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### Subject 4. The CAL and CML

The efficient frontier only considers the investments in risky assets. However, investors may choose to invest in a risk free asset, which is assumed to have an expected return commensurate with an asset that has no standard deviation (i.e. zero variance) around the expected return. That is, a risk-free asset's expected return is entirely certain, and is known as the risk-free rate of return (RFR). Therefore, a risk-free asset lies on the vertical axis of a portfolio graph.

The risk free asset is important to the capital asset pricing model. When a risk-free asset is combined with a risky portfolio (assume the proportion of the portfolio the investor places in the tangency portfolio T is wT):

• The expected rate of return for the new portfolio is the weighted average of the two returns: E(R) = (1 - wT) Rf + wT E(RT)
• The standard deviation of the new portfolio is the linear proportion of the standard deviation of the risky asset portfolio T: σportfolio = wT σT

A graph of possible portfolio risks-return combinations becomes a straight line between the two assets.

• First, pick a risky stock or risky portfolio A. Hint: start with one that is already on the efficient frontier since you know that these portfolios dominate everything below them in terms of return offered for risk taken.

• Now combine the risk-free asset with portfolio A. Remember, the combination of the risk-free asset and portfolio A will be a straight line. Observe that any combination on the line RfA dominates the portfolios below it. But any combination on the line RfB will dominate RfA. Why? Because you always get more return for a given amount of risk.

• You can keep getting better portfolios by moving up the efficient frontier...

• At point M you reach the best combination. The RfM line dominates everything else in terms of return offered for the level of risk taken. The introduction of a risk-free asset changes the efficient frontier into a straight line. This straight efficient frontier line is called the Capital Market Line (CML) for all investors, and Capital Allocation Line (CAL) for one investor. Since the line is straight, the math implies that any two assets falling on this line will be perfectly positively correlated with each other. Note: When p(a,b) = 1 then the equation for risk changes to σportfolio = WAσA + WBσB.

• Investors at point Rf have 100% of their funds invested in the risk-free asset.
• Investors at point M have 100% of their funds invested in portfolio M.
• Between Rf and M investors hold both the risk-free asset and portfolio M. This means investors are lending some of their funds (buying the risk-free asset).
• To the right of M, investors hold more than 100% of portfolio M. This means they are borrowing funds to buy more of portfolio M. This represents a levered position.

Now, the line Rf-M dominates all portfolios on the original efficient frontier. Thus, the CML becomes the new efficient frontier.

• Market risk premium: E(RM) - Rf.
• Slope of the CAPM: (E(RM) - Rf) / σM. It is also called the market price of risk because it indicates the market risk premium for each unit of market risk.

Portfolio M is a completely diversified portfolio that includes all risky assets in proportion to their market value. It is referred to as the market portfolio. It includes all risky assets.

The CML represents all the possible portfolio combinations by investing in the risk-free asset and the market portfolio.

• The risk of an alternative portfolio on the CML comes entirely from the market portfolio.
• The difference between the risks of various portfolios on the CML is caused by the weight of the market portfolio in each portfolio.

Note for CAL, the portfolio used in the equation does not necessarily need to be a fully diversified portfolio. It is the best portfolio available to one single investor.

The CML leads all investors to invest in the same risky portfolio, the market portfolio. That is, all investors make the same investment decision. They can, however, attain their desirable risk preferences by adjusting the weight of the market portfolio in their portfolios.

• A strongly risk-averse investor will lend some fund at the risk-free rate and invest the remainder in the market portfolio.
• A less risk-averse investor will borrow some fund at the risk-free rate and invest all the fund in the market portfolio.

Therefore, investors make different financing decisions based on their risk preferences. The separation of the investment decision from the financing decision is called the separation theorem.

#### Practice Question 1

Assume the risk free asset is F, the tangency risky portfolio is P, the market portfolio is M, and the combined portfolio is C. If an investor includes F and P into C, the slope of the CAL is

A. (E(RP) - RF) / σP.
B. (E(RP) - RF) / σC.
C. (E(RP) - RF) / σM.

Note this is CAL and it has nothing to do with the market portfolio.

#### Practice Question 2

Select the correct statement(s).

