PORT 2 – Analysis of Active Portfolio Management
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Active management is the attempt to beat a benchmark. Everything in this reading follows from taking that sentence literally. The investor is not judged on total return but on return relative to a passive alternative that could have been bought instead at very low cost. Where delegated portfolio management is involved, that comparison is also the natural discipline on the principal–agent problem: the client can always buy the index fund, so the manager has to justify the fee by doing something the index fund does not.
Value added is positive when the managed portfolio beats the benchmark and negative when it lags. A negative figure means the investor would have been better off holding the benchmark over the measurement period, and that gap widens once fees and expenses are counted. Typical benchmarks are broad market indexes such as the MSCI All Country World Index for global equities and the Bloomberg Barclays Global Aggregate Bond Index for global bonds.
Active management theory begins from an assumption that must be stated plainly: markets are not perfectly efficient. If they were, no forecasting advantage would exist and the only rational portfolio would be the index. The theory then asks a narrower and more tractable question. Given some assumed skill in predicting returns, how should a portfolio be built so that the skill is converted into value added as efficiently as possible? The modern answer runs through Grinold (1989), Black and Litterman (1992), and Clarke, de Silva and Thorley (2002).
What makes a benchmark usable
A benchmark is only a fair comparison if it satisfies three requirements.
- Representative. It has to be drawn from the same opportunity set the investor selects from. Comparing a small-cap manager with a large-cap index measures a style tilt, not skill.
- Replicable at low cost. The investor must genuinely be able to hold the benchmark as an alternative. A benchmark containing illiquid positions that cannot be bought cheaply is not a real alternative.
- Verifiable in advance and timely afterwards. Benchmark weights must be knowable ex ante, and benchmark returns must be available ex post without long delay.
Most benchmarks are security market indexes weighted by market capitalisation. Capitalisation weighting has a useful mechanical property: such indexes are largely self-rebalancing, because a security whose price rises automatically takes a larger share, and they can be held simultaneously by every investor. Float adjustment refines this by excluding the portion of a security that is privately held and therefore not available to the investing public.
One consequence of using a float-adjusted capitalisation-weighted market index deserves emphasis. When the benchmark spans all relevant assets, active management becomes a zero-sum game against it. The market portfolio is by construction the average holding of all owners of those securities, so before costs the active investors as a group must earn exactly the market return. One investor can only be overweight a security because another is underweight it. For a benchmark narrower than the total market, the zero-sum property breaks, because investors can hold assets outside the benchmark entirely.
From returns to active weights
The benchmark return is the weighted sum of the individual security returns, and the managed portfolio return has the same form with different weights.
Value added, also called active return, is the plain difference between the two.
A risk-adjusted version of value added is the portfolio alpha, which scales the benchmark return by the portfolio beta before subtracting it.
Now combine the two return equations. What survives is the difference between the weights, and those differences are the active weights.
Because the active weights sum to zero, a constant can be subtracted from every return without changing the sum. Subtracting the benchmark return converts total returns into active security returns, RAi = Ri − RB, and gives the form that carries all the intuition.
Read that equation as a statement about agreement. Value added is positive when securities that beat the benchmark were overweighted and securities that lagged it were underweighted. It is negative when the pairing goes the other way. Nothing else enters. If every asset sits at its benchmark weight, every active weight is zero and there is no value added at all, whatever the market does.
A composite benchmark is 60 percent stocks and 40 percent bonds. An investor who expects stocks to beat bonds over the coming year holds 70 percent stocks and 30 percent bonds, an active weight of +10 percentage points on stocks and −10 percentage points on bonds. After the fact the stock market returns 14.0 percent and the bond market returns 2.0 percent.
Benchmark: 0.60(14.0) + 0.40(2.0) = 9.2%.
Value added: 10.4 − 9.2 = 1.2%.
RA = 0.10(14.0 − 9.2) − 0.10(2.0 − 9.2) = 0.5 + 0.7 = 1.2%.
The overweight in stocks contributed 0.48 percentage points and the underweight in bonds contributed 0.72 percentage points, which the source reports rounded to 0.5 and 0.7. Note that both legs are positive. Being underweight the weaker asset adds value just as surely as being overweight the stronger one, and here the underweight actually contributed more.
Benchmark: 0.60(−14.0) + 0.40(2.0) = −7.6%.
Using the active weights directly, RA = 0.10(−14.0) − 0.10(2.0) = −1.4% − 0.2% = −1.6%. The same single overweight decision that added 1.2 percent in the first case destroys 1.6 percent here. The decision was identical; only the outcome changed.
The MSCI EAFE Index is the benchmark for a portfolio that allocates across countries but does no security selection within them. Weights are those at the start of 2018 and returns are for the 2018 calendar year.
| Country | Benchmark weight | Portfolio weight | 2018 return |
|---|---|---|---|
| United Kingdom | 17% | 16% | −7.6% |
| Japan | 25% | 14% | −9.0% |
| France | 11% | 8% | −3.5% |
| Germany | 9% | 24% | −15.8% |
| Other countries | 38% | 38% | −0.1% |
Data from MSCI.
−0.01(−7.6) − 0.11(−9.0) − 0.03(−3.5) + 0.15(−15.8) = −1.2%.
The Other Countries active weight is zero, so that block contributes nothing regardless of its return. The same answer follows from using active returns rather than total returns, because the active weights sum to zero. The damage is concentrated in one decision: a 15 percentage point overweight in the worst-performing market of the five.
A single value added number tells the client whether the year went well. It does not say which decisions worked. Performance attribution systems break the total into sources, and the most common split is between asset allocation and security selection.
Consider a composite portfolio in which the weights across asset classes differ from a policy benchmark, and within each asset class an active manager selects securities. Total value added is still the difference between the actual portfolio return and the benchmark return, but now both weights and returns differ.
Rewriting this so that the two effects separate gives the standard attribution equation, in which the value added from security selection inside class j is RA,j = RP,j − RB,j.
The first term applies active class weights to benchmark class returns, which is the pure allocation decision. The second term applies actual portfolio class weights to the selection value added inside each class. The arrangement is not unique: it assigns the interaction between the two effects to security selection, because the second term uses portfolio weights rather than policy weights. That is a convention, not a theorem, and it is worth stating when presenting attribution results.
With two asset classes the equation reads more concretely:
The active weights in the first bracket are deviations from the policy portfolio, the long-run strategic allocation, not from the holdings of the previous year. A fund whose policy is 60/40 and which runs 68/32 this year has taken an +8 percentage point allocation bet on equities.
An investor holds two actively managed funds for the 2018 calendar year: 68 percent of the total portfolio in Fidelity Magellan, benchmarked to the S&P 500 Index, and 32 percent in PIMCO Total Return, benchmarked to the Bloomberg Barclays US Aggregate Index. The policy portfolio is 60 percent equities and 40 percent bonds.
| Fund | Fund return (%) | Benchmark return (%) | Value added (%) |
|---|---|---|---|
| Fidelity Magellan | −5.6 | −4.5 | −1.1 |
| PIMCO Total Return | −0.3 | 0.0 | −0.3 |
| Portfolio return | −3.9 | −2.7 | −1.2 |
0.68(−1.1%) + 0.32(−0.3%) = −0.8%.
0.08(−4.5%) − 0.08(0.0%) = −0.4%.
Overweighting equities in a year when equities underperformed bonds cost the portfolio 0.4 percentage points.
Checking directly, the portfolio return was 0.68(−5.6%) + 0.32(−0.3%) = −3.9% and the policy portfolio returned 0.60(−4.5%) + 0.40(0.0%) = −2.7%, a difference of −3.9% − (−2.7%) = −1.2%. The two routes agree.
Attribution generalises without difficulty. A four-class system covering stocks, bonds, real estate and cash simply sets M = 4. Within any class the analysis can be pushed further: an equity sleeve can be decomposed into industry sector over- and underweights and then stock selection inside sectors, and a fixed-income sleeve into the sovereign against corporate mix and then individual bond selection. The principle never changes. Every layer of decomposition is another set of active weights multiplied by the returns they were applied to.
Risk-adjusted performance can be measured in absolute terms or in benchmark-relative terms. The Sharpe ratio is the absolute measure: reward per unit of total risk, where reward is return in excess of the riskless rate.
The same formula serves two purposes. Used ex ante it takes an expected return, E(RP), and a forecast of volatility, both of which are subjective and will differ between investors. Used ex post it takes an average realised return, an average risk-free rate and a sample standard deviation over the measurement period.
