expected rate of return vs required rate of return in asset pricing

Question

From Wikipedia, I read that "expected rate of returns" have two different meanings:

1: The expected return (or expected gain) on a financial investment is the expected value of its return (of the profit on the investment). The expected rate of return is the expected return per currency unit (e.g., dollar) invested. It is computed as the expected return divided by the amount invested.

2: The required rate of return is what an investor would require to be compensated for the risk borne by holding the asset; "expected return" is often used in this sense, as opposed to the more formal, mathematical, sense above.

So the first meaning is a statistical expected value which can be thought of as a type of forecast. The second meaning is only an investor requirement for compensation of taking risk, nothing about future expectations.

So, in the CAPM formula $r_{i} = r_f +b_{i}(r_m-r_f)$, which of the above is the correct interpretation for the asset return $r_{i}$? Does it compute a future expectation that can be used for forecasting, or is it simply a requirement for investors? Finally, are the two views related somehow?

Answer 1

So, in the CAPM formula $r_i=r_f+b_i(r_m−r_f)$, which of the above is the correct interpretation for the asset return $r_i$? Does it compute a future expectation that can be used for forecasting, or is it simply a requirement for investors? Finally, are the two views related somehow?

In real life, CAPM is not exactly 100% accurate model, however, $r_i$ could be both depending on how you model things.

$r_i=r_f+b_i(r_m−r_f)$ could be viewed as a required rate of return, because if all assumptions of CAPM were satisfied then $r_i$ would the rate that investors require to be compensated for market risk. A rational investor would not invest if $r_i$ would be less.

However, at the same time you can see it as an expected rate of return because you can view the CAPM model as a regression model where:

$$\underbrace{r_i}_{\text{dep variable}} = \underbrace{r_f}_{\text{intercept}} - \underbrace{b_i}_{\beta_1}(\underbrace{r_m−r_f}_{\text{independent variable}})$$

So basically you can reinterpret it as regression model and in regression model what you are estimating is:

$$E(y) = α + βE(x)$$

or in this case after taking expectations of $r_i=r_f+b_i(r_m−r_f)$ you will get:

$$E[r_i]=r_f+b_iE[(r_m−r_f)]$$

Also there is no expectation around $r_f$ as CAPM treats it as constant not variable.

So you can reinterpret exactly the same CAPM formula as an expected return if $r_i$ and $r_m$ are random variables as you can reinterpret it as regression model.

Answer 2

Let us consider a risky investment project/asset $i$ between time $t$ and $t+1$. Its return $r_{i,t+1}$ is unknown in advance (as of time $t$). (I will suppress the time subscripts as we will only ever consider the period between $t$ and $t+1$, thus $r_{i}$.) An investor can model this future return as a random variable, e.g. $R_{i}\sim N(\mu_i,\sigma_i^2)$*.

Suppose the investor already holds the market portfolio. Then we should also take the market return $r_{m}$ into consideration. We could model $\pmatrix{R_{i} \\R_{m}}$ together, e.g. as $\pmatrix{R_{i} \\R_{m}}\sim N\left(\pmatrix{\mu_{i} \\\mu_{m}},\pmatrix{\sigma_{i}^2 & \sigma_{i,m} \\ \sigma_{i,m} & \sigma_{m}^2}\right)$. Having specified the joint distribution (which does not have to be Normal; it is just an example) we can find the return distribution of a portfolio consisting of the market portfolio $m$ and the risky asset $i$ with some given weights. From the return distribution it is straightforward to arrive to price distribution at time $t+1$, given that prices at $t$ are known.

Given a utility function that takes wealth as an argument, the investor can determine which of the following two wealth distributions is more attractive in terms of expected utility: one arising from the market portfolio alone vs. one arising from the market portfolio plus the risky asset $i$ minus the price of the risky asset at $t$.

If we vary $\mu_i$, we can find a value that makes the expected utilities of the two alternatives equal. This value can be denoted $\bar r_{i}$ and called the required return.** We can then state that the investor will only be willing to invest into the risky asset $i$ if $\bar r_{i,t+1}\leq\mu_i$, i.e. the expected value of the actual return matches or exceeds the required return.

How does CAPM come into play? We could use it to model the expected return $\mu_i$ given the risk-free rate $r_f$, $\mu_m$, $\sigma_{i,m}$ and $\sigma_{m}^2$: $\mu_i=r_f+\frac{\sigma_{i,m}}{\sigma_{m}^2}(\mu_m-r_f)$. It may not be all that helpful here, though, as we do not have anything (except for $r_f$) on the right hand side of the equation as of $t$.

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*If we were to use a statistical model to arrive at this distribution, the result would contain estimates rather than true values, e.g. $\hat\mu_i$ instead of $\mu_i$. This applies to the rest of the exposition as well.
**I have come up with this definition of required return by myself, so take it with a pinch of salt. I wonder if it is in line with standard finance theory, and I have posted a question about that here.