modern or, as in our case, post-modern portfolio theory—suffer from certain inherent draw- backs. A particular drawback associated with traditional frameworks is the sheer number of constraints applied to the optimization process. Constraints reduce the credibility of the optimi- zation process as, by applying constraints, the practitioner is guiding the model towards a particular solution—rather than allowing the model to calculate one itself independently. In deriving our optimal portfolios, we do not apply any constraints. hence, we let the model dictate the direction and magnitude of results.
33 All optimizations are based on J.P. Morgan Long Term Capital Market Return Assumptions.
Optimal Portfolio Solution for our hypothetical Investor
In this section, we ask ourselves how incorporating non-normality into our return distributions affects our optimal portfolio solution for our hypothetical investor.
Exhibit 49 compares our current hypothetical investor with a portfolio allocation using an identical target arithmetic return of 9.1%, but optimized using our revised non-normal framework33.
For illustrative purposes, we also show the optimized allocation derived from a traditional-mean variance framework. We do not impose any constraints on our optimization process.
ExhIBIT 49: OPTIMIzATION RESULTS
Source: J.P. Morgan Asset Management. For illustrative purposes only. Sharpe ratio calculated assuming risk free return of 4.0%.
Current allocation Optimized, Unconstrained Normal allocation Optimized, Unconstrained Non-normal allocation
Total bonds 30.0% 0.0% 34.5%
U.S. Aggregate Bonds 30.0% 0.0% 34.5%
Total equity 55.0% 18.3% 36.0%
U.S. Large Cap Equity 40.0% 8.9% 21.7%
International Equity 10.0% 7.6% 5.8%
Emerging Markets Equity 5.0% 1.9% 8.5%
Total alternatives 15.0% 81.7% 29.5%
REITs 5.0% 11.9% 11.1%
Hedge Fund of Funds 5.0% 64.1% 7.0%
Private Equity 5.0% 5.6% 11.5%
Total 100.0% 100.0% 100.0%
key statistics
Target expected arithmetic return 9.1% 9.1% 9.1%
Expected volatility 10.0% 8.6% 9.5%
Expected compound return 8.7% 8.6% 8.7%
Sharpe ratio 0.51 0.59 0.54
CVaR95 ($ million) allowing for non-normality $168 mm $206 mm $148 mm
CVaR95 vs. current allocation - 23% 12%
34 Minimizing standard deviation for a given expected return target is the equivalent of
maximizing Sharpe ratio.
Our results indicate that the ‘optimal’ portfolio using a tradi- tional mean-variance framework actually increases (rather than decreases) risk by 22.6% relative to the current allocation— as defined by CVaR95. As a traditional framework minimizes standard deviation34—which we argue is an inadequate risk
measure—it inadvertently exposes our investor to even worse scenarios on the downside than the current allocation. However, the more significant issue by far—and the biggest drawback of the traditional approach—is that it produces a highly concentrated and impractical asset allocation. This is because in the absence of formal constraints, it over allocates to asset classes based on small differences in assumptions. This limits our ability to draw useful insights into the portfolio construction process, using such a framework.
On the other hand, our non-normal CVaR95 based framework produces a diversified solution with allocations across the asset class spectrum. No single asset class significantly dominates the portfolio. Despite the fact that our investor already holds quite a diversified portfolio, our framework suggests that there is still scope for our investor to improve portfolio efficiency further. The “optimal” portfolio identified by the non-normal frame- work improves both the expected Sharpe ratio and reduces the CVaR95 relative to the current allocation. This signifies that our optimal portfolio is more efficient (in Sharpe ratio and CVaR95 space) than the current hypothetical portfolio. Based on our Long Term Capital Market Return assumptions, the allocations to fixed income and alternatives increase, while the allocation to equities decreases.
SErIAl CorrElAtIon
The impact of serial correlation is negative for International Equity, Emerging Markets Equity, hedge Fund of Funds and Private Equity as these asset classes test positive for first order serial correlation. Recall that the presence of serial correlation increases the risk associated with the particular asset class— hence its negative marginal impact. As U.S. Bonds, U.S. Equity and REITs do not test positive for serial correlation, the marginal impact for these asset classes is positive (relative to a traditional mean-variance framework). “FAt” lEFt tAIlS
Our second form of non-normality—“fat” left tails—has a negative marginal impact on optimal allocations to U.S. Equi- ties, REITs and hedge Fund of Funds (relative to a traditional mean-variance framework). This is driven by the asset classes’ distributional properties—its negative skewness, excess kurtosis and extreme negative values (all of which are bad for investors). In particular, our model indicates higher excess kurtosis for U.S. Equity, REITs and hedge Fund of Funds compared to U.S. Bonds, International Equity, Emerging Markets Equity and Private Equity (after allowing for serial correlation). This implies that these asset classes have “fatter” left tails than the other asset classes, when compared to a normal distribution. hence, our model reduces allocations to U.S. Equity, REITs and hedge Fund of Funds (relative to a traditional framework) due to the prevalence of this phenomenon.
ConVErgIng CorrElAtIonS
Our calibration results for the copula and optimization results suggest correlation convergence during periods of market stress is much more pronounced for International Equity and hedge Fund of Funds (after allowing for serial correlation and “fat” left tails). hence, the model reduces allocations to these asset classes relative to traditional mean-variance frameworks. In other words, the degree of non-linearity in correlations is much more severe for International Equity and hedge Fund of Funds, compared to the other asset classes. This implies that diversification benefits from these asset classes do not materialize to the extent implied by linear correlation matrices (during periods of market stress). This detracts from relative allocations.
35 Note the impact of the form of non-normality on the optimal portfolio solution varies
along the efficient frontier. The analysis in this section is broadly representative of a portfolio with an expected (arithmetic) return of 9.0%—based on our Long Term Capital Market Return Assumptions.