Sat, 01/10/2005 - 22:00
OPTIMAL NET EXPOSURE
In this section we calculate numerically the optimal net market exposure as a function of the market return, volatility, alpha and the long short correlation.
The following chart shows sensitivity analysis for the optimal net market exposure as a function of the long short correlation and market return given the alpha of 10% and the market volatility of 15% as in the previous example.
CONCLUSION
We derived and analyzed the theoretical result for the information ratio as a function of the net market exposure and other main parameters. We also derived the optimal net exposure as a function of the long short correlation and the market return. Our analysis has the following important implications on managing a long short investment strategy:
THE APPENDIX
We construct a long short strategy by overlaying a directional strategy over the market neutral strategy. We start from conventional definitions of return and return volatilities for the market neutral and the market separately and then derive the combined (long short) returns and volatilities as well as the information ratio.
R_{ls = }R_{mn }+ kR_{m}
Where:
R ls denotes the long short return
R mn denotes the market neutral return and
R m denotes the market return
k denotes the Net market exposure and k>0 implies net long position, whilst
k<0 implies net short position
We can express the long short strategy information ratio IR=R_{ls }/ ó_{ls} in terms of the Net
market exposure, market neutral as well as market returns, volatilities and the correlation
between the market and the market neutral strategy.
IR_{ls }=R_{ls }/ ó_{ls }= (R_{mn }+kR_{m}) / (ó^{2}_{mn }+ k^{2}ó^{2}_{m }+ 2ñkó_{mn}ó_{m})^{1/2}
Where:
ó _{mn }denotes the market neutral volatility
ó _{m} denotes the market volatility
ñ denotes the correlation between the market and the market neutral strategy
It is helpful to decompose the market neutral return and its volatility into the long and the short side returns volatilities and the correlation.
R_{mn} = R_{l }- R_{s}ó ^{2} _{mn }= ó_{1}^{2 }+ ó_{s}^{2 }- 2 ñ_{ls} ó_{l} ó_{s}
Where:
R_{l}, R_{s }denote the long-side and the short-side return respectively,
ó_{l}, ó_{s} denote the long-side and the short-side return volatilities and
ñ_{ls} denotes the long-side and the short-side return correlation
So the information ratio becomes
IR_{ls }= (R_{l }- R_{s }+ kR_{m}) / [(1+2ñkó_{m}) ó_{1}^{2}+ (1+2ñkó_{m}) ó_{s}^{2}-2ñ_{ls}ó_{l}ó_{s }+ k_{s}^{2}ó^{2}_{m}- 4ññ_{ls }kó_{l}ó_{s}ó_{m}]^{1/2}
The returns from the long and the short side can be further decomposed as follows:
R_{l }= R_{m} + á_{l}R_{s}= R_{m }- á_{s}
Where:
á_{l }, á_{s }denote long and short alphas (measures of long-side out-performance and short-side under-performance against the market)
So that the information ratio finally becomes
IR_{ls }= (á_{l }+ á_{s} + kR_{m}) / [(1+2ñkó_{m}) ó_{1}^{2}+ (1+2ñkó_{m}) ó_{s}^{2}-2ñ_{ls}ó_{l}ó_{s }+ k_{s}^{2}ó ^{2} _{m}- 4ññ_{ls }kó_{l}ó_{s}ó_{m}]^{1/2}
The effects of all nine parameters under realistic assumptions are summarized in the table below:
Parameter |
Effect on information ratio when increasing |
Effect on information ratio when decreasing |
á_{l} |
Positive |
Negative |
á_{s} |
Positive |
Negative |
á_{m} |
Negative |
Positive |
ó_{l} |
Negative |
Positive |
ó_{s} |
Negative |
Positive |
ñ_{ls} |
Positive |
Negative |
ñ |
Negative |
Positive |
R_{m} |
Positive |
Negative |
k |
Can be either |
Can be either |
The net market exposure can have either positive or negative effect as it features both in the numerator as well as the denominator of the above formula. In order to investigate the impact of taking net long / short position, it is helpful to make some realistic assumptions
Which simplify the above formula:
If á_{l }= á_{s} = á / 2, ó_{l }= ó_{s}= ó_{m }and ñ = ()^{3 }we can condense alphas and volatilities so that the information ratio can be simplified.
These assumptions are rather realistic and robust in terms of how they affect the information ratio
For additional information, please contact: Melissa Hill,
Sabre Fund Management Ltd, Windsor House, 55 St James's Street, London SW1A 1LA
Tel: +44 (0)20 7316 2801 Fax: +44(0)20 7316 0180
Email: [email protected] Website: www.sabrefund.com
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