Convention annualises both parts. Monthly data are annualised by multiplying the average return by 12 and the standard deviation by the square root of 12. The square root appears because variance, under the assumption that returns are independent across periods, grows in proportion to time.
| Item | MSCI World | S&P 500 | Russell 2000 | MSCI EAFE | Bloomberg Barclays US Aggregate |
|---|---|---|---|---|---|
| Average annual return | 7.9% | 9.9% | 10.3% | 6.3% | 5.0% |
| Return standard dev. | 14.5% | 14.4% | 19.1% | 15.8% | 3.5% |
| Sharpe ratio | 0.38 | 0.53 | 0.41 | 0.25 | 0.77 |
Annualised monthly US dollar returns, not compounded, over the 25-year period. Small differences between the tabulated ratios and the ratio of the two rounded rows above them come from rounding in the reported averages.
Long-run realised Sharpe ratios for equity benchmarks usually land somewhere between 0.20 and 0.60. Over shorter windows they range far more widely and can easily be negative. The fixed-income figure of 0.77 in the table is unusually high, and the reason is specific to the sample: interest rates declined secularly across those 25 years, which lifted realised bond returns well above what a forward-looking investor would assume.
| Item | Fidelity Magellan | Growth Fund of America | Templeton World | T. Rowe Price Small Cap | JPMorgan Bond |
|---|---|---|---|---|---|
| Average annual return | 8.5% | 11.1% | 7.9% | 11.6% | 5.2% |
| Return standard dev. | 16.5% | 15.7% | 15.2% | 16.7% | 3.6% |
| Sharpe ratio | 0.38 | 0.56 | 0.37 | 0.56 | 0.80 |
These funds are shown as illustrations only, with no implication about their merits relative to other choices.
Two rules of hygiene follow. Compare Sharpe ratios only across the same measurement period, never one fund over one window against another fund over a different window. And compare only portfolios whose risk is measured on the same basis.
Cash and leverage leave the Sharpe ratio alone
Combine an actively managed portfolio, weight wP, with risk-free cash, weight (1 − wP). The combined return is RC = wPRP + (1 − wP)RF, and because the cash leg has no volatility the combined risk is simply wPσP. Substituting both into the Sharpe ratio, the weight cancels.
Nothing in the derivation requires wP to be below 1. If it exceeds 1, then (1 − wP) is negative, which is borrowing at the riskless rate to buy more of the risky portfolio. Leverage does not change the Sharpe ratio either.
That invariance is the whole content of two-fund separation. Whatever the risk preference of the investor, the right approach is to hold two things: the riskless asset, and the risky portfolio with the highest Sharpe ratio available. If that risky portfolio is more volatile than the investor wants, hold cash alongside it. If it is less volatile than the investor is willing to bear, lever it. The choice of the risky portfolio is separated entirely from the choice of how much risk to take.
To make that concrete, take an investor who believes the Growth Fund of America record above will repeat but who tolerates only 10 percent volatility. Investing 64 percent in the fund and 36 percent in cash gives an expected return of 0.64(11.1%) + 0.36(2.3%) = 7.9% and a volatility of 0.64(15.7%) = 10.0%. The Sharpe ratio is (7.9% − 2.3%)/10.0% = 0.56, exactly the ratio of the fund on its own.
An investor is choosing between a large-cap and a small-cap stock portfolio and forecasts the following statistics. The risk-free rate is 2.3 percent. The small-cap portfolio has the better forecast Sharpe ratio, but the investor does not want its 19.2 percent volatility.
| Item | Large cap | Small cap |
|---|---|---|
| Expected return | 8.2% | 10.3% |
| Expected volatility | 14.6% | 19.2% |
| Sharpe ratio | 0.40 | 0.42 |
Value added is a relative quantity, so the natural risk adjustment for it is also relative. The information ratio divides active return by active risk, where active risk is the volatility of the active return and is also called benchmark tracking risk.
The ratio is best understood as a measure of the consistency of value added. Two managers who both add 1.5 percent a year are not equally good if one does it in a steady trickle and the other in violent swings around a large tracking error. Most investors prefer the smooth version, and the information ratio is what prices that preference.
As with active return and alpha, there is a beta-adjusted counterpart to active risk, which Grinold and Kahn (1999) called residual risk. Everything in this reading assumes the managed portfolio has a beta of exactly 1.0 against its benchmark, so that active risk and residual risk coincide. The assumption can be relaxed, and Fischer and Wermers (2013) set out an information ratio that does not require it.
Like the Sharpe ratio, the information ratio can be ex ante, using expected active return E(RA) = E(RP) − E(RB) over expected active risk, or ex post, using the realised average active return over the realised sample standard deviation of active return.
| Fund | Benchmark | Active return | Active risk | Information ratio |
|---|---|---|---|---|
| Fidelity Magellan | S&P 500 | −1.4% | 5.1% | −0.27 |
| Growth Fund of America | S&P 500 | 1.2% | 6.2% | 0.20 |
| Templeton World | MSCI World | 0.0% | 5.0% | 0.00 |
| T. Rowe Price Small Cap | Russell 2000 | 1.4% | 4.7% | 0.29 |
| JPMorgan Bond | Bloomberg Barclays US Aggregate | 0.2% | 1.0% | 0.19 |
Active returns are obtained by subtracting the average benchmark return from the average fund return over 1994 to 2018; small residuals against the rounded figures in the two Sharpe ratio tables are rounding effects. Active risk is the annualised standard deviation of the return differences and cannot be reconstructed from summary averages alone.
Three observations belong with that table. Realised information ratios are negative whenever active return is negative, and under the zero-sum property the average realised information ratio across funds sharing a benchmark should be roughly zero. The values here sit in a band of about −0.30 to +0.30, which is typical of a long window; over shorter windows the spread is far larger. Ex ante the calculus is different: an investor who did not expect a positive information ratio would simply buy the benchmark.
Note also that ranking by active risk is not the same as ranking by total risk. Fidelity Magellan has slightly lower active risk than the Growth Fund of America, 5.1 percent against 6.2 percent, yet slightly higher total risk, 16.5 percent against 15.7 percent. Relative and absolute risk are genuinely different measurements.
Two instructive edge cases
A closet index fund advertises itself as active but holds something close to the index. Its excess return and volatility resemble the benchmark, so its Sharpe ratio is close to the benchmark Sharpe ratio and looks respectable. Its active risk is small but strictly positive, being a volatility estimate, and its active return after fees is small or negative, so its information ratio sits near zero or slightly below. The Sharpe ratio hides the problem and the information ratio exposes it. Where holdings are available, closet indexing is easy to detect using active share, defined by Cremers and Petajisto (2009) as one half of the summed absolute active weights.
A market-neutral long–short equity fund holds offsetting long and short positions and has a beta of zero to the market. If the riskless rate is treated as its benchmark, then excess return and active return are the same calculation, as are total risk and active risk, so its Sharpe ratio and information ratio are identical.
What moves the information ratio and what does not
Unlike the Sharpe ratio, the information ratio is affected by adding cash. Cash dilutes the active positions relative to the benchmark, so the ratio generally shrinks.
What the information ratio of an unconstrained portfolio is not affected by is the aggressiveness of the active weights. Multiply every active weight by a constant c. Active return scales by c because it is a linear function of the weights, and active risk scales by c for the same reason, so the ratio is unchanged.
An outside investor cannot reach inside an existing fund to rescale its active weights. The same effect is achieved from outside by taking a position in the benchmark. If a fund has active risk of 5.0 percent, holding it in an 80/20 mix with the benchmark portfolio gives combined active risk of 0.80(5.0%) = 4.0%, with the active return reduced in the same proportion and the information ratio preserved. To go the other way, short the benchmark and put the proceeds into the fund. In institutional practice the short leg is usually unnecessary: the investor simply reduces the amount that would otherwise have gone into the benchmark or into another active fund.
Basic portfolio theory says that with a riskless asset available, the best risky portfolio is the one on the efficient frontier that is tangent to a ray from the risk-free rate, because that portfolio has the highest attainable Sharpe ratio. Since cash and leverage then adjust the risk level without touching the ratio, the risk aversion of the investor never enters the choice of which risky portfolio to hold.
Active management theory has an exact counterpart. Given that active risk and active return can be adjusted freely by mixing the active fund with the benchmark, the squared Sharpe ratio of an actively managed portfolio decomposes into the squared Sharpe ratio of the benchmark plus the squared information ratio.
