In simple terms — A portfolio is not just “how much money is in each position”. This chapter shows, step by step, where risk is concentrated and how value responds when the market, time, volatility or an interest rate changes.
Risk allocation connects three different decisions: where capital is deployed, how much of a chosen risk measure is assigned to each component, and how the resulting sensitivities are controlled. A 20% weight, a 20% contribution to volatility, a delta-equivalent exposure and a DV01 are numbers with different units and purposes. They may appear in the same report, but not in the same sum.
This chapter organises established knowledge from portfolio theory, risk management, options and fixed income. It does not present a model portfolio, prescribe universal thresholds or introduce rules of the Emiciclo Method. The Emiciclo Method remains a later, autonomous layer. Any future comparison will live on a dedicated page and will not retrospectively rewrite the sources reconstructed here.
Interactive map: from capital to sensitivities
| Node | The right question | Confusion to avoid |
|---|---|---|
| Which resources are deployed, under which weights, reserves and constraints? | calling every invested unit “risk” | |
| Which issuers, factors, markets, currencies or counterparties dominate? | merely counting positions | |
| How much does a component contribute to the selected aggregate measure? | treating contribution as an immutable property of the instrument | |
| How much risk is assigned ex ante, under which measure and horizon? | confusing it with available capital or margin | |
| Which weights approach the declared contribution targets? | always reducing it to inverse-volatility weights | |
| How does exposure change when estimated volatility changes? | promising constant realised volatility | |
| How have returns responded to a defined benchmark? | reading it as total risk or a forecast | |
| What is the local slope of value with respect to the underlying? | automatically equating it with probability | |
| How does delta change when the underlying moves? | assuming a constant sign and magnitude | |
| How does value respond to a conventional change in implied volatility? | confusing it with realised volatility or certain P&L | |
| How does theoretical value change as time passes, with other inputs fixed? | calling it a guaranteed daily loss | |
| Which relative yield sensitivity describes the instrument or portfolio? | confusing it with maturity alone | |
| How much does curvature matter beyond the linear duration approximation? | assuming it is always positive or favourable | |
| What local money change corresponds to one basis point? | treating it as numerically synonymous with duration |
1. Capital, risk and contributions are different layers
Capital allocation describes how resources and weights are distributed among instruments, strategies, cash and reserves. A risk budget instead begins with a stated measure — such as volatility, scenario loss or another metric consistent with the mandate — and sets ex-ante targets or limits. Two components may receive the same capital and absorb very different amounts of risk; they may also receive the same budget while requiring different nominal weights.
Concentration does not disappear when the number of tickers increases. One position may dominate by weight, while several apparently different positions may depend on the same issuer, equity factor, currency, yield curve, source of liquidity or counterparty. Control therefore needs a declared scope, unit, look-through and observation time. Prudential large-exposure standards provide one institutional example of concentration control. Their thresholds belong to that regime and do not become universal limits for every trader.
Risk contribution asks a further question: given an aggregate measure, which part is associated with each component? Marginal contribution observes the local change in the measure when a position changes; component contribution combines that response with the position size. For some homogeneous measures, Euler allocation provides a consistent decomposition of the total. That does not make the same formula valid for every metric, payoff or constraint.
Risk parity is a family of constructions that assigns targets to contributions under a selected measure. Equal risk contribution is an important case, not the only possible meaning. Inverse volatility weights coincide with broader solutions only under particular assumptions about dependencies and constraints. Change the covariance matrix, leverage, limits, currency or measure, and both weights and contributions may change.
Volatility targeting primarily acts along the time axis: it compares a volatility target with an estimate and changes exposure under a documented rule. Window, frequency, lag, smoothing, floor, cap, leverage and costs belong to the model. Realised volatility may differ from the target because of jumps, estimation error, correlations, non-linearities and execution constraints. Empirical research on volatility-managed portfolios is evidence about the method studied, not proof that every implementation produces the same outcome.
2. A sensitivity is a local coordinate
A sensitivity measures how value changes with respect to one variable while other inputs are held fixed under a convention. At least seven coordinates are needed to interpret it:
- the position and scope included;
- the perturbed factor and shock size;
- the money or relative unit;
- the valuation point and timestamp;
- the inputs held fixed;
- the pricing model or repricing procedure;
- the sign convention and aggregation rules.
These coordinates prevent misleading comparisons. A beta of 1.2, a delta of 0.40 and a DV01 of 500 do not form a total: the first depends on returns and a benchmark, the second is a derivative with respect to the underlying, and the third translates a one-basis-point move on a specified curve or yield into money.
Market beta is an estimated slope relative to a benchmark over a defined sample. It depends on the series, frequency, window, currency, handling of missing observations and model specification. It helps describe an observed linear relation, but does not automatically include idiosyncratic risk, non-linearity, liquidity, tails or regime change. Historical beta is not a certain forecast of the next move.
