# if the​ risk-free interest rate​ increases, and nothing else changes, is the market portfolio still efficient?

I think the response is no but I don't know why If so, Stocks with betas greater than one will be buying opportunities and stocks with betas less than one will be selling opportunities because I can take more risk here ?

Assuming the CAPM*, the expected return $$r_i$$ of stock $$i$$ equals $$E[r_i] = r_f + \beta(r_m - r_f)$$ with $$r_f$$ as the risk-less rate of interest and $$r_m$$ as the return of the market portfolio. The expected market return $$E[r_m]$$ remains unaffected by changes in $$r_f$$ and still equals $$r_m$$. The expected return of a single stock $$i$$ from the formula above after minor calculus is

$$E[r_i] = \beta \cdot r_m + r_f \cdot (1- \beta)$$

An increase of $$r_f$$, i.e. $$\Delta r_f>0$$, decreases the expected return $$E[r_i]$$ of stocks with $$\beta > 1$$ and increases $$E[r_i]$$ of stocks with $$\beta < 1$$. On average, these changes dissolve on average, as the market portfolio per definition has $$\beta = 1$$.

So how does an increase of $$r_f$$ affect the market portfolio?

Assume a mean-variance framework and lets calculate the efficient frontier when short-sales are allowed and there is a risk-less lending and borrowing** rate $$r_f$$. Following the Tobin-separation, investors hold a linear combination of the risk-less asset and the risky portfolio with $$N$$ assets, which generates a return of $$r_p$$. A rational investor maximizes the Sharpe-Ratio $$\Theta = \frac{\bar{r_p} - r_f}{\sigma_p}$$ with $$\sigma_p$$ as the standard deviation of $$r_p$$. The maximization problem is constrained to $$\sum_{i=1}^N{X_i} = 1$$, i.e. the sum of all weights $$X_i$$ on each asset $$i$$ have to equal 1. Rewriting the Sharpe-Ratio leads to $$\Theta = \frac{\sum_{i=1}^{N}{X_i \left( \bar{r_i} - r_f \right)}}{\sigma_p}$$

Maximizing the Sharpe-Ratio and solving for the asset weights $$X_i$$ leads to a system of simultaneous equations with $$N$$ terms and the expression $$\bar{r_i} - r_f$$ for all $$1 \le i \le N$$ on the "left side". So an increase of $$r_f$$ decreases all these terms, changing the weights $$X_i$$ for all assets. So in conclusion, the market portfolio, consisting of all assets $$i$$ is still efficient after an increasing $$r_f$$, as it is the unique solution of the above maximization problem (facing all rational investors). The increasing $$r_f$$ however changes the weights $$X_i$$ for all investors, i.e. investors will rebalance their investments. This arises from the changes in future expected stock returns described in the first paragraph (which depend on $$\beta_i$$).

*Be aware of the CAPM. Plenty of empirical research does not support the CAPM, it is long ago "shot dead" by academics.

**This case is the simplest case you can consider. See chapter 6 "Techniques for calculating the efficient frontier" in Elton et al. (2014) for further analysis.

Reference:

Elton/Gruber/Brown/Götzmann (2014), Modern Portfolio Theory and Investment Analysis, ed. 9.

• To clarify, do investors need to trade or do the prices just adjust? I think it is the latter: a float weighted portfolio only needs to trade when share counts (or amount issued) changes. – Charles Fox Oct 17 '18 at 16:33
• You are absolutely right @CharlesFox. As the portfolio is value-weighted, there is no need for the investor for rebalancing, as long as the number of shares outstanding remains the same. – skoestlmeier Oct 17 '18 at 17:09

If you take the classic monkey-model of a two-asset market. Something like stocks = 6% returns for 18% vol; bonds = 2% returns for 6% vol; with 0 correlation.

This gives you the classic/cliched efficient frontier you'd expect:

Below are Sharpe Ratios, given %-in-stocks assuming different different risk-free rates. Clearly, as risk-free increases, the Max Sharpe portfolio shifts to the right, into proportionately higher stock allocations.

However, this sort-of makes sense. Raising risk-free without raising bond yields means a flatter yield curve, lowering the carry and roll that drive bond returns. In a sense, when "nothing else changes", duration does become less of an attractive source of returns! Go too far the other way and BY rises 1:1 with risk-free, then equities become less attractive but bonds stay as attractive as before. The SR curve will then left-shift. So the feedthrough of your rates into bonds does matter.

But let's forget that for a moment. The so-called "tangency point" on our efficient frontier has shifted to the right... The CAPM theory says that if we mix market and cash, then we should be mixing a market made up of more stock and less bonds with our cash.

This tries to float a massive assumption under the radar screen! Namely that the tangency point and the market portfolio are the same thing in the first place! Mixing the market portfolio with cash assumes that investors want to run a portfolio that is lower-return, lower-risk than the market portfolio. And alternatively, if they want higher-risk, then they leverage up the market portfolio.

The obvious point is that most investors are more leverage-constrained than theory assumes in reality. Most cannot borrow at riskless, and are heavily biased against leverage by a combination of regulation, tradition, or mandate. So if they want to run more risk than the 30-40% stock weightings that the models tend to describe as "optimal", they simply run up the efficient frontier (rather than leveraging up the tangency point).

Their Sharpe Ratio will still fall; but if the portfolio is "efficient" in the sense of running the right level of risk that the investor is happy to run, then that wouldn't change (in your scenario). The same portfolio would still be "efficient", using that definition of efficiency that causes investors to deviate from the model in the first place.

If the market portfolio and the tangency point were never the same thing in the first place, there is no need for any structural re-pricing here. [And if you try to force them to be the same for theory's sake, then the re-pricings you would require would neither be internally consistent nor realistic in any market history context]