Two consequences follow immediately. First, among funds sharing a benchmark, the one with the highest squared information ratio delivers the highest squared Sharpe ratio, so the expected information ratio is the single best criterion for selecting an active manager. Second, the equation is useless for ranking managers with negative information ratios, because squaring destroys the sign. An investor should choose the manager with the highest expected information ratio for each asset class, because that choice maximises the Sharpe ratio of the resulting total portfolio.
How much active risk to take
The next question is how aggressive the active positions should be. Remarkably, this can be answered without any utility function or risk aversion parameter. The mean–variance optimality condition requires that the ratio of expected active return to active return variance equal the ratio of expected benchmark excess return to benchmark return variance.
Solving that for the level of active risk gives the aggressiveness that maximises the Sharpe ratio in Equation 7.
The formula is worth reading before it is used. Optimal active risk rises with the information ratio, falls as the benchmark itself becomes more attractive, and scales with benchmark volatility. As the information ratio approaches zero, whether because constraints bite or because the manager is judged unskilled, optimal active risk goes to zero and the optimal portfolio collapses into the passive benchmark. That is the correct answer, not a degenerate one.
Take a worked instance. Suppose an active portfolio has an information ratio of 0.30 with active risk of 8.0 percent, while the benchmark has an expected excess return of 6.4 percent and total risk of 16.0 percent, so its Sharpe ratio is 6.4/16.0 = 0.40. The optimal aggressiveness is (0.30/0.40)16.0% = 12.0%. At that level the Sharpe ratio becomes (0.402 + 0.302)1/2 = 0.50.
The arithmetic can be verified from the ground up. At 12.0 percent active risk the expected active return is 0.30(12.0%) = 3.6%, so total expected excess return is 6.4% + 3.6% = 10.0%. Total risk follows from the fact that portfolio variance is the sum of benchmark variance and active variance.
That gives (16.02 + 12.02)1/2 = 20.0%, and 10.0/20.0 = 0.50, confirming the maximum. Since the fund as it stands carries only 8.0 percent active risk, reaching 12.0 percent requires managing it 12.0/8.0 = 1.5 times more aggressively, or equivalently investing 1.5 times the capital in the fund and funding the difference with a 0.5 times short position in the benchmark.
Reading the same picture in absolute and relative space
The source illustrates this with a pair of charts. In absolute space, individual risky assets are plotted with forecast excess return on the vertical axis and total risk on the horizontal axis, so the slope of a line from the origin to any point is the Sharpe ratio of that asset. The benchmark portfolio combines those assets and, thanks to diversification, sits on a steeper ray than most individual assets: an expected excess return of 5.0 percent against volatility of 10.8 percent, a Sharpe ratio of 5.0/10.8 = 0.46. But the benchmark is not the best available combination. The maximum-Sharpe-ratio portfolio built from the same assets has an expected excess return of 8.7 percent and volatility of 14.2 percent, a ratio of 8.7/14.2 = 0.61. Sliding down that steeper ray with cash to match benchmark risk of 10.8 percent would yield an expected excess return of 0.61(10.8%) = 6.6%, against the 5.0 percent offered by the benchmark itself.
Re-plot the same assets in relative space, with forecast active return against active risk. The benchmark now sits exactly at the origin, since it has zero active return and zero active risk by construction. Individual assets scatter on both sides of zero active return. The optimal portfolio has an active return of 3.8 percent and active risk of 9.4 percent, an information ratio of 3.8/9.4 = 0.40, and that slope exceeds the information ratio of every individual asset. In fact the optimal information ratio is the square root of the sum of the squared individual asset information ratios, including assets whose information ratios are negative, which is the same quadrature structure as Equation 7. Assets with negative information ratios enter the optimum with negative weights, so short positions may be required.
The two pictures are consistent. The active risk needed to construct that optimal portfolio is (0.40/0.46)10.8% = 9.4%, exactly Equation 8, and the resulting Sharpe ratio is (0.462 + 0.402)1/2 = 0.61, exactly the tangency ratio found in absolute space. Optimal active risk levels in real equity mandates are usually far below 9.4 percent, because long-only constraints depress achievable information ratios well below 0.40.
Assume the historical records below are indicative of future performance. Fund I has the Fidelity Magellan profile and Fund II the Growth Fund of America profile, both benchmarked to the S&P 500. The risk-free rate is 2.3 percent. In practice these would be forecast values set by the investor rather than history.
| Item | S&P 500 | Fund I | Fund II |
|---|---|---|---|
| Average annual return | 9.9% | 8.5% | 11.1% |
| Return standard dev. | 14.4% | 16.5% | 15.7% |
| Sharpe ratio | 0.53 | 0.38 | 0.56 |
| Item | Fund I | Fund II |
|---|---|---|
| Active return | −1.4% | 1.2% |
| Active risk | 5.1% | 6.2% |
| Information ratio | −0.27 | 0.20 |
Checking the result: at 5.4 percent active risk the expected active return is 0.20(5.4%) = 1.1%. The benchmark excess return is 9.9% − 2.3% = 7.6%, so total expected excess return is 7.6% + 1.1% = 8.7%. Total risk is (14.42 + 5.42)1/2 = 15.4%, and the Sharpe ratio is 8.7/15.4 = 0.57, matching question 2. Dividing the rounded 8.7 by the rounded 15.4 literally gives 0.565, which rounds up to 0.57. Equation 7 applied to the rounded benchmark Sharpe ratio, (0.532 + 0.202)1/2, gives 0.5665, which also rounds to 0.57. Carrying every intermediate through without rounding instead gives 0.564, which is the more defensible figure; the two rounding routes above land on 0.57 only because rounding the benchmark Sharpe ratio to 0.53 nudges the result upward.
In summary, the information ratio measures relative reward per unit of relative risk, it is invariant to the aggressiveness of active weights, and the improvement it can deliver to a Sharpe ratio depends on its square. That makes the expected information ratio the right criterion for building an active portfolio, and the realised information ratio the right criterion for judging one after the fact.
The fundamental law is a framework for thinking about how much value active management can be expected to add. It can be used to size individual active weights, to forecast the value added of a strategy, or to measure what a strategy actually delivered. Its most common use is descriptive: it lets an analyst take a strategy apart into a small number of interpretable parameters and ask which of them is doing the work.
The starting assumption is that the investor wants to maximise active return subject to a ceiling on active risk. To do that the investor forms a forecast for each candidate security of its active return.
The forecast itself is written μi, a subjective expectation of that active return rather than anything produced by an equilibrium model. Other definitions of active security return are available depending on the risk model in use. A single-factor version defines it as a residual, RAi = Ri − βiRB, where the beta is the sensitivity of the security to the benchmark. This resembles the CAPM but is not the CAPM: the benchmark need not be the market portfolio, and the fundamental law does not require the CAPM, the arbitrage pricing theory or any other equilibrium theory to hold. A multi-factor version removes K market-wide factors.
The three corners of the problem
Three sets of numbers determine everything: the forecast active returns μi, the active weights Δwi the investor actually takes, and the realised active returns RAi. Realised value added is the sum product of the last two, and expanding that sum shows exactly what it depends on.
That expansion uses the fact that when the mean of a variable is zero, its population covariance reduces to the plain average of the products, and its variance to the average of the squares.
So value added is driven by a correlation coefficient: the cross-sectional correlation between the active weights taken and the active returns that actually happened. That single number can be broken into two more useful ones.
- Information coefficient (IC). The correlation between forecast active returns and realised active returns. This is signal quality, the ability of the investor to tell winners from losers. There is no hope of adding value if forecasts bear no relation to outcomes.
- Transfer coefficient (TC). The correlation between forecast active returns and the active weights actually taken. This is implementation quality: how much of the forecast survives portfolio construction. A perfect forecast that constraints prevent you from acting on adds nothing.
Value added therefore requires both. Skill without transmission is wasted, and transmission without skill merely converts noise into tracking error.
Sizing the active weights
Assuming active returns are uncorrelated across securities, mean–variance optimisation subject to a limit on active portfolio risk produces this set of optimal active weights.
Every term does what intuition demands. A larger forecast active return justifies a larger deviation from the benchmark weight. Higher forecast volatility for that security shrinks the deviation, because the bet is less reliable. And the whole vector scales with σA, so wanting more active risk means taking bigger positions everywhere.
Proofs of the fundamental law also assume that the return forecasts have been scaled before optimisation using the Grinold (1994) rule, often summarised as alpha equals volatility times IC times score.
The scores carry the ranking and the two multipliers restore the correct magnitude. The volatility term is security specific and the IC term is a single number applied to all securities. Insisting that the scores have unit cross-sectional variance is what makes the scaling meaningful: if the assumed IC is low, the resulting spread of expected active returns is correspondingly narrow, which is exactly right for a weak signal. Real forecasting processes may need something more elaborate than this rule, depending on how views are originally expressed.
