Benchmark Timing Introduction In Chap. 4 we separated active management into benchmark timing and stock selection and postponed consideration of benchmark timing until a later chapter. We can postpone it no longer. In this chapter we will explore benchmark timing as another avenue for adding value. The main conclusions of this chapter are as follows: • Successful benchmark timing is hard. The potential to add value is small, although it rises with the number of independent bets per year. • Exceptional or unanticipated benchmark return is the key to the benchmark timing problem. Forecasts of exceptional benchmark return lead to active beta positions. • We can generate active betas using futures or stocks with betas different from 1. There is a cost— measured in unavoidable residual risk and transactions costs—associated with relying on stocks for benchmark timing. • Performance measurement schemes exist to evaluate benchmark timing skill. We'll start with the definitions. Defining Benchmark Timing As discussed in Chap. 4, benchmark timing is an active management decision to vary the managed portfolio's beta with respect
Page 542 to the benchmark. If we believe that the benchmark will do better than usual, then beta is increased. If we believe the benchmark will do worse than usual, then beta should be decreased. Notice the relative nature of our expectations—better than usual and worse than usual. We will need some feeling for what we should expect in the usual circumstances. In its purest sense, we should think of benchmark timing as choosing the correct mixture of the benchmark portfolio and cash. This is a one-dimensional problem, and variations along that dimension should not cause any active residual bets in the portfolio; i.e., all the active risk will come from the active beta. This type of benchmark timing is akin to buying or selling futures contracts1 on the benchmark. Benchmark timing is not asset allocation. As we saw in Chap. 18, asset allocation focuses on aggregate asset classes rather than specific individual stocks, bonds, etc. In the simplest case, the aggregates may be domestic equity, domestic bonds and cash. In more complicated cases, the asset allocation may include several kinds of equity and bonds as well as international equities and bonds, real property, and precious metals. International managers betting on several country indices are engaged in global asset allocation, not benchmark timing. The motivation for asset allocation is to simplify an extremely complicated problem. While tactical asset allocation involves 5 to 20 assets, benchmark timing involves only 1. This makes adding value through benchmark timing very difficult, as we can see from the fundamental law of active management. Remember that the information ratio for benchmark timing arises from a combination of forecasting skill, the benchmark timing information coefficient ICBT, and breadth BR, the number of independent bets per year: An independent benchmark timing forecast every quarter leads to a breadth of only 4. Then, according to Eq. (19.1), to generate a 1A forward contract is equivalent to being long the benchmark and short cash, i.e., borrowing to buy the benchmark. A futures contract is very similar to a forward contract.
Page 543 benchmark timing information ratio of 0.5 requires an information coefficient of 0.25—extremely high! The fundamental law of active management captures exactly why most institutional managers focus on stock selection. Stock selection strategies can diversify bets cross-sectionally across many stocks. Benchmark timing strategies can diversify only serially, through frequent bets per year. The fundamental law quantifies this. Significant benchmark timing value added can arise only with multiple bets per year. To keep this point clear, this chapter will monitor the forecast frequency: first once per year and later multiple times per year. Futures versus Stocks Benchmark timing is choosing an active beta. We can implement benchmark timing with futures. We can also implement an active beta without modifying the cash/benchmark mix. For example, if we think the benchmark will be exceptionally strong this month, then we might overemphasize the higher-beta stocks in our portfolio. However, this has three drawbacks. First, we have to take on residual risk as a result of emphasizing one group of stocks over another. Second, we must have faith that we have identified the betas of the stocks correctly. Even the best forecasts of beta are subject to error. There is no error in the pure cash/benchmark trade-off. The beta of cash is exactly 0, and the beta of the benchmark is exactly 1. Third, the transactions costs involved in trading many individual securities are generally much greater than for a forward or futures contract. We can push the residual risk problem a bit further with the following analysis. Let's build the minimum-risk, fully invested portfolio with beta constrained to be βP. Given the beta constraint, the portfolio which minimizes total risk will also minimize residual risk. As shown in the technical appendix, the optimal portfolio is a weighted combination of the benchmark and portfolio C. Its residual variance—the lowest possible residual variance for a fully invested portfolio with specified active beta βPA = βP – 1—is
