Risk budgets and associated metrics can be used to analyze the contributions made by individual assets to a portfolio’s overall risk. We consider three three methods for analyzing this on both an End-of-Horizon and Within-Horizon basis.

- Risk Budget
- Risk Attribution
- Value-at-Risk Sensitivity

**STANDARD DEVIATION**

Standard deviation is a measure of dispersion used to measure an asset’s risk. Standard deviation is the square root of the variation, and is equal to the square root of the average of the squared deviation from the mean. Roughly sixty-eight percent of the observations under a normal distribution are within the mean plus and minus one standard deviation of the mean.

**VALUE-AT-RISK (VaR)**

Value-at-Risk (VaR) is the maximum loss or lowest gain which could occur at a given confidence level over a specified horizon, and is based on both expected return and standard deviation. For more on Value-at-Risk, visit our Loss page here.

### RISK BUDGETS

Risk budgets are a tool to help us answer the question of how much risk each asset adds to the portfolio. Looking at the individual value at risk of each asset in isolation from the remaining assets, we can see if we have a big allocation to a less risky asset or a small allocation to a high risk asset. The risk budgets of each asset class will not sum to the total portfolio value at risk because risk budgets look at each asset class in isolation neglecting the diversification benefit of the portfolio as a whole. The individual risk budgets are displayed in the column labeled “End-Of-Horizon” and “Within-Horizon”.

To read more about risk budgets, check out our Insights post!

### RISK ATTRIBUTION

Risk Attribution is the proportion of total portfolio capital at risk attributable to an asset. Unlike the individual Risk Budgets, the Percent of Portfolio column will sum to 100% because it considers the correlations amongst the asset classes in explaining the total portfolio value at risk.

### VALUE AT RISK SENSITIVITY

If we are looking for the most efficient way to alter the risk of a portfolio while minimizing the change in allocation, value at risk sensitivity can show us how much a change to each asset class will impact the total portfolio risk. Value at risk sensitivity is, in a sense, a form of “What If” analysis that allows us to see how a change in allocation to one asset class will impact the total portfolio value at risk. It can be used to measures the impact of a re-allocation across all asset exposures. While a specific asset’s exposure is increased by the increment, the remaining portfolio’s components are reduced proportionally to preserve the initial investment value of the portfolio throughout the process. The ranking of the portfolio’s value at risk sensitivities will not necessarily match the ranking of the portfolio’s percentage exposure or the ranking of the individual components’ value at risk. The WCA displays the VaR sensitivity of each asset class at the same time. It does not display the adjusted weight to each asset class in the portfolio.

*Example*

Let’s investigate the Risk Budgets of a simple $1,000 portfolio, consisting of 50% US Stocks and 50% US Bonds. We set our Time Horizon at three years and our Probability Threshold at 5%. Looking at the individual value at risk assignment of these two asset classes we can see that at the 5% threshold, US Stocks could lose $60 or more at the end of three years. At the same threshold, US Bonds could lose $37 or more at the end of three years. Additionally, we can see that US Stocks account for approximately 208% of the portfolio capital at risk while the proportion attributable to Bonds is negative 109%.

If we are looking to decrease total portfolio risk, value at risk sensitivity shows us how a re-allocation to US Bonds will change total portfolio risk. We can see that an allocation increase of 5% to Stocks assuming a proportional decrease to the remaining asset classes (only bonds in this example) would increase total portfolio value at risk by $6.50. However, that same increase to Bonds (assuming the proportional decrease to Stocks) would actually decrease total portfolio value at risk by about $5.