Beta and WACC with pensions. Operating = total or not? Or when?

There is one concept I am missed in.

It is about including pensions into balance sheet which modifies beta and WACC.

Let’s assume they give such data:

| Market value | Beta

Operating assets | 800 | ?

Pension assets | 200 | 0,5

Total assets | 1000 | 0,4

Now, the question is “calculate estimated overall WACC, which include operating and pension assets”.

Now, for me overall WACC means total assets WACC which is: Rf + 0,4 * RiskPremium

But the answer is rather: estimate operating assets beta:

0,4 = [(200/1000)*0,5] + [(800/1000)*Beta] —> Beta=0,375

And then overall WACC is to be Rf + 0,375 * RiskPremium.

This is not clear for me as what they counted I would call “operating assets WACC”, not “overall WACC”.

In fact in some questions in CFAI books it is ambigous. They ask for “total asset beta” or “operating asset beta” interchangeably and I don’t get what they mean. I noticed even one question stating to estimate “total (operating) asset beta”. What da… come on. TOTAL or OPERATING?

How do you approach this?

I haven’t seen any calculation of WACC that involves total assets. It’s always

WACC = Rf + beta(ao) * ERP

Is there a WACC calculation in CFAI books that involves the total asset beta?

Total asset beta = weight of equity times equity beta

if pension assets are not considered then same as operating asset beta

You need to understand the following formula first:

  1. RHS: Total Asset Beta = Weight of Equity X Equity Beta

  2. LHS: Total Asset Beta = Weight of Operating Asset X Operating Asset Beta + Weight of Pension Asset X Pension Asset Beta

  3. WACC = Risk Free Rate + Operating Asset Beta X Equity Risk Premium

Result of (1) Depends on whether you include pension liabilities or ignore. Including pension liabilities will reduce total asset beta

(2) Also depends on whether on whether you include pension asset. Otherwise if you ignore pension asset/liability, RHS total asset beta = LHS operating asset beta

Finally (3) if you include pension asset/liabilities you will be left with lower total asset beta, lower operating asset beta and thus a lower WACC.