I. After adding the risk-free asset to a risky portfolio, the shape of efficient frontier changes from a curve to a line.
II. CML assumes all investors have the same risk aversion level.
III. An investor's CAL is a straight line.

The CAL intersects the y-axis at the risk free rate and lies tangent to the efficient frontier.

#### Practice Question 3

The risk-free asset has a return of 0.05. A risky portfolio, X, has an expected return of 0.12 and a standard deviation of 0.20. For a portfolio that is 60% X and 40% risk-free asset:

I. the expected return is 8.5%
II. the standard deviation is 12%.
III. the standard deviation is 20%.

E(Rp) = (0.6)(0.12) + (0.4)(0.05) = 0.092 or 9.2%. wp σ 2X = (0.6)(0.20) = 0.12 or 12%.

#### Practice Question 4

If I can borrow at the risk-free rate and use these funds to purchase portfolio X (from the previous problem), then:

A. I will generate a rate of return equal to X's
B. I will create a portfolio with less risk than X
C. I will create a portfolio with more risk than X
D. my new portfolio will have an expected return less than X's

The portfolio's risk is equal to the weight in X times X's standard deviation. Borrowing at the risk-free rate allows us to establish a weight in X that is greater than 100%. Therefore, the risk level will exceed that of X alone.

#### Practice Question 5

According to capital market theory, the market portfolio, M:

I. contains all risky assets
II. is efficient
III. is the preferred combination of risky assets for all investors

Correct Answer: I, II and III

Investors will agree on M but will disagree on how much M to combine with the risk-free asset.

#### Practice Question 6

According to capital market theory, which of the following statements is true?

A. M is not efficient by itself, but must be combined with the risk-free asset
B. Portfolios formed by borrowing at the risk-free rate and investing in M are less risky than M alone
C. The Capital Market Line connects the zero risk, zero expected return point with M
D. A portfolio with 50% M and 50% risk-free asset will have 1/2 the risk of M

M is efficient - it lies on the EF. Borrowing portfolios are more risky because they have added financial risk (leverage). The CML goes to the RF, which dominates holding currency (zero risk, zero return).

#### Practice Question 7

The market portfolio plays an important role because:

A. it is the most efficient of all portfolios.
B. it is the one portfolio that all investors will choose to get the highest CML.
C. it dominates all portfolios on the CML.

#### Practice Question 8

The dominant line is the one that is tangent to the efficient frontier. All investors should target points along this line depending on their risk preferences. This line is referred to as the:

A. capital allocation line.
B. efficient frontier.
C. capital market line.

#### Practice Question 9

Select the correct statement(s).

I. When combined with the risk-free asset, no other portfolio along the efficient frontier provides a higher expected reward-to-risk ratio than the tangency portfolio.
II. Investors can have different CALs.
III. When combined with the risk-free asset, the optimal risky portfolio should always be the tangency portfolio.

Correct Answer: I, II and III

II: The intercept and slope of the CAL depends on the asset expectations (mean returns, variance of returns, and correlations) of an individual investor.
III: This is because the portfolio maximizes the investor's reward-to-risk ratio.

#### Practice Question 10

Consider the following three statements. Which are true?

I. The risk-free asset has a variance of 0
II. Any portfolio containing the risk-free asset has zero risk
III. The risk-free asset has a correlation of 0 with any risky asset

By definition, the risk-free asset has a variance and a standard deviation of 0. Portfolios containing the risk-free asset will have risk proportionate to their exposure in risky assets. Since its return does not vary, the risk-free asset is uncorrelated with any risky asset.

#### Practice Question 11

In the presence of a risk-free security, the efficient frontier

A. consisting of risky assets moves down parallel to itself.
B. consisting of risky assets moves up parallel to itself.
C. changes and is replaced by a straight line.

In the presence of a risky asset, the efficient frontier of risky assets is dominated by a new efficient frontier consisting of the risk-free security and the tangent portfolio found by joining the risk-free security with a point on the efficient frontier of risky securities where the line has the highest slope.

#### Practice Question 12

The Capital Market Line reflects risk in terms of

A. beta.
B. Sharpe ratio.
C. standard deviation.