Page 544 Assuming a benchmark risk of 18 percent and a portfolio C risk of 12 percent, and remembering that the beta of portfolio C is: an active beta of 0.1 leads to a residual risk of at least 1.6 percent. With a moderate level of residual risk aversion (λR = 0.1), this corresponds to a penalty of about 0.25 percent. This analysis of the benefits of using futures rather than stocks to implement benchmark timing strategies has clear implications for situations where the benchmark has no closely associated futures contract. In that case, the potential for adding value through benchmark timing is very small. Value Added In Chap. 4 we derived an expression for the value added by benchmark timing. The key ingredients in this formula are βPA = the portfolio's active beta with respect to the benchmark. This is the decision variable. ΔfB = the forecast of exceptional benchmark return. This is the departure, positive or negative, from the usual level of benchmark return. If µB is the usual annual expected excess return on the benchmark and fB is our refined forecast of the expected excess return on the benchmark over the next year, then ΔfB = fB – µB is the exceptional benchmark return in the next year. σB = the volatility of the benchmark portfolio. λBT = a measure of aversion to the risk of benchmark timing. We will start with the simple case in which we only make one benchmark timing decision per year. In Chap. 4 we determined that the value added by benchmark timing is
Page 545 TABLE 19.1 Active Beta Aversion to Timing Risk λBT Exceptional Forecast ΔfB High, 0.14 Medium 0.09 Low, 0.06 4.00% 0.05 0.08 0.12 2.00% 0.02 0.04 0.06 0.00% 0.00 0.00 0.00 –2.00% –0.02 –0.04 –0.06 –4.00% –0.05 –0.08 –0.12 The optimal level of active beta, , is determined by setting the derivative of Eq. (19.4) with respect to beta equal to zero. We find Table 19.1 shows how , will vary with changes in the exceptional forecast ΔfB and the aversion to benchmark timing risk λBT. Table 19.1 assumes a 17 percent annual volatility for the benchmark. The value added at the optimal beta is Table 19.2 displays this value added, assuming a benchmark volatility of 17 percent. Given only one active decision per year, this corresponds to basis points per year. We can take this analysis a step further by looking in depth at the forecast deviation ΔfB and the risk aversion λBT. In particular, we want to reformulate this analysis to • Avoid the need to forecast the usual expected excess return on the market µB • Make it easier to build up a forecast of exceptional return ΔfB • Avoid having to determine the risk aversion parameter λBT
Page 546 TABLE 19.2 Value Added Aversion to Timing Risk λBT Exceptional Forecast ΔfB High, 0.14 Medium 0.09 Low, 0.06 4.00% 9.9 15.4 23.1 2.00% 2.5 3.8 5.8 0.00% 0.0 0.0 0.0 –2.00% 2.5 3.8 5.8 –4.00% 9.9 15.4 23.1 To begin with, we can see from Eq. (19.5) that the difference ΔfB between the forecast fB and the usual µB drives the optimal active beta. Hence, we can greatly simplify matters by not worrying about either µB or fB and forecasting the exceptional return ΔfB directly. However, we must adjust our thinking from an absolute framework (e.g., fB) to a relative framework (e.g., ΔfB). This view is completely consistent with the approach to forecasting discussed in Chap. 10. Recall that the refined forecast of exceptional benchmark return is where IC = the information coefficient, the correlation between our forecasts and subsequent exceptional benchmark returns. It's a measure of skill. S= score, a normalized signal with mean 0 and standard deviation equal to 1 over time. What is an appropriate level of benchmark timing skill? With sufficient data on past forecasts, you can calculate the IC directly. Without those data, or if you think that the past is not an accurate guide to the future, then reasonable IC estimates are 0.05, 0.1, or 0.15 depending on whether you are good, very good, or terrific. This is where humility should enter the game. Benchmark timing skill is rare. If you assume that you have this skill that most others lack, you may be misleading yourself. As a crude test, consider your ability to forecast whether the benchmark will do better than
Page 547 TABLE 19.3 Scores for Benchmark Timing View Probability Score Very positive 0.11 1.73 Positive 0.22 0.87 No view 0.33 0.00 Negative 0.22 –0.87 Very negative 0.11 –1.73 average.2 With a correlation of IC = 0.1, you would expect to be correct 55 percent of the time. Table 19.3 shows one way to translate qualitative views into quantitative scores. The scores in Table 19.3 have an average of 0 and a standard deviation of 1. The probability column indicates that we should be, on average, very positive one time in nine. Using ΔfB = σB · IC · S, we can calculate the optimal active beta and value added as a function of the score: Table 19.4 displays these relationships, assuming an IC of 0.10, a 17 percent benchmark volatility, and a risk aversion of 0.06. To make the benchmark timing process more transparent, we would like to ignore the risk-aversion parameter and find a more direct way to determine aggressiveness. We can do this using κ, defined in Eq. (19.8). Assuming that the score is normally distributed, a κ of 0.06 implies that the portfolio's beta will lie between 0.94 and 1.06 two-thirds of the time, falling above 1.06 one time in six and below 0.94 one time in six. If that seems too aggressive, then decrease κ. This implies an increase in risk aversion and/or a decrease in information coefficient, but we can also deal with κ 2This does not mean better than the risk-free rate.
