A Simple Error-Proof Approach to Find Composite Discount Rates Such as CoLE, WACC and the Like

The interrelationships between cash flows, corresponding discount rates and values follow certain rules, knowing which one can quite easily and correctly find out value of given cash flow using the discounting-by-components framework, or to find the correct formulation of the discount rate for the compound cash flow. The approach is easy to apply in practice and in theory. In my papers “How to Avoid Mistakes in Valuation: guidelines for practitioners” and “Capital Structure Irrelevance: The Modigliani-Miller Theorem” I suggested a simple algorithm to find complicated discount rates for composite cash flows, such as CoLE, WACC and even more complex ad hoc discount rates. The papers claimed that there is a major rule in discounting cash flows, which consists in that discount rates for any composite cash flow can be expressed as weighted average of discount rates for components of given composite cash flow with weights determined as values of components divided by the value of given composite cash flow being evaluated. This paper provides mathematical proof of the statement that all composite costs of capital are weighted averages of cash flow specific discount rates