A risk factor is an uncertain driver used to explain or model changes in the value or returns of a position. For a bond, a yield is a possible factor; for a foreign asset, an exchange rate may be another. Identifying the driver makes a risk report actionable: it tells you which market move to stress and which exposure a hedge should address.
There is no single universal list of factors. A valuation model might use individual yield-curve points, exchange rates and implied volatility. An equity return model might instead use market, industry and style factor returns. The factor definition belongs to the model and the position being analysed.
The driver is not the exposure
A factor answers what can change? A sensitivity answers how does this position respond? A scenario supplies how much the factor changes. Multiplying sensitivity by shock gives a local estimate of the resulting value change.
Consider a synthetic bond position with a stipulated signed sensitivity of −USD 45,000 per 1.00 change in decimal yield. Here, the factor is a single yield coordinate representing a parallel shift. Ignore the passage of time, cash flows, other factors and convexity.
A yield increase of 10 basis points is an additive change of 0.001 in decimal yield. Writing for that coordinate and for the position's USD value:
The estimated value change is −USD 45; under a positive-loss convention the estimated loss is USD 45. This is a first-order approximation, not an exact repricing or a statement about how likely the shock is. The negative sensitivity is stipulated for this example, not a property of every bond position.
Original synthetic sensitivity illustration. Positions and time are fixed; the line is a local approximation, not exact bond repricing. Yield shocks are additive and expressed in basis points; value changes are in USD.
Units travel together. The very same response can be quoted three ways:
| Yield-change unit | Signed sensitivity in USD per stated unit | Same upward shock | Approximate value change |
|---|---|---|---|
| Decimal yield | −45,000 | +0.001 | −USD 45 |
| Percentage point | −450 | +0.10 | −USD 45 |
| Basis point | −4.50 | +10 | −USD 45 |
Changing the quote unit changes the numerical sensitivity, not the economic exposure. Using −45,000 with a shock entered as 10 would mix decimal-yield and basis-point units. Duration gives a related way to describe a bond's local yield sensitivity; it still needs a consistent yield convention.
For several differentiable factor coordinates , the corresponding fixed-time approximation is
A single parallel-shift exposure compresses a curve into one direction. It can hide losses from curve twists; adding separate maturity-point factors makes those directions visible. The Risk Factor Mapping and the Market Risk Loss Operator lesson develops the valuation map and its loss operator. It has a public preview; the full lesson requires membership.
When “factor” means a return series
In an equity factor model, the inputs can be return series rather than levels such as a yield. A schematic model for an asset's excess return is
Here is a factor return, is its loading, and is its model-attributed return contribution. The residual is the part left unexplained by this specification. If both returns use decimal units, the loading is dimensionless.
For example, Kenneth French's definitions construct SMB from small-stock portfolio returns minus big-stock portfolio returns, and HML from value-stock portfolio returns minus growth-stock portfolio returns. These are specified return constructions. A company's size characteristic, its loading on SMB and the realised SMB return are different objects. A label such as “value factor” is incomplete without its construction and normalization.
What a factor report can establish
Exposure is not risk contribution. Sensitivity says how a position responds to a move. To estimate market risk, a model also needs the possible moves and their joint behaviour. A hedge based on offsetting typical factor moves can fail when those factors stop moving together as expected. Correlation and the chosen scenarios matter.
Attribution is conditional on the model. An exposure-times-move calculation allocates an approximate change to a defined driver. It does not prove that the driver caused the change, nor does it establish an expected return premium. Nonlinear effects, omitted factors, trading and cash flows can leave a difference between attributed and actual P&L. Large shocks call for checking full revaluation rather than extending a local sensitivity mechanically.
Before using a factor number, identify its definition, unit, observation horizon and valuation convention. Then keep the factor level, factor change, loading or sensitivity, and resulting P&L in separate columns.
Sources
- McNeil, Frey and Embrechts, Quantitative Risk Management: Concepts, Techniques and Tools, revised edition, Princeton University Press, 2015, §2.2.1, pp. 47–49. The valuation map, factor changes and local approximation underpin the bond example.
- Basel Committee, MAR21, §§21.1 and 21.16. This banking framework distinguishes risk-factor variables from positions measured through sensitivities; its regulatory factor definitions apply within that framework.
- Kenneth R. French, Description of Fama/French Factors. Primary construction definitions for SMB, HML and the market excess return.