# 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!

• What is wrong with taking the weighted sum of log returns accross the portfolio ? The main question here is what it is uou are trying to do with this. This will drive the assesment of whether a definition is pertinent or not. – Ezy Jan 1 '19 at 23:02
• It's not linear: $$r_p = \log(\sum_i w_i \exp(r_i))$$. Which means it's hard to get $\sigma(r_p)$ out of individual $\sigma(r_i)$. – Vim Jan 2 '19 at 1:28
• why not simply define the portfolio log return as the linear combination of each asset log return ? What is wrong with that ? – Ezy Jan 2 '19 at 1:41
• @Ezy it is not "defined", it is computed instead. Log return has its original definition which one cannot simply alter just for their own convenience. – Vim Jan 2 '19 at 4:50
• your original question is precisely to define an alternative version of the portfolio return and i just gave you one. Call it a different name, “weighted log ret” eg if “log ret” makes you confused with the log of the actual portfolio return. My point is that linear weighted sum of the stocks actual ret seems a reasonable definition for your purpose. – Ezy Jan 2 '19 at 6:31