Substituting the Grinold rule into the optimal weights replaces the awkward square root of a sum with two clean parameters, IC and breadth, BR, where breadth has taken the place of N.
Information coefficient
IC is the ex ante cross-sectional correlation between the N forecast active returns and the N realised active returns. Stated precisely it is a risk-weighted correlation.
As a correlation it can in principle run from −1.00 to +1.00, but small positive values below 0.20 are the norm in practice. The anticipated IC must be positive, otherwise there is no case for active management at all and the investor should hold the benchmark. The realised information coefficient, discussed later, can of course come out negative, and when it does the value added is negative.
Breadth
Breadth is the number of genuinely independent decisions the investor makes per year. The simplest case is a single-factor risk model in which the only source of correlation between securities is the common market factor and forecasts do not persist from one year to the next. Then each security is one independent decision and breadth equals the number of securities.
Real risk models are richer, and breadth departs from the security count in both directions.
- Cross-sectional dependence lowers breadth. If the risk model says every security in an industry is positively correlated, then a view that the whole industry will outperform is one opinion, not thirty. Breadth is intuitively lower than the number of securities.
- Negative correlation raises breadth. If the risk model assigns negative active return correlations to some pairs, breadth can exceed the number of securities. In these cases breadth is generally not a whole number.
- Persistence over time lowers breadth. If a forecast rests on a characteristic that barely changes from quarter to quarter, then a fresh forecast each quarter is not a fresh decision.
- Genuine rebalancing raises breadth. If quarterly or monthly forecasts really are independent over time, breadth can rise to the number of securities multiplied by the number of rebalancing periods per year.
Four securities have active returns that are uncorrelated with each other and active return volatilities of 25.0 percent and 50.0 percent. An investor believes the first two will outperform the other two over the coming year and assigns scores accordingly.
| Security | Score | Volatility |
|---|---|---|
| #1 | 1.0 | 25.0% |
| #2 | 1.0 | 50.0% |
| #3 | −1.0 | 25.0% |
| #4 | −1.0 | 50.0% |
#2: 0.20(50.0%)(1.0) = 10.0%. #3: 0.20(25.0%)(−1.0) = −5.0%. #4: 0.20(50.0%)(−1.0) = −10.0%.
Securities with identical scores receive different forecasts because their volatilities differ. The more volatile security has the larger expected active return for the same conviction.
Δw1* = (0.05 ÷ 0.252) × [0.09 ÷ (0.20 × 41/2)] = 0.80 × 0.225 = 18.0%.
The remaining three follow the same route.
| Security | Expected active return | Active return volatility | Active weight |
|---|---|---|---|
| #1 | 5.0% | 25.0% | 18.0% |
| #2 | 10.0% | 50.0% | 9.0% |
| #3 | −5.0% | 25.0% | −18.0% |
| #4 | −10.0% | 50.0% | −9.0% |
Expected value added is the sum product of active weights and forecast active returns.
Substituting the optimal active weights and the Grinold-scaled forecasts collapses that sum into three parameters. This is the basic fundamental law.
Dividing through by active risk gives the same statement in ratio form, and this version is the one worth memorising because it strips out the aggressiveness decision entirely.
The unconstrained information ratio depends only on skill and on the number of independent chances to apply it. It does not depend on how hard the strategy is pushed, which is the invariance property established earlier. The algebra behind Equation 13 assumes breadth equals the number of securities; Clarke, de Silva and Thorley (2006) provide the general proof where the two differ.
The square root is the most important feature of the formula. Doubling breadth does not double the information ratio, it multiplies it by about 1.41. Quadrupling breadth doubles it. Skill, by contrast, enters linearly. A manager who doubles IC doubles the information ratio; a manager who wants the same improvement from breadth alone must find four times as many independent decisions.
Take the four securities from the previous example, with active returns uncorrelated across securities and forecasts independent over time.
| Security | Expected active return | Active return volatility | Active weight |
|---|---|---|---|
| #1 | 5.0% | 25.0% | 18% |
| #2 | 10.0% | 50.0% | 9% |
| #3 | −5.0% | 25.0% | −18% |
| #4 | −10.0% | 50.0% | −9% |
| Security | Total weight | Total return forecast |
|---|---|---|
| #1 | 43% | 15.0% |
| #2 | 34% | 20.0% |
| #3 | 7% | 5.0% |
| #4 | 16% | 0.0% |
| Total | 100% |
0.43(15.0) + 0.34(20.0) + 0.07(5.0) + 0.16(0.0) = 13.6%.
Active return is 13.6 − 10.0 = 3.6%. The same answer comes from active weights times active returns:
0.18(5.0%) + 0.09(10.0%) + (−0.18)(−5.0%) + (−0.09)(−10.0%) = 0.9 + 0.9 + 0.9 + 0.9 = 3.6%.
All four positions contribute 0.9 percentage points each, which is what optimal sizing under this scaling rule produces.
[0.182 × 25.02 + 0.092 × 50.02 + (−0.18)2 × 25.02 + (−0.09)2 × 50.02]1/2 = 9.0%,
which is exactly the active risk constraint that was imposed when the weights were sized.
In ratio form, IR* = 3.6/9.0 = 0.40, and IC × BR1/2 = 0.20 × 41/2 = 0.40. The law is an identity here, not an approximation, because the weights were built to be optimal.
The optimal weight formula assumes nothing stands in the way of taking the positions it prescribes. Reality intervenes. A large negative optimal active weight may imply an absolute weight below zero, that is, a short sale, and many investors are long only either by regulation or by choice, because shorting adds cost and complexity. Add turnover limits, position caps, ESG screens and country ceilings, and closed-form optimal weights disappear. Quantitative investors then use a numerical optimiser; less quantitative investors set weights judgmentally. Either way the fundamental law framework can still be applied after the fact to analyse what those weights imply.
Write Δwi, without an asterisk, for the actual constrained active weights. The transfer coefficient is the risk-weighted cross-sectional correlation between the forecast active returns and those actual weights.
Because optimal weights are themselves proportional to the scaled forecasts, the same quantity can be written as the risk-weighted correlation between the optimal weights and the actual ones.
As a correlation, TC could range from −1.00 to +1.00. In practice it is positive and typically falls between about 0.20 and 0.90. At TC = 0.00 the active weights bear no relation to the forecasts and there is no expectation of value added whatever the skill level. At TC = 1.00 no constraint binds and the full forecast is expressed in the portfolio. A negative TC is conceivable, for instance in a portfolio badly in need of rebalancing, where current relative weights have drifted into opposition to current expected returns.
Inserting the transfer coefficient gives the expanded law, and this is the version used from here on.
These four parameters are the whole analytical vocabulary of the reading. Skill and implementation enter linearly, breadth enters as a square root, and aggressiveness scales the return without touching the ratio. It is worth noting that only the first, skill, normally lies within the control of the manager. Breadth is largely a property of the mandate, aggressiveness is often set by an investment policy tracking error budget, and the transfer coefficient is set by the constraints imposed from outside.
Two simplifications are baked into these equations. They assume a single-index model, so active security returns are residual returns that are uncorrelated with each other. If one goes further and assumes every security has the same residual volatility, the risk weighting inside the IC and TC correlations becomes unnecessary. Moving the other way, towards realism, one can use a single-factor model with security betas, RAi = Ri − βiRB, or the multi-factor risk models used in quantitative practice. The parameter values become harder to compute but the form of the law survives intact.
The same four securities are now managed by a numerical optimiser subject to several constraints in addition to the 9.0 percent active risk limit. The optimal weights from before are shown alongside the actual constrained weights that resulted.
| Security | Expected active return | Active return volatility | Optimal active weight | Actual active weight |
|---|---|---|---|---|
| #1 | 5.0% | 25.0% | 18% | 6% |
| #2 | 10.0% | 50.0% | 9% | 4% |
| #3 | −5.0% | 25.0% | −18% | 7% |
| #4 | −10.0% | 50.0% | −9% | −17% |
For the optimal weights the transfer coefficient is 1.0 by definition, which the same correlation calculation confirms. There the risk-weighted value for Security #1 is 0.18(25.0) = 4.5% against the same 20.0%.
The reason the constrained TC falls so far is visible in the table: Security #3 has a forecast active return of −5.0 percent but carries a positive actual weight of +7 percent. The constraint has forced a position that contradicts the forecast outright.
