Aggregating Portfolio Returns Across Assets and Time

# How to reasonably aggregate returns across both different assets and different time-horizons?


# How to reasonably aggregate returns across both different assets and different time-horizons?












This might be a somewhat open question, so any suggestion of improvement is welcome.

Suppose at time $t=0$, we have $N$ different assets whose weights are $w_1,\cdots,w_n$ ($\sum w_i = 1$), and they are all invested over the horizon $[0,T]$. Let $\{r_{i}^t\}_{i=1,\cdots,N}^{t=1,\cdots,T}$ be "returns" of asset $i$ over the interval $[t-1,t]$.

Question: I'm looking for a reasonable definition of "returns" that enables one to nicely represent/aggregate the total portfolio return from $\{r_{i}^t\}_{i=1,\cdots,N}^{t=1,\cdots,T}$.

At this point, I'm at the dilemma between choosing the ordinary return (something like $(S_{t+\Delta t}/S_t - 1)$) and the log return (something like $\log(S_{t+\Delta t}/S_t)$.) The two choices are kind of complementary in terms of the following:

- The ordinary return enables easy linear aggregation across different assets. For example, if we know all assets' weights $w_i$ and ordinary returns $r_i^t$ over year $t$, then the portfolio (ordinary) return over year $t$ is just a simple linear sum $$r_p^t=\sum_i w_ir_i^t$$ However, a very serious problem is that ordinary returns do NOT linearly add up over the time dimension (note that my $T$ might be large so the naive linear approximation $\log(1+x)\approx x$ cannot apply). For example, for a certain asset $i$, if we know its ordinary return over each year $t$, then what is its total ordinary return over $[0,T]$? The answer is $r_i^{[0,T]}=\Pi_t (1+r_i^t) - 1$ which is quite ugly in terms of tractability (say, it's clearly impossible to compute the volatility of $r_i^{[0,T]}$ if only given those of $r_i^t$.).

- The log return is more or less the contrary. It enables simple linear summation across the time dimension: suppose the log returns of asset $i$ over year $t$ are $r_i^t$, then the total log return is simply $r_i^{[0,T]}:=\sum_t r_i^t$. However, it completely loses the compatibility with the asset dimension: given the log return $r_i$ of each asset $i$, the total log return of the portfolio would be a hideous expression $r_p = \log(\sum_i w_i \exp(r_i))$.

Is there any other version of "returns" that would hopefully allow simple aggregation (linear best) over both the time and the asset dimension?

Thanks!