Risk Budgeting

Setting your monthly budget can be tricky. You need to factor in certain priorities and also random expenses that can occur unexpectedly. A sound plan can go a long way toward ensuring that you have money to meet your needs and enjoy your life. Similarly, asset allocation can involve a budgeting process that's focused on risk allocations. But in reality, risk allocation is just like regular budgeting in that there's a dual purpose; it's not just about risk. What's the primary goal of risk allocation?

No. You'd want to own the risk-free asset if that's the case.

Not so. No standard deviation indicates limited return.

Exactly! Just like you want to maximize your spending on certain things, you want to maximize the benefits of taking risk to earn the maximum return. So the overall goal of risk allocation is maximizing the return per unit of risk. Really, you can call this risk allocation process __risk budgeting__, which is the identification and then efficient allocation of the total amount of optimal risk.

In theory, risk budgeting takes a similar approach to allocations as a regular budget does. So what information do you need to know to allocate risk?

Yes! You need to know how much risk a security brings to the overall portfolio so you can accurately allocate that risk. That's called the marginal contribution to total risk (MCTR), and it's the partial derivative of risk that's defined by the type of portfolio holding you're identifying. For example, total risk is associated with asset allocation holdings, active risk is a part of active holdings, and residual risk is a part of residual holdings. By knowing this information, you can approximate the change in portfolio risk due to a change in an individual position, determine the optimal positions for risk, and create a risk budget.

Not quite. That's not an overall budget approach; that's an excess risk approach.

That's not it. Risk isn't focused on cash flows in comparison to the total return.

In practice, risk budgeting starts by calculating the MCTR using this equation. $$\displaystyle \mbox{Asset beta relative to portfolio} \times \mbox{Portfolio standard deviation} = \mbox{MCTR}$$ So essentially, the MCTR is the rate of risk prior to introducing asset class weights. What's another way to think of the MCTR?

That's not it. The risk-free rate isn't included in the calculation.

Indeed! It's the marginal contribution to total risk, so it's really the rate of change in total portfolio risk per a change in the asset class weights. That's because it's based on the asset beta times the standard deviation. To include the asset class weights, you'll need to calculate the absolute contribution to total risk (ACTR), which is $$\displaystyle \mbox{Asset weight in portfolio} \times \mbox{MCTR} = \mbox{ACTR}$$. The ACTR measures how much the asset class contributes to portfolio return volatility.

No. It's a risk measurement, not a weighting measurement.

You can also calculate the ratio of excess return to the MCTR. $$\displaystyle \frac{(\mbox{Expected return} - \mbox{Risk-free rate})}{\mbox{MCTR}}$$ This equation should actually look really familiar, especially given the usage and location of the risk-free rate. What ratio does this excess return equation look like?

No. That doesn't include the risk-free rate.

Yes! The Sharpe ratio is a similar risk measurement of the excess return to the MCTR because it's also a measurement of returns over the risk-free rate. So they're pretty similar adjusted return measurements. This is important to note because the optimal asset allocation portfolio from a risk-budgeting perspective happens when the ratio of excess return to MCTR is the same for all asset classes and matches the Sharpe ratio of the tangency portfolio. That shouldn't be a surprise given that the excess return ratio is essentially the Sharpe ratio, so the two adjust risk measures should match.

Not quite. The standard deviation doesn't include the risk-free rate.

To sum it up: [[summary]]

Maximum volatility reduction

Minimum standard deviation

Maximum return per unit of risk

A security's risk in relation to the risk-free rate

A security's contribution to overall portfolio risk

A security's cash flows in relation to overall return

The volatility of the portfolio in excess of the risk-free rate

The rate of change for risk per change in the current portfolio weights

The rate of change in the portfolio's weights to equal a certain risk level

Beta

Sharpe ratio

Standard deviation

Continue

Continue

Continue

Continue