Page 548 TABLE 19.4 View Probability Score Forecast Active Beta Value Added Very positive 0.11 1.73 2.94% 0.09 0.12% Positive 0.22 0.87 1.47% 0.04 0.03% No view 0.33 0.00 0.00% 0.00 0.00% Negative 0.22 –0.87 –1.47% –0.04 0.03% Very negative 0.11 –1.73 –2.94% –0.09 0.12% directly. Table 19.5 shows how κ depends on risk aversion and information coefficient. Using κ, we can also examine value added in more detail. Equation (19.9) writes the value added conditional on the score S. The unconditional value added then is using the condition that the scores have mean 0 and standard deviation 1. A very good forecaster, with an IC = 0.10, given a benchmark volatility of 17 percent and a κ of 0.05, can produce a not very impressive expected value added of 4.2 basis points.3 And TABLE 19.5 κ Aversion to Timing Risk Skill Level IC High, 0.14 Medium, 0.09 Low, 0.06 Good 0.05 0.01 0.02 0.02 Very good 0.10 0.02 0.03 0.05 World class 0.15 0.03 0.05 0.07 3The situation is even worse if the forecaster implements the timing bet using stock selection as opposed to futures. The technical appendix will show that even a low aversion to the unavoidable residual risk of that approach will shave 2.9 basis points off that 4.2 basis points.
Page 549 with only one forecast per year, this is 4.2 basis points per year. However, we shouldn't give up yet. The way to add more value with benchmark timing is to make high-quality forecasts more frequently. Forecasting Frequency The analysis to this point has assumed a 1-year investment horizon. That 1-year horizon is mainly responsible for the vanishing contribution of benchmark timing to value added. The strategy's information ratio and value added depend on skill and breadth, according to the fundamental law of active management, and benchmark timing once per year sets the lower positive bound on breadth (1 bet per year). To add more value, we must forecast more frequently.4 Assume that we can make T forecasts per year. Divide the year into T periods, indexed by t = 1, 2, . . . , T, with each period of length 1/T years. For quarterly forecasts, T = 4; for monthly forecasts, T = 12; for weekly forecasts, T = 52; and for daily forecasts, T = 250 trading days. The volatility of the benchmark over any period t will be Period by period, the forecasting rule of thumb still applies: Now the IC is the correlation of the forecast and return over the period of length 1/T. Since we ultimately keep score on an annual basis, we must analyze the annual value added generated by these higher4Consider the plight of a gambler who has a 65 percent chance to ''beat the spread" on the Super Bowl (once per year) compared to another gambler who has a 55 percent chance to beat the spread on each of the 480 (as of 1999) regular season and playoff games.
Page 550 frequency forecasts. It is the sum of the value added for each period. So, appropriately extending Eq. (19.4), we find Using Eq. (19.12), this becomes and therefore the optimal active beta in period t becomes If we forecast once per year, this reduces to Eq. (19.8). If we forecast more frequently, we can be more aggressive. So, according to Eq. (19.15), other things being equal, our active betas will double if we forecast quarterly instead of annually. However, we will also see later that the IC may shrink as we move to shorter time periods. Given the optimal active betas in each period, the annual value added conditional on the sequence of scores {S(1), S(2), . . . , S(T)} is and the unconditional expected annual value added is This is a form of the fundamental law of active management: The optimal value added is proportional to the breadth T of the strategy. Table 19.6 shows this potential for value added for various numbers of forecasts per year and various IC levels. We have assumed a medium aversion to risk of λBT = 0.09. These results assume that each forecast is based on new information. The forecasts must be independent. If you make one yearly forecast, then divide it by 4 and use that for the four quarterly forecasts, you have added no new information. It still counts as only one forecast per year.
Page 551 TABLE 19.6 Value Added Number of Forecasts per Year IC 1 4 12 52 0.1 2.78 11.11 33.33 144.44 0.05 0.69 2.78 8.33 36.11 0.02 0.11 0.44 1.33 5.78 0.01 0.03 0.11 0.33 1.44 To see this concretely, let us represent the benchmark's exceptional return over the year using a binary model: where the θj are independent and equally likely to be +1 or –1. We will further specify this model in the following particular way: Of those 400 components, the first 100 occur in the first quarter, the second 400 occur in the second quarter, etc.5 According to this model, the benchmark exceptional return has annual variance of 400 and annual risk of 20 percent, with quarterly variance of 100 and quarterly risk of 10 percent. First assume that we make only one forecast g per year: where g includes elements of signal, the θk; and elements of noise, the ηj, which are independent of the θk and of each other. Each ηj is equally likely to equal +1 or –1. The variance of this raw forecast is 16; its standard deviation is 4 percent. Using Eq. (19.19), we can calculate both an annual and a quarterly information coefficient 5Alternatively, we could label these binary elements as θij, with i = 1, . . . , 4 and j = 1, . . . 100. The label i would denote the quarter. We prefer the notation in the text because it emphasizes that without additional information, we will not know which binary element influences which quarter.
