1. Framework

2. Notation r=vector of excess returns –Bold signifies vectors and matrices. We denote individual stock excess returns as r n. –Excess above risk free return, i F. –These returns are stochastic. We will also use mean and standard deviation: – f=E{r} –  =StDev{r}

3. Notation, continued  =vector of active returns = Once again, we also use mean and standard deviation: –  =E{  } –  =StDev{  } Covariance matrix, V. Portfolios h P, h B, h PA :

5. Portfolio Basics Portfolio returns Portfolio risk

7. Fully-Invested Portfolios Some of the prior relationships assumed fully invested portfolios. What does this mean? It means that the holdings sum to 1: This is different from specifying that the portfolio is long-only. In fact, fully-invested portfolios will typically have short positions. When we construct portfolios, we will have to include explicit instructions if we want long-only and/or fully-invested results.

8. Active Management Utility As active managers, we seek to consistently outperform a benchmark. The natural utility function for active management is: We trade off expected active return against active risk. The risk aversion parameter,, measures our individual preferences for that trade-off. This utility is a measure of certainty-equivalent active return. For a given, we can convert an active return and active risk to an equally desirable—and certain—active return. The book (Chapter 4) goes through a long hand-waving derivation of this, starting from mean/variance utility in terms of total returns and risk. So if you accept mean/variance at the total return/total risk level, it tends to lead to mean/variance at the active return/active risk level. We will simply assert this result and move on.

9. Utility: Intuition As we change the positions in our actively managed portfolio, the portfolio alpha and active risk vary. There will be one set of positions that maximizes utility. If we also want to impose full-investment or long- only constraints, we must explicitly add these.

10. Simplified Portfolio Construction Goal: make some simplifying assumptions and then solve for optimal active holdings. How do these depend on alpha and risk? Key step: simplify active risk: Assume active returns are uncorrelated. Hence:

11. Simplified Portfolio Construction Under that simple assumption, the active utility function becomes: And the optimal holdings are: So we take active positions in proportion to alphas and inverse proportion to variance. The positions scale inversely with risk aversion. More generally:

13. Utility and Information Ratios Define IR as ratio of expected active return to active risk: –Aside: how is this a measure of performance consistency? Use this as a budget constraint: We know this works if we increase risk by scaling positions. It will break down if we can’t do that—e.g. due to the long-only constraint.

14. Utility and Information Ratios We can rewrite the utility as:

15. Optimal Risk and Utility We can solve for the optimal risk level: And the maximum achievable utility: Why is this a critical result?

16. Information Ratios and Consistency of Performance

17. Information Ratios Typical Numbers: Study Results* *Mutual Funds (300, 1/91-12/93; Institutional Portfolios (367, 10/95-12/96)

18. Active Risk Levels

19. Annualization To annualize monthly alphas, multiply by 12. To annualize monthly risk, multiply by sqrt(12). To annualize IR, multiply by sqrt(12).

20. Risk Aversion Parameter Prior result provides intuition for risk aversion. Intuition 1: Risk aversion is not dimensionless. Intution 2: Typical levels:

21. Fundamental Law of Active Management We have seen that IR is critical measure for active management. The fundamental law can provide intuition into the IR. We will prove the simple case. Chapter 6 in the book proves the general case.

22. Background econometrics* Best linear unbiased estimate of  conditional on information z is: If  and z are normally distributed, then the variance of  conditional on information z is: *We will spend much more time on this topic in the forecasting section.

23. Simpler Case Assume N assets and N signals (one for each asset), active returns uncorrelated, active risks identical, and correlations between signals and returns identical. Signals are N[0,1]. Optimal holdings: What are predicted alpha and risk?

24. Predicted Alpha, Risk Predicted Alpha given z: Predicted Risk given z: What do these imply for our optimal portfolio?

25. Portfolio Alpha and Risk Conditional holdings: Portfolio Alpha: Portfolio Risk:

26. Conditional Information Ratio We can now combine previous results to calculate the information ratio conditional on the signals {z}: Taking the unconditional expectation of the above, we find:

27. Additivity We can see from the proof how IR’s add. For instance, instead of assuming all signals have same strength, we assume strength IC 1 for BR 1 of the stocks, and IC 2 for BR 2 of the stocks. Then: This is generally how IR’s add.

28. Fundamental Law Information Ratio depends on skill and breadth: We measure skill by the information coefficient. To consistently outperform requires skill and breadth. We measure breadth as the number of independent bets per year.

29. Breadth/Skill Tradeoff

30. Understanding IC’s Simple IC model: fraction fr of correct calls (i.e. correctly predicting sign of  ): Typical IC’s:

31. Non-investment Example: The Roulette Wheel Consider an American roulette wheel, with numbers from 1-36, plus 0 and 00. There are 38 numbers in all. We are the casino. Bet on Evens. This pays off if the ball lands on an even number from 2, … 36. Assume that gamblers bet $2 million on such bets on this wheel over the course of a year. The casino has, therefore, $2 million at risk. Two cases: –One $2 million bet. –One million $2 bets.

32. One $2 Million Bet Casino wins with 20/38=52.63% probability. - $2 million -100% + $2 million +100% 47.23% 52.63%

33. One Bet Analysis

35. Million Bet Analysis Invest 1 one-millionth of capital on each bet:

38. Investment Applications of the Fundamental Law More important for intuition and general direction than exact calculations. Example: stock picking versus market timing. –Stock picker follows 200 stocks, with quarterly rebalances based on new information. Correctly calls stocks about 51% of the time. This leads to IR of 0.02*sqrt(800)=0.57. This is top quartile. –Market timer makes 4 independent calls per year and is correct 55% of the time. This leads to IR of 0.1*sqrt(4)=0.20. This is just above the median. Lower IR doesn’t mean market timer will not have some great years.

39. Implementation Efficiency What happens if constraints and costs keep us from the optimal tradeoff between return and risk, Portfolio Q: It’s also true that:

40. But we own Portfolio P What is its information ratio? We call this correlation the transfer coefficient, TC. It measures the efficiency of our implementation, and has a maximum value of 1. Our extended fundamental law is:

41. Transfer Coefficients Prior example showed benefits of stock-picking over market-timing. Now let’s compare long-only stock picking with IC=0.05, BR=500 x 4, TC=0.3; to asset allocation with IC=0.10, BR=12 x 4, TC=0.3: What happens if we convert that asset allocation fund to a global macro hedge fund. We follow exactly the same markets, but invest long-short using (low trading cost) futures contracts. Our TC goes from 0.3 to 0.95: