Plotting Efficient Frontiers

Question

Lets make an assumption that short-selling is forbidden: w₁ ≥ 0;w₂ ≥0
Suppose also that we can invest risk-free at r= 2% but borrow at R= 4%. On
the (Risk and Return)-plane plot the efficient portfolios, which can be attained under such
restrictions.

I have been having difficulties with this question.

Answer 1

Your question is a portfolio optimization problem, where the goal is to either maximize return for a given level of risk, or minimize risk for a given level of return (these are effectively equivalent).

What do we know?

1. No short-selling, thus the borrowing rate is irrelevant.
2. Risk-free asset is 2%. If you invest 100% in the risk-free asset, you have the minimum variance portfolio (MVP). Thus, you have your point on the far right of the frontier.
3. In a two asset portfolio, the other asset must be the risky asset (e.g. not the risk-free asset). If you invest 100% in this asset, you have the maximum return portfolio (MRP). Thus, you have your point on the far left of the frontier.

Now we have to fill in the points along the frontier between the minimum variance portfolio and the maximum return portfolio. Given only two assets, this is an straightforward exercise.

We'll calculate 9 other portfolios by incrementally allocating 10% from the risk-free to the risky asset with the total weight equal to 100% (remember the constraints of no leverage or short selling).

1. Allocate 10% from the risk-free to the risky asset. Risk-free weight (w1) is equal to 100% - 10% = 90% and the risky asset weight (w2). Portfolio weights (w1 + w2) equal 100 (90 + 10).
2. Repeat by allocating 10% again form the risk-free (w1) to the risky asset (w2). Risk-free weight (w1) is equal to 90% - 10% = 80% and the risky asset weight (w2) equal 10 + 10 = 20. Portfolio weights (w1 + w2) total 100 (80 + 20).
3. Repeat process in 10% increments until finish with 90% in the risky asset.

Before calculating the portfolio statistics, you require the expected returns, expected risk, and the covariance between the two assets (e.g. the risk-free and risky assets). They are either calcuated form the time-series data or provided in the question. We'll assume they are provided (numerical examples are used below).

w1 = weight of the risk-free asset

w2 = weight of the risky asset

E(R1) = expected return the risk-free asset (e.g. 2%)

E(R1) = expected return the risky asset (e.g. 8%)

E(V1) = expected variance the risk-free asset (e.g. 0.0001)

E(V2) = expected variance the risky asset (e.g. 0.0225)

E(C12) = expected covariance of asset 1 and asset 2 (e.g. 0.000375)

Calculate the expected return for each portfolio:

Expected return (Px) = w1 * E(R1) + w2 * E(R2)

Expected Return (P1) = 0.9 * 0.02 + 0.1 * 0.8 = 0.026 or 2.6%

Calculate the expected risk for each portfolio:

Expected risk (Px) = w1 * variance1 + w2 * variance2 + 2 * w1 *w2 * covariance12

Expected risk (P1) = 0.9 * 0.001 + 0.1 * 0.0225 + 2 * 0.9 * 0.1 * 0.000375 = 0.0032175

Since we express risk in term of standard deviation rather than variance, we find the square root of the variance:

sqrt(0.0032175) = 0.0567230 or simply 5.67%

After calculating return and risk for each of the 11 portfolios (MRP, MVP, and the 9 others), plot them on a graph with return on the y=axis and risk on the x-axis. You now have the efficient frontier for a 2 asset portfolio.

Of course, this is more involved when there are move than 2 assets, which requires a solver engine to find the optimal portfolio weights and a covariance matrix to calculate risk, but that is a question for another time!

Hope this helps clarify.