<p>Some textbook presentations of the capital asset pricing model (CAPM) take returns on stocks as a primitive and proceed as if agents derive utility from asset returns. Assuming a concave utility function and a normal return distribution one can then derive the CAPM. However, it is not obvious to me that anyone would derive utility from the asset returns <em>per se</em>. I would find it more intuitive if agents derived utility from consumption. (Consumption is based on wealth, so wealth-based utility would be intuitive enough for me.)</p>
<p>What is the simplest way to go from consumption-based or wealth-based utility to return-based utility? I would appreciate either an explicit answer or a reference, the less technical, the better. (I have seen something like this* covered in Chapter 9 of Cochrane's &quot;Asset Pricing&quot;, but I wonder if there is an even simpler derivation.)</p>
<p>*The notation is a bit confusing, as I think he is mostly working with <span class="math-container">$R_{t+1}:=\frac{P_{t+1}}{P_t}$</span> instead of <span class="math-container">$ret_{t+1}:=\frac{P_{t+1}-P_t}{P_t}$</span>.</p>

CC BY-SA 4.0
2023-02-26T15:32:47.630Z
Somewhat related: https://quant.stackexchange.com/questions/74139
Richard Hardy
CC BY-SA 4.0
2022-12-20T10:17:27.360Z
There are certainly simpler ways but you sacrifice some rigor in the process, simplest way I think of is just to point out that utility is increasing in income/wealth as it expands the budget constraint $m=\bf{p_ix_i}$ so increasing returns will allow you to attain higher utility and hence you can work with income. It is simple story but its not as rigorous as deriving CAPM from utility based on consumption, but it is also not technical
1muflon1
CC BY-SA 4.0
2022-12-20T10:20:44.320Z
@1muflon1, I suppose this works with ordinal utility but probably not with cardinal utility, and I guess we need the latter to derive the model.
Richard Hardy
CC BY-SA 4.0
2022-12-20T10:25:55.833Z
why wouldnt it work with cardinal utility? Here the objective is to maximize the utility, so it does not matter if you work with different function as long as the utility is maximized in the end, you do not want to find the exact number of utils. Also you could work with money metric utility function which allows you to use expenditure function as utility function given additional assumption that expenditure equals income/wealth which is innocuous
1muflon1
CC BY-SA 4.0
2022-12-30T18:02:26.917Z
Maybe the problem is not ordinal vs cardinal but the fact that % return does not map to wealth 1-to-1. E.g. investing \$100 with 5% return or \$200 with 2.5% return would yield the exact same gain in wealth. Also, a \$5 gain may feel quite differently when starting from 0 vs. starting from 1,000,000.
Richard Hardy
<utility><finance><expected-utility>
Deriving the CAPM: going from utility of consumption to utility of asset returns
Richard Hardy
# Deriving the CAPM: going from utility of consumption to utility of asset returns

(Asked by: username_0 on 2022-12-20)

Some textbook presentations of the capital asset pricing model (CAPM) take returns on stocks as a primitive and proceed as if agents derive utility from asset returns. Assuming a concave utility function and a normal return distribution one can then derive the CAPM. However, it is not obvious to me that anyone would derive utility from the asset returns *per se*. I would find it more intuitive if agents derived utility from consumption. (Consumption is based on wealth, so wealth-based utility would be intuitive enough for me.)

What is the simplest way to go from consumption-based or wealth-based utility to return-based utility? I would appreciate either an explicit answer or a reference, the less technical, the better. (I have seen something like this\* covered in Chapter 9 of Cochrane's "Asset Pricing", but I wonder if there is an even simpler derivation.)

\*The notation is a bit confusing, as I think he is mostly working with $R\_{t+1}:=\frac{P\_{t+1}}{P\_t}$ instead of $ret\_{t+1}:=\frac{P\_{t+1}-P\_t}{P\_t}$.

**Tags:** utility, finance, expected-utility

> **Comments:**
> **username_1**: There are certainly simpler ways but you sacrifice some rigor in the process, simplest way I think of is just to point out that utility is increasing in income/wealth as it expands the budget constraint $m=\bf{p_ix_i}$ so increasing returns will allow you to attain higher utility and hence you can work with income. It is simple story but its not as rigorous as deriving CAPM from utility based on consumption, but it is also not technical  
> **username_0**: @username_1, I suppose this works with ordinal utility but probably not with cardinal utility, and I guess we need the latter to derive the model.  
> **username_1**: why wouldnt it work with cardinal utility? Here the objective is to maximize the utility, so it does not matter if you work with different function as long as the utility is maximized in the end, you do not want to find the exact number of utils. Also you could work with money metric utility function which allows you to use expenditure function as utility function given additional assumption that expenditure equals income/wealth which is innocuous  
> **username_0**: Maybe the problem is not ordinal vs cardinal but the fact that % return does not map to wealth 1-to-1. E.g. investing \$100 with 5% return or \$200 with 2.5% return would yield the exact same gain in wealth. Also, a \$5 gain may feel quite differently when starting from 0 vs. starting from 1,000,000.  
> **username_0**: Somewhat related: https://quant.stackexchange.com/questions/74139  

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## 0 answers

