Suppose the risky portfolio of your 401(k) retirement fund is currently in an S&P 500 index fund, and you are pondering whether you should take some extra risk and allo- cate some funds to Target’s stock, the high-performing discounter. You know that, absent research analysis, you should assume the alpha of any stock is zero. Hence, the mean of your prior distribution of Target’s alpha is zero. Downloading return data for Target and the S&P 500 reveals a residual standard deviation of 19.8%. Given this volatility, the prior mean of zero, and an assumption of normality, you now have the entire prior distribution of Target’s alpha.
One can make a decision using a prior distribution, or refine that distribution by expend- ing effort to obtain additional data. In jargon, this effort is called the experiment. The experiment as a stand-alone venture would yield a probability distribution of possible out- comes. The optimal statistical procedure is to combine one’s prior distribution for alpha with the information derived from the experiment to form a posterior distribution that reflects both. This posterior distribution is then used for decision making.
A “tight” prior, that is, a distribution with a small standard deviation, implies a high degree of confidence in the likely range of possible alpha values even before looking at
S&P 500 Active Pf A HP Dell WMT Target BP Shell
s2(e) 0.0705 0.0572 0.0309 0.0392 0.0297 0.0317 25.7562 a/s2(e) 2.0855 3.0641 6.2544 7.1701 6.0566 1.1255 1.0000 w0(i ) 0.0810 0.1190 0.2428 0.2784 0.2352 0.0437 [w0(i )]2 0.0066 0.0142 0.0590 0.0775 0.0553 0.0019
aA 0.2018
s2(eA) 0.0078
w0 7.9116
w* 0.5661 0.4339 0.0351 0.0516 0.1054 0.1208 0.1020 0.0190
Overall Portfolio
Beta 1 0.9538 0.9800 0.0351 0.0516 0.1054 0.1208 0.1020 0.0190
Risk premium 0.06 0.2590 0.1464 0.0750 0.1121 0.0689 0.0447 0.0880 0.0305 Standard
deviation
0.1358 0.1568 0.1385 0.3817 0.2901 0.1935 0.2611 0.1822 0.1988
Sharpe ratio 0.44 1.65 1.0569
M-square 0 0.1642 0.0835
Benchmark risk 0.0385
the data. In this case, the experiment may not be sufficiently convincing to affect your beliefs, meaning that the posterior will be little changed from the prior. 2 In the con- text of the present discussion, an active forecast of alpha and its precision provides the experiment that may induce you to update your prior beliefs about its value. The role of the portfolio manager is to form a posterior distribution of alpha that serves portfolio construction.
Adjusting Forecasts for the Precision of Alpha
Imagine it is June 1, 2006, and you have just downloaded from Yahoo! Finance the analysts’
forecasts we used in the previous section, implying that Target’s alpha is 28.1%. Should you conclude that the optimal position in Target, before adjusting for beta, is a / s 2 ( e ) 5 .281/.198 2 5 7.17 (717%)? Naturally, before committing to such an extreme position, any reasonable manager would first ask: “How accurate is this forecast?” and “How should I adjust my position to take account of forecast imprecision?”
Treynor and Black 3 asked this question and supplied an answer. The logic of the answer is quite straightforward; you must quantify the uncertainty about this forecast, just as you would the risk of the underlying asset or portfolio. A Web surfer may not have a way to assess the precision of a downloaded forecast, but the employer of the analyst who issued the forecast does. How? By examining the forecasting record of previous forecasts issued by the same forecaster.
Suppose that a security analyst provides the portfolio manager with forecasts of alpha at regular intervals, say, the beginning of each month. The investor portfolio is updated using the forecast and held until the update of next month’s forecast. At the end of each month, T, the realized abnormal return of Target’s stock is the sum of alpha plus a residual:
u(T)5RTGT(T)2bRM(T)5a(T)1e(T) (27.4) where beta is estimated from Target’s security characteristic line (SCL) using data for periods prior to T,
SCL: RTGT(t)5a1bRM(t)1e(t), t,T (27.5) The 1-month, forward-looking forecast a f ( T ) issued by the analyst at the beginning of month T is aimed at the abnormal return, u ( T ), in Equation 27.4. In order to decide on how to use the forecast for month T, the portfolio manager uses the analyst’s forecasting record.
