### Capital Allocation Line (CAL) and Capital Market Line (CML)

When we combine a risky asset portfolio with a risk-free asset, we form a capital allocation line. This represents the allocation between the risk-free asset and the risky asset based on investor risk preferences. The capital market line is a special case of the CAL where the portfolio of risky assets is the market portfolio.

## Passive and Active Portfolios

If the market is informationally efficient, then the quoted price of a security in the market is an unbiased estimate of all the future discounted cash flows and reflects all publicly known information about the security. If all security prices reflect all publicly available information, then, in theory, there is no way to outperform the market. If this is the investor belief, then the most simple and convenient approach is to invest in a passive portfolio. A passive portfolio will track and replicate the market.

However, many investors do not believe the market price accurately reflects valuations. They have confidence in their ability to determine these mispricings based on their models of valuation. Such an investor would take an active approach to investing, overweighting undervalued assets and underweighting (or shorting if allowed) overvalued assets. This style of investing is called active management.

## The Market

The market includes all risky assets or anything that has value – stocks, bonds, real estate, human capital, commodities – these are all assets defined in “the market.” Not all market assets are tradeable or investable. If global assets are considered, there are hundreds of thousands of individual securities that make up the market that are considered tradable and investable. A typical investor is likely to be more concerned with their local or regional stock market as a measure of “the market”.

## The Capital Market Line (CML)

The Capital Market Line (CML) is a special case of the CAL – the line which makes up the allocation between a risk-free asset and a risky portfolio for an investor. In the case of the CML, the risk portfolio is the market portfolio. Where an investor has defined “the market” to be their domestic stock market index, the expected return of the market is expressed as the expected return of that index. The risk-return characteristics for the potential risk asset portfolios can be plotted to generate a Markowitz efficient frontier. Where the line from the risk-free asset touches, or is tangential, to the Markowitz portfolio, this is defined as the market portfolio. The line connecting the risk-free asset with the market portfolio is the CML.

The expected return and variance for the portfolio can be represented as follows:

$$\text{Expected return} = E(R_P) = wR_f + (1-w)E(R_m)$$

$$\text{Variance} = \sigma_P^2 = w^2\sigma_f^2 + (1-w)^2\sigma_m^2 + 2w(1-w)Cov(R_f,R_m)$$

Where

$$R_f$$ is the return on the risk-free asset;

$$R_m$$ is the return on the market;

$$w$$ is the weight of the risk-free asset in the portfolio; and

$$1-w$$ is the weight of the market asset in the portfolio.

Theoretically, the standard deviation of the risk-free asset is zero, and the term, $$w^2\sigma_f^2$$ falls out of the equation. Equally, the risk-free asset is assumed to have no covariance with the market portfolio which means the portfolio standard deviation is written as:

$$\sigma_p = (1 – w)\sigma_m$$

By substitution, we can write:

$$E(R_p) = R_f + \frac {E(R_m) – R_f}{\sigma_m} × \sigma_p$$

This is in the form of an equation of a straight line where the intercept is Rf and the slope is $$\frac {E(R_m) – R_f}{\sigma_m}$$. This is the CML line which has a positive slope as the market return is greater than the risk-free return.

## Question

What happens to the portfolio risk and return respectively as an investor moves up the CML?

A. Risk decreases, return decreases.

B. Risk increases, return decreases.

C. Risk increases, return increases.

Solution

As an investor moves up the CML, the overall portfolio risk and return increase.

Explain the capital allocation line (CAL) and the capital market line (CML)

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