&2.1 Trading off Expected return and risk

How to invest our wealth?

(a) To maximize the expected return;

(b) To minimize the risk=Variance return.

&2.2 One risky asset and one risk-free asset

Suppose that there is one risky asset, e.g., a mutual fund with expected return 0.15 and sd (standard deviation) of the return 0.25 and one risk-free asset, a 30-day T-bill with expected return 0.06 and sd 0. If a fraction w of our wealth is invested in the risky asset, then what is the expected return and risk?

Question1: Suppose you want an expected return of 0.10, what should w be?

Question 2: Suppose you want sd=0.05, what should w be?

What is the conclusion can be drawn from this simple example? Finding an optimal portfolio can be obtained by :

1. Finding the optimal portfolio of risky assets;

2. Finding the appropriate mix of the risk-free asset and the optimal portfolio from step one.

Example: In Feb 2001 issue of Paine Webber’s Investment Intelligence, he said that “the chart shows that a 20% municipal /80% S&P 500 mix sacrificed only 0.42% annual after-tax return relative to a 100% S&P 500 portfolio, while reducing risk by 13.6% from 14.91% to 12.88%”. Webber’s point is correct, but for a investor, what is over-emphasize?

Question: usually, the risk-free rate is known. (Treasury bill rates are published in most newspapers.) But, how to estimate E(R) and Var(R)?

&2.3 Two risky assets

Suppose the two risky assets have returns R1 and R2 and we mix them in proportion w and 1-w.

Example:

Questions: How to estimate the means, variances and covariance of R1 and R2?

(Under stationary assumption.)

&5.4 Combining two risky asset with a free asset Recall Fig 5.3, we see that the dotted line lies

above the dashed line. This means that the dotted lines gives a higher expected return than the dashed line under given risk. The bigger the slope of the line (Sharpe ratio) the better, why? The point T on the parabola represents the portfolio with the highest Sharpe ratio. It is the optimal portfolio for the purpose of mixing with the risk-free asset. This portfolio is called “tangency portfolio” since its line is tangent to the parabola.

Key result: The optimal or efficient portfolios mix the tangency portfolio of two risky assets with the risk-free asset. Each efficient portfolio has two properties:

it has a higher expected return than any other portfolio with the same (or smaller) risk.

It has a smaller risk than any other portfolio with the same (or smaller) return.

&5.4.1 Tangency portfolio with two risky assets

How to find the tangency portfolio?

&2.5 Risk-efficient portfolio with N risky assets

mu=(0.08, 0.03, 0.05), Sigma=[ 0.3, 0.02, 0.01, 0.02, 0.15, 0.03 0.01, 0.03, 0.18]