Return Rate Aggregation - Investments under Uncertainty

Joachim Kuczynski, 09 February 2023

In many books you can read that the return rate of a set of several assets is the weighted average of the single asset’s return rates. But up to now I did not found any proof for this statement. In this post I provide a derivation of that relationship. An additional benefit of that calculation is to understand the conditions under which that relationship is valid basically.

Let us start with an asset value at time $t$ , $C(t)$ , which is the sum of different assets values $C_i(t)$ :

$C(t)=\sum_{i}^{}C_i(t)$

At time $t=0$ the asset values are $C(0)$ and $C_i(0)$ with $C(0)=\sum_{i}^{}C_i(0)$ . The asset value $C_i$ is developing in time $t$ with its specific return rate $r_i$ , that means:

$C_i (t)=C_i(0)exp(r_i t)$

Now we are searching an aggregated return rate $r$ , that describes the development of the aggregated asset value $C$ . Setting $C(t)=C(0)exp(rt)$ we obtain:

$r=\frac{1}{t}ln\frac{C(t)}{C(0)}=\frac{1}{t}ln\left( \sum_{i}^{} \left \frac{C_i(0)}{C(0)} exp \left( r_i t \right) \right \right)$

This is the exact relationship between the aggregated return rate $r$ and the differential return rates $r_i$ . This expression cannot be simplified any more. Now we develop the exponential and logarithmic functions using Taylor series and take the polynomial approximation only up to its first oder. That means $\text{exp}\left( x \right)\simeq 1+x$ and $\text{ln}\left( x \right)\simeq x-1$ . Hence we get a first order approximation of $r$ :

$r\simeq \frac{1}{t}\left( \sum_{i}^{} \left( \frac{C_i(0)}{C(0)} \left( 1+ r_i t \right) \right) -1 \right)$

This simplyfies to:

$r\simeq \sum_{i}^{} \frac{C_i(0)}{C(0)} r_i$

This is the result, that many authors present and use in their books. Also the calculation of the WACC, or aggregated return / discount rate respectively, is told to be the weighted average of debt $D$ return rate $r_D$ and equity $E$ return rate $r_E$ :

$WACC=r=\frac{D}{E+D}r_D+\frac{E}{E+D}r_E$

But take care, that all is only an approximation. And in some cases is can be an inaccurate approximation. With increasing differential return rates $r_i$ and increasing time $t$ the approximation becomes more and more inaccurate. If you require an exact calculation, take the formula presented above.

Note that it does not matter whether you take $C_i (t)=C_i(0)exp(r_i t)$ or $C_i (t)=C_i(0)(1+r_i^*)^t$ . With a substitution of $r=ln(1+r^*)$ , you can transform these two return rates into each other.