Page 552 by correlating the forecast g with the annual and quarterly return, respectively. We find We can substitute these results into Eq. (19.17) and see that the value added is identical whether we consider the forecast g annually or quarterly. In contrast, suppose that we receive the same information, but in parcels each quarter. The quarterly forecasts are The IC in each quarter is and we get the full benefit of the four separate forecasts, according to Eq. (19.17). We can also observe that breaking the annual forecastg into four appropriate quarterly forecasts g1 through g4 requires information on precisely which components of our signal apply to which quarters. The signal g in Eq. (19.19) contains only the sum over all the quarterly information. Performance Analysis We have already discussed performance analysis generally in Chap. 17, where we even presented approaches to benchmark timing performance analysis. If we are limited to returns-based performance analysis, Chap. 17 showed how to estimate benchmark timing skill by distinguishing up-market betas from down-market betas.
Page 553 For portfolio-based performance analysis (or for benchmark timing so long as we have ex ante estimates of portfolio betas), we can separate the achieved active systematic return into three components: expected active beta return, active beta surprise, and active benchmark timing return. To do this, we require two parameters: the expected benchmark return µB and the average active beta. The ex ante analysis takes µB as given and assumes that the average active beta is 0. In an ideal world, these parameters would be part of the prior agreement between the manager and the client. With these two parameters, we can attribute the systematic active return, βPA(t), · γB(t), over the time interval {t, t + Δt} as We can interpret these components as 1. The expected active return βPA(t) · µB · Δt 2. Benchmark timing βPA(t) · [γB(t) – µB · Δt] The benchmark timing component measures whether the portfolio's active beta is positive (negative) when the benchmark's excess return is greater (less) than µB · Δt. This benchmark timing term is the realization of exactly what we are hoping for in the benchmark timing utility, Eq. (19.4). The ex post approach to portfolio-based performance attribution is very similar, except that it establishes the average market return and beta target ex post. Let Then we separate the active systematic return as follows:
Page 554 This ex post approach is similar in spirit to the ex ante approach. The two approaches would be identical if we had specified ex ante an average return of and an average beta of . Over the entire period, the first term averages to . The second term averages to zero. The third term, the benchmark timing contribution, when averaged, captures the in-sample covariance between the active beta and the benchmark returns. We can also invent hybrid approaches, with one of the parameters set ex ante and the other ex post. As a final, general comment, the forecasting frequency can also affect this ex post component of benchmark timing. A one-forecast-per-year strategy will exhibit not only a low information ratio and value added, but also a low t statistic. It may require many years of observations to prove with statistical confidence the existence of any benchmark timing skill. Summary Benchmark timing strategies adjust active portfolio betas based on forecasts of exceptional benchmark returns. Benchmark timing is a one-dimensional problem, so whereas stock selection strategies can benefit from diversifying bets cross-sectionally across stocks, benchmark timing strategies can diversify bets only serially, by frequent forecasts per year. Benchmark timing can realistically generate significant value added only through such frequent forecasts per year. The most efficient approach to implementing benchmark timing is through the use of futures, as opposed to the use of stocks with betas different from 1. Performance analysis techniques exist that can measure benchmark timing contributions. Problems 1. Given a risk aversion to benchmark timing of 0.09, an exceptional market return forecast of 5 percent, and market risk of 17 percent, what is the optimal portfolio beta? 2. Bob is a benchmark timer. His IC is 0.05, he bets once per year, and he has a low aversion to benchmark timing risk λBT = 0.06. What is his value added? What is his optimal level of active risk?
