Allocating the Risk Budget
If you've ever worked on a financial budget, you probably know how important it is to be accurate and thorough and to use the proper expense allocations. If there is even one wrong budget number, the entire balance can be thrown off dramatically. Essentially, the goal is to maximize your income across your expenses. What's another way to state this budgeting goal?
No.
Maximum debt indicates that there's not a proper balance between income and expenses.
Not quite.
Savings should also ideally be part of the budget process, so spending isn't the ultimate goal.
Right.
Maximum efficiency is the ultimate goal in financial budgeting, and it's the same approach for risk budgeting as well. For portfolio managers, risk budgeting seeks to optimize the portfolio's exposures relative to the benchmark to ensure that the allocation is the most efficient use of the active risk budget. An effective risk budget requires that the portfolio manager determines the most appropriate risk measure, understands how each aspect of the strategy contributes to overall risk, determines what level of risk budget is appropriate, and properly allocates risk among individual positions/factors.
Recall that there are two ways to measure risk—absolute and relative. This measurement decision depends upon the manager's mandate and the investor's goals. In addition, a manager's own beliefs about how they add value can influence the choice between an absolute or a relative risk measure. Think about why this is the case. If a manager is good at benchmark outperformance, then the risk measure should be focused around active risk because that's the manager's perceived skill. But what should happen with the other risks within the portfolio?
Incorrect.
This would mean that the manager is taking potentially greater risk in having higher correlations to the benchmark while also using an active strategy.
No.
The manager's main skill is market outperformance, so that should be the main risk of the portfolio.
Correct.
Other risks outside of the manager's perceived skill should be minimized or diversified so that the main portfolio risk is associated directly with the manager. For this reason, it's important to understand the causes and sources of absolute risk. There are two fundamental principles to consider. First, if a manager adds a new asset to the portfolio that has a higher covariance with the portfolio than most current securities, total portfolio risk will rise. Second, if a manger replaces an existing security with another security that has a higher covariance with the portfolio than the security being replaced, the total portfolio risk will rise.
Obviously, adding or replacing a security can directly impact the portfolio's risk in either direction. To calculate total portfolio variance, you can use the following equation.
$$V_{p} = \sum^{n}_{i=1} \sum^{n}_{j=1} x_{i} x_{j} C_{ij}$$
To determine the contribution of each asset to portfolio variance:
$$CV_{i} = \sum^{n}_{j=1} x_{i} x_{j} C_{ij} = x_{i} C_{ip}$$
where $$x_{j}$$ is the asset's weight in the portfolio, $$C_{ij}$$ is the covariance of returns between asset $$i$$ and asset $$j$$, and $$C_{ip}$$ is the covariance of returns between asset $$i$$ and the portfolio.
The contribution of an asset-to-total-portfolio variance is equal to the product of the weight of the asset and its covariance with the entire portfolio. You can apply this single asset approach to other management philosophies where the weights are specific exposures.
Essentially, this equation breaks down the manager's factor exposure risk and unexplained risk. But suppose that risk is fully explained by the manager's portfolio equaling the market portfolio. If that's the case, then what would the beta of the market portfolio be in relation to the market factor?
Exactly.
If the manager's portfolio is equal to the market portfolio, then the portfolio returns would be explained by a beta of 1 to the market factor. This also means that idiosyncratic risk would be fully diversified. But as the manager moves away from the market portfolio, other factors and unexplained variance are introduced.
Incorrect.
The manager's portfolio is equal to the market portfolio, so it's not an inverse relationship.
No.
The market portfolio is equal to the manager's portfolio, so there's a natural correlation.
Similarly, managers that have mandates relative to a benchmark must use relative risk measures. In this case, the manager may use the variance of the portfolio's active return:
$$AV_{p} = \sum^{n}_{i=1} \sum^{n}_{j=1} (x_{i} - b_{i})(x_{j} - b_{j}) RC_{ij}$$
where $$x_{i}$$ is the asset's weight in the portfolio, $$b_{i}$$ is the benchmark weight in asset $$i$$, and $$RC_{ij}$$ is the covariance of relative returns between asset $$i$$ and asset $$j$$.
