Index creation from multiple time-series and variable weights

I am trying to compose one index out of several (three) indices with variable weights, 50%, 25% and 25%.

After normalizing and calculating the log returns, what would be the best way to create the final benchmark index. We would also require to re-balance daily?

• Hi SolitonK, welcome to Quant.SE! What do you mean with the final benchmark index? – Bob Jansen Nov 4 '14 at 15:22
• Hi Bod, the final composite index. – SolitonK Nov 4 '14 at 15:41
• Are you saying the weights change daily? Do the indices have approximately the same value? If so, INDEX1/2 + INDEX2/4 + INDEX3/4 works. If not, you should decide how to normalize (set them all to 100 on day 0?) – barrycarter Nov 4 '14 at 20:44
• Hi, no the indices are quite different in value but I normalize to a common base say 100. – SolitonK Nov 4 '14 at 21:47

You can't really combine the assets' log returns. You should calculate percentage returns for the three assets. Then at each time step, the portfolio's total return is:

$r(i) = 0.5 \times \text{asset1_return}(i) + 0.25 \times \text{asset2_return}(i) + 0.25 \times \text{asset3_return}(i)$

Once you've calculated the time series of the portfolio's returns, you create another time series by adding 1 to each return, $r(i)$ -- let's call this series "$s$". It is the daily return of a portfolio with 50% in asset 1, 25% in asset 2 and 25% in asset 3.

Then the portfolio index is created by multiplying each successive element of "$s$" with all the previous elements. This can be written recursively as:

$\text{Index}(i) = \text{Index}(i-1) \times s(i)$

This index then represents what a daily rebalanced portfolio would perform as.

Importantly, the index weights are constant, so if you had \$100 invested the first day, the portfolio would consist of \$50 in the first asset and \$25 each in assets 2 and 3. If on the second day, the portfolio value was \$103, then the procedure above assumes that the portfolio is rebalanced to consist of \$51.50 in the first asset and \$25.75 in the second and third assets.

Note that I've just asked some questions about this procedure elsewhere on this site -- this is the classic way to backtest a strategy but in reality few portfolios rebalance daily like this.