# P1.T4.24.1. The efficient frontier and asset return distributions

#### Nicole Seaman

##### Director of CFA & FRM Operations
Staff member
Subscriber
Learning objectives: Describe the mean-variance framework and the efficient frontier. Compare the normal distribution with the typical distribution of returns of risky financial assets such as equities.

Questions:

24.1.1. Given a universe of investible (but risky) securities, Peter the analyst is able to render the portfolio possibilities curve (PPC) as displayed below. The efficient frontier is plotted with the darker line.

Peter is asked to update his analysis by including the risk-free rates. However, he is told the risk-free borrowing rate, r(B), exceeds the risk-free lending rate, r(L), such that r(B) > r(L). He doesn't know the exact values, but he does know they both lie in the shaded region, from 1.0% to 5.0% (on the y-axis, by definition).

After the introduction of the risk-free rates, which of the following statements is TRUE about the updated efficient frontier; aka, capital market line?

a. It will be a straight line represented by a linear equation of the form Y=aX+b
b. It will be fully defined by two straight-line segments
c. It will be defined by linear and non-linear segments or regions
d. It will be parallel to the incumbent efficient frontier

24.1.2. Given a universe of 500 stocks and an initial assumption for the risk-free rate, an analyst generates the efficient frontier. She assumes the mean-variance framework. The investment committee compliments her analysis and makes a request: they ask her to decrease the risk-free rate (the rate at which an investor can both borrow and lend) but otherwise make no changes (aka, ceteris paribus) to the expected returns or variance matrix for the equities. Which of the following is most likely to be correct?

a. The expected return on optimal portfolios with low risk & high risk will both decrease
b. The expected return on optimal portfolios with low risk will decrease, and the expected return on optimal portfolios with high risk will increase
c. The expected return on optimal portfolios with low risk will increase, and the expected return on optimal portfolios with high risk will decrease
d. The expected return on optimal portfolios with low risk & high risk will both increase

24.1.3. An analyst is fitting a distribution to the actual (aka, empirical) returns of a financial variable. His benchmark is the normal distribution. He defines three regions of both (fitted-to-actual and normal) distributions as follows:
• The head (aka, peak) lies within one standard deviation of the mean; i.e., μ +/- 1σ or within the interval {μ - σ, μ + σ}
• The shoulders extend from one to three standard deviations from the mean; i.e., left shoulder {μ - 3σ, μ - 1σ} and right shoulder {μ + 1σ, μ + 3σ}
• The tail is greater than three standard deviations from the mean; i.e., less than μ - 3σ, or greater than μ + 3σ
The mass is the probability of return falling into one of the three regions. To contrast the distributions, he defines the following ratio: the actual (i.e., empirical distribution's) mass to the normal distribution's mass, where their standard deviations are equal. Which of the following patterns should the analyst expect?

a. All ratios are roughly equal to one (≅ 1)
b. The head's ratio exceeds one (>1); the shoulder's is less than one (<1); and the tail's exceeds one (>1)
c. The head's ratio is less than one (<1); the shoulder's exceeds one (>1); and the tail's is less than one (<1)
d. All ratios significantly exceed one (>1)