On averaging
You swim across the lake. It is three km and takes one hour. You return to where you started. It takes you another hour. Together it takes you two hours.
You swim down the river for three km, then return to where you started. When you swim down, you go with the flow. It will take less than an hour. When you swim up , you go against the flow. It will take more than an hour. The average speed, though, is still three km per hour. So the total time of swimming down and up is still two hours.
Some people might feel a little bit uncomfortable about the above argument. Let’s do some detailed calculation. Suppose the speed of water flow is two km per hour. When one swims down, the total speed of moving is 3 + 2 = 5 km/hour. It will take 3/5 hour to swim three km downstream. When one swims up, the total speed of moving is 3 - 2 = 1 km/hour. It will take 3/1 = 3 hour to swim three km upstream. Together, it will take more than three hours to finish the round trip. From the calculation, and from the practical experience as well, the faster the river flow, the longer it takes to finish the round trip.
Mathematically speaking, we cannot average denominators. It is not a linear relation. One might say this is obvious and trivial. It is obvious and trivial, except that averaging denominators is widely practiced in finance. It is called WACC (weighted average cost of capital), one of the most important concepts in finance. It is taught to every finance student and used in valuing billions after billions of assets worldwide.
The theoretical foundation of WACC was first developed by Modigliani and Miller. They both received Noble prizes. They proved that when expected return from the investment is constant to perpetuity, WACC does provide the correct answer. This is equivalent to prove that when the speed of water flow is zero, the time to take a round trip in the river is the same as the time to take a round trip in the lake with the same distance. But this is the only case when the statement is valid, in finance or in swimming.
Modigliani and Miller wrote the paper in 1958, sixty years ago. Their paper has become a bible in finance. Is it sin to discuss the validity of bible? You be the judge!
For more information, please refer to
On the scope of validity of Modigliani and Miller propositions
http://web.unbc.ca/~chenj/papers/WACC.pdf