0.06(5.0) + 0.04(10.0) + 0.07(−5.0) + (−0.17)(−10.0) = 0.3 + 0.4 − 0.35 + 1.7 = 2.1%.
Active risk with the actual weights:
[(0.06)2(25.0)2 + (0.04)2(50.0)2 + (0.07)2(25.0)2 + (−0.17)2(50.0)2]1/2 = 9.0%, as required by the active risk constraint.
This is the essential comparison. The constrained portfolio spends exactly the same 9.0 percent risk budget as the optimal portfolio but earns 2.1 percent instead of 3.6 percent. The risk is paid for in full and only part of the return is collected.
Optimal aggressiveness once constraints exist
The transfer coefficient also changes how much active risk should be taken. Equation 8 generalises by simply multiplying through by TC.
Running at that level of aggressiveness delivers the best Sharpe ratio the constrained portfolio can reach.
Notice what happens at TC = 0.00: optimal active risk is zero and the portfolio should simply be the benchmark. Constraints severe enough to sever the link between forecasts and weights make active management pointless rather than merely inefficient.
A numerical illustration. Suppose a portfolio has a transfer coefficient of 0.50 and an unconstrained information ratio of 0.30, while the benchmark has a Sharpe ratio of 0.40 and risk of 16.0 percent. Optimal aggressiveness is 0.50(0.30/0.40)16.0 = 6.0%, half the 12.0 percent that the same unconstrained information ratio justified earlier. The resulting Sharpe ratio is (0.402 + 0.502 × 0.302)1/2 = 0.43, against 0.50 without constraints. If the constrained portfolio as delivered carries 8.0 percent active risk, the investor lowers it to 6.0 percent by holding 1 − 6.0/8.0 = 25% in the benchmark and 75.0 percent in the active fund.
Everything so far has been about expected value added. Realised performance in any period will scatter around that expectation within a range set by the tracking risk. The natural next question is how to interpret a realised result: how much of it reflected the forecasting process working or failing, and how much was an accident of implementation.
The sign and size of realised value added depend on whether positive active weights landed on securities with positive relative returns. That relationship is captured by the realised information coefficient, ICR, which is simply the ex post version of IC. Conditional on knowing ICR, the expected value added follows the same law.
Actual active return will still differ from that conditional expectation, and the gap is attributed to noise.
Realised value added therefore splits into two pieces: the part explained by how skilful the forecasts turned out to be, and the part caused by constraints pushing the portfolio away from the structure the forecasts called for. Clarke, de Silva and Thorley (2005) showed that the variance of realised active return divides in the same way, with a share TC2 attributable to variation in the realised information coefficient and 1 − TC2 attributable to constraint-induced noise.
That split is unforgiving at low transfer coefficients. At TC = 0.60, only TC2 = 36% of the variation in performance traces back to forecasting, and 1 − TC2 = 64% comes from constraints. A manager in that position will regularly live through periods when the research process worked and the fund still lagged, and periods when the research process failed and the fund still beat its benchmark. Neither outcome tells the investor much about skill, which is precisely why short-run performance evaluation of a heavily constrained manager is close to uninformative.
An active strategy involves BR = 100 investment decisions, for example 100 individual stocks with uncorrelated active returns and annual rebalancing. The expected information coefficient is IC = 0.05, the transfer coefficient is TC = 0.80, and annualised active risk is σA = 4.0%. Expected value added under the fundamental law is therefore 0.80 × 0.05 × 1001/2 × 4.0% = 1.6%.
Noise has an expected value of zero, so conditional on the forecasting process having gone wrong, the investor should expect a negative active return of this size.
−2.6 − (−3.2) = +0.6%.
Constraints happened to soften the damage this time. They could just as easily have deepened it. Note also that a realized active return of −2.6 percent sits comfortably inside the range implied by a 4.0 percent tracking error, so nothing about this period is statistically remarkable.
The framework is easier to trust once it has been run against a realistic portfolio. This example benchmarks an actively managed fund to the MSCI All Country World Index and treats the individual MSCI market indexes as the investable assets: the 21 EAFE markets, the United States, Canada and the Emerging Markets Index, 24 assets in total. The valuation date is the start of calendar 2019. Active risk estimates assume the future resembles the past and are built from US dollar returns on those indexes over 2009 to 2018. In a live process, judgement or a commercial risk model would supply the forecasts. The rankings shown are hypothetical.
Active risk for each market is the annualised standard deviation of the beta-adjusted difference between that market return and the ACWI return. The United Kingdom carries 6.4 percent and Japan 9.1 percent. These are active risks, not total risks; total risk for each market would be higher once benchmark risk and beta are added back.
| Market | Score | Active vol. | Forecast active return | Active weight | ACWI weight | Total weight |
|---|---|---|---|---|---|---|
| United States | 0.0 | 3.8% | 0.0% | −5.3% | 54.3% | 48.9% |
| Emerging | 0.0 | 9.0% | 0.0% | 0.1% | 11.9% | 12.0% |
| Japan | 0.0 | 9.1% | 0.0% | −2.0% | 7.6% | 5.7% |
| United Kingdom | 2.0 | 6.4% | 1.3% | 15.7% | 5.2% | 20.9% |
| France | −2.0 | 8.5% | −1.7% | −12.0% | 3.4% | −8.5% |
| Canada | −1.0 | 9.2% | −0.9% | −6.7% | 3.0% | −3.7% |
| Germany | 0.0 | 8.4% | 0.0% | 2.8% | 2.7% | 5.6% |
| Switzerland | 2.0 | 7.8% | 1.6% | 9.7% | 2.7% | 12.3% |
| Australia | 0.0 | 11.1% | 0.0% | −1.0% | 2.1% | 1.1% |
| Hong Kong SAR | 1.0 | 12.0% | 1.2% | 2.3% | 1.2% | 3.5% |
| Netherlands | 0.0 | 7.9% | 0.0% | 1.8% | 1.1% | 2.9% |
| Spain | −2.0 | 16.0% | −3.2% | −5.3% | 1.0% | −4.3% |
| Sweden | 0.0 | 10.1% | 0.0% | −1.2% | 0.8% | −0.4% |
| Italy | −1.0 | 15.1% | −1.5% | −0.8% | 0.7% | −0.1% |
| Denmark | 0.0 | 12.3% | 0.0% | −1.2% | 0.5% | −0.7% |
| Singapore | 0.0 | 11.8% | 0.0% | −1.8% | 0.4% | −1.3% |
| Belgium | 1.0 | 10.2% | 1.0% | 4.4% | 0.3% | 4.7% |
| Finland | −1.0 | 13.7% | −1.4% | −2.5% | 0.3% | −2.2% |
| Israel | 1.0 | 15.3% | 1.5% | 1.4% | 0.2% | 1.6% |
| Norway | 0.0 | 13.7% | 0.0% | 0.1% | 0.2% | 0.3% |
| Ireland | 0.0 | 14.9% | 0.0% | 0.3% | 0.2% | 0.5% |
| Austria | −1.0 | 14.6% | −1.5% | −1.6% | 0.1% | −1.6% |
| New Zealand | 1.0 | 14.2% | 1.4% | 1.8% | 0.1% | 1.8% |
| Portugal | 0.0 | 15.2% | 0.0% | 0.9% | 0.0% | 0.9% |
| Total | 0.0 | 0.0% | 100.0% | 100.0% |
Markets are shown here in descending order of ACWI weight. Fundamental law parameters for this portfolio: transfer coefficient 0.995, information coefficient 0.099, breadth 24.5, active return 0.98%, active risk 2.00%, information ratio 0.49. Risk statistics based on MSCI returns from 2009 to 2018.
Scores take one of five values: 2.0 for strong outperformance, 1.0 for weak outperformance, 0.0 for neutral, −1.0 for weak underperformance and −2.0 for strong underperformance. How many markets fall into each category is not free. The scores must sum to zero and must have a cross-sectional standard deviation of 1, which is what makes the Grinold scaling valid.
Expected active returns follow the Grinold rule of IC times volatility times score, with an assumed IC of 0.10. The United Kingdom, scored 2.0, gets 0.10(6.4)(2.0) = 1.3%. Japan, scored 0.0, gets 0.0 percent regardless of its volatility. Note the IC used for fundamental law accounting at the foot of the table is 0.099 rather than the 0.10 used to build the forecasts, a point explained shortly.
Active weights come from a numerical optimiser maximising expected active return subject to a 2.00 percent active risk constraint. They correlate strongly with the forecasts but are not proportional to them, for two reasons.
