The relationship between Breadth, Strategy risk and Information ratio. Example From South-East Asian
Updated: Mar 18, 2021
IR and Breadth
The Fundamental Law of Active Management (FLAM) introduced by (Grinold & Kahn, 2000) is mostly analytical tool for constructing and evaluating active trading strategies. The original version is (1):
The result of FLAM is the Information ratio (IR) which is the value added by the active portfolio management on risk-adjusted basis. In (1) IR depends only on the forecasting ability of the investment strategy described by the Information Coefficient (IC) and the Breadth (Br), which is the number of independent active bets that are used while implementing the investment strategy. It is a great tool for comparing complex investment strategies or evaluating performance of active managers. Subsequently FLAM is developed further in papers like (Qian & Hua, 2004), (Clarke, de Silva, & Thorley, 2005), (Ye, 2008) and culminated in (Ding & Martin, 2017).
However the relationship between the IR and Breadth is still debatable. There are serious disagreements around how exactly to estimate Breadth. Most papers simply assume the number of assets (N) that are included in the active portfolio. Such an approach completely disregards the correlation between forecasts or uncovered dependencies between residual returns. While it is important and interesting problem we will assume that Breadth is equal to N and will focus on the impact of Breadth on the IR. Using (1) the relation between N and IR is exhibit (1):
Exhibit 1 shows that IR is increasing with the increase of the number of assets included in the active portfolio. Even more, IR will continue increasing to infinity if N keeps increasing. Such a direct relationship give very clear insight for active managers – increase the size of your portfolio!
Introducing strategy risk the relationship between the two variables takes another form. According to (Ye, 2008) strategy risk is the variation of the forecasting skill. It is a relaxation of the assumption that IC is constant through time. Taking this into account the relationship becomes evident in exhibit (2):
The difference between exhibits (1) and (2) is the rate at which IR is increasing when we include more securities into the active portfolio. When standard deviation of IC is taken into account, IR is increasing at diminishing rates. As it is evident in such a case there is no big difference in value added if 1200 or 1500 securities are included in the managed portfolio. However the positive relationship between Breadth and IR still holds.
It is interesting to examine where this relationship stems from. To get more insight we need to dig deeper into Ye’s model. The expected IR can be obtained using (2):
The three main determinants of IR are:
The IC – measuring the accuracy of the forecasts. Usually for IC is used the cross-section parameter of regression between historical residual returns (realized alphas) and factor scores (forecasts). When residual returns and forecast are properly standardized, then it is equal to the correlation between them.
The Breadth (N) – as it was stated it is “the number of independent bets” and we assume (as many other researchers) it is the number of assets included in the portfolio. Also can be thought as the size of the universe on which the forecasting skill is applied.
Standard deviation of IC – the volatility of the forecasting skill through time. It is a measure of the “strategy risk” and it is specific for each strategy or alpha factor.
In order get more insight into the effect of Breadth on active portfolio management let’s examine its relationship with the other two determinants and the IR itself. What we will do is rank the stocks from the universe by market capitalization and construct portfolios from the 100 largest stocks, then expanded it to 200, 300 etc. Then plotting the estimated mean IC , standard deviation of IC and IR of each portfolio will uncover the effects of Breadth.
We apply the proposed methodology on investment universe from the South-East Asian capital markets (Malaysia, Thailand, Philippines, Indonesia and Vietnam). The universe is comprised of publically traded stock that have:
At least 10 years of continuous data set of prices;
At least 10 years of information on quarterly financial statements.
After applying these criteria, we end up with sample of 1200 stocks. As alpha factors, we use Earnings Yield, Return on Equity (ROE), Net margin and Sales to EV (SEV). We begin with constructing active portfolio with the first 100 largest stocks then add next 100 and until we utilize 1100 stocks. This algorithm is applied for each alpha factor so in the end we have 44 portfolios. After that we extract the information on mean IC, standard deviation of IC and IR and then summarize.
Empirical results are presented in order to validate the theoretical findings. Additionally, examining the performance of active portfolios of various sizes we find interesting relation between strategy risk and the number of assets included in the portfolio. First we examine the impact of Breadth over the value added (IR) by increasing the number of assets and optimizing. The direct positive relation between the Breadth and the expected Information ration is evident by exhibit 3:
Exhibit 3 shows positive impact of breadth on the expected IR over all four alpha factors that are tested. In the case of the Return on Equity factor the lowest expected IR is 0.62 with only 100 stocks included in the portfolio, while the highest is at 0.86 with all 1100 stocks. In the case of the factors Margin and Sales to EV we also see positive growth of IR with increasing Breadth. However the marginal addition to IR by increasing Breadth is declining. This is interesting finding because it raises the notion that the relationship between Breadth and IR may not be linear. Apparently, we need to investigate where this relationship stems from. The expected IR from eq. 2 is a function of two variables: a) expected IC and the b) volatility of this expectation. Next from our tested portfolio, we extract both of these variables and illustrate the impact of the increasing number of assets included in the portfolios.
Starting with the Information coefficient the expectation is that if we increase the number of assets, the quality of the forecasts will drop and therefore lower the IR. On exhibit 4 is shown the expected IC from our tested portfolios:
The expected IC in the case of the factors Earnings Yield and Net margin appears to be completely random. When using the factor Earnings Yield on only 100 stocks we get IC of 0.09 , when it is applied over 300 stock then IC is 0.115 and when all stocks are used we get 0.105 therefore no relationship can be validated. In the case of Net margin factor, the lowest IC is of portfolio with 400 stocks, while the highest is when the factor is applied over 300 stocks. Slight positive relationship between IC and Breadth can be found in the case of the Sales to EV factor, but it is almost negligent. The lowest IC is at 100 stocks included and it is 0.038 while the highest is 1100 stocks and it is equal to 0.048 giving us a range of mere 1%. Similar results can be drawn from the case of using ROE as alpha factor. We can conclude that there is no empirical evidences of any relation between increasing the number of assets in the portfolio and the Information Coefficient. Therefore, we must examine the other variable in the expected IR – strategy risk or the variation of the signal quality.
The relationship between standard deviation of IC and Breadth from our empirical observations is plotted on exhibit 5.
From exhibit 5 we uncover that surprisingly when managers increase the number of assets held in the portfolio the volatility of their forecasting skill decreases and gets more focused around its mean value. One reason for this is the diversification effect of increasing assets on the errors of the forecasting model. Other explanation is with the “Law of Large Numbers”, each IC is a parameter of cross-sectional regression. As such parameter when it is estimated by larger sample then it gets closer to the mean IC and thus reduce the variation from period to period.
Given exhibits 3, 4 and 5 we conclude that the positive relation between expected IR and the Breadth stems from reducing the strategy risk. Increasing the number of assets in the active portfolio seems not to impact the quality of the forecasting signal (IC). However in the case of strategy risk we can confirm that more assets in the portfolio lead to smaller and from thereon higher expected Information Ratio.
1. Clarke, R., de Silva, H., & Thorley, S. (2005). Portfolio Constraints and Fundamental law of active management. Financial Analysts Journal, 58(5).
2. Ding, Z., & Martin, D. (2017). The Fundamental Law of Active Management: Redux. Retrieved from https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2730434
3. Grinold, R., & Kahn, R. (2000). Active portfolio management 2nd edition. New York: McGraw-Hill.
4. Qian, E., & Hua, R. (2004). Active Risk and Information Ratio. The Journal of Investment Management, vol 2(N3), 20-34.
5. Ye, J. (2008). How Variation in Signal Quality Affects Performance. Financial ANalysts Journal, vol. 64(N4), 48-61.