<p>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:</p>
<p><span class="math-container">$\overline{r_i} = r_f + \beta (r_m - r_f)$</span></p>
<p>where <span class="math-container">$\overline{r_i}$</span> is the required return, <span class="math-container">$r_m$</span> is the market return and <span class="math-container">$r_f$</span> is the risk free rate of return.</p>
<p>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 <span class="math-container">$r_i$</span>’s as they are more risk averse).</p>
<p>These different behaviors according to risk profiles can be modeled through wealth-based utility functions:</p>
<ul>
<li>If the utility function <span class="math-container">$u(w)$</span> is strictly concave, the individual is risk averse</li>
<li>If the utility function <span class="math-container">$u(w)$</span> is strictly convex, the individual is risk loving</li>
<li>If the utility function <span class="math-container">$u(w)$</span> is linear, the individual is risk neutral</li>
</ul>
<p>The Arrow-Pratt absolute risk aversion coefficient is given by:</p>
<p><span class="math-container">$r(w) = - \frac{u’’(w)}{u’(w)}$</span></p>
<p>Since utility is increasing in wealth, the sign of <span class="math-container">$r(w)$</span> is determined by the con cavity/convexity of <span class="math-container">$u(w)$</span>.</p>
<ul>
<li>For risk averse individuals, <span class="math-container">$r(w) &gt; 0$</span></li>
<li>For risk loving individuals, <span class="math-container">$r(w) &lt; 0$</span></li>
<li>For risk neutral individuals, <span class="math-container">$r(w) = 0$</span></li>
</ul>
<p>Utility functions that take into account risk tolerance are taught in Game Theory/Advanced Microeconomics.</p>
<p>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.</p>
<p>So to answer your question, yes, your approach makes sense and aligns itself with mainstream finance/economics theory.</p>
<p>I just found this article concerning utility based CAPM’s:</p>
<p><a href="https://www.tandfonline.com/doi/full/10.1080/23322039.2017.1343230" rel="nofollow noreferrer">https://www.tandfonline.com/doi/full/10.1080/23322039.2017.1343230</a></p>

CC BY-SA 4.0
2023-02-25T15:29:15.110Z
Thank you for your answer. I think I am familiar with all of the things you mention. My problem is, I have never seen an explicit derivation based on first principles (such as maximization of expected utility) of what the required return needs to be. I have supplied such a derivation, and I wonder if anyone else has done that before me. Could you supply any references to that? Or am I indeed the first one to provide a rigorous derivation? (I would find that very hard to believe.)
Richard Hardy
CC BY-SA 4.0
2023-02-25T15:33:05.770Z
Also, note that the CAPM does not imply the regression model you mention (though it seems to be a common misconception). What it implies is a regression model where the right-hand-side variable is the *expected value* of the market return, not the market return itself; see e.g. [this thread](https://quant.stackexchange.com/questions/74526). The equation needs to have an error term, too, otherwise it is not a regression nor is it compatible with the CAPM.
Richard Hardy
CC BY-SA 4.0
2023-02-25T17:07:46.543Z
@RichardHardy I just found an article that solves for CAPM equations from different types of utility functions, is this what you wanted? The link is at the end of my now edited answer
Nicolas Torres
CC BY-SA 4.0
2023-02-26T15:27:39.857Z
Hmm, I do not tink that is what I want. The CAPM characterizes the cross section of assets' expected (excess) returns by their relationship to the market's expected (excess) return. Meanwhile, my question is about why the expected value is the relevant distributional characteristic for an agent to require for holding a risky asset. But perhaps that boils down to the derivation of the CAPM from "first principles" such as a utility function? (I had a [question](https://economics.stackexchange.com/questions/53951) about that, too.) I am not sure.
Richard Hardy
CC BY-SA 4.0
2023-02-25T15:07:36.033Z
2023-02-25T17:02:22.133Z
2023-02-25T17:02:22.133Z
Nicolas Torres
<p>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. <strong>Please let me know if my approach makes sense</strong> and if it aligns itself with any mainstream finance theory.</p>
<ul>
<li>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 <strong>reference</strong>?</li>
<li>Otherwise, I am looking for pointers to mainstream explanations (approximately on the level of detail as the analysis below).</li>
</ul>
<hr />
<p>One interesting feature of the required return is that it is a scalar <strong>constant</strong>, even though for most assets the actual return is a <strong>random variable</strong>, hence more naturally characterized as a distribution than a scalar. How do we bridge this gap?</p>
<p>Let us consider a <strong>risky asset</strong> <span class="math-container">$i$</span> between time <span class="math-container">$t$</span> and <span class="math-container">$t+1$</span>. Its return <span class="math-container">$r_{i,t+1}$</span> is unknown in advance (as of time <span class="math-container">$t$</span>). (I will suppress the time subscripts as I will only ever consider the period between <span class="math-container">$t$</span> and <span class="math-container">$t+1$</span>, thus <span class="math-container">$r_{i}$</span>.) An investor can model this future return as a random variable, e.g. <span class="math-container">$R_{i}\sim N(\mu_i,\sigma_i^2)$</span>*.</p>
<p>Suppose the investor already holds the <strong>market portfolio</strong>. Then we should also take the market return <span class="math-container">$r_{m}$</span> into consideration. We could model <span class="math-container">$\pmatrix{R_{i}\\ R_{m}}$</span> together, e.g. as <span class="math-container">$\pmatrix{R_{i} \\ R_{m}}\sim N\left(\pmatrix{\mu_{i} \\ \mu_{m}},\pmatrix{\sigma_{i}^2 &amp; \sigma_{i,m} \\ \sigma_{i,m} &amp; \sigma_{m}^2}\right)$</span>. 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 <span class="math-container">$m$</span> and the risky asset <span class="math-container">$i$</span> with some given weights. From the return distribution it is straightforward to arrive to price distribution at time <span class="math-container">$t+1$</span>, given that prices at <span class="math-container">$t$</span> are known.</p>
<p>Given a utility function that takes wealth as an argument**, the investor can determine which of the following two wealth distributions has higher <strong>expected utility</strong>: one arising from the market portfolio alone vs. one arising from the market portfolio plus the risky asset <span class="math-container">$i$</span> minus the price of the risky asset at <span class="math-container">$t$</span>.</p>
<p>If we could vary <span class="math-container">$\mu_i$</span>, we could find a value that makes the expected utilities of the two alternatives equal. Let us denote this value <span class="math-container">$\bar r_{i}$</span> and call it the <strong>required return</strong>. 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 <span class="math-container">$i$</span> if <span class="math-container">$\bar r_{i,t+1}\leq\mu_i$</span>, i.e. if the expected value of the actual return matches or exceeds the required return.</p>
<hr />
<p>*If we were to use a statistical model to arrive at this distribution, the result would contain estimates rather than true values, e.g. <span class="math-container">$\hat\mu_i$</span> instead of <span class="math-container">$\mu_i$</span>. This applies to the rest of the exposition as well.<br />
**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 <a href="https://economics.stackexchange.com/questions/53951">here</a>.</p>
<hr />
<p>Other related questions are <a href="https://economics.stackexchange.com/questions/53689">1</a> and <a href="https://quant.stackexchange.com/questions/73820">2</a>.</p>

