Portfolios

Abstract

What investors want to see are positive returns that lack variability, skew to the upside and don’t have extreme changes. Delivering consistently favourable returns requires not just getting your main views right, but making sure losses are relatively limited. Doing this encourages the use of mathematical models to combine views and control risk. When targeted investments are heavily correlated, the real risk attached to a view might be unintentionally large, even if prevailing investment risk limits permit the position. Models need not merely provide a cross-check on exposures, though, as they can also be used to drive the investment decisions. The combination of views into constructed portfolios will be addressed in this chapter, along with the main assumptions underlying portfolio theory.

Appendices

Main Messages

• Investors want positive returns that lack variability, skew to the upside and don’t have extreme changes. In other terms, they love the odd statistical moments (return and skew) but hate the even ones (variance and kurtosis).

• Assets that are inherently riskier are usually expected to ultimately yield higher returns as fair compensation to the investors taking those risks.

• The Capital Asset Pricing Model (CAPM ) and Arbitrage Pricing Theory (APT) both use simple linear regressions to estimate the risk factors for an asset relative to the overall market, but CAPM is limited to the beta of returns relative to the market.

• As the number of assets in the portfolio increases, the portfolio variance should reduce. Idiosyncratic risks can be diversified away, but systemic risks cannot because there are many non-vanishing correlations.

• Portfolios comprised of assets with lower absolute correlations should experience less variance. Attention should, therefore, be paid to the correlation of assets when constructing a minimum risk portfolio .

• Most portfolio managers don’t formally optimise where they are investing in the feasible set of potential portfolios. Many will have a benchmark portfolio where their views are aiming to add value (“alpha”) relative to that.

• The most popular approach to optimising a portfolio is to set an objective function for minimising the variance of the portfolio subject to the constraint of expecting to exceed a predefined level of return.

• Value at Risk (VaR ) attempts to quantify the loss (α) that is supposed to occur with a specified probability (β). Meanwhile, Conditional Value at Risk (CVaR) is the mean expected loss beyond the value at risk (α), so it contains more information about the adverse tail.

• Options have nonlinear return distributions by design, so selling out of the money ones picks up a little premium without much variance in the current value under normal circumstances, but as the option nears the strike price, losses can balloon. VaR does not capture this risk.

• Shocks after a period of calm can cause VaR models to abruptly incorporate additional downside risks, thereby causing many investors to reduce risk to stay within permitted levels. That is a “VaR shock”, which can sometimes be anticipated among others and sidestepped by assuming a more realistic distribution of returns in the model.

• CVaR lends itself to optimisation far better than VaR , which is a non-smooth, non-convex and multi-extremal function with respect to positions. In other words, the optimal solution depends on the starting weights and need not lead to the actual optimum at all.

• Portfolio performance will naturally depend on the investment universe, period, constraints and assumptions, but in general, optimised portfolios tend to deliver more stable performance.

• Instead of minimising risk, the investor’s utility might be maximised instead. However, optimisation of approximated utility functions is computationally intensive and has only recently come within the scope of computing.

• The Black-Litterman model adds value by allowing the user to tilt towards views that the investor wants to incorporate. It then minimises variance subject to a returns target given the model’s augmented baseline expectations .

• Covariance between one view and either others or the baseline portfolio will constrain the weight assigned to that opinion in the overall portfolio . Conviction levels will also influence the relative weighting of view portfolios and the baseline.

• For investors seeking to express nonlinear views, perhaps on the lower tail of the distribution or its codependence, a more complex solution like entropy pooling is needed.

• Less sophisticated models will be wide of the mark when normal distributions are assumed in environments when nonlinear moves are likely, perhaps because of a nonlinear economic shock or when inherently nonlinear options are in the portfolio .

Further Reading

• Fabozzi, Frank. Kolm, Petter. Pachamanova, Dessislava. Focardi, Sergio. 2007. Robust Portfolio Optimization and Management. John Wiley & Sons.

• Uryasev, Stanislav. 2000. Conditional value-at-risk: Optimization algorithms and applications. Financial Engineering News, 14. February, 1–5.

• He, Guangliang. Litterman, Robert. 2002. The Intuition Behind Black-Litterman Model Portfolios.

• Meucci, Attilio. 2008. Fully Flexible Views: Theory and Practice. Risk. 21 (10), 97–102.