Risk contributions and risk budgeting

A portfolio with 60% in global equities and 40% in government bonds sounds balanced. Measured by risk, it is not: with the module's numbers, equities supply 96% of its volatility and bonds 4%. Weights tell you where the money is. Risk contributions tell you where the risk is, and that is what a risk manager has to answer for.

L3.1.06: Many assets and the covariance matrix gave one number for a portfolio's risk: $\sigma_p$. This lesson splits that number into pieces, one per asset, so you can see which positions drive it.

Marginal contribution: what one more pound adds

Ask first: if the group put a little more into one asset, paid for out of cash, how much would the portfolio's volatility rise? The answer per unit of weight is that asset's marginal contribution to riskHow much the portfolio's volatility rises per unit of extra weight in one asset, all other weights held fixed: (Σw)ᵢ / σp. (MCR). Try it with £10m, 1% of the portfolio:

The MCR of property is 15.6%: each extra 1% of weight adds about 0.156 percentage points of volatility. Government bonds add only 0.017 points per 1%. The marginal contribution depends on the asset's own volatility and on how it moves with the rest of the portfolio: it equals the asset's volatility times its correlation with the portfolio. Property (18% volatility, correlation 0.87 with the portfolio) adds a lot; government bonds (6%, correlation 0.28) add little.

Component contribution: the risk each asset owns

Multiply each marginal contribution by the asset's weight and you get its component contributionThe part of the portfolio's volatility that belongs to one asset: its weight times its marginal contribution. The parts add up to the whole. (CR). The striking fact: the components add up exactly to the portfolio's volatility. Dividing each by the total gives each asset's share of the risk.

Property is a quarter of the money and 41% of the risk. Government bonds are also a quarter of the money and only 4% of the risk. If the group wants to cut risk, property is the place to start; if it wants to add return without much extra risk, the bond allocation is cheap in risk terms.

Since each share is the weight times the asset's beta against the portfolio (the same beta as in L3.1.09: CAPM, beta and the security market line, with the group's portfolio in place of the market), a share can be negative. An asset that moves against the rest of the portfolio is a hedgeA position whose contribution to risk is negative: adding a little more of it lowers the portfolio's volatility.: it lowers the total, so it gets a negative share. Nothing in the group's portfolio does that today, though government bonds, with a beta of 0.17 against the portfolio, come close to free.

Risk parity: equal shares of risk

A risk-parityA portfolio in which every asset contributes the same share of the total risk. portfolio sets every asset's share of the risk equal. The idea is to diversify the risk, not the money. With two assets there is a neat rule: equal shares need $w_1\sigma_1 = w_2\sigma_2$, so $w_1 = \sigma_2 / (\sigma_1 + \sigma_2)$, whatever the correlation. For global equities (16%) and government bonds (6%) that is 27.3% in equities and 72.7% in bonds. Move the correlation below and check that the shares stay at 50/50.

With more than two assets the correlations matter and there is no formula. Inverse-volatility weights ($w_i \propto 1/\sigma_i$) give equal shares only when all correlations are equal. For the group's five asset classes they leave government bonds with 13.5% of the risk, not 20%. The true risk-parity weights come from a solver (the formal section shows how):

Risk parity puts 45% in government bonds, because they are the calmest asset. Its volatility (7.19%) sits between the minimum-variance portfolio (5.41%) and equal weights (10.62%), and so does its expected return. It also has a slightly higher Sharpe ratio than the current mix (0.265 against 0.258). Risk-parity funds therefore often borrow to scale up: levered 1.32 times to match the current 9.50% volatility, it would expect 5.52% against 5.45% today. That small edge rests on these made-up assumptions and on borrowing at the risk-free rate; it is not a law.

Risk budgeting

Risk parity is one risk budgetA target share of the total risk for each asset or strategy, set in advance; the weights are then chosen to meet it.: 20% each. A board can set any budget it likes. RiskAtoms Group's investment committee wants equities to carry 40% of the risk (30% global, 10% emerging), government bonds 15%, corporate bonds 20% and property 25%. The solver finds the weights:

Meeting the budget means selling property down from 25% to 13.5% and putting the money mostly into government bonds. Volatility falls from 9.50% to 7.49% and expected return from 5.45% to 4.97%.

From volatility to component VaR

L2.5.06: Value at risk: the idea turned volatility into parametric VaR: $z \times \sigma_p \times V$. Multiply every component contribution by the same $z$ and portfolio value $V$, and you have component VaROne position's share of the portfolio VaR: z × its component contribution × portfolio value. The pieces add up to the VaR.: each asset's share of the VaR, adding up to the total. At 99% over a year the group's VaR is £221.0m, of which property accounts for £90.8m.

Component VaR answers “whose VaR is it?”. It does not answer “what if we sold it all?”. That is incremental VaRThe change in portfolio VaR when a position is removed entirely (or added). Unlike component VaR, it is not a share and the pieces do not add up.: selling all the property would cut the VaR by £80.8m, less than its component VaR. Component VaR uses today's marginal rate, and that rate falls as property is sold: the less property the group holds, the less each further pound of it adds to the risk.