- The optimiser also uses the estimated correlations between market active returns, so a market whose active return is correlated with another already-held position is sized differently.
- The active weights are constrained to sum to zero, a budget constraint that ties every weight to every other.
| Market | FR | DE | ES | CH | GB | SE | AU | JP |
|---|---|---|---|---|---|---|---|---|
| FR | 1.000 | 0.30 | 0.34 | 0.16 | 0.21 | 0.15 | −0.03 | −0.08 |
| DE | 0.30 | 1.000 | 0.19 | 0.10 | 0.08 | 0.15 | −0.07 | −0.03 |
| ES | 0.34 | 0.19 | 1.000 | 0.11 | 0.18 | 0.11 | 0.01 | 0.01 |
| CH | 0.16 | 0.10 | 0.11 | 1.000 | 0.13 | 0.10 | 0.06 | 0.04 |
| GB | 0.21 | 0.08 | 0.18 | 0.13 | 1.000 | 0.14 | 0.00 | −0.02 |
| SE | 0.15 | 0.15 | 0.11 | 0.10 | 0.14 | 1.000 | 0.06 | −0.07 |
| AU | −0.03 | −0.07 | 0.01 | 0.06 | 0.00 | 0.06 | 1.000 | −0.08 |
| JP | −0.08 | −0.03 | 0.01 | 0.04 | −0.02 | −0.07 | −0.08 | 1.000 |
Eight largest EAFE markets only; the full 24-market matrix is not shown. Based on MSCI returns from 2009 to 2018.
These are correlations of active returns, which is why they are small and take both signs. The United Kingdom and Japan sit at −0.02, essentially unrelated, while France and Germany sit at 0.30. Correlations of total market returns would all be positive and much larger, in the region of 0.4 to 0.9. Removing the common benchmark return is what strips out the shared market factor and leaves something close to independent bets.
Reading the parameters
The transfer coefficient is 0.995, almost perfect, because the optimisation is essentially unconstrained. It is not exactly 1.0 only because of the budget constraint forcing active weights to sum to zero. Were the sum allowed to be non-zero, effectively permitting cash or leverage inside the equity portfolio, the transfer coefficient would be exactly 1.0.
Breadth is 24.5, slightly more than the 24 assets, because the risk model includes non-zero active return correlations. Had every off-diagonal correlation been exactly zero, breadth would have been exactly 24.0.
The portfolio itself is relatively unconstrained. Positive total weights sum to about 120 percent and negative total weights to about −20 percent, which practitioners would describe as a 120/20 long–short structure. France, with a total weight of −8.5 percent, is genuinely short. The largest active weight is the United Kingdom at 15.7 percent and the smallest is France at −12.0 percent. The United States shows how active and total weights come apart: its active weight is −5.3 percent, but because the ACWI benchmark weight is 54.3 percent, the total weight is still 48.9 percent, the largest single holding in the fund.
Now check the law. E(RA) = 0.995 × 0.099 × (24.5)1/2 × 2.00 = 0.98%, which is exactly the active return the optimiser produced. In ratio form, IR = 0.995 × 0.099 × (24.5)1/2 = 0.49, matching 0.98/2.00 = 0.490. Because this portfolio is essentially unconstrained, raising the active risk target to 3.00 percent would raise the forecast active return proportionally to 1.47 percent, leaving the information ratio at 1.47/3.00 = 0.49.
A more ambitious set of scores
Now swap two pairs of scores: Germany and the United Kingdom exchange scores, and so do Australia and Switzerland. Breadth is unchanged at 24.5, but the information coefficient used in the accounting rises from 0.099 to 0.105, even though the IC used to scale the forecasts is still 0.10.
The reason is that the new scores represent a more ambitious forecast relative to the risk model. France and Germany are now forecast to move strongly in opposite directions, yet the risk model says their active returns are positively correlated at 0.30. Betting against a positive correlation is a bolder claim than betting with it, and the accounting IC recognises that.
| Parameter | Original scores | Switched scores |
|---|---|---|
| Transfer coefficient | 0.995 | 0.997 |
| Information coefficient | 0.099 | 0.105 |
| Breadth | 24.5 | 24.5 |
| Active return | 0.98% | 1.04% |
| Active risk | 2.00% | 2.00% |
| Information ratio | 0.49 | 0.52 |
Under the switched scores the United Kingdom active weight falls to 1.7% and Germany rises to 14.5%, while Australia rises to 7.7% and Switzerland falls to −2.0%.
Applying the law to the switched scores gives IR = 0.997 × 0.105 × (24.5)1/2 = 0.52, matching the 0.52 reported for the portfolio and equal to 1.04/2.00. The source text records this product as 0.532; the three inputs as printed multiply to 0.518, which rounds to the 0.52 that the exhibit itself reports, so 0.52 is the figure to carry forward.
What constraints do to the same forecasts
Keep the original scores and forecasts and now impose two constraints. First, the portfolio must be long only, so no negative active weight can exceed the benchmark weight in absolute size. France, with a benchmark weight of 3.4 percent, can carry an active weight no more negative than −3.4 percent, giving a total weight of zero. Second, no market may be more than 10.0 percentage points over or under its benchmark weight, which binds on the United Kingdom and Switzerland at +10.0 percent and on the United States at −10.0 percent.
| Parameter | Active risk target 2.00% | Active risk target 3.00% |
|---|---|---|
| Transfer coefficient | 0.694 | 0.567 |
| Information coefficient | 0.099 | 0.099 |
| Breadth | 24.5 | 24.5 |
| Active return | 0.68% | 0.76% |
| Active risk | 2.00% | 2.74% |
| Information ratio | 0.34 | 0.28 |
Long-only and maximum 10 percentage point over- or underweight constraints, same score assignment as the unconstrained fund. Under the higher risk target the New Zealand active weight rises from 4.7% to the 10.0% cap and Israel likewise reaches 10.0%.
The constraints cut the transfer coefficient from 0.995 to 0.694. Through the law, E(RA) = 0.694 × 0.099 × (24.5)1/2 × 2.00 = 0.68%, against 0.98 percent unconstrained, and the information ratio falls from 0.49 to 0.694 × 0.099 × (24.5)1/2 = 0.34. Skill and breadth are unchanged. Roughly 31 percent of the expected value added has been removed purely by portfolio construction rules.
The second column shows something more subtle and more important. In the unconstrained fund, raising the active risk target from 2.00 to 3.00 percent scaled active return proportionally and left the information ratio alone. Here it does not. Larger target active risk means larger unconstrained active weights, which means the caps bind on more markets, which drags the transfer coefficient down from 0.694 to 0.567. The information ratio falls to 0.567 × 0.099 × (24.5)1/2 = 0.28, matching the reported 0.76/2.74 = 0.28.
Compare two active strategies. The first selects individual stocks from a benchmark of 100 securities. The second selects industrial sectors from a benchmark of nine sectors. In both cases the active asset returns are residuals from a risk model, so they are essentially uncorrelated, and the forecasts do not carry over from one year to the next. The stock selector has an information coefficient of 0.05; the sector selector has a higher information coefficient of 0.15.
Sector selection: IR = 0.15 × 91/2 = 0.45.
The sector investor is three times as skilful and still finishes second, because the security investor has eleven times the breadth. That is the square root at work: three times the skill needs nine times the breadth to be matched, and 100 against 9 is slightly more than eleven times.
The global equity example was purely cross-sectional: many assets, one decision date. Fixed-income strategies bring in the time-series side of the law, where breadth comes from repeated decisions rather than from many securities.
Timing credit exposure quarter by quarter
Consider a benchmark composed of 70 percent investment-grade corporate bonds and 30 percent high-yield bonds. Once a quarter the investor makes a single yes-or-no decision: overweight investment grade and underweight high yield, or the reverse. That is the entire strategy.
The quarterly return volatility of the investment-grade asset is 2.84 percent and that of the high-yield asset is 4.64 percent, with an estimated correlation of 0.575. The active risk of the decision is the volatility of the difference between the two portfolios.
The investor expects to call the market correctly 55 percent of the time, that is, 11 quarters out of 20. For a dichotomous signal the time-series information coefficient is simply the percentage correct minus the percentage incorrect: 0.55 − 0.45 = 0.10.
With no limit on active risk, the expected active return per quarter is a probability-weighted average of the two outcomes: 0.55(3.80) + 0.45(−3.80) = 38 bps. The Grinold rule reproduces the same figure, 0.10(3.80)(1.0) = 38 bps, which is a useful confirmation that the two ways of thinking agree.
