Why Risky-Asset Allocation and Capital Allocation Are Separate
Summary
The document distinguishes choosing a portfolio of risky assets from deciding how much to invest in that portfolio versus a risk-free asset. Asset allocation forms an optimal risky portfolio by weighing expected returns against covariance risk in a mean-variance framework. Capital allocation then sets the investor’s exposure to that portfolio according to its expected excess return, variance, and the investor’s risk aversion; the remainder goes to the risk-free asset.
This separation clarifies why portfolio theory often solves the risky-assets problem independently rather than treating cash as just another risky holding with zero variance. Investors can share a chosen risky portfolio while selecting different total risk exposures based on their risk preferences. The answer also identifies security selection as a further, more granular stage. The explanation is conceptual and presents formulas, but it does not work through numerical inputs, constraints, or cases with multiple risk-free assets.
Key ideas
- Asset allocation selects the composition of the risky portfolio using expected returns and covariance risk.
- Capital allocation divides wealth between the risky portfolio and the risk-free asset.
- An investor’s risk aversion helps determine the share allocated to the risky portfolio.
- Different investors may choose different exposure levels while using the same optimal risky portfolio.
- Security selection is a separate, more granular decision stage.
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# Why seperate portoflio between only risky and risky+risk-free asset
# Why seperate portoflio between only risky and risky+risk-free asset
I am currently studying basic portfolio theory. I observed that, for minimizing the variance of a portfolio, one usually does the case of $n$ risky assets, and then $n$ risky assets together with one risk-free asset (hence $n+1$ assets).
I don't understand why there is this distinction, unless it is to do computations only with matrices of size $n$ corresponding to the risky part. I don't see why the computation for $n+1$ risky assets is not valid for just $n+1$ assets, including a risk-free one, or several risk-free ones, plugging in the zeros where needed.
Could someone explain me why they explain this this way ?
## Answer by KaiSqDist (score 3, accepted)
https://quant.stackexchange.com/a/81610
In Investments by Bodie, Kane & Marcus (Chapter 6), the authors talk about capital allocation (CA) and asset allocation (AA) as separate processes, mainly because of the individual risk aversion being different for different investors. The former (CA) talks about allocating between the riskless asset and the risky portfolio, the latter (AA) talks about allocating across risky assets to form the (optimal) risky portfolio.
If we want the optimal risky portfolio, an optimization occurs whereby one has to maximize the utility of a mean-variance investor:
$$\mathop{max}_{w}w'_{1*n}r_{n*1}-w'_{1*n}\Sigma_{n*n} w_{n*1}$$
where all the variables above are vectors with the subscript as the dimensions.
This is different from capital allocation, where the allocation to the risky portfolio is given by:
$$y=\frac{E(r_p)-r_f}{A\sigma^2_p}$$
such that allocation to the riskless asset is $1-y$. As you can see, AA and CA are done in different stages. There is a third stage, called security selection, done at a more granular level than AA, but that is for another time.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.