This section proposes using so-called replicating portfolios for benchmark purposes. Replicating portfolios are optioned portfolios designed to repro- duce the risk characteristics of a given asset by combining a benchmark asset with options and the risk-free asset.
The expected return on a replicating portfolio for a given risk shape (of a particular asset) will be called the replicating return. The replicating return can be interpreted as the alternative return an investor may achieve if, hold- ing the risk exposure (defined here in terms of return variance, skewness, and a number of higher moments) constant, she chooses to invest in the optioned market instead of investing in a given portfolio.
The efficiency gain/loss measure or excess replicating return is simply the difference between the expected return of the asset under investigation and that of its synthetic benchmark asset. The expected return of this repli- cating benchmark asset will be termed as the replicating return.1
This asset-specific replicating return embodies, at the same time, the minimum acceptable return on investments having the same risk structure, and serves thus as a natural benchmark. That is, investors always have the
1Certainly the replicating benchmark asset will have to be computed to achieve maximum expected return within the set of possible replicating assets with the same (moment-based) risk characteristics.
alternative of being paid the return of the replicating optioned portfolio.
Consequently, this replicating return has to be exceeded by other investments exhibiting similar risk characteristics.
The efficiency gain/loss measure (the excess replicating return) takes the form
ERRp=E(rp)−RR(rp) (16.1) whereE(rp) =expected return on portfolio p
RR(rp)=expected replicating return
The excess replicating return can be directly interpreted as an efficiency gain, if it is positive, or an efficiency loss, if it is negative. If the replicating optioned return is higher than the expected return for an arbitrary CTA port- folio, this underlying asset offers an inferior performance (compared to the benchmark asset). That is, the comparable investment in form of an optioned portfolio offers a higher expected return for the same risk characteristics of returns. The fund’s shareholder would do better with a different fund (of course, as stand-alone investment only). The excess replicating return pro- vides a simple measure in assessing whether a portfolio outperformed others on an ex-post basis.
This measure is in a close relationship with the excess return measure proposed by Ang and Chau (1979), which is an alpha-like composite per- formance ratio. An important distinction is that, in the replicating case, individual portfolios do not have to possess the same systematic risk char- acteristics as the benchmark asset. It is sufficient if both share the same return distribution shapes.
Construction of Replicating Portfolios
As defined earlier, replicating portfolios are portfolios that have the same risk structure in terms of some statistical moments (of order three and higher) as the portfolio being assessed. The foundation for including repli- cating portfolios in the performance assessment is the assumption that port- folios can be created to “mimic” the risk structure of the underlying asset as benchmarks.
The present replicating framework will be termed partial, because only a reduced set of the return characteristics (the moments) is used for de- scribing any return distributions, thereby reducing the return distribution’s dimensionality.
It is very important to note that the term “replication” as we use it does not intend to create the same payoff profile in terms of identical probabil- ity distributions, nor does it intend to create portfolios having the same pay- off in every possible state of nature.
For the construction of individual replicating portfolios, we used the Standard and Poor’s (S&P) 500 index as underlying. Based on the assump- tion that returns from the index are independent and follow a lognormal distribution—a simplification that greatly facilitates the use of contingent claim–based performance evaluation but is not essential—we calculated prices for a specified number of Black-Scholes call options. Considering only call options ensures that asset returns are not linearly dependent. For the sake of simplicity, a holding period of one year is assumed.
In the next step, we used nonlinear programming for generating returns on replicating portfolios, with variance and (a predefined number of) higher statistical moments being set to that of the CTA under investigation.
This approach provides a relatively simple and robust means for calcu- lating individual benchmark returns. This idea parallels the work of Amin and Kat (2001). They propose a point-by-point optimization algorithm with a 500-pins-setting, that is, they match 500 separate points of the return distribution, to calculate hedge fund efficiency gains/losses.
The optimization algorithm that produces replicating portfolio weights xi, can be formulated:
(16.2) subject to
target variance
target skewness (16.3)
and the constraints on the portfolio weights
whereE(ri) =expected return on asset i
s02s03=target values for variance and skewness, respectively.
The constraints in equation 16.3 can be expanded to include moments of order higher than three.
xi
∑i =1
s3 = s03 σ2 = σ02
Max
xi i
i
Z = ∑x E r( )i
Choosing the Right CTA 299