The Basel sensitivities-based framework supplies an institutional application: delta, vega and curvature are calculated for specified factors and risk classes, then aggregated under prudential rules. It is useful as an example of discipline around factors and units. Its regulatory shocks, weights and correlations should not be transferred automatically to a non-bank portfolio.
3. Options: slope, curvature, volatility and time
For an option, underlying price, implied volatility and time act together. The Greeks locally isolate one direction of movement in the model:
- delta is the first derivative with respect to the underlying price, holding the other declared inputs fixed;
- gamma measures the change in delta as the underlying moves and exposes the limitation of a linear hedge;
- vega measures the response to a conventional change in implied volatility;
- theta isolates the effect of time passing in the theoretical valuation.
These measures change with the underlying, strike, expiry, volatility and model. A Greek for one leg does not automatically describe a multi-leg strategy: the legs must be aggregated with consistent quantities and multipliers. Even a net Greek remains local. Large moves, exercise, assignment, jumps, a moving smile or simultaneous changes in inputs require scenario repricing.
Delta is not generally a probability; gamma is not positive for every position; vega does not guarantee a profit when observed volatility rises; and theta does not equal the P&L that will be recorded the next day. The Options Industry Council describes the Greeks as theoretical guides to sensitivity, not exact predictions of option premiums.
4. Rates: duration, convexity and DV01
Bonds turn cash flows across time into a price that responds to yields, curves, credit, optionality and liquidity. Bond duration has several definitions. Macaulay duration organises the weighted timing of cash flows; modified duration approximates the relative price change for a small change in yield; effective duration may be used when cash flows depend on the scenario. The variant and yield convention must accompany the number.
The price–yield relation is curved. Convexity adds a second-order term and improves the duration approximation when the move is not infinitesimal. Instruments with embedded options may exhibit different behaviour and signs. “More convexity” is not a complete assessment without its price, scenario and other risks.
DV01 expresses in money the local change associated with a one-basis-point move in a declared yield or curve. It is related to modified duration through an approximation, but it is not the same quantity: position value, currency, price and convention also matter. A portfolio spanning maturities often needs DV01 by curve node, rather than one net total that hides offsetting exposures.
Duration, convexity and DV01 describe interest-rate risk. They do not replace analysis of credit, spread, inflation, currency, calls, liquidity or non-parallel curve scenarios.
5. A verifiable workflow
A legible process connects decision, measurement and control:
- define mandate, capital, liquidity, collateral and reserves;
- reconcile positions, multipliers, currencies and prices;
- state the measure used for concentration, contributions and budgets;
- estimate dependencies and preserve alternative scenarios;
- compare observed contributions, targets and limits without conflating them with weights;
- measure beta, Greeks and rate risk in their respective units;
- reprice under coherent shocks to move beyond local approximations;
- include liquidity, funding, margin, costs and operational risk;
- document data, model, versions, exceptions and responsibility;
- rerun measurement and reconciliation after moves, expiries and portfolio changes.
The output is not one “risk score.” It is a reconciled collection of views: weights and capital, concentrations, contributions, budgets, local sensitivities and scenario losses. When two views diverge, that divergence is information to explain, not an error to erase by choosing the more favourable number.
Four reading routes
Portfolio allocation — capital allocation → concentration → risk contribution → risk budget → risk parity → volatility targeting. This route separates resources, measure, target and observed use.
Market and benchmark — beta. This route reads historical sensitivity to a declared benchmark without treating it as total risk or a forecast.
Options — delta → gamma → vega → theta. This route reads a valuation surface through slope, curvature, implied volatility and time before moving to scenarios in which several inputs change.
Fixed income — duration → convexity → DV01. This route separates relative sensitivity, curvature and the money change per basis point.
The common prerequisite is Risk measurement and control. For sizing, leverage, margin and capital survival, start with Risk management in trading.
Scope
This chapter provides general education. It does not optimise a specific portfolio, prescribe weights or hedges, or replace contractual, regulatory or professional documentation. Banking, exchange and self-regulatory examples remain within their institutional scope.
Sources
- Investor.gov — Asset Allocation and Diversification
- Harry Markowitz — Portfolio Selection, The Journal of Finance (1952)
- Basel Committee — Supervisory framework for measuring and controlling large exposures
- Basel Committee — Principles for effective risk data aggregation and risk reporting
- Dirk Tasche — Capital Allocation to Business Units and Sub-Portfolios: the Euler Principle
- Maillard, Roncalli and Teïletche — On the Properties of Equally-Weighted Risk Contributions Portfolios
- Moreira and Muir — Volatility-Managed Portfolios, The Journal of Finance (2017)
- William F. Sharpe — Capital Asset Prices, The Journal of Finance (1964)
- Investor.gov — Glossary: Beta
- Options Industry Council — Understanding Options Greeks
- Options Industry Council — Volatility & the Greeks
- Basel Framework — MAR21, Standardised approach: sensitivities-based method
- FINRA — Brush Up on Bonds: Interest Rate Changes and Duration
- CME Group — How Can You Measure Risk in Treasuries?
- Basel Committee — Interest rate risk in the banking book