## Portfolio Management - Security Market Line

Following with our previous example, you now have enough assumptions to compute the security market line (SML), the equation that shows the relationship between the risk and the expected return of a particular investment that you might want to recommend to a client.

Here's how to build the model:

- Establish the risk-free rate (RFR), which was done previously.
- Establish the market return rate which, again, was done above; it is usually designated R
_{M}. - Compute the variance of R
_{M}(σ²_{M}). - Compute the covariance of the market with the return on a particular investment (Cov
_{i,M}). - Divide the Cov
_{i,M}from step 4 by the σ²_{M}from step 3. This quantity is known as the investment's beta (β_{i}). - Subtract step 1's RFR from step 2's R
_{M}. - Multiply step 5's β
_{i}by the result of step 6. - Add step 7's result to step 1's RFR to compute the estimated return on the investment.

Here is the SML formula:

E(R_{i}) = RFR + βi (R_{M} - RFR) |

**
**

Let's go through that step-by-step. Start by assuming a 3% RFR and a 15% R

_{M}.

Now let's start with some "dummy" data and calculate the σ²

_{M}the same way we calculated variance in Section 1. Here, the X quantities are annual rates of return. In other words, the first year, the broadest market portfolio yielded 15 cents for every dollar invested, the following year it yielded 13 cents, and so on:

Observation |
X |
X - μ |
(X - μ)^{2} |

1. | 15 | 0 | 0 |

2. | 13 | -2 | 4 |

3. | 14 | -1 | 1 |

4. | 12 | -1 | 9 |

5. | 12 | -3 | 9 |

6. | 14 | -1 | 1 |

7. | 16 | 1 | 1 |

8. | 17 | 2 | 4 |

9. | 18 | 3 | 9 |

10. | 19 | 4 | 16 |

Sum (Σ) |
150 | 54 | |

Number (N) |
10 | 10 | |

Mean (μ) |
15 | σ² = |
5.4 |

The variance of the market is 5.4. You can also tell by looking at the mean why the market rate of return in this example is said to be 15%.

Now let's compute the covariance of the market and the particular investment:

Year |
R_{M} |
R_{i} |
(rStep 2 _{M }- r_{Mave}) |
(rStep 3 _{i} - r_{iave}) |
Step 4 and 5(r_{M }- r_{Mave}) * (r_{i} - r_{iave}) |

1. | 15 | 35 | 0 | -8 | 0 |

2. | 13 | 33 | -2 | -10 | 20 |

3. | 14 | 37 | -1 | -6 | 6 |

4. | 12 | 41 | -3 | -2 | 6 |

5. | 12 | 40 | -3 | -3 | 9 |

6. | 14 | 48 | -1 | 5 | -5 |

7. | 16 | 48 | 1 | 5 | 5 |

8. | 17 | 50 | 2 | 7 | 14 |

9. | 18 | 48 | 3 | 5 | 15 |

10. | 19 | 50 | 4 | 7 | 28 |

r , _{1ave}r_{2ave } |
15 | 43 | 98 |
||

Number (N) |
10 | Cov (R_{M},R_{i}) |
9.8 |

Again, just based on numbers that we simply plugged in, you can come up with a covariance of 9.8.

- To find the β
_{i }for this investment, simply divide the covariance, 9.8, by the market variance, 5.4. Your result will be 1.8. - The next step is even simpler: subtract the RFR, 3%, from the market rate of return, 15%, and you will get 12%.
- Multiply that 12% by 1.8, and you will get 21.6%.
- Finally, add the RFR, 3%, to 21.6% and you will get a final result of 24.6%. That is the rate of return you would expect from that investment given its correlation with the market, notated E(R
_{i}).