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I have been thinking about the notion of required return lately. I am not familiar with a formal definition, but I have tried to reason my way towards one. Please let me know if my approach makes sense and if it aligns itself with any mainstream finance theory.

  • If it does, then I do not believe I am the first one to provide a rigorous derivation; people must have done that before. Could I get a reference?
  • Otherwise, I am looking for pointers to mainstream explanations (approximately on the level of detail as the analysis below).

One interesting feature of the required return is that it is a scalar constant, even though for most assets the actual return is a random variable, hence more naturally characterized as a distribution than a scalar. How do we bridge this gap?

Let us consider a risky 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 I 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 multivariate 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 has higher 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 could vary $\mu_i$, we could find a value that makes the expected utilities of the two alternatives equal. Let us denote this value $\bar r_{i}$ and call it the required return. We could then state that the investor would only be willing to invest (a given amount corresponding to the analysis above) into the risky asset $i$ if $\bar r_{i,t+1}\leq\mu_i$, i.e. if the expected value of the actual return matches or exceeds the required return.


*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.
**Consumption is perhaps the most natural argument of a utility function, but wealth translates into consumption pretty easily. It is harder to show how and under what additional assumptions one could define utility directly on returns; a related question is here.


Other related questions are 1 and 2.

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  • $\begingroup$ Richard: To comment on the question you linked to: A simple utility function is for example, the one used in the markowitz portfolio framework :$ w^{\prime} r - \lambda \times w^{\prime} \Sigma w$, where $\Sigma$ is the covariance matrix of the portfolio returns. So, the return minus a risk aversion parameter multiplied by the resulting variance. $\endgroup$
    – mark leeds
    Commented Jan 1, 2023 at 15:57
  • $\begingroup$ @markleeds, if by the linked question you mean this one, how does your comment help to answer it? $\endgroup$ Commented Jan 1, 2023 at 16:44
  • $\begingroup$ yes, in the question, the OP said that it is not obvious to him how one would derive utility from returns "per se". I think that my answer makes that pretty obvious, no ? $\endgroup$
    – mark leeds
    Commented Jan 2, 2023 at 17:15
  • $\begingroup$ @markleeds, it would be nice if it were that simple. The point is, unlike wealth, you cannot consume % returns, so what kind of utility do they yield? The question is not about mathematics but about humans. $\endgroup$ Commented Jan 2, 2023 at 18:47
  • $\begingroup$ okay. I'm not sure how they convert returns to wealth but I've definitely seen the function I used referred to as a utility function. Of course, the term "utility function" probably is used in various contexts and could have different meanings in different fields. I'm glad that we cleared it up and my apologies for useless ( atleast for you ) comment. $\endgroup$
    – mark leeds
    Commented Jan 3, 2023 at 19:19

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There is a very famous model taught in corporate finance courses called the Capital Asset Pricing Model (CAPM) which consists of a linear regression between the minimum expected return needed to consider that risky asset’s investment worthwhile (required return), and the excess return from the market portfolio:

$\overline{r_i} = r_f + \beta (r_m - r_f)$

where $\overline{r_i}$ is the required return, $r_m$ is the market return and $r_f$ is the risk free rate of return.

Investopedia mentions that the requiere rate of return can be different for each person depending on their needs/risk profile (people approaching retirement would have higher $r_i$’s as they are more risk averse).

These different behaviors according to risk profiles can be modeled through wealth-based utility functions:

  • If the utility function $u(w)$ is strictly concave, the individual is risk averse
  • If the utility function $u(w)$ is strictly convex, the individual is risk loving
  • If the utility function $u(w)$ is linear, the individual is risk neutral

The Arrow-Pratt absolute risk aversion coefficient is given by:

$r(w) = - \frac{u’’(w)}{u’(w)}$

Since utility is increasing in wealth, the sign of $r(w)$ is determined by the con cavity/convexity of $u(w)$.

  • For risk averse individuals, $r(w) > 0$
  • For risk loving individuals, $r(w) < 0$
  • For risk neutral individuals, $r(w) = 0$

Utility functions that take into account risk tolerance are taught in Game Theory/Advanced Microeconomics.

There are in fact more complex required rate of return linear regression based models such as the 3 factor Fama-French model for a portfolio which takes into account if stock as are growth/value stocks and big/small cap, since these have different risk/return profiles.

So to answer your question, yes, your approach makes sense and aligns itself with mainstream finance/economics theory.

I just found this article concerning utility based CAPM’s:

https://www.tandfonline.com/doi/full/10.1080/23322039.2017.1343230

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