Risk Budgets

There are two common definitions of risk budgets that seem to prevail in the industry.

1. A plan for converting a portfolio’s monetary allocations to various categories into Value at Risk (VaR) assignments

2. The sensitivities of a portfolio’s VaR to a small change in the portfolio’s exposure to each component.

Efficient Portfolio Allocations

The first perception of a risk budget is appropriate as long as it follows from mean-variance optimization. It would be inefficient to plan the value at risk independently of mean-variance optimization, assuming lognormally distributed returns. Any portfolio that is efficient with respect to VaR must lie on the mean-variance efficient frontier.

The intuition here is that, for any given expected return, a portfolio located on the efficient frontier has the lowest standard deviation. Thus, for any given expected return, the portfolio with the lowest VaR must also lie on the efficient frontier. It follows, therefore, that any risk budget that isn’t mean-variance efficient has a higher portfolio VaR for a given confidence level than a corresponding portfolio that is mean-variance efficient.

We should not regard risk budgeting as a process for determining efficient portfolio allocations. We should instead regard it as a means for converting efficient portfolio allocations into VaR assignments. Portfolio choice based on minimizing VaR implies an improbable attitude towards risk, which often leads to overlooking other factors. Investors should consider all portfolios along the efficient frontier, weigh all factors and all possible outcomes—not just a single threshold.

We should consider risk budgets an extension of mean-variance optimization that enables us to decouple allocations from fixed monetary values. Suppose the optimal allocation of a $100 million fund calls for the percentage allocations shown in the table below.

Asset	Percentage Allocation
1	20.70%
2	30.73%
3	48.57%

An Alternative Approach

The traditional approach for implementing these allocations would be to invest $20.70 million in asset 1, $30.73 million in asset 2, and $48.57 million in asset 3. The risk budget alternative would instead view these allocations as Value at Risk assignments of $4.9567 million for asset 1, $4.5429 million for asset 2, and $4.6664 million for asset 3—the respective amounts that each assignment could lose with a 5% probability over a one-year horizon.

Note that the sum of the individual VaRs is nearly twice as large as the portfolio VaR. This is because the assets are less that perfectly correlated with each other, and therefore introduce diversification to the portfolio.

This interpretation of a portfolio’s allocation offers the flexibility to allocate and leverage a smaller monetary amount to each asset. This leveraged investment would contribute the same marginal utility to the portfolio, which is tantamount to preserving the portfolio’s optimality—if two conditions prevail:

• The leveraged investment preserves the expected return, volatility, and correlation with the balance of the portfolio as assumed by the original percentage allocation.

• The balance of the portfolio preserves the expected return and risk attributes assumed in the original optimization.

The economic equivalence of a VaR assignment and a monetary allocation prevails for any confidence level used to measure VaR, if returns are lognormally distributed. Thus a risk budget, in effect, simply maps a portfolio’s percentage allocations onto VaR assignments. This offers the benefit of freeing portfolio allocations from monetary constraints. However, a risk budget is efficient only if determined by mean-variance optimization.

Sensitivities to Changes in Exposures

The second definition recognizes that the VaR of the individual categories won’t sum up to the portfolio’s total VaR unless all categories are perfectly positively correlated. Thus, the independent VaRs could mislead an investor. Some investors thus define a risk budget as the sensitivities of a portfolio’s VaR to a small change in the portfolio’s exposure to each component.

This definition suffers from a problem of semantics; a budget implies a plan or action. Due to this notion, we propose the label RISK ATTRIBUTION for this definition of risk budget.

Risk Attribution

Let us once again consider a portfolio of three assets. To calculate its risk attribution, we take the partial derivative of its VaR with respect to each of the assets. First, we write the partial derivative of a portfolio’s percentage loss in continuous units, with respect to exposure to Asset 1 as the sum of the derivatives of the two components of a portfolio’s percentage loss:

$$\frac{\partial{L_c}}{\partial{w_1}}=\frac{\partial{\mu_{pc}}}{\partial{w_1}}+\frac{\partial{Z \sigma_{pc}}}{\partial{w_1}}$$

In this equation, $L_c$ is percentage loss in continuous units, $w_1$ is the exposure to Asset 1, $\mu_{pc}$ is the expected portfolio return in continuous units, $Z$ is the Normal deviate, and $\sigma_{pc}$ is the expected portfolio standard deviation in continuous units.

