Strategic Asset Allocation

Utility and Maximization

Strategic asset allocation involves deciding among the various securities within a portfolio and the relative weightings of each. Asset allocation differs from Investor to Investor as the optimal mix depends on the investor’s preferences. The utility theory helps formalize these human preferences into a mathematical equation. According to utility theory, the optimal asset allocation is the one that is expected to provide the highest utility to the investor, given the investor’s investment time horizon.

The Optimal Choice – Capturing Human Preferences in a Math Formula

The most straightforward asset allocation decision would be to mix risky and risk-free assets. The formula below allows for the calculation of the weight of the risky asset in the portfolio:

$$ w^\ast=\frac {1}{\lambda} \left[\frac { (\mu-r_f) }{ \sigma^2 } \right] $$

Where:

• \(w^\ast\) = Weight of risky asset in the portfolio.
• \(\lambda\) = Investor’s degree of risk aversion.
• \(r_f\) = Risk-free rate of return.
• \(\mu\) = Mean expected return of a risky asset.
• \(\sigma^2\) = Variance of the return.
• Weight of the risk-free asset = 1 – (Weight of the risky asset).

The Nine-step Process

Asset allocations are the result of a process of analysis and study. Portfolio managers can use various methods, which can frequently change with technology. However, at the highest level, managers must decide on a trusted process and go through the necessary iterations to determine the asset allocation that best enhances the client’s wealth.

1. Determine and quantify the investor’s objectives. Begin with broad thinking about the investor’s investor’s intentions and objectives regarding the portfolio.
2. Determine the investor’s risk tolerance and how risk should be expressed and measured. A savvy analyst must choose how to quantify and analyze the investor’s risk tolerance. Various frameworks are provided in the curriculum.
3. Determine the investment horizon(s). Over what horizon(s) should the objectives and risk tolerance be evaluated?
4. Determine other constraints and their requirements on asset allocation choices. The analyst will look at the IPS document to determine what other constraints may be relevant. These can range from legal to ESG, to tax status issues.
5. Determine the best approach to asset allocation. Again, the analyst must determine the method of arriving at the asset allocation that will be most suitable for the investor.
6. Specify asset classes. This is necessary to develop a set of capital market expectations for the specified asset classes.
7. Develop a range of potential asset allocation choices for consideration. These choices are often developed through optimization exercises, MVO, and Monte Carlo simulation.
8. Test the robustness of the potential choices. This testing often involves simulations to evaluate the perspective allocations’ potential outcomes.
9. Repeat Step 7. Repeat until an appropriate and agreed-upon asset allocation is constructed and put into the IPS.  

Question

According to the optimal choice equation, increasing the variance of the returns on the risky asset most likely affects the weight of the risky asset in the portfolio?

1. Reduce weight.
2. Not enough information.
3. Increase weight.

Solution

The correct answer is A:

Mathematically, the formula shows the returns’ variance in the fraction’s denominator. Increasing this denominator will decrease the quotient, and the resulting weight afforded to the risky asset.

Theoretically, this also makes sense because more significant variances mean the asset is subject to more powerful swings and significant unknowns. Leaving all other factors of the decision equal, the risky asset should be afforded less weight in the portfolio since it is more volatile, a method of taming the volatility by increasing the presence of the risk-free asset. This will keep the original outcome the same.

B is incorrect. The optimal choice equation provides enough information to determine the effect of increasing the variance of the returns on the risky asset. The equation shows how the optimal weight of the risky asset depends on the expected return, the risk-free rate, the risk aversion coefficient, and the variance of the risky asset. Therefore, if we know these parameters, we can calculate the optimal weight of the risky asset for any given level of variance.

C is incorrect. It contradicts the optimal choice equation. The equation implies that the optimal weight of the risky asset is inversely proportional to the variance of the returns on the risky asset. Therefore, if the variance increases, the optimal weight decreases, and vice versa. Intuitively, this means that the investor will increase the exposure to the risky asset as its riskiness decreases, and allocate less wealth to the risk-free asset.

Asset Allocation: Learning Module 3: Overview of Asset Allocation; Los 3(g) Recommend and justify an asset allocation based on an investor’s objectives and constraints