The contribution of each asset to the portfolio active variance is:
$$CAV_{i} = (x_{i} - b_{i})RC_{ip}$$
where $$RC_{ip}$$ is the covariance of relative returns between asset $$i$$ and the portfolio.
Here in this equation, it's crucial to note that risk is measured *relatively*. While that might sound obvious, it's important because it changes the relationship between how an asset's own risk changes the risk of the entire portfolio. That's because risk is measured against something else—namely, a benchmark. Think about how adding cash to a portfolio changes the risk structure. Clearly, it lowers downside risk, but in comparison to the benchmark, it's a different story. How would the active risk change if cash were added to the portfolio?
That's right!
Active risk actually increases when cash is added to a portfolio that's compared to a fully invested benchmark. Put another way, adding a low-volatility asset within a portfolio benchmarked against a high-volatility index would increase active risk. Likewise, adding a high-volatility asset to a portfolio might lower active risk if the asset has a high covariance with the benchmark.
No, actually.
Even though the portfolio's downside risk is lower, its active risk in relation to the benchmark isn't lower.
No way.
If the benchmark is fully invested, then the active risk will change by adding cash relative to the benchmark.
After determining the type of risk measure and how each strategy contributes to risk, you'll want to determine the appropriate level of risk. Again, this really comes down to the manager's investment style and their conviction in their ability to add value using various strategies. So how might you describe the appropriate risk level?
No.
Right!
The appropriate risk level is subjective in that it depends upon the manager's ability. Even managers with similar approaches can have different opinions on risk. This risk level is important because it also impacts the portfolio structure, turnover, and implementation. So managers must clearly communicate to investors their overall risk orientation, and investors must understand implications of this risk process.
From the manager's perspective, it's crucial to understand that just because a strategy can be executed at a level of risk doesn't mean that it should be. For example, in the presence of constraints (like transaction costs or no short selling), portfolios may face implementation issues that degrade the information ratio if active risk increases beyond a specific level. There's also the factor that portfolios with higher absolute-risk targets face limited diversification opportunities. Essentially, investors seeking a high growth portfolio only have so many options before running out of opportunities. So if investors lose the ability to effectively diversify, how does this impact the graph of risk and return?
Incorrect.
At a certain point, the ability to earn additional returns per a level of risk diminishes.
Incorrect.
At a certain point, investors lose the ability to diversify risk, which diminishes further increases in returns.
Right!
This is the Markowitz theory, where the relationship between risk and return is concave. This means that expected returns increase with risk but at a declining pace because portfolios seeking higher returns eventually run out of high-return investment opportunities.
A third critical point to understand is that there's a level of leverage beyond which volatility reduces the expected compounded returns. This applies to multi-period settings. For example, first take the following equation:
$$R_g = R_a - \frac{\sigma^2 }{2}$$
where $$R_g$$ is the expected compounded/geometric return, $$R_a$$ is the expected arithmetic/periodic return, and $$\sigma$$ is the expected volatility.
Adding leverage to both the return and the standard deviation (expected volatility) will eventually lead to lower returns as the standard deviation is squared. Plus, the equation doesn't consider the cost of funding. So putting it all together, the use of leverage can lead to reduced returns. How might this excess leverage then impact the Sharpe ratio?
Clearly, yes.
At a certain point, the Sharpe ratio will be lower as the returns are offset by an increasing standard deviation. So adding leverage has some benefits initially but can quickly become an issue as managers continue to add risk.
Incorrect.
The Sharpe ratio factors in returns and the portfolio's standard deviation, and the standard deviation is increasing faster than the returns.
Definitely not.
The Sharpe ratio uses returns in the numerator and standard deviation in the denominator, so it's definitely impacted.
At this point, after analyzing which risk measure is appropriate, understanding how the strategy contributes to risk, and determining what level of risk is appropriate, you're finally ready to properly allocate risk among individual positions. Essentially, this process is a culmination of the preceding points and the manager's personal market beliefs.
To sum it up:
[[summary]]
Maximum debt
Maximum spending
Maximum efficiency
They should be magnified
They should be diversified or minimized
They should be correlated to the benchmark
Continue
-1
0
+1
Continue
It would increase
It would decrease
There would be no impact
Objective
Subjective
It's linear
It's convex
It's concave
Continue
Lower ratio
Higher ratio
No impact
Continue
