What does beta mean when the policy changes with the state?

Beta is the most common single-number summary of a strategy's market exposure. For a portfolio held at constant weights it is a meaningful parameter: the sensitivity of the portfolio to the market. For a systematic strategy whose policy changes its positions as conditions change, it is something else, a statistic that averages over states. What it averages, and with what weights, is less obvious than it looks.

A simple model

Let $x_t$ be the market state, observed before the period's return, and suppose the strategy's policy sets its market exposure to $\beta(x_t)$. The strategy's return is then

where $r_{m,t}$ is the market return and $\varepsilon_t$ is unrelated to it. Write $\sigma^2(x)$ for the variance of the market return in state $x$. If, for simplicity, the expected market return is zero in every state, the beta estimated by regressing the strategy's returns on the market's over the full sample converges to

Full-sample beta is a variance-weighted average of the state betas. A state contributes in proportion to how often it occurs multiplied by how volatile the market is when it does. Rare, turbulent states carry far more weight than their share of days.

When expected returns also vary with the state, additional terms appear, including one that reflects whether exposure tends to be high when the market's expected return is high. Separating that timing component from average exposure is the subject of the market-timing literature (Treynor and Mazuy 1966; Henriksson and Merton 1981). It adds to the averaging problem rather than removing it.

Two strategies, one beta

Consider four market states with the shares of days and annualized market volatilities below, and two illustrative strategies. Strategy A reduces its exposure as volatility rises. Strategy B behaves like a seller of convexity: its exposure is small in calm markets and grows as prices fall, as a short put position's does.

Both strategies report a full-sample beta of 0.30. The number is correct for both and describes neither.

• Strategy A's average exposure across days is 0.51. On the 92% of days outside the stress state it carries between 0.40 and 0.60. An allocator who sizes it as a 0.30-beta holding will hold more equity risk than intended on most days.
• Strategy B's average exposure across days is 0.17, and in the stress state it is 0.42. Its full-sample beta overstates its equity exposure on ordinary days and understates it in the state in which it matters most.

The stress state is 8% of days and 42.5% of the weight. Whatever happens there dominates the estimate, and that is also where the two strategies differ most.

More data does not fix it

A longer sample produces a more precise estimate of the same average. It does not turn the average into a description.

Estimating exposure within states is the natural remedy, and it has its own difficulty: the states that matter most are rare and clustered. At 8% of trading days, a decade contains roughly two hundred stress days, and they arrive in a few episodes rather than independently. Conditional estimates in those states will be noisy, and they should be reported with their uncertainty.

This is one reason to anchor exposure descriptions in the policy and the positions, not only in returns. A strategy's mechanical exposure, implied by what it holds, and its policy exposure, how its intended positioning changes with the state, can be read directly from the strategy. Statistical exposure estimated from returns is then a check on them rather than the only source. Where these disagree, the disagreement is itself evidence about the strategy.

Conditional evaluation is not new

Ferson and Schadt (1996) showed that allowing a fund's beta to vary with publicly available information can materially change conclusions about its performance. The market-timing literature has long treated the relationship between exposure and market conditions as the object of interest rather than a nuisance. What is less common is reporting conditional exposure as the primary description of a systematic strategy, with the states defined in advance and justified by the strategy's mechanism.

What to report instead

For a strategy whose policy changes with the state, a more useful exposure description includes:

1. the states, defined by variables observable before each decision and justified by the mechanism;
2. the share of time and the share of market variance in each state;
3. the policy's intended exposure in each state;
4. the estimated exposure in each state, with its uncertainty and the estimation method named; and
5. the exposure the strategy actually carried in production, compared with what the description predicted.

Full-sample beta can stay in the report as one row. It answers a narrow question, the average linear co-movement over the sample, and for a strategy with a state-dependent policy that is rarely the question an allocator is asking. The more useful object is the function: how exposure responds as conditions change, and in which states it is concentrated. That is the question our work on conditional exposure is organized around.

References

Ferson, W. E., and R. W. Schadt (1996). "Measuring Fund Strategy and Performance in Changing Economic Conditions." Journal of Finance 51 (2): 425–461.

Henriksson, R. D., and R. C. Merton (1981). "On Market Timing and Investment Performance. II. Statistical Procedures for Evaluating Forecasting Skills." Journal of Business 54 (4): 513–533.

Treynor, J. L., and K. K. Mazuy (1966). "Can Mutual Funds Outguess the Market?" Harvard Business Review 44 (4): 131–136.