The marginal contribution. Write the variance as a quadratic form (M-E02: Matrix multiplication and quadratic forms) and differentiate one weight at a time:

The second line uses the symmetry $\sigma_{ik} = \sigma_{ki}$; the third is the chain rule. In vector form, $\nabla_w\, \sigma_p = \Sigma w / \sigma_p$.

Three ways to read it. The $k$-th entry of $\Sigma w$ is the covariance of asset $k$ with the portfolio: $(\Sigma w)_k = \sum_j \sigma_{kj}w_j = \operatorname{cov}(R_k, \textstyle\sum_j w_j R_j) = \operatorname{cov}(R_k, R_p)$. So

The Euler allocation. Volatility is homogeneous of degree one: scaling every weight by $\lambda > 0$ scales it by $\lambda$. Differentiate both sides with respect to $\lambda$ and set $\lambda = 1$:

For volatility you can also check it directly: $\sum_i w_i (\Sigma w)_i / \sigma_p = w^\top\Sigma w / \sigma_p = \sigma_p$. The argument works for any risk measure that scales with the position size, including VaR and expected shortfall. For parametric VaR, $\mathrm{VaR} = z\,\sigma_p V$, so component VaR is $z\,\mathrm{CR}_i\,V$ and the pieces add up to the VaR. Dividing by $\sigma_p$, the shares are $w_i\beta_{i,p}$ and add up to one.

Risk parity with two assets. Equal contributions mean $w_1(\Sigma w)_1 = w_2(\Sigma w)_2$. Write out both sides:

The cross term is the same on both sides and cancels, which is why the correlation drops out. With three or more assets each cross term involves a different pair, so nothing cancels.

Any risk budget, any number of assets. Given target shares $b_1, \dots, b_n$ (positive, adding up to one), solve

The first-order condition for each $y_i$ is $(\Sigma y)_i - b_i / y_i = 0$, that is $y_i(\Sigma y)_i = b_i$: each contribution is proportional to its budget. The function is convex (a positive definite quadratic plus minus-logs), so there is exactly one solution. Rescale with $w = y / \sum_j y_j$: contributions scale by the same factor, so their shares stay at $b_i$. The lesson's solver updates one $y_i$ at a time, solving the quadratic $\sigma_i^2 y_i^2 + c_i y_i - b_i = 0$ with $c_i = \sum_{j\ne i}\sigma_{ij}y_j$ for its positive root, and repeats until nothing moves.

Start from the current allocation and watch the coloured bars: they show each asset's share of the risk next to its weight. Move the weights towards the target ticks, or press Solve to jump there. Then try equal weights and inverse volatility, and see how far each is from equal risk.

The report the committee sees: property, a quarter of the money, carries 41% of the group's £221.0m VaR. Moving to the group's risk budget would lower the VaR to £174.3m.

No single rule prescribes this method, but the Euler allocation is the standard way to split a firm's capital between business lines so that the pieces add up and reward the right behaviour. Banks and insurers use it with VaR or expected shortfall to set desk limits and to measure return on allocated capital. M7.1: Enterprise risk and economic capital builds on it. The covariance matrix tool at Portfolio risk: covariance matrix computes the same contributions for your own inputs.

• The marginal contribution MCRᵢ = (Σw)ᵢ / σp is how fast volatility rises with asset i's weight. It equals ρᵢ,p σᵢ.
• The component contribution CRᵢ = wᵢ × MCRᵢ. The components add up exactly to σp (the Euler allocation); each share of risk is wᵢ βᵢ,p.
• Weights and risk can be very different: a 60/40 equity-bond mix takes most of its risk from equities.
• Risk parity sets equal shares of risk; with two assets w₁σ₁ = w₂σ₂. Any risk budget can be met with a solver.
• Component VaR = z × CRᵢ × V splits the VaR into pieces that add up. Incremental VaR, the change from removing a position, is different.

Marginal contribution to risk (MCR)

How much the portfolio's volatility rises per unit of extra weight in one asset, all other weights held fixed: (Σw)ᵢ / σp.

Component contribution (CR)

The part of the portfolio's volatility that belongs to one asset: its weight times its marginal contribution. The parts add up to the whole.

Euler allocation

The rule that splits a risk measure into weight × marginal contribution for each position. It adds up exactly for any measure that doubles when every position doubles.

Hedge (in a risk decomposition)

A position whose contribution to risk is negative: adding a little more of it lowers the portfolio's volatility.

Risk parity

A portfolio in which every asset contributes the same share of the total risk.

Risk budget

A target share of the total risk for each asset or strategy, set in advance; the weights are then chosen to meet it.

Component VaR

One position's share of the portfolio VaR: z × its component contribution × portfolio value. The pieces add up to the VaR.

Incremental VaR

The change in portfolio VaR when a position is removed entirely (or added). Unlike component VaR, it is not a share and the pieces do not add up.