Next, we define portfolio expected return and standard deviation measured in continuous units as a function of the portfolio’s exposure to the component assets

$$\mu_{pc}=\mu_{1c} w_1+\mu_{2c} w_2+\mu_{3c} w_3$$

$$\sigma_{pc}=\left( \sigma_{1c}^2w_1^2+\sigma_{2c}^2w_2^2+\sigma_{3c}^2w_3^2+2\rho_{1,2}\sigma_{1c}w_1\sigma_{2c}w_2+ 2\rho_{1,3}\sigma_{1c}w_1\sigma_{3c}w_3+2\rho_{2,3}\sigma_{2c}w_2\sigma_{3c}w_3 \right)^{1/2}$$

where $\mu_{ic}$ is the expected return of asset $i$ in continuous units, $\sigma_{ic}$ is the expected return of asset $i$ in continuous units, and $\rho_{i,j}$ is the correlation of asset $i$ and $j$ in continuous units.

The derivative of the portfolio expected return with respect to exposure to Asset 1 is straightforward:

$$\frac{\partial \mu_{pc}}{\partial w_1}=\mu_{1c}$$

The derivative of $Z\sigma_{pc}$ with respect to exposure to Asset 1 is slightly more complicated. We need to invoke the chain rule, by first taking the partial derivative of $Z\sigma_{pc}$ with respect to portfolio variance. We then need to multiply it by the partial derivative of portfolio variance with respect to exposure to Asset 1:

$$\frac{Z \sigma_{pc}}{\partial w1}=\frac{Z \sigma_{pc}}{\partial \sigma_{pc}^2} \left( \frac{\partial Z\sigma_{pc}^2}{\partial w_1} \right)$$

We convert this sensitivity of percentage loss measured in continuous units to the sensitivity of the portfolio’s VaR measured in monetary units:

$$\frac{\partial \text{VaR}}{\partial w_1}=-e^{\frac{\partial L_c}{\partial w}}-1$$

The following table shows the risk attribution of the same portfolio described earlier.

Asset	Allocation	Value at Risk (per $100)	VaR Sensitivity
1	20.70%	4.9567	0.1846
2	30.73%	5.5429	0.0866
3	48.57%	4.46663	0.0387
Portfolio	100.00%	7.1453

Features of Risk Attribution

Two interesting features of risk attribution:

1. It is mathematically impossible to partition a portfolio’s total VaR into the fractions associated with the individual components. VaR is a function of standard deviation (which is not additive), and it is impossible to disentangle the interactions of the various components.

2. The ranking of the portfolio’s VaR sensitivities will not necessarily match the ranking of the portfolio’s percentage exposures or the ranking of the individual components’ values at risk. Note that Asset 1 has the smallest percentage allocation, yet has the greatest impact on the portfolio’s VaR. Also, although Asset 2’s VaR is lower than Asset 3’s, an additional $1 allocation to Asset 2 increases portfolio VaR by $0.0866, while the same increase in allocation to Asset 3 raises portfolio VaR by less than half as much ($.0.0387). Although Asset 2 has a lower individual VaR, the higher correlation with the portfolio offsets this advantage.

This risk attribution shows a very important message. If we wish to limit the potential loss of our portfolio, we should focus not on its largest holding. We shouldn’t focus on its most volatile asset, or on the asset with the greatest value at risk. Instead, we should focus on the asset to which the portfolio is most sensitive.

Summary

Risk budgets have gained widespread attention in the last few years, but are often applied inappropriately or mislabeled. Regarding risk budgets as a plan for assigning VaR to categories independently of mean-variance optimization will likely have inefficient results.

Investors who regard a risk budget as a process to choose a portfolio by minimizing value at risk implicitly care only about a single outcome. These investors are no more averse to losses far greater than VaR than a loss equal to VaR. Portfolio choice, instead, should be based on the consideration of every portfolio on the efficient frontier and their probability distributions. This process produces efficient VaR results, and considers the pleasure or disutility associated with all possible outcomes.

The proper conception of a risk budget is the conversion of optimal percentage allocations from mean-variance optimization into VaR assignments. This conversion preserves the economic exposures of the portfolio and enables investors to achieve these exposures with much greater latitude.