Figure 11-13. Minimum risk portfolios for given return level (crosses)
The efficient frontier is comprised of all optimal portfolios with a higher return than the
absolute minimum variance portfolio. These portfolios dominate all other portfolios in
terms of expected returns given a certain risk level.
Capital Market Line
In addition to risky securities like stocks or commodities (such as gold), there is in general
one universal, riskless investment opportunity available: cash or cash accounts. In an
idealized world, money held in a cash account with a large bank can be considered riskless
(e.g., through public deposit insurance schemes). The downside is that such a riskless
investment generally yields only a small return, sometimes close to zero.
However, taking into account such a riskless asset enhances the efficient investment
opportunity set for investors considerably. The basic idea is that investors first determine
an efficient portfolio of risky assets and then add the riskless asset to the mix. By adjusting
the proportion of the investor’s wealth to be invested in the riskless asset it is possible to
achieve any risk-return profile that lies on the straight line (in the risk-return space)
between the riskless asset and the efficient portfolio.
Which efficient portfolio (out of the many options) is to be taken to invest in optimal
fashion? It is the one portfolio where the tangent line of the efficient frontier goes exactly
through the risk-return point of the riskless portfolio. For example, consider a riskless
interest rate of rf = 0.01. We look for that portfolio on the efficient frontier for which the
tangent goes through the point ( f,rf) = (0,0.01) in risk-return space.
For the calculations to follow, we need a functional approximation and the first derivative
for the efficient frontier. We use cubic splines interpolation to this end (cf. Chapter 9):
In [ 66 ]: import scipy.interpolate as sci
For the spline interpolation, we only use the portfolios from the efficient frontier. The
following code selects exactly these portfolios from our previously used sets tvols and
trets:
In [ 67 ]: ind = np.argmin(tvols)
evols = tvols[ind:]
erets = trets[ind:]
The new ndarray objects evols and erets are used for the interpolation:
In [ 68 ]: tck = sci.splrep(evols, erets)
Via this numerical route we end up being able to define a continuously differentiable
function f(x) for the efficient frontier and the respective first derivative function df(x):
In [ 69 ]: def f(x):
”’ Efficient frontier function (splines approximation). ”’
return sci.splev(x, tck, der= 0 )
def df(x):
”’ First derivative of efficient frontier function. ”’
return sci.splev(x, tck, der= 1 )
What we are looking for is a function t(x) = a + b · x describing the line that passes
through the riskless asset in risk-return space and that is tangent to the efficient frontier.
Equation 11-4 describes all three conditions that the function t(x) has to satisfy.
Equation 11-4. Mathematical conditions for capital market line
