In a previous post, we introduced the concept of the efficient frontier. The efficient frontier concept is frequently presented in a graph, such as the following.

Capital Market Line

Now comes the insight of a single “efficient risky portfolio.” The assumption is that an investor can choose to allocate his or her portfolio between the riskless asset and the efficient portfolio (that is, in our example some combination of stocks and bonds).

But what combination of stocks and bonds is “optimal” in the sense of providing maximum return for minimum risk?

Without the riskless asset, the investor must select a portfolio somewhere along the efficient frontier, which is the blue curve in the graphs. Each point along the curve represents a different combination of stocks and bonds.

The solution (in the abstract world of academic finance) is to draw a line that begins at the riskless asset and is tangent to the efficient frontier. An example is shown below.

The line—called the “Capital Market Line”—is tangent at about the point that corresponds to a portfolio of 60% stocks and 40% bonds. With our assumptions, the expected return on this portfolio is about 8%, and the standard deviation is about 12.65%.

Tangency Portfolio

In this example, the capital market line is tangent to the efficient frontier at a portfolio that consists of 60% stocks and 40% bonds. That is the “tangency portfolio.”

Note that the above model is based on only two assets: “stocks” and “bonds.” The entire stock market is represented by a single set of return and variance assumptions, and the entire bond market is represented by a single, different, set of return of and variance assumptions, and we have assumed a covariance (correlation) between them.

In addition, this model assumes that these two risky assets, and the so-called riskless asset, comprise the entire universe of possible investments.

Given these restrictive assumptions, all optimal investment portfolios must lie somewhere on the capital market line.

Remember that optimal is defined as portfolios that offer the maximum expected return for a given amount of risk (where risk is defined as standard deviation of return).

Investing Implications

If an investor accepts the assumptions underlying the Capital Market Line, the investment implication is that investments should consist only of portfolios that are on the capital market line.

Risky Assets

The model leads to the conclusion that optimizing investors should only hold risky assets in the ratio of the tangency portfolio. In this example, that means that an optimizing investor will always hold the risky portion of his or her portfolio in the ratio of 60% stocks and 40% bonds. Again, this ratio is the tangency portfolio.

The investor, according to the model, can select different levels of risk and expected return by forming portfolios consisting of different weightings of the riskless asset and the tangency portfolio.

Those two assets – the riskless asset and the tangency portfolio – would be the only two assets an investor would hold. And by holding differing weights, the investor would be exposed to more or less risk, and therefore expected return.

Example

Let’s now look at this in slightly more real-world terms. Let’s assign the names “cash” to the so-called riskless asset, and “balanced fund” to the tangency portfolio.

Again, given the strict assumptions of the model, an optimizing investor would hold only cash and the balanced fund.

In the above example, if the investor held only cash, the expected return would be 4%, with no risk.

If the investor held only the balanced fund, the expected return would be 8%, with a standard deviation of 12.65%

Linearity

If the investor wants a different level of expected return, he or she finds it by dividing the portfolio between cash and the balanced fund.

For example, a portfolio 50% in cash and 50% in the balanced fund would have an expected return of 6%. We can find the return, and associated risk, along the Capital Market Line.

Because the Capital Market Line is a line (as opposed to a curve), the calculations are simple. The expected return on the 50/50 cash and balanced fund portfolio is 50% of 4% (the expected return on cash) and 50% of 8% (the expected return on the balanced fund). The standard deviation of the portfolio is found the same way. So, in this case, that expected standard deviation of the 50/50 portfolio is 6.325%.

Leverage

If you’ve followed the logic so far, you realize even if you invest 100% of your portfolio in the tangency portfolio, the highest expected return you can get is 8%. If you want a higher expected return, you could hold a higher percentage of stocks, but then you would drop below the Capital Market Line. That is correct.

The theoretical answer is that to stay on the capital market line, and therefore own the most efficient portfolio possible, you could borrow at the risk-free rate, and invest the borrowed money in the tangency portfolio.

It is a bit easier to illustrate this by using dollars rather than portfolio weightings. Assume a portfolio with a value of $1000. To obtain an expected return of 10%, you would borrow $500 at the risk-free rate of 4%, and invest $1500 in the tangency portfolio. Here are the expected returns, and cost of borrowing, in dollars:

The expected return on the tangency portfolio is 8% of $1500, or $120. The cost of borrowing $500 at 4% is $20. Therefore, the expected return on the portfolio is $120 minus $20, which is $100. That $100 is the 10% on your $1000 of investment capital.

Risk

The leverage also increases your risk. The riskless asset, by assumption, contributes nothing to the risk. But the 12.65% standard deviation of the tangency portfolio is multiplied by 1.5, so that the standard deviation of the leveraged portfolio is 18.98%.

Recall that (by assumption) the expected return on the stock market is 10%, and the standard deviation is 20%.

So, in this example, by incurring this large amount of leverage (equal to 50% of your capital) you theoretically reduce your risk by about 1% of standard deviation. That is, if you ignored the Capital Market Line, didn’t borrow anything, and simply invested 100% of your capital ($1000 in this example) in the stock market, you’d have an expected return of 10% and a standard deviation of 20%. By following the Capital Market Line approach, you’d have a 10% expected return, and a standard deviation of 19%.

Many theorists are guided in part by the perceived beauty of proposed theories.

For example, physicist Paul Dirac said, “it is more important to have beauty in one’s equations than to have them fit experiment.”[1]

The view seems to be particularly prominent among physicists. Richard Feynman echoed Dirac: “You can recognize truth by its beauty and simplicity.”[2]

The ideas of market efficiency, mean-variance optimization, the efficient frontier and the Tangency Portfolio might be simple and even to some beautiful. But how practical are they?

We will address that question in a future post. In the meantime, click here to request one of our advisor guides.

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[1]The evolution of the Physicist’s Picture of Nature Scientific American 208 (5) (1963)

[2] Cited in Sympathetic Vibrations: Reflections on Physics as a Way of Life, 1985