CC BY-SA 4.0
2023-01-07T12:30:53.453Z
This is an exact duplicate of [this question](https://quant.stackexchange.com/questions/73981) on Quantitative Finance SE. It has not received any answers there despite a bounty, so I am trying my luck at Economics SE now. Should you be interested, the bounty over at QF SE is good for another two days. It would be a shame if it went to waste.
Richard Hardy
<finance>
Trying to understand the notion of required return
Richard Hardy
# Trying to understand the notion of required return

(Asked by: username_0 on 2023-01-07)

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](https://economics.stackexchange.com/questions/53951).

---

Other related questions are [1](https://economics.stackexchange.com/questions/53689) and [2](https://quant.stackexchange.com/questions/73820).

**Tags:** finance

> **Comments:**
> **username_0**: This is an exact duplicate of [this question](https://quant.stackexchange.com/questions/73981) on Quantitative Finance SE. It has not received any answers there despite a bounty, so I am trying my luck at Economics SE now. Should you be interested, the bounty over at QF SE is good for another two days. It would be a shame if it went to waste.  

---

## 1 answers

### Answer by username_1 on 2023-02-25. Score: 1

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>

> **Comments:**
> **username_0**: Thank you for your answer. I think I am familiar with all of the things you mention. My problem is, I have never seen an explicit derivation based on first principles (such as maximization of expected utility) of what the required return needs to be. I have supplied such a derivation, and I wonder if anyone else has done that before me. Could you supply any references to that? Or am I indeed the first one to provide a rigorous derivation? (I would find that very hard to believe.)  
> **username_0**: Also, note that the CAPM does not imply the regression model you mention (though it seems to be a common misconception). What it implies is a regression model where the right-hand-side variable is the *expected value* of the market return, not the market return itself; see e.g. [this thread](https://quant.stackexchange.com/questions/74526). The equation needs to have an error term, too, otherwise it is not a regression nor is it compatible with the CAPM.  
> **username_1**: @RichardHardy I just found an article that solves for CAPM equations from different types of utility functions, is this what you wanted? The link is at the end of my now edited answer  
> **username_0**: Hmm, I do not tink that is what I want. The CAPM characterizes the cross section of assets' expected (excess) returns by their relationship to the market's expected (excess) return. Meanwhile, my question is about why the expected value is the relevant distributional characteristic for an agent to require for holding a risky asset. But perhaps that boils down to the derivation of the CAPM from "first principles" such as a utility function? (I had a [question](https://economics.stackexchange.com/questions/53951) about that, too.) I am not sure.  

---

