Asset allocation

Question

Hi all,I am looking for help on asset attribution of a portfolio over a period of time with some top level additions and withdrawals. If for example we had an investor deposit $1mn at the start of month 1 and this purchased four underlying assets for 40%/30%/20%/10% of the proceeds. Each asset has a 10% return in resulting in a profit of $100k and 10% portfolio return. On a weighted basis asset a would have made a 4% contribution (400k / 1mn) * 10% ; asset b 3% / asset c 2% and asset d 1%. The weighted attribution is additive for one month in month 2 the investor adds another $1mn adding asset e, leaving opening assets as $2.1mn in the portfolio. Each asset again makes 10% which leaves us with the following at the end of month 2asset a $440k * 10% = $44k ; $44k / $2.1mn = 2.10%asset b $330k * 10% = $33k ; $33k / $2.1mn = 1.57%asset c $220k * 10% = $22k ; $22k / $2.1mn = 1.05%asset d $110k * 10% = $11k ; $11k / $2.1mn = 0.52%asset e $1mn * 10% = $100k ; $100k / $2.1mn = 4.76%portfolio $210k / $2.1mn = 10%the time weighted return of the portfolio is 21% for the two months but the underlying asset returns are asset a (100 * (1 + 4%) * (1 + 2.10%)) /100 - 1 = 6.18%asset b (100 * (1+3%) * (1+1.57%))/100 - 1 = 4.62%asset c (100 * (1+2%) * (1+1.05%))/100 - 1 = 3.07%asset d (100 * (1+1%) * (1+0.52%))/100-1 = 1.53%asset e 100 * (1+4.76%))/100 - 1 = 4.76%the sum of each of the asset ytd is 20.16% which doesn’t match the portfolio twr calc of +21% and I’m trying to understand how best to break the portfolio ytd number into each sub strategy as I have come to the conclusion that because the weighted returns monthly are additive, compounding the component parts won’t equal the top number but how else would you go about breaking this down? Without the additional $1mn at the start of month 2 I think you get there by summing total $ p&l over fund initial value but it gets distorted when there is rebalancing any help is much appreciated thanks j

cfachris · Answer

hey J, not sure if I'm right here, but time weighted return should have a square root in this case, as it is over 2 period, i.e. ^1/2Time weighted return for portfolio should be (1.1 * 1.1)^(0.5) - 1 = 10%Time weighted return for underlying assets over 2 months:asset A: (1.04*1.021)^(0.5) - 1 = 3.05%asset B: (1.03*1.0157)^(0.5) - 1 = 2.28%asset C: (1.02*1.0105)^(0.5) - 1 = 1.52%asset D: (1.01*1.0052)^(0.5) - 1 = 0.76%asset E: (1.0475)^(0.5) - 1 = 2.35%The total is 9.96%, probably due to rounding as I used 2 d.p. input rather than the usual 4 decimals.

jackT · Answer

Hey Chris,Many thanks for your time on this one, I'm still a little stuck. I think the total time weighted return over the two periods is +21%, as we made 10% in month1 and 10% in month 2, so compounded that would be 21%. If we are then trying to decompose the contribution of each of the assets to that overall return, i would need to keep 21% as the figure from which i would say asset E has contributed x%. But the sum i had so far would add up asset A-E and equal a YTD overall return of only 20.16%, so i am missing nearly 1% of performance which i am not explaining. Just wondered if there was an adjustment that is needed to account for the step up in assets from month 1 to month 2 and how that keys in to the overall return. Thanks again!J