stocks is too much risk, you always have a choice as to how you can reduce that risk. Combinations of stocks and bonds get you to places on the curve. If you are at the yellow dot, and you have too much risk--you are too far right on the curve--and you want to move left, you have a choice of two ways to do it. You can add more bonds, or you can add the risk-free asset. Adding bonds moves you left and down on the curve, adding the risk-free asset moves you left and down on the straight line. When you are at the optimum point--the yellow dot--it is better to adjust your risk by adjusting your cash holdings than by changing the stock-bond allocation.
In other words,
assuming you are trying to keep your risk at a desired level, when the risk-free rate increases, you should hold more stocks in your stock-bond portfolio,
because you should be using more of the risk-free asset as well. You want to dilute your stock risk with a mixture bonds
and cash, and when the risk-free rate increases, you should be using more cash and less bonds to make the dilution.
Three other points that should be made. The first is that inflation doesn't change the theory, it just changes the values you plug in. Just use real rates for everything instead of nominal rates.
The second, and this is one that I think people get conceptually wrong, is that the optimum does
not change if the return of stocks, bonds, and the risk-free asset all change together by the same amount. That just slides the whole diagram, curve, straight line, and all, up and down together, and the tangent point--the optimum stock/bond allocation--doesn't change. It is often assumed that something changes if the risk-free rate or the return from bonds becomes negative, but it is not so. Even though it is making the best of a bad situation, the best allocation remains the same.
I don't want to try to explain the framework, except to say that 1) it is neither realistic nor crazy. And that 2) if you are going to take it literally as a way of finding an "optimum," then you have a choice of thinking. Either
- it tells you what would have been the optimum over some particular past pariod of time, or
- it requires you to have accurate future predictions of the return, volatility, and correlation between the assets in the future.
Either way, it is much less interesting in any practical sense than words like "optimization" suggest.
The third is that if you are a risk-seeker--and, of course, if you can borrow at the risk-free rate--nothing changes except that some signs turn negative. If you are at the yellow dot and you want
more risk, it is better to move right along the straight line--meaning to use leverage on the stock-bond portfolio--than to increase stock allocation. I'm just pointing that out as math. In real life that might not be a good idea because of the cost of borrowing, and also because the curve, which is a hyperbola, straightens out so much that the difference between the tangent line and the hyperbola is something that happens with perfect mathematical data and probably is lost in the noise in real life. Actually even in the picture you don't see any visible air between the straight line and the curve as you move up and to the right of the yellow dot.
*Modern portfolio theory gets you the "efficient frontier" curve, CAPM gets you the Sharpe ratio as the measure of risk-adjusted return to be optimized, and Tobin's Separation Theorem gets you the "capital markets line," the straight line that you make tangent to the curve.
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