DeMiguel, Garlappi, and Uppal (2009) summarize the current state of academic thinking about applied portfolio optimization:
In about the fourth century, Rabbi Issac bar Aha proposed the following rule for asset allocation: “One should always divide his wealth into three parts: a third in land, a third in merchandise, and a third ready to hand.” After a “brief” lull in the literature on asset allocation, there have been considerable advances starting with the pathbreaking work of Markowitz (1952), who derived the optimal rule for allocating wealth across risky assets in a static setting when investors care only about the mean and variance of a portfolio’s return. Because the implementation of these portfolios with moments estimated via their sample analogues is notorious for producing extreme weights that fluctuate substantially over time and perform poorly out of sample, considerable effort has been devoted to the issue of handling estimation error with the goal of improving the performance of the Markowitz model… of the 14 models evaluated, none is
consistently better than the naive 1/N benchmark…
The naive 1/N — or Talmudic—asset allocation strategy is to weigh all stocks equally. For example, if you have a population of 10 stocks, you invest 10% of your portfolio in each stock.
Starting around 2009, academics started noting that a naive 1/N approach often performs better than the intricate approach introduced by Markowitz, which entails estimating means (μ) and standard deviations (σ) (of asset returns) from historical data. The key intuition is that estimates of μ, σ, and ρ introduce estimation bias large enough to invalidate the optimal weight computations. Moreover, small changes in μ/σ/ρ estimates can lead to large changes in optimal portfolio weights.
To be sure, more research is needed in this field, but in the meantime, real-world portfolio managers have to make decisions.
The rest of this post summarizes my approach using the Marginal Sharpe Ratio, which I introduced in 2021.
An Example
This article was inspired by this counterintuitive example:
In this setup, we currently own the stock market (e.g., an index fund) and want to add two additional stocks. Stock 2 has a higher Sharpe Ratio and a higher Alpha, but the classical Markowitz optimization algorithm leads to the result that 81.5% of the portfolio should be in Stock 1 and just 9.8% should be in Stack 2. Why is this happening?
Adding One Asset to the Market Portfolio
Consider the case where we have the market portfolio (m) and decide to add just a single new asset (p). Then
We’re trying to find the optimal portfolio weight (w), assuming all other variables are constant. To simplify the problem, let
Then
We’re trying to find the optimal h for any value of μₚ. We can take the first derivative of SR with respect to h and set it to 0.
After a few steps of algebra, we have
This equation shows that a high value of σₚ is associated with a lower optimal portfolio weight. This is consistent with the earlier example. Moreover,
The interesting thing to note about this equation is that the volatility factors drop out and we have an expression that only depends on Sharpe Ratios and correlations.
Asset Power
The last equation suggests the following definition: An asset’s “Power” is the difference between the portfolio-optimal Sharpe Ratio and the Sharpe Ratio of m alone. In other words, it’s the potential contribution of an asset to the market portfolio.
The following table adds the asset power to the earlier example.
Even though the weights of stocks 1 and 2 are quite different, the Marginal Sharpe Ratios and Asset Powers are the same. The above equation for the optimal portfolio Sharpe Ratio confirms that there should be a tight relationship between the Marginal Sharpe Ratio and Asset Power.
Moreover, we shouldn’t be too surprised about the different weights. What matters is Asset Power, not the specific optimal weights.
Taylor Series Approximation
To see this more clearly, we can use a Taylor Series approximation of the portfolio Sharpe Ratio.
In other words, the portfolio Sharpe Ratio will increase if the asset’s Marginal Sharpe Ratio is positive. The factor σₚ/σₘ can be interpreted as implied leverage. In other words, a higher σₚ gives you more of that asset.
As an approximation, we can now write
To find the maximum value, we can write
Finally, we can estimate the Asset Power as
Here, we see again that Asset Power primarily depends on the Marginal Sharpe Ratio i.e., μ and σ drop out.
The above shows us two things about the Marginal Sharpe Ratio:
- For small h, the portfolio Sharpe Ratio increases linearly with the asset’s Marginal Sharpe Ratio.
- The Marginal Sharpe Ratio — and not alpha or standard deviation — is closely associated with Asset Power.
These observations confirm that the Marginal Sharpe Ratio is a potent metric of asset quality. This is especially true for small values of h.
My Portfolio Optimization Heuristic
While we can solve for optimal weights, the naive 1/N research suggests this isn’t the best approach. Rather, I propose the following:
- Compute Marginal Sharpe Ratios for N assets and put them in quartile buckets. Each quartile indicates the amount of equity we’d like to associate with each asset.
- In f′(0), the factor σₚ/σₘ indicates the risk level of the asset. Moreover, leverage associated with the asset should be scaled up or down based on this ratio. For example, if two assets have the same Marginal Sharpe Ratio but one has double the σₚ, its weight should be half that of the low σₚ asset.
This approach blends the best aspects of classic Markowitz and naive 1/N optimization algorithms.