The analyst’s record is the paired time series of all past forecasts, a f ( t ), and realizations, u ( t ). To assess forecast accuracy, that is, the relationship between forecast and realized alphas, the manager uses this record to estimate the regression:
u(t)5a01a1af(t)1 e(t) (27.6) Our goal is to adjust alpha forecasts to properly account for their imprecision. We will form an adjusted alpha forecast a ( T ) for the coming month by using the original forecasts a f ( T ) and applying the estimates from the regression Equation 27.6, that is,
a(T)5a01a1af(T) (27.7)
The properties of the regression estimates assure us that the adjusted forecast is the
“best linear unbiased estimator” of the abnormal return on Target in the coming month, T.
2 In application to debates about social issues, you might define a fanatic as one who enters the debate with a prior that is so tight that no argument will influence his posterior, making the debate altogether a waste of time.
3 Jack Treynor and Fischer Black, “How to Use Security Analysis to Improve Portfolio Selection,” Journal of Business, January 1973.
“Best” in this context means it has the lowest possible variance among unbi- ased forecasts that are linear functions of the original forecast. We show in Appendix A that the value we should use for a 1 in Equation 27.7 is the R -square of the regression Equation 27.6. Because R -square is less than 1, this implies that we “shrink” the forecast toward zero.
The lower the precision of the original forecast (the lower its R -square), the more we shrink the adjusted alpha back toward zero. The coefficient a 0 adjusts the forecast upward if the forecaster has been consistently pessimistic, and down- ward for consistent optimism.
Distribution of Alpha Values
Equation 27.7 implies that the quality of security analysts’ forecasts, as mea- sured by the R -square in regressions of realized abnormal returns on their fore-
casts, is a critical issue for construction of optimal portfolios and resultant performance.
Unfortunately, these numbers are usually impossible to come by.
Kane, Kim, and White 4 obtained a unique database of analysts’ forecasts from an investment company specializing in large stocks with the S&P 500 as a benchmark port- folio. Their database includes a set of 37 monthly pairs of forecasts of alpha and beta values for between 646 and 771 stocks over the period December 1992 to December 1995—in all, 23,902 forecasts. The investment company policy was to truncate alpha forecasts at 1 14% and 2 12% per month. 5 The histogram of these forecasts is shown in Figure 27.3 . Returns of large stocks over these years were about average, as shown in the following table, including one average year (1993), one bad year (1994), and one good year (1995):
1993 1994 1995 1926–1999 Average SD (%)
Rate of return, % 9.87 1.29 37.71 12.50 20.39
The histogram shows that the distribution of alpha forecasts was positively skewed, with a larger number of pessimistic forecasts. The adjusted R -square in a regression of these forecasts with actual alphas was .001134, implying a tiny correlation coefficient of .0337. As it turned out, the optimistic forecasts were of superior quality to the pessimistic ones. When the regression allowed separate coefficients for positive and negative fore- casts, the R -square increased to .001536, and the correlation coefficient to .0392.
4 Alex Kane, Tae-Hwan Kim, and Halbert White, “Active Portfolio Management: The Power of the Treynor-Black Model,” in Progress in Financial Market Research, ed. C. Kyrtsou (New York: Nova, 2004).
5 These constraints on forecasts make sense because on an annual basis they imply a stock would rise by more than 380% or fall below 22% of its beginning-of-year value.
Figure 27.3 Histogram of alpha forecast
−15 −10 −5 0 5 10 15
0 4,000 3,000 2,000 1,000 6,000 5,000 7,000
These results contain “good” and “bad” news. The “good” news is that after adjusting even the wildest forecast, say, an alpha of 12% for the next month, the value to be used by a forecaster when R -square is .001 would be .012%, just 1.2 basis points per month. On an annual basis, this would amount to .14%, which is of the order of the alpha forecasts of the example in Spreadsheet 27.1 . With forecasts of this small magnitude, the problem of extreme portfolio weights would never arise. The bad news arises from the same data:
the performance of the active portfolio will be no better than in our example—implying an M -square of only 19 basis points.
