What modern portfolio theory reveals about PPM
Why should you select projects via portfolio optimization? Many experts justify optimization by citing the Nobel Prize-winning modern portfolio theory (MPT) from finance. MPT's principle of diversification is the foundation of every financial portfolio, but its portfolio optimization, which estimates the efficient frontier, is so sensitive to errors in data, called estimation errors, the optimization creates portfolios that are horribly suboptimal, often to the point of being obvious nonsense.
Financial portfolio optimization has a quadratic objective function, which causes its extreme sensitivity, so tiny errors in data can cause dreadful results. You've heard the phrase, "garbage in, garbage out." With MPT, it's often "great data in, garbage out."
Knowing of this sensitivity, one may ask about optimization in project portfolio management (PPM). It is as sensitive to uncertainty? When creating financial portfolios, how do experts cope with optimization's extreme sensitivity? Can these techniques teach fundamental principles to PPM?
Addressing the first question, most PPM optimizations are linear models, which are much less sensitive than MPT's optimization. However, uncertainty is still an issue. For examples, see my discussions:
- Revenue forecasting errors dominate decision trees
- Overconfidence and underestimating risk
- Estimating probabilities of success - it's not so successful
- How erroneous data causes project selection errors
Additionally, PPM suffers from another source of uncertainty. The optimization models, in addition to data, can differ from reality. For some examples, see my discussions:
Let's see how MPT handles its sensitivity to estimation error and develop insight for improving PPM. Three common methods of managing the sensitivity are (1) simplifying the objective function, (2) simplifying the correlation matrix and (3) replacing optimization with the 1/n allocation rule.
Simplify the objective function: To improve MPT's performance, one can replace its quadratic objective function with a linear one. This solution may seem odd because the linear function violates the physics of the problem; it's the wrong model. However, compared to quadratic functions, linear objective functions are much less sensitive to errors in data. The benefits of reducing this sensitivity exceed the errors created by using the wrong model. This example suggests provocative questions for PPM. Can sophisticated portfolio optimization models be too complex? If so, can simplifying a portfolio optimization model improve performance?
Simplify the correlation matrix: In MPT, a correlation matrix contains the correlations of every asset's value with every other asset's value. Unfortunately, future correlations are notoriously difficult to predict and errors in predicting the correlations wreck optimization. Let's review three methods for managing this problem: single-index models, multi-index models and constant correlation matrices.
- Single-index model: A single-index model estimates an asset's return by using just one variable: the asset's beta, Β, which is its responsiveness to the market. A single-index model predicts the covariation of two assets' returns by using their betas: Cov(Ri, Rj) = ΒiΒjσ2 , where σ is the variance of the market.
- Multi-index model: A multi-index model is
similar to a single asset model, but in addition to beta, it
includes other factors that affect assets' values such as
inflation and energy prices. If a multi-index model has
m factors, the co-variation between two assets' returns is
predicted by:
Cov(Ri, Rj) = Βi,1 Βj,1 σ12 + Βi,2 Βj,2 σ22 +...+ Βi,m Βj,m σm2
The equation is similar to the single-index model, except it contains one term for each factor that affects the assets' returns. - Constant correlation matrix: A constant correlation matrix assigns the same correlation to every pair of assets. To construct the matrix, the technique first calculates the correlation between each pair of assets by using historical data. It then averages all of these correlations together to create a grand average. Finally, it fills the entire correlation matrix with the grand average.
The relative performance of these methods is startling. One might expect multi-index models to outperform single-index models. Multi-index models use more variables and information so they might better approximate the behavior of financial assets. Meanwhile, single-index models ignore important and impactful variables. Yet single-index models significantly outperform multi-index models .
Now consider the constant correlation matrix. By assigning the same correlation to every pair of assets, it's guaranteed to be the wrong matrix. Surely, it can't work! Actually, it works the best, outperforming both multi-index models and single-index models.
How can this be? Here are the reasons:
- Theoretically, the multi-index models are the best ones, since they better match reality. However, their numerous variables provide more opportunities for uncertainty to infect the model. Single-index models admit less uncertainty, so they are less corrupted.
- Theoretically, the constant correlation matrix is completely wrong, with every entry being erroneous. Yet the grand average is a reasonable guess for any correlation. Filling the matrix with the grand average ensures that no entry is wrong by too much. In contrast, by trying to predict every correlation, the single-index and multi-index models make bigger errors.
Once again one can see that the benefits from managing uncertainty exceed the costs of using a simpler and theoretically imperfect model. One might wonder whether sophisticated optimization models in PPM, such as those that include project interactions, are helpful or hurtful.
Remove optimization: The most surprising result in MPT come from the paper:
- DeMiguel, V., L. Garlappi and R. Uppal (2009), "Optimal versus naive diversification: how inefficient is the 1/n portfolio strategy," The Review of Financial Studies, vol. 22, no. 5, pp. 1915-1953.
Here is a simple rule: If there are n assets, assign 1/n of the capital to each one. The 1/n rule replaces optimization and it eschews all historical data, forecasts, theoretical models and experts estimating variables.
How does 1/n perform? When tested against fourteen strategies of portfolio optimization including complex methods such as Bayesian approaches, 1/n performed best.
Why is 1/n so successful? By eschewing all data and models, its decision method is unaffected by uncertainty. Meanwhile, by assigning an average amount of capital to each asset, it produces a reasonable portfolio, generating reasonable results. All of its allocation decisions are suboptimal but the 1/n approach commits smaller errors than optimization models. By striving for perfection, portfolio optimization creates large errors and large losses.
Insights from MPT: The linear objective function, simpler correlation matrices and 1/n rule are suggestive. Perhaps, managing uncertainty well is the pathway to success, and when facing sufficient uncertainty, simpler models outperform sophisticated ones. With these possibilities in mind, consider some questions.
- Can models for project portfolio optimization be too complex?
- Which features of optimization models are robust to uncertainty? Which ones are sensitivity to uncertainty?
- Can projects, drug discovery and drug development face so much uncertainty that simpler methods, like rankings and cutoff values, outperform optimization models?
These questions are subjects of my PPM research. You can read some thoughts about them from my discussions "Action flexibility and state flexibility in PPM" and "Why some C-level executives are skeptical of PPM."
Finally, financial portfolio optimization receives excellent feedback. The results of one's decisions, and of portfolios that were not chosen, are unambiguous and forceful. Unfortunately, the executives who perform PPM receive poor feedback. Lacking feedback, problems with PPM, such as sensitivity to uncertainty, can persist unnoticed and unchecked. For some thoughts on feedback and PPM, see my discussion, "Where's the feedback?"
After reading my discussions, many managers wish to share their experiences, thoughts and critiques of my ideas. I always welcome and reply to their comments.
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