Question

How can I find out the tangency portfolio base on the info state.

Answer 1

A portfolio is defined by 2 parameters - expected return(y axis) and standard deviation(x axis). Using these 2 parameters, we can find a combination of all the portfolios which give the highest return for the lowest possible risk. Portfolios outside this frontier are not possible to achieve, and portfolios below the frontier are irrational, because you could get higher return with less risk.

The portfolio which gives you the highest return per unit of risk is the tangency portfolio. Of course, it is the most efficient portfolio as well and is a tangent to the above frontier that was mentioned in the last paragraph. Return per unit of risk is measured by Sharpe Ratio.

In constructing portfolios, investors often combine risky assets with risk-free assets (such as government bonds) to reduce risks. A complete portfolio is defined as a combination of a risky asset portfolio, with return R_p, and the risk-free asset, with return R_f.

The expected return of a complete portfolio is given as:

E(R_c) = w_pE(R_p) + (1 − w_p)R_f

And the variance and standard deviation of the complete portfolio return is given as:

Var(R_c) = w²_pVar(R_p), σ(R_c) = w_pσ(R_p),

where w_p is the fraction invested in the risky asset portfolio.

While the expected excess return of a complete portfolio is calculated as:

E(R_c) – R_f,

if we substitute E(R_c) with the previous formula, we get: w_p(E(R_p) − R_f).

The standard deviation of the complete portfolio is σ(R_c) = w_pσ(R_p), which gives us:

w_p = σ(R_c)/σ(R_p)

Therefore, for each complete portfolio:

Or E(R_c) = R_f + S_pσ(R_c), where S_p =

The line E(R_c) = R_f + S_pσ(R_c) is the capital allocation line (CAL). The slope of the line, S_p, is called the Sharpe ratio, or reward-to-risk ratio. The Sharpe ratio measures the increase in expected return per unit of additional standard deviation.

The optimal portfolio consists of a risk-free asset and an optimal risky asset portfolio. The optimal risky asset portfolio is at the point where the CAL is tangent to the efficient frontier. This portfolio is optimal because the slope of CAL is the highest, which means we achieve the highest returns per additional unit of risk.

The tangent portfolio weights are calculated as follows