From risk budget to position size: The driver is the contribution to risk

The question and the route to an answer

The most frequent question about Bitcoin in a portfolio is how much Bitcoin belongs in it. There is no universally valid answer, and this page does not specify an allocation. It describes the route by which a position size can be justified. A risk budget first sets what share of portfolio risk a position may claim. Under a risk-budgeting approach, position size is set so that the estimated risk contribution remains within the allocated risk budget and the position complies with the applicable constraints.

The page deliberately contains no empirical analysis, no allocation ratios and no recommendation. Episode 3 describes a mechanism. The underlying assumptions are stated below but are not empirically tested here. This page accompanies episode 3 of the topic series “Volatilität im Portfoliokontext” in the format “21Nexus | Portfoliofragen” (video in German) and can be read without the video.

From budget to size

A risk budget is a decision made in advance on what share of total portfolio risk a position or a group of positions may claim. It shifts the question from “What portfolio weight?” to “How much of the risk?”.

The risk contribution of a position depends on three quantities: its weight in the portfolio, its stand-alone volatility and its dependence on the rest of the portfolio. With volatility as the risk measure, the contribution of a position in the usual decomposition is the product of weight, stand-alone volatility and correlation with the total portfolio. The contributions of all positions add up to the volatility of the portfolio. Dividing a contribution by this volatility gives the position’s share of risk.

It follows that a position’s portfolio weight and its share of risk are different quantities. A position with high stand-alone volatility can claim a considerably higher share of risk than its portfolio weight. A lower dependence on the rest of the portfolio can reduce the risk contribution; how large the difference between portfolio weight and risk share turns out to be has to be calculated case by case. The sketch in episode 3 shows this difference schematically with two bars for the same position, without an axis, without a scale and without a number. It is not a measurement and is not to be read as a value.

Risk contributions are not necessarily positive parts of a whole. If a position has a negative correlation with the rest of the portfolio, its contribution can be negative, and individual relative contributions can then exceed 100%. Because the correlation with the portfolio and the portfolio volatility themselves depend on the weight, the size cannot in practice be read from a single formula. It is determined numerically by choosing the weight so that the estimated risk contribution remains within the budget and the position complies with the applicable constraints.

What the derivation presupposes

Every derived size rests on decisions. The table names the most important ones and their effect. The assumptions and modelling choices are stated explicitly but are not empirically tested here.

Budget and risk contribution over time

The budget can remain unchanged while the currently estimated risk contribution changes. If the correlation with the other positions rises, the same position can contribute more to portfolio risk, even with unchanged weight and unchanged stand-alone volatility. The same applies to higher stand-alone volatility. Whether the size has to be adjusted is decided by the comparison with the budget and the defined rules: if the contribution is within the range, nothing needs to be done; if a defined threshold is exceeded, the rule applies.

How the dependence between Bitcoin and other positions can change is covered by the analysis in episode 2 of the topic series, using the S&P 500 as an example. This page takes no values from it.

The view of total wealth

The first step is to clarify which portfolio the risk budget covers. If the overall allocation is to be assessed, holdings outside the mandate must also be taken into account. If part lies outside the mandate, for example in the client’s own wallet, the budget depicts only a section and says little about the whole. The correlation of a position with the mandate portfolio can differ from its correlation with total wealth, and total risk cannot be read from the sub-portfolio. For wealth managers responsible for the overall allocation, this delineation therefore belongs before any calculation.

Limits of the conclusion

The episode describes a mechanism, not results. Four points matter for the interpretation.

No size given. This page specifies neither a position nor a ratio. The outcome of the derivation can be a position or none. The difference lies not in the outcome but in the reasoning, which follows from budget, risk measure and constraints.

Estimation uncertainty. Risk contributions are estimates. The correlation with the other positions in particular is not stable, and a value estimated from historical data need not hold in the future. The budget itself then remains unchanged, the currently estimated contribution does not.

Risk measure. The decomposition by volatility describes risk in a particular way. It captures losses in extreme market phases only incompletely. Other risk measures are not interchangeable variants of the same formula: they require their own definition of the decomposition and additional parameters, and an Euler risk decomposition presupposes suitable mathematical properties of the risk measure. The ranking of contributions may change as a result.

Simplification. The decomposition model starts from the composition of the portfolio at one point in time. Leverage, options, liquidity risks or custody issues are not captured by a simple risk budget on their own; they have to be addressed through constraints and rules.

Placing this in a portfolio context

The risk budget follows on from the two preceding questions: whether a single metric describes the effect within a portfolio on its own, and how the dependence on the other positions changes. It does not answer whether a position makes sense, but how large it may be if one holds it. Where the analysis of return would have to begin, and why this topic series ends there, is covered by episode 4.

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Legal notice

This publication serves solely for general information and professional commentary. It constitutes neither investment advice nor an investment recommendation, financial analysis or an offer to buy or sell financial instruments. Individual investment objectives, financial circumstances or the risk tolerance of individual investors are not taken into account. Information on historical performance and metrics permits no conclusions about future developments. The data used originate from sources that we consider reliable; no guarantee is given for accuracy or completeness. The information reflects the position at the date of publication; there is no obligation to update. Publisher: 21Nexus Capital GmbH, Neuer Wall 54, Hamburg.