Asset-Only Asset Allocations and Mean–Variance Optimization

Sometimes combining two unrelated things can be very beneficial. For instance, bread and cheese, when combined, can make a fantastic pizza. Also, arabica beans and boiling water can produce fabulous espresso. You get the point—combining items can result in valuable benefits. It's the same thing with two different assets, which can lead to a reduction in risk when combined the right way. But there's a factor that needs to be considered to make sure that benefits are realized. What characteristic of the two assets must be considered?

No, actually. Returns aren't going to be reduced due to returns.

Not really. That's not a requirement of assets that reduce risk.

You got it! The correlation of the two assets needs to be less than 1 so that risk-reduction benefits can be achieved. The risk can be reduced while returns are maintained. This is the basic premise of the Markowitz theory of __mean–variance optimization__, which focuses on what amount to allocate to each asset in order to maximize the expected return of the portfolio for an expected level of risk.

Essentially, Markowitz focuses on the final product, the overall portfolio, when combining assets, instead of individual asset risk levels. So you can think of mean–variance optimization as a risk assignment tool. Its objective function can be written as $$\displaystyle U_m = E(R_m) - 0.005 \lambda \sigma^{2}_{m}$$ where _U__{_m_} is the investor's utility for the asset mix, _R__{_m_} is the return for asset mix _m_, the investor's risk aversion coefficient is $$\lambda$$, and $$\sigma^{2}_{m}$$ is the expected variance of return for asset mix _m_.

It's important to note that the investor's risk aversion coefficient captures the risk-return tradeoff. So since the coefficient is a part of the variable that's subtracted from the expected return, what do you think happens to the investor's risk level because of a larger investor risk aversion coefficient?

No, actually. Since the risk aversion coefficient is subtracted after being multiplied by 0.005, it doesn't result in a more aggressive portfolio.

Yes! A larger risk aversion coefficient means a more risk-averse investor because the investor's utility is reduced. That means that a completely risk-tolerant investor would have a risk aversion coefficient of 0 and a completely risk-averse investor would have a coefficient of 10. That forms a range for you to remember as you calculate the investor's utility mix. The investor's risk aversion coefficient typically runs from 1–10, with 4 representing a moderately risk-averse investor. It's also important to note that the value 0.005 is presented with the assumption that the expected return and the expected variance are in percentage terms. If these terms aren't percentages, then the value is 0.5.

That's not it. Although the risk aversion coefficient is multiplied by 0.005, it still impacts the investor's utility.

Overall, the mathematical calculation of the investor's utility asset mix is pretty straightforward, but the application clearly isn't. Besides, balancing risk and return, there are other constraints to consider. That's because in a perfect world, investors would be able to implement the entire range of allocation weightings, both positive and negative. So what strategies does a full mean–variance optimization include?

That's not it. Long positions are allowed in the mean–variance optimization.

Not quite. Negative allocation weights are allowed in the full mean–variance optimization.

You got it! A full mean–variance optimization strategy allows for holding long and short positions because there are positive and negative allocation weights. Basically, there would be a wide range of asset allocations that would have very extreme long and short weightings. But a shorting or short-selling strategy isn't always available to all investors, and that can act as a constraint to the portfolio. It's known as the budgeting constraint or the utility constraint because it forces you to optimize asset allocation numerically.

Here, it's also important to remember that the risk aversion coefficient isn't a precise process. It's an estimation based on the investor's capacity to take risk and the investor's preference to take risk. For example, suppose an investor has several million CNY and has total spending needs of CNY 75,000. That clearly indicates a capacity to take risk, so capacity is based on hard numbers. But this investor could also have a clear preference to avoid risk because preference is emotionally based. So which risk factor would you describe as subjective?

That's not it. Capacity is a numerical evaluation.

That's it! Preference for risk taking is a subjective measure because it involves an evaluation of the client, which can be influenced by biases and previous experiences. So it's important to remember that for a given level of expected return to achieve the client's goals, you'll want to err on the side of risk aversion. Essentially, this means that you don't want to take more risk than the client wants to take at a given level of expected return. So really, there's a delicate balance for clients who have a significant need for return to meet future requirements but don't have any risk tolerance.

No. One is numerical and one is emotional.

To sum it up: [[summary]]

Here's a quick example: with an expected return of 12%, standard deviation of 18%, and risk aversion factor of 1.6, what utility do you calculate?

Good work!

Using the formula: $$\displaystyle U_m = E(R_m) - 0.005 \lambda \sigma^{2}_{m}$$ You can substitute the values using percentages: $$\displaystyle U_m = 12 - 0.005 (1.6)(18^2) \approx 9.408$$ which tells you that the utility is about 9.4%, or else you can use decimals to calculate the same value, noting the 0.5 coefficient this time: $$\displaystyle U_m = 0.12 - 0.5 (1.6)(0.18^2) \approx 0.094 = 9.4 \%$$

No, that's mixing the 0.005 coefficient with decimals. Here's the calculation:

Not quite. Here's the calculation:

Return

Cash flows

Correlation

It's more aggressive

It's more risk averse

There's minimal impact

Only short selling

Holding only long positions

Holding both long and short positions

Capacity

Preference

Both risk factors

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Other responses

0.094

0.12

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9.4