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# What is Capital Allocation Line?

## Capital Allocation Line (CAL) and Optimal Portfolio

The Capital Allocation Line (CAL) is a line that shows the risk-and-reward profile of assets and can be utilized to build an optimal portfolio. The process of building the CAL for a set of portfolios is given below.

## Portfolio Expected Return and Variance

Let’s construct a portfolio with only two risky assets for the sake of simplicity and understanding. The portfolio’s expected return is the weighted average of each individual assets’ expected returns, and is calculated as −

$$\mathrm{𝐸(𝑅_{𝑝}) = 𝑤_{1}\:𝐸(𝑅_{1}) + 𝑤_{2} \:𝐸(𝑅_{2})}$$

Where

$𝑤_{1}$ and $𝑤_{1}$ are the weights for the two assets, and

$𝐸(𝑅_{1})$ and $𝐸(𝑅_{2})$ are the respective expected returns.

Levels of variance usually translate directly with levels of risk of the assets; so, higher variance means higher levels of risk. The variance of a portfolio does not just depend on the weighted average of the variance of individual assets but also on the correlation and covariance of the two assets. The formula for the variance of portfolio is −

$$\mathrm{Var(𝑅_{𝑝}) = (𝑤_{1})^{2}\:Var(𝑅_{1}) + (𝑤_{2})^{2} Var(𝑅_{2}) + 2𝑤_{1}𝑤_{2}\:Cov(𝑅_{1}\:, \:𝑅_{2})}$$

Where

$Cov(𝑅_{1}\:,\:𝑅_{2})$ represents the covariance of the two assets’ returns.

The equation can be written as −

$$\mathrm{(σ_{ρ})^{2} = (𝑤_{1})^{2}(σ_{1})^{2} + (𝑤_{2})^{2}(σ_{2})^{2} + 2ρ(𝑅_{1}, 𝑅_{2})𝑤_{1}𝑤_{2}σ_{1}σ_{2}}$$

using ρ(R_{1}, R_{2}) as the correlation of R_{1} and R_{2}.

The relationship between correlation and covariance is given as −

$$\mathrm{ρ(𝑅_{1}\:, \:𝑅_{2}) =\frac{Cov(𝑅_{1}\:, \:𝑅_{2})}{σ_{1}σ_{2}}}$$

The variance of portfolio return is higher when the covariance of the two assets is positive, and the variance is less when the covariance is negative.

Since variance shows the risk, the portfolio risk will be lower when its asset components have negative covariance. Diversification can be used to lower the risk. Diversification is a process that reduces portfolio risk by investing in numerous assets with negative covariance.

The returns and standard deviations of individual assets are unknown in practice, but these values can be estimated depending on these assets’ historical values.

## Conclusion

Capital Allocation Line shows all possible combinations of risky and risk-free assets. The line provides an idea of how much returns would be possible for a risky and risk-free asset portfolio. Investors use the Capital Allocation Line to achieve different sets of risk and return based on their requirements. As is obvious, the CAL offers investors the base to assume how much to invest in risky assets and how much in risk-free assets.

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