The investor caps annual active risk at 2.00 percent, so the deviation from the 70/30 benchmark weights is set at 2.00/7.60 = 26.3%. Assuming active returns are uncorrelated over time, breadth is 4.0, one decision per quarter.
| Position | Investment grade | High yield |
|---|---|---|
| Benchmark | 70.0% | 30.0% |
| Credit risk expected to pay off | 43.7% | 56.3% |
| Credit risk expected not to pay off | 96.3% | 3.7% |
Applying the law, E(RA) = 0.10 × (4.0)1/2 × 2.00 = 40 bps a year, or 10 bps a quarter. The direct route agrees: an active weight of 26.3 percent applied to a 38 bps per quarter opportunity gives 0.263 × 38 = 10 bps. The annual information ratio is IR = 0.10 × (4.0)1/2 = 0.20.
That is a thin result for a strategy that is right 55 percent of the time, and the reason is breadth. Four decisions a year is almost nothing. The same problem afflicts any quarterly market-timing strategy, such as switching between equities and cash. With so few chances to be right, forecasting accuracy has to be extraordinary before the information ratio becomes respectable. A useful rule of thumb for breadth in the presence of correlation is given later; here, with the average correlation between active returns equal to zero and four decisions, breadth is exactly 4.0.
Does rebalancing more often help
Yes, but only under conditions that are rarely met. If the information coefficient of 0.10 can be maintained and the credit decisions are genuinely independent from month to month, monthly decisions give IR = 0.10 × 121/2 = 0.35. If the same investor made truly independent daily decisions across 250 trading days and remained correct 55 percent of the time, the information ratio would reach IR = 0.10 × 2501/2 = 1.58.
The qualifier matters more than the arithmetic. Making monthly calls that do not actually change within a quarter, a signal of +1.0 in January, February and March, adds no breadth at all and leaves the information ratio at 0.20. Breadth counts independent decisions, not rebalancing dates.
An information ratio of 1.58 implies an expected active return of 3.16 percent at only 2.00 percent active risk, which would tempt any investor to push harder. Doubling to 4.00 percent active risk would in principle give 6.32 percent. Two things stand in the way. Transaction costs are the obvious one. The subtler one is constraints. At 4.00 percent active risk the required active weight is 4.00/7.60 = 52.6%, so a tilt away from credit implies an investment-grade holding of 70% + 52.6% = 122.6%, paid for by a short position of −22.6% in the high-yield asset.
The same logic survives if the signal is continuous rather than plus or minus one. Under the 4.0 percent active risk target, a signal of −0.57 calls for an active weight of −0.57(4.0)/7.6 = −30.0%, which against a 30 percent high-yield benchmark weight means 100 percent investment grade and nothing in high yield. A signal of +1.32 calls for 1.32(4.00)/7.60 = 69.5%, which means essentially everything in high yield and almost nothing in investment grade. Under a long-only constraint neither position is fully attainable, so the transfer coefficient falls below 1 and the expanded law takes over. For scores drawn from a normal distribution, the transfer coefficient of this strategy is the probability mass between those two signals, Φ(1.32) − Φ(−0.57) = 0.62, where Φ is the cumulative standard normal distribution function.
The consequence is large. IR = 0.62 × 0.10 × 2501/2 = 0.98, not 1.58, and at 4.00 percent active risk the expected active return is 0.98 × 4.00% = 3.92%, not 6.32 percent. A further practical note: with long-only limits binding, the realised active risk of the constrained portfolio would come in below 4.0 percent, so an optimiser would have to take larger positions than the simple formula suggests in order to spend the full risk budget.
Cross-sectional again: the Treasury maturity ladder
The second fixed-income application returns to a cross-sectional setting. The assets are five US Treasury bond portfolios, and the neutral benchmark is an equally weighted composite, 20 percent in each, rebalanced annually.
| Item | Treas. 0–1 | Treas. 1–3 | Treas. 3–7 | Treas. 7–10 | Treas. 10–20 |
|---|---|---|---|---|---|
| Average return | 0.40% | 0.90% | 2.21% | 3.15% | 3.89% |
| Volatility | 0.17% | 0.85% | 3.20% | 5.86% | 7.95% |
Bloomberg Barclays US Treasury indexes, return statistics from 2009 to 2018.
| Item | Treas. 0–1 | Treas. 1–3 | Treas. 3–7 | Treas. 7–10 | Treas. 10–20 |
|---|---|---|---|---|---|
| Active volatility | 3.45% | 2.85% | 1.05% | 2.40% | 4.57% |
| Treas. 0–1 | 1.000 | 0.49 | 0.21 | −0.49 | −0.47 |
| Treas. 1–3 | 0.49 | 1.000 | 0.26 | −0.49 | −0.49 |
| Treas. 3–7 | 0.21 | 0.26 | 1.000 | −0.19 | −0.33 |
| Treas. 7–10 | −0.49 | −0.49 | −0.19 | 1.000 | 0.46 |
| Treas. 10–20 | −0.47 | −0.49 | −0.33 | 0.46 | 1.000 |
Active volatility is measured against the equally weighted benchmark. Return statistics from 2009 to 2018.
Two features of these tables are worth stopping on. Total volatility rises monotonically with maturity, from 0.17 percent to 7.95 percent. Active volatility does not: it is highest at the two ends of the curve, 3.45 percent for the shortest bucket and 4.57 percent for the longest, and lowest in the middle at 1.05 percent for the 3 to 7 year bucket. That is because the equally weighted benchmark sits in the middle of the curve, so the middle bucket barely differs from it while both extremes do.
The active return correlation matrix also carries both signs, unlike a matrix of total return correlations, which for Treasury portfolios would be uniformly large and positive. Adjacent short maturities move together, 0.49 between the 0 to 1 and 1 to 3 buckets. Distant maturities move apart, −0.49 between the 0 to 1 and 7 to 10 buckets. Because these off-diagonal values are substantially different from zero, breadth for this five-asset strategy is 9.4, not 5.
| Item | Treas. 0–1 | Treas. 1–3 | Treas. 3–7 | Treas. 7–10 | Treas. 10–20 | IC | Active return |
|---|---|---|---|---|---|---|---|
| Score, first set | 0.63 | 0.67 | 0.92 | −0.46 | −1.76 | 0.12 | 0.37% |
| Active return | 0.43% | 0.38% | 0.19% | −0.22% | −1.61% | ||
| Active weight | −1.6% | −2.1% | 15.4% | 7.4% | −19.1% | ||
| Total weight | 18.4% | 17.9% | 35.4% | 27.4% | 0.9% | ||
| Score, second set | −0.22 | 1.20 | 0.23 | 0.57 | −1.77 | 0.18 | 0.55% |
| Active return | −0.15% | 0.68% | 0.05% | 0.27% | −1.62% | ||
| Active weight | −11.3% | 17.0% | −12.8% | 24.3% | −17.2% | ||
| Total weight | 8.7% | 37.0% | 7.2% | 44.3% | 2.8% |
Active returns scaled by the Grinold rule with an assumed information coefficient of 0.20. Return statistics from 2009 to 2018.
Active returns come from the Grinold rule at an assumed IC of 0.20. For the 10 to 20 year bucket in the first set, 0.20 × 4.57% × (−1.76) = −1.61%. Weights come from an optimiser subject to the 1.00 percent active risk constraint. The active weight of −19.1 percent on the longest bucket, against a benchmark weight of 20 percent, leaves a total weight of 20 − 19.1 = 0.9%, so the strategy has all but sold out of long Treasuries.
Now compare the two score sets, which is the point of the exhibit. Both were scaled using the same IC of 0.20, yet the IC that enters the fundamental law accounting differs: 0.12 for the first set and 0.18 for the second.
- First set, IC = 0.12. Positive scores on every short maturity and negative scores on every long maturity is really one view expressed five times: rates will rise. Because the risk model already knows short and long active returns move in opposite directions, the forecast is not adding much information, and the accounting IC is cut sharply from 0.20 to 0.12. Expected active return is 0.12 × (9.4)1/2 × 1.00 = 37 bps a year.
- Second set, IC = 0.18. Here the scores describe a change in the shape of the yield curve rather than a parallel move, with a positive score in the 7 to 10 year bucket sitting alongside a negative score at the very short end. That is a genuinely more ambitious claim, so the accounting IC falls only slightly, from 0.20 to 0.18. Expected active return is 0.18 × (9.4)1/2 × 1.00 = 55 bps a year, and the information ratio is 0.18 × (9.4)1/2 = 0.55, equivalently 55/100 = 0.55.