Page 555 3. How many years of active returns would you require in order to determine that Bob has statistically significant (95 percent confidence level) benchmark timing skill? 4. How would the answers to problems 1 and 2 change if Bob bet 12 times per year? References Ambachtsheer, Keith P. "Pension Fund Asset Allocation in Defense of a 60/40 Equity/Debt Asset Mix." Financial Analysts Journal, vol. 43, no. 5, 1987, pp. 14–24. Brocato, Joe, and P. R. Chandy. "Does Market Timing Really Work in the Real World?" Journal of Portfolio Management, vol. 20, no. 2, 1994, pp. 39–44. ———. "Market Timing Can Work in the Real World: A Comment." Journal of Portfolio Management, vol. 21, no. 3, 1995, pp. 39–44. Cumby, Robert E., and David M. Modest. "Testing for Market Timing Ability." Journal of Financial Economics, vol. 19, no. I, 1987, pp. 169–189. Gennotte, Gerard, and Terry A. Marsh. "Variations in Economic Uncertainty and Risk Premiums on Capital Assets." Berkeley Research Program in Finance Working Paper 210, May 1991. Henriksson, Roy D., and Robert C. Merton. "On Market Timing and Investment Performance II. Statistical Procedures for Evaluating Forecasting Skills." Journal of Business, vol. 54, no. 4, 1981, pp. 513–533. Larsen, Glen A., Jr. and Gregory D. Wozniak. "Market Timing Can Work in the Real World." Journal of Portfolio Management, vol. 21, no. 3, 1995. Modest, David. "Mean Reversion and Changing Risk Premium in the U.S. Stock Market: A Survey of Recent Evidence." Presentation at the Berkeley Program Finance Seminar, April 3, 1989. Rudd, Andrew, and Henry K. Clasing, Jr. Modern Portfolio Theory, 2d ed. (Orinda, Calif.: Andrew Rudd 1988). Sharpe, William F. "Likely Gains from Market Timing." Financial Analysts Journal, vol. 43, no. 2, 1975, pp. 2–11. ———. "Integrated Asset Allocation." Financial Analysts Journal, vol. 43, no. 5, 1987, pp. 25–32. Wagner, Jerry, Steve Shellans, and Richard Paul. "Market Timing Works Where It Matters Most . . . in the Real World." Journal of Portfolio Management, vol. 18, no. 4, 1992, pp. 86–92. Technical Appendix This technical appendix will investigate how to implement a benchmark timing strategy using stocks instead of futures. It will show that such an approach leads to unavoidable residual risk. Consider the problem of constructing a fully invested portfolio with beta βBT and minimum residual risk.
Page 556 Proposition 1 1. The portfolio BT, is the minimum-risk, fully invested portfolio with β = βBT. As is clear from Eq. (19A.1), it is a linear combination of the benchmark and portfolio C. 2. Portfolio BT is also the minimum-residual-risk, fully invested portfolio with β = βBT. 3. Portfolio BT has residual risk ωBT: where βPA is portfolio BT's active beta: βPA = βBT – 1. Proof To prove item 1, start with the observation that portfolio BT is clearly fully invested, since the weights in Eq. (19A.1) sum to 1, and the benchmark and portfolio C are fully invested. We can also quickly verify that portfolio BT has β = βBT. More generally, we can show that portfolio BT is the solution to the problem It is the minimum-risk portfolio, subject to the constraints of full investment and β = βBT. To derive Eq. (19A.1), we must solve the minimization problem, use the definition of portfolio C, and use the definition of the vector β in terms of the benchmark portfolio. To prove item 2, for any portfolio P, we can decompose total risk as Among the universe of all portfolios with β = βP, the minimum-total-risk portfolio is also the minimum-residual-risk portfolio.
Page 557 To prove item 3, we can calculate the residual holding for portfolio BT: Using Eq. (19A.8), we can directly calculate residual variance and verify Eq. (19A.2). We can use this result to analyze further the value-added implications of this unavoidable residual risk. Assuming T forecasts per year, we will incur expected residual risk over the year of The expected residual variance each period is where the benchmark total variance over period . Using Eq. (19.14) from the main text, which solves for the active beta each period, Eq. (19A.10) becomes Subtracting the value-added cost of this expected unconditional residual variance from the expected unconditional value added from benchmark timing [Eq. (19.16)] leads to As we discussed in the main text, remembering that and assuming σB = 18 percent and σC = 12 percent leads to βC = 4/9. Equation (19A.13) then shows that benchmark timing via stock selection leads to positive net value added only if the investor's
Page 558 aversion to residual risk is significantly less than his or her aversion to benchmark timing risk. Exercise 1. Prove Eq. (19A.1), the formula for the minimum-risk, fully invested portfolio with β = βBT. Applications Exercises Using the MMI stocks, build portfolio BT, the minimum-risk, fully invested portfolio with β = 1.05 relative to the (benchmark) CAPMMI. Also build portfolio C from MMI stocks. 1. What is the beta of portfolio C? 2. Compare portfolio BT to the linear combination of portfolio C and portfolio B (the CAPMMI) according to Eq. (19A.1). 3. What is the residual risk of portfolio BT? Compare the result to Eq. (19A.2).
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