The document discusses asset allocation and measures for evaluating portfolio performance over multiple time periods. It proposes using wavelet analysis to decompose portfolio returns into different time scales or frequencies, representing different investment horizons. This allows calculation of a Sharpe ratio at each time scale to assess performance over various periods. As an example, it calculates 6-period Sharpe ratios for different asset classes and ranks their performance over those periods. Wavelet analysis provides a tool for measuring multi-period portfolio performance while addressing issues like non-normal returns and non-stationary data.

1. Anna Zamojska
Chair of Econometrics
University of Gdansk

2.  Asset allocation is a process that identifies the
optimal portfolio for a given investor over a
particular investment horizon.
 An investment horizon depends on when and
how much profit the investor expects in the
future. This assumption implies that the length
of the investment horizon influences the
optimal investment strategy.
 In general, the shorter the investors' horizon,
the less risk she/he should be willing to accept
for given level of profit.

3.  One of the oldest and the most popular
measure of the performance of a portfolio in
assets allocation is the Sharpe ratio [Sharpe
1966]:
S = E[ rp – rf]/sd(rp)
 After Sharpe’s 1994 revision:
S = E[ rp – rb]/sd(rp – rb)
 Ratio is sensitive for window of the selected
sample, but a common approach in portfolio
evaluation is calculation one-period investment
Sharpe ratio [Levy 1972].

4.  It is suggested in the literature that for n-period
investment scaling factor should be taken into
account, but in empirical analysis researchers
and experts usually calculate only the one-
period Sharpe ratio [Kim and In 2005].
 The portfolio managers make decisions over
different time periods and especially they
concentrate on the performance at the end of
the clearing period. So the problem is how to
create the performance measure resist from
above restrictions.

5.  Given the importance of multihorizon portfolio
performance measure we finded precious little
econometric tools and empirical analysis.
 Kim and In [2005] proposed the wavelet analysis
as a tool for measurement of portfolio
performance. That approach allows to decompose
the unconditional variance into different time
scale, which means different sample windows. The
advantage of the approach is possibility of analysis
of the non-normally distributed portfolio returns
and non-stationary series.

6.  We compare the traditional Sharpe ratio and the
Sharpe ratio at each scale (from 1 to 6). The Sharpe
ratios at different scales represent the performance
measure of portfolio at various frequencies
(various time scales).
 We use a simple Haar wavelet, which is the first
wavelet filter. That simply wavelet allows to
receive two types of elements of the time series.
The first one, which capture the higher-frequency
oscillations, represent increasingle fine scale
devations from the smooth trend, and the second,
represents the smooth coefficients that capture the
trend.

7. w avelet scale
1 2 3 4 5 6
0.000000
0.000252
0.000504
0.000756
0.001008
0.001260 MIXED
STOCK
BOND
MONEY

8. Scale
Ranking
Mixed Stock Bond Money
1 0,0009
1
0,0007
2
-0,0013
3
-0,0042
4
2 -0,0018
3
0,0034
1
-0,0014
2
-0,0143
4
3 -0,0036
4
-0,0004
3
0,0016
2
0,0082
1
4 0,0132
3
0,0050
4
0,0429
2
0,2887
1
5 0,0049
3
0,0024
4
0,0145
2
0,1029
1
6 -0,0564
2
-0,0378
1
-0,2768
3
-1,1521
4