The Basic and Full Fundamental Laws

Suppose you're picking out equity portfolio managers to add to your portfolio. Clearly, you want the best of the best, the managers who add the most value. To find that manager, you need to decompose the manager's return into key criteria. What criteria would you look for?

Yes! You would actually need all three to find the best portfolio manager based on expected return: the portfolio's active risk, the ability to forecast returns (information coefficient), and the number of holdings in the portfolio (breadth). When you combine the three, you can calculate the expected active portfolio return. $$\displaystyle E(R_A)^* = IC \sqrt{BR} \sigma _A$$. That's the value you'd want to maximize to find the best manager, and it's the basic fundamental law of active management.

The basic fundamental law of active management should naturally contain the information ratio. What term represents the IR in the equation?

No. The information coefficient isn't the information ratio.

That's not it. Breadth multiplied by the risk coefficient isn't going to capture the information risk because the active management aspect isn't captured.

You got it! The $$\displaystyle IC \sqrt{BR}$$ captures the information ratio of the unconstrained portfolio because it reflects the information coefficient and breadth that the active manager can control to drive returns. Notice, however, that the portfolio is unconstrained, meaning that short selling is not only allowed, but it can be done in large quantities. Unfortunately, that's not always the case for most investors. In addition, other constraints, like social investing or turnover limits, can also be imposed. So, a numerical optimizer is used to help calculate the necessary active portfolio weights.

Think about how the numeral optimizer will work to provide the necessary constraints. It would have to impact the portfolio prior to actual performance. Which fundamental building block is going to immediately reflect the optimizer?

Not quite.

Yes!

In analyzing the equation, you can see that with the correlation measurement, the TC should range from -1 to 1, although values are typically positive from 0.20 to 0.90. That should be no surprise, though, because this is an active management measurement. What would be the expectation at a level of 0?

Bingo! If the TC is 0, no value is being added, so the correlation between the forecasted and active weights isn't generating a positive return. That means that the goal of using the TC optimizer is to create a correlation of 1 so the full benefit of active management is realized. To include the TC component, the full fundamental law is $$\displaystyle E(R_A) = (TC)(IC) \sqrt{BR} \sigma _A$$.

No. The correlation wouldn't add positive value at 0.

Not quite. The correlation wouldn't negatively impact the portfolio at 0.

Notice here that there's no * included, which means that this portfolio has constrained active security weights, and as you can see, the fundamental law has four inputs: the transfer coefficient, the assumed information coefficient, the square root of breadth, and the portfolio's active risk. In addition, these formulas assume a single-index model, which means that the breadth component is based on a 1 to 1 relationship with the number of securities in the portfolio. Obviously, breadth will vary based on the correlation of the securities in the portfolio.

Given these assumptions, you can also use the TC to calculate the optimal amount of active risk. $$\displaystyle \sigma _A = TC \frac{IR^*}{SR_B} \sigma _B$$ Here, _IR_* is the information ratio of the unconstrained portfolio. Or, you can use the transfer coefficient to find the maximum possible value of the constrained portfolio's Sharpe ratio. $$\displaystyle SR^2_P = SR^2_B + (TC)^2(IR^*)^2$$ The equation is similar to the calculation of optimal active risk, but it introduces the TC in order to add the constraints. Again, just like the previous optimal risk equation, you can use it to adjust the portfolio holdings to reflect the necessary risk level.

To sum it up: [[summary]]

The active security weights are going to be impacted by the optimizer because the weights will impact the return. But the actual active realized returns won't be impacted by the numerical optimizer because returns are a result of the optimizer. That means that the numerical optimizer puts constraints on the relationship between the active security weights and the forecasted active security returns. It's called the transfer coefficient, which is basically the risk-weighted correlation between the two. $$\displaystyle TC = COR( \frac{\mu}{\sigma _i}, \Delta w_i \sigma _i)$$

The portfolio's active risk

The ability to forecast returns

The number of holdings in the portfolio

$$IC$$

$$\sqrt {BR} \sigma _A$$

$$IC \sqrt {BR}$$

Active security weights

Actual active realized returns

No value added

Positive value added

Negative value added

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