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._{T}):*straight line* between the two assets.**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 σ_{portfolio} = W_{A}σ_{A} + W_{B}σ_{B}._{f}-M dominates all portfolios on the original efficient frontier. Thus, **the CML becomes the new efficient frontier.***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*.*all* the possible portfolio combinations by investing in the risk-free asset and the market portfolio.*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.*different financing decisions* based on their risk preferences. The separation of the investment decision from the financing decision is called the **separation theorem**.

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 w

- The expected rate of return for the new portfolio is the weighted average of the two returns: E(R) = (1 - w
_{T}) R_{f}+ w_{T}E(R_{T}) - The standard deviation of the new portfolio is the linear proportion of the standard deviation of the risky asset portfolio T: σ
_{portfolio}= w_{T}σ_{T}

A graph of possible portfolio risks-return combinations becomes a

- 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 R
_{f}A dominates the portfolios below it. But any combination on the line R_{f}B will dominate R_{f}A. 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 R
_{f}M 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

- Investors at point R
_{f}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 R
_{f}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 R

- Market risk premium: E(R
_{M}) - R_{f}. - Slope of the CAPM: (E(R
_{M}) - R_{f}) / σ_{M}. 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

The CML represents

- 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

- 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

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

Correct Answer: A

A. (E(R_{P}) - R_{F}) / σ_{P}.

B. (E(R_{P}) - R_{F}) / σ_{C}.

C. (E(R_{P}) - R_{F}) / σ_{M}.

Correct Answer: A

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

Select the correct statement(s).

Correct Answer: I and III

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.

Correct Answer: I and III

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

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:

Correct Answer: II

I. the expected return is 8.5%

II. the standard deviation is 12%.

III. the standard deviation is 20%.

Correct Answer: II

E(R_{p}) = (0.6)(0.12) + (0.4)(0.05) = 0.092 or 9.2%. w_{p} σ ^{2}_{X} = (0.6)(0.20) = 0.12 or 12%.

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

Correct Answer: C

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

Correct Answer: C

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.

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

Correct Answer: I, II and III

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.

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

Correct Answer: D

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

Correct Answer: D

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).

The market portfolio plays an important role because:

Correct Answer: B

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.

Correct Answer: B

The other answers are untrue.

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:

Correct Answer: C

A. capital allocation line.

B. efficient frontier.

C. capital market line.

Correct Answer: C

Select the correct statement(s).

Correct Answer: I, II and III

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.

Consider the following three statements. Which are true?

Correct Answer: I and III

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

Correct Answer: I and III

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.

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

B. consisting of risky assets moves up parallel to itself.

C. changes and is replaced by a straight line.

Correct Answer: C

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.

Correct Answer: C

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.

The Capital Market Line reflects risk in terms of

B. Sharpe ratio.

C. standard deviation.

Correct Answer: C

A. beta.

B. Sharpe ratio.

C. standard deviation.

Correct Answer: C

The CML is the line that is tangent to the efficient frontier and has the intercept equal to the risk-free rate. It is plotted on the expected return versus the standard deviation plane.