## 12.4 Computing the Mean-Variance Efficient Frontier

The analytic expression for a minimum variance portfolio (12.21) can be used to show that any minimum variance portfolio can be created as a linear combination of any two minimum variance portfolios with different target expected returns. If the expected return on the resulting portfolio is greater than the expected return on the global minimum variance portfolio, then the portfolio is an efficient frontier portfolio. Otherwise, the portfolio is an inefficient frontier portfolio. As a result, to compute the portfolio frontier in $$(\mu_{p},\sigma_{p})$$ space (Markowitz bullet) we only need to find two efficient portfolios. The remaining frontier portfolios can then be expressed as linear combinations of these two portfolios. The following proposition describes the process for the three risky asset case using matrix algebra.

Proposition 12.1 (Creating a frontier portfolio from two minimum variance portfolios)

Let $$\mathbf{x}$$ and $$\mathbