# Expected vs required return in valuation

This is a rather simple question, so this is maybe not the right place, but...

I have done quite a bit of reading on predicting asset returns, i.e. determining return expectations. I have now started reading some fundamental literature on DCF valuation. For example, one might perform company valuation by doing as follows:

$$$$EquityValue= \Sigma^T_t\frac{FTE_{t}}{(1+k^e_{t})^t}$$$$

where $$FTE_i$$ denotes the expected free cash flow to equity holders in period $$t$$, $$k^e_t$$ denotes the required rate of return on equity (of that firm) for the respective period $$t$$.

What I do not fully understand is: Usually textbooks will say that one estimates $$k^e_t$$ by means of the CAPM. In other words, required return on equity is assumed to be equal to the expected return on equity (derived via the CAPM). I am not trying to start a discussion on whether the CAPM is an appropriate model for deriving expected returns. What I do not get is: Why is the required return assumed to be equal to the expected return? What I expect and what I require are conceptually completely different to me.

• In equilibrium expected and required are the same. And equilibrium or close to equilibrium are usually assumed in this kind of calculation of asset returns. Nov 17, 2022 at 16:11
• Thank you for your quick response. Why do the two things have to be the same in equilibrium? I have not yet come across a statement why this equivalence holds in any basic corporate finance textbook. If too broad of a question, could you point me to any paper/book chapter which explicitly explains this? Nov 17, 2022 at 16:15
• If the question gets closed here, try Economics Stack Exchange instead. Nov 20, 2022 at 16:09
• Dec 1, 2022 at 20:25

Why is the required return assumed to be equal to the expected return?

Required return in a model depends on how you model agent's behavior and beliefs.

If you assume agents in a model consider CAPM the correct model of asset pricing, that is they all believe CAPM is the correct description of how returns should be related to market risk and opportunity cost of money, then no agent would accept return lower than the return given by CAPM. If the return would be lower they would simply not invest according to the model.

From economic perspective when CAPM is used to model agent's behavior, CAPM can be viewed as:

$$\underbrace{r_i}_{\text{required return}} = \underbrace{r_f}_{\text{compensation for impatience }} + \underbrace{b_i(r_m−r_f)}_{\text{compensation for risk}}.$$

If we introduce randomness into the model by making $$r_i$$ and $$r_m-r_f$$ random variable and people believe this is the correct model then this will also be return given by rational expectations (you can just take expectations of both sides).

Alternatively if we assume that CAPM is the 'correct' statistical model to model returns the we also can reinterpret the CAPM model as a regression model where you regress $$r_i$$ on $$r_m-r_f$$ making $$r_i$$ the expected return.

• And the expected return derived via CAPM being negative would imply that the required return is negative (assuming that the CAPM holds). Correct? I can not really wrap my head around negative required returns making sense when I expect the respective asset to yield a negative return. Dec 2, 2022 at 10:28
• @shenflow the expected rate of return can be negative if supply and demand for funds intersect in II quadrant. Its the same as price of futures of oil that was temporary negative. There is a lot of empirical evidence for rare but real negative interest rates that are still accepted by investors Dec 2, 2022 at 10:33
• @shenflow also by the way CAPM is not necessary correct model that people believe in. But if inside of your mathematical model you say that all people take it as a fact then yes it would be their expected rate of return Dec 2, 2022 at 10:36

Expected and required have to be equal in equilibrium. These are market expectations and requirements, not yours, though (but we assume that everyone shares the same expectations). So if the market requires more than it expects, the demand for the stock will go down, taking the price with it, so that the return will grow – until the expected value matches the required one. And if the market expects more than it requires, the demand for the stock will go up, taking the price with it, so that the return will shrink – until the expected value matches the required one.

• Okay I get that. Now what if I expect some stock to yield a negative return. This would imply that I am requiring a negative return for holding that stock, i.e. the present value of the expected cash flow is higher than its face value. How does this make sense in this case? Nov 21, 2022 at 7:40
• @shenflow, you may be willing to do that if the stock delivers excellent diversification benefits (considerably reduces the systematic risk of your portfolio). This is a bit like buying insurance: the expected return is negative, yet it is nice to know you are protected from the worst-case scenario. Nov 21, 2022 at 13:09
• I understand that this might be the case in some of the cases in which I expect negative returns. However, the DCF-model implies that this would be the case in every of the cases in which I expect negative returns (assuming what you are saying is the reasoning behind it). However, I do not see why this should be the case. E.g. when I expect a negative market return and a risk free rate of 0, this would imply that any stock with positive beta delivers diversification benefits, because in every one of these cases present value > face value. Why should this hold true in every case? Nov 21, 2022 at 15:17
• @shenflow, this sounds like a problem. However, I doubt you can assume that expected market return is below the risk-free rate in equilibrium. If this were the case, nobody would be willing to hold the market portfolio, given that they could instead hold a risk-free asset and earn a higher expected (or actually guaranteed) return while being exposed to less (in fact zero) systematic risk. This would drive the demand for – and thus price of – the market portfolio so far down that the expected return becomes greater than the risk-free return. Nov 21, 2022 at 15:57
• You are right, my example is not logical. However, my issue still holds: Not in every case in which I expect a stock return to be below 0, I would necessarily expect it to contribute to diversification. I do not see why the former necessarily leads to the latter. (I can imagine it being the case sometimes). Nov 21, 2022 at 17:35