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Practical Portfolio Performance Measurement and Attribution, Second Edition by Carl R. Bacon

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8
Multi-period Attribution
An expert is a person who has made all the mistakes that can be made in a very narrow field.
Niels Bohr (1885-1962)
We observed in Chapter 3 that the sum of the arithmetic excess return for each finite period does not sum to the total arithmetic excess return for the total period:
612
Therefore over multiple periods we should not expect our arithmetic attribution factors which add over single periods to add up for the total period under analysis.

SMOOTHING ALGORITHMS

Accepting that it is desirable for multiple period arithmetic attribution factors to add up over time, a number of methodologies (known as smoothing algorithms) have been developed to achieve this:

Carino

Carino (1999) suggests that the results for the single period can be transformed into results that naturally cumulate over time. Continuously compounded returns may be summed as demonstrated in Equation (2.30).
Using this relationship Carino introduces the factor:
613
if
614
Since from Chapter 2 for continuously compounded returns we known that:
615
and similarly for the benchmark:

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