An investment company that delivers such limited performance will not be able to cover its cost. However, this performance is based on an active portfolio that includes only six stocks. As we show in Section 27.5, even small information ratios of individual stocks can add up (see line 11 in Table 27.1 ). Thus, when many forecasts of even low precision are used to form a large active portfolio, large profits can be made.
So far we have assumed that forecast errors of various stocks are independent, an assumption that may not be valid. When forecasts are correlated across stocks, precision is measured by a covariance matrix of forecasting errors, which can be estimated from past forecasts. While the necessary adjustment to the forecasts in this case is algebraically messy, it is just a technical detail. As we might guess, correlations among forecast errors will call for us to further shrink the adjusted forecasts toward zero.
Organizational Structure and Performance
The mathematical property of the optimal risky portfolio reveals a central feature of invest- ment companies, namely, economies of scale. From the Sharpe measure of the optimized portfolio shown in Table 27.1 , it is evident that performance as measured by the Sharpe ratio and M -square grows monotonically with the squared information ratio of the active portfolio (see Equation 8.22, Chapter 8, for a review), which in turn is the sum of the squared information ratios of the covered securities (see Equation 8.24). Hence, a larger force of security analysts is sure to improve performance, at least before adjustment for cost. Moreover, a larger universe will also improve the diversification of the active port- folio and mitigate the need to hold positions in the neutral passive portfolio, perhaps even allowing a profitable short position in it. Additionally, a larger universe allows for an increase in the size of the fund without the need to trade larger blocks of single securities.
Finally, as we will show in some detail in Section 27.5, increasing the universe of securi- ties creates another diversification effect, that of forecasting errors by analysts.
The increases in the universe of the active portfolio in pursuit of better performance naturally come at a cost, because security analysts of quality do not come cheap. However, the other units of the organization can handle increased activity with little increase in cost.
All this suggests economies of scale for larger investment companies provided the organi- zational structure is efficient.
Optimizing the risky portfolio entails a number of tasks of different nature in terms of expertise and need for independence. As a result, the organizational chart of the portfolio management outfit requires a degree of decentralization and proper controls. Figure 27.4 shows an organizational chart designed to achieve these goals. The figure is largely self- explanatory and the structure is consistent with the theoretical considerations worked out in previous chapters. It can go a long way in forging sound underpinnings to the daily work of portfolio management. A few comments are in order, though.
The control units responsible for forecasting records and determining forecast adjust- ments will directly affect the advancement and bonuses of security analysts and estimation experts. This implies that these units must be independent and insulated from organiza- tional pressures.
An important issue is the conflict between independence of security analysts’ opinions and the need for cooperation and coordination in the use of resources and contacts with corporate and government personnel. The relative size of the security analysis unit will further complicate the solution to this conflict. In contrast, the macro forecast unit might become too insulated from the security analysis unit. An effort to create an interface and channels of communications between these units is warranted.
Finally, econometric techniques that are invaluable to the organization have seen a quantum leap in sophistication in recent years, and this process seems still to be accelerat- ing. It is critical to keep the units that deal with estimation updated and on top of the latest developments.
Figure 27.4 Organizational chart for portfolio management
Source: Adapted from Robert C. Merton, Finance Theory, Chapter 12, Harvard Business School.
Product Final Risky Portfolio Customers
Control Is final portfolio super-efficient?
Product
Active Portfolio
Product Passive Portfolio
Data Feedback Feedback
Feedback
Control
Is alpha positive?
Data Feedback
Data
Passive Portfolio Manager Form passive portfolio at minimum cost
Feedback
Control Is correlation with market 1?
Feedback
Feedback Data
Macro Analyst Estimate E(RM), σ(M)
Forecasts
Quality Control
1. Keep forecast records 2. Estimate quality 3. Adjust forecasts Micro Analyst
Estimate alpha values Econometric Unit
1. Estimate betas 2. Estimate σ(e) 3. Help other units
Forecasts Forecasts
Data Forecasts
Estimates
Estimates and Help with Statistics Feedback Feedback
Overall Portfolio Manager 1. Mix active and passive portfolios 2. Monitor performance of active portfolio 3. Monitor performance of passive portfolio
1. Form active portfolio
2. How good are alpha forecasts?
3. How good are beta + residual variance estimates?
Active Portfolio Manager