By Yuriy Smirnov Ph.D.

Capital market line (CML) is a graph that reflects the expected return of a portfolio consisting of all possible proportions between the market portfolio and a risk-free asset. The market portfolio is completely diversified, carries only systematic risk, and its expected return is equal to the expected market return as a whole. In general terms, the expected return of a particular portfolio (E(R_{C})) can be calculated as follows

E(R_{c}) = y × E(R_{M}) + (1 – y) × R_{F}

where y is a proportion of a market portfolio, E(R_{M}) is an expected return of a market portfolio, (1-y) is a proportion of a risk-free asset, and RF is a risk-free rate.

The return of nonleveraged portfolios can be less than or equal market return (if the proportion of the market portfolio equals 1 or 100%), but the return of a leveraged portfolio can significantly exceed market return.

The capital market line equation can be written as follows.

E(R_{c}) = R_{F} + SD_{c} |
E(R_{M}) - R_{F} |

SD_{M} |

where, SD_{C} is a standard deviation of portfolio C return, SD_{M} is a standard deviation of a market return.

The slope of CML is defined by reward to variability ratio (RVR).

RVR = | E(R_{M}) - R_{F} |

SD_{M} |

Let’s assume that current risk-free rate is 5%, expected market return is 20% and standard deviation of a market portfolio is 10%. Thus CML equation is as follows.

E(R_{c}) = 5 + SD_{c} |
20 - 5 | = 5 + 1.5 × SD_{c} |

10 |

Suppose there are two portfolios:

- Lending (non-leveraged) Portfolio A with standard deviation 5%
- Borrowing (leveraged) Portfolio B with standard deviation 15%

The expected return of Portfolio A is 12.5% and the expected return of Portfolio B is 27.5%.

E(R_{A}) = 5 + 1.5 × 5 = 12.5%

E(R_{B}) = 5 + 1.5 × 15 = 27.5%

The key problem of capital market line in real markets conditions is that CML is based on the same assumptions as capital asset pricing model CAPM).

- There are taxes and transaction costs, which can significantly differ for various investors.
- It is supposed that any investor can ether lend or borrow unlimited amount at risk-free rate. In real market conditions investors can lend at lower rate than borrow, that brings to bend of CML like on figure below.
- Real markets don’t have strong form of efficiency, so investors have unequal to information.
- Not all investors are rational and risk-averse.
- Standard deviation isn’t the only risk measurement, because real markets are subject to inflation risk, reinvestment risk, currency risk etc.
- There are no risk-free assets.

Thus CML in real market condition looks like a fuzzy area rather than a precise line.

As was mentioned above capital market line represents all possible combinations of portfolios consisting in various proportions between risk-free asset and market portfolio. On the other hand, an efficient frontier represents all possible combinations of efficient portfolios, including only risky assets in various proportions.

The intercept point of CML and efficient frontier calls market or tangency portfolio. If investor is rational and risk-averse it will accept higher risk only when return increase proportionally. From this standpoint tangency portfolio is most efficient portfolio.

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