The lesson carries far beyond bonds. A forecast that lines up with what the risk model already expects is worth less than a forecast that cuts across it.
At an information ratio of 0.55 the investor might want to raise active risk from 1.00 percent to 2.00 percent. But the longest maturity bucket already has a total weight of just 2.8 percent in the second set, so doubling the active positions would require shorting it. If short sales are prohibited, the transfer coefficient falls below 1.00 and the extra risk does not buy a proportional return, exactly as in the constrained global equity fund.
An active strategy takes overweight and underweight decisions on four assets, for example four country equity ETFs or four fixed-income ETFs. The active returns of Assets #1 and #2 are positively correlated, as are those of Assets #3 and #4, and the risk model has no other non-zero correlations. Breadth for this correlation structure is 3.2. Decisions are dichotomous: each year two assets are forecast to outperform and two to underperform.
| Correlations | #1 | #2 | #3 | #4 |
|---|---|---|---|---|
| #1 | 1.00 | 0.25 | 0.00 | 0.00 |
| #2 | 0.25 | 1.00 | 0.00 | 0.00 |
| #3 | 0.00 | 0.00 | 1.00 | 0.25 |
| #4 | 0.00 | 0.00 | 0.25 | 1.00 |
First, the monthly decisions must be genuinely uncorrelated over time; a signal that changes slowly delivers twelve copies of one decision, not twelve decisions.
Second, the information coefficient must survive the shorter horizon at its original level.
Third, the decisions must be fully implementable, that is, TC = 1.00. Turnover constraints in particular tend to bite hardest exactly when rebalancing frequency rises, pulling the transfer coefficient down and offsetting the breadth gain.
Taking the applications together: IC measures the strength of the return-forecasting signal and must be positive for active management to be justified; BR counts independent decisions per year, equal to the number of securities only when active returns are cross-sectionally uncorrelated, and rising with rebalancing frequency only when decisions are uncorrelated over time; and TC measures how much of the forecast survives the constraints. In the absence of constraints TC is close to 1.00 and the basic form of the law applies. In practice TC values between 0.20 and 0.80 are common, which means realised performance is routinely only 20 percent to 80 percent of what the basic form would predict.
The fundamental law is a good framework and a poor oracle. Its strength is that it takes something vague, the potential of an active strategy, and expresses it as four measurable parameters, so that a strategy can be diagnosed rather than merely admired. Its weaknesses are of two kinds: practical items it ignores, such as transaction costs and taxes, and conceptual items it handles badly, above all the measurement of skill and the assumption that decisions are independent.
Because the law extends mean–variance optimisation into relative space, it inherits the weaknesses of mean–variance optimisation. Those are not the subject here, and neither are the technical difficulties of estimating a risk model, such as choosing the right factors or coping with nonlinearity and non-stationary returns. The law simply takes as given that optimising risk against return relative to a benchmark is the right objective, and that the investor can model risk adequately.
Measuring skill before the fact
The information coefficient is the correlation between forecasts and outcomes. Using it requires the investor to assume a level of skill, and that is a fragile input. Active investors already assume the market is not fully efficient with respect to public information and that they hold some edge over other active investors, otherwise active management is not justified at all. Even setting aside the behavioural tendency to overrate personal ability, an honest investor still faces a genuine estimation problem, and forecasting ability plainly varies across asset segments and over time.
Qian and Hua (2004) formalised this by adding uncertainty about the level of skill. They showed that realised active risk depends both on the tracking risk the risk model predicts, σRM, and on the extra risk induced by the fact that the realised information coefficient itself varies, σIC.
Substituting that into the law gives a more accurate expression for expected active return, in which the relevant quantity is not skill itself but skill relative to the uncertainty about skill.
The practical consequence is blunt. Actual information ratios are substantially lower than a naive application of the original law predicts. For individual stock selection strategies the achievable figure can be confirmed analytically and empirically at 45 percent to 91 percent of the original estimate. This strategy risk works in the same direction as the transfer coefficient: both reduce expected and realised information ratios, and the greater the uncertainty about forecasting ability, the smaller the likely value added.
Independence of decisions
The number of assets is not an adequate measure of breadth whenever active returns are correlated in the risk model or forecasts persist from period to period. Deciding to overweight every stock in an industry, or every country in a region, because they all respond to the same influence is one decision wearing many hats, and breadth is below the asset count. The reverse also happens: when the law is applied to hedging strategies using derivatives or to arbitrage, breadth can rise far above the number of securities.
Clarke, de Silva and Thorley (2006) offer a workable approximation in which every off-diagonal element of the risk model carries the same correlation.
The formula behaves sensibly at both ends. With ρ = 0 breadth equals N, the independent case. With ρ positive, breadth falls below N. And with ρ strongly negative, breadth explodes. Take a near-arbitrage: one country equity market ETF listed on two exchanges, so N = 2 and the active returns are almost perfectly opposed, say ρ = −0.8. Then BR = 2/[1 − (2 − 1)0.8] = 10.0, five times the number of securities. That is why an arbitrage strategy can produce a high information ratio from an unimpressive information coefficient.
Fixed income is the harder case. Most descriptions of the law are built around stock selection, where a risk model separates systematic from idiosyncratic returns and the residuals are close to independent, so breadth is at least approximately knowable. Bonds do not behave that way. Nearly every bond carries duration risk, and most carry credit risk and optionality as well, so returns are correlated through several subtle channels at once. On top of that, the implicit assumption that realised returns are normally distributed is clearly wrong for bonds with default risk and embedded options.
Time-series independence is the mirror image of the same problem. Raising the rebalancing frequency raises the realised information ratio only to the extent that successive forecasts are truly independent. Refinements such as Buckle (2004) have improved the cross-sectional handling of breadth, but a genuinely satisfactory treatment of the law in a multi-period, multi-asset setting is still outstanding.
None of this makes the law useless. It remains a workable conceptual framework across many applications and can produce real operational measurements of the parts of a strategy. But an analyst who quotes a fundamental law information ratio without saying something about the uncertainty in the assumed skill and the way breadth was counted has not finished the job.
An investor selects individual stocks from the S&P 500 on a monthly basis and applies the fundamental law with an information coefficient of IC = 0.05 and breadth of BR = 12 × 500 = 6,000. That gives an information ratio of IR = 0.05 × 6,0001/2 = 3.87, so at an active portfolio risk of 3.0 percent the expected active return would be 3.87(3.0) = 11.6%.
1. Cross-sectional dependence. The active returns of 500 S&P 500 stocks are certainly correlated with each other, so the number of independent decisions each month is well below 500. The investor may in reality be forecasting that one industrial sector will outperform and another will underperform, which is a handful of decisions, not hundreds.
2. Time-series dependence. Decisions on a given stock are likely to be correlated month to month. If the forecasting process rests on something slow-moving such as earnings yield, a stock forecast to outperform in January will probably still be forecast to outperform in February and March. The multiplier of 12 is then unearned.
3. Uncertainty about skill. An information coefficient of 0.05 sounds modest, but the basic law takes it as a known constant. It ignores uncertainty in the coefficient, the likelihood that it drifts over time, and the possibility that it differs across groups of stocks.
4. Constraints. Long-only rules and turnover limits would reduce the transfer coefficient below 1.00. This is a weaker answer than the first three, because the effect of constraints is a well-established refinement of the law rather than a limitation of the framework itself, even though it has clearly been left out of this calculation.
Pulling the reading together
Value added is the return on the managed portfolio less the return on a passive benchmark, and it is driven entirely by active weights, which sum to zero. It is positive exactly when realised active security returns are positively correlated with the active weights set at the start of the period, and it can be decomposed into asset allocation, security selection, and finer cuts such as sector or country effects.
The Sharpe ratio measures reward per unit of absolute risk and the information ratio measures reward per unit of benchmark-relative risk. Both work ex ante and ex post. The information ratio is the key criterion for evaluating active portfolios, because a higher information ratio translates into a higher Sharpe ratio for the investor once the active fund is optimally combined with the benchmark. Active risk is adjusted to its desired level by taking a position in the benchmark, and total volatility is then adjusted by combining with cash, which is two-fund separation.
The fundamental law separates expected value added into skill (IC), portfolio structuring (TC), breadth (BR) and aggressiveness (σA). The last three of these are frequently outside the control of the manager, being set by investment policy or by regulation. The law has been applied to country selection in global equity funds and to credit and duration timing in fixed-income funds, and it has real limitations, chiefly uncertainty about the ex ante information coefficient and the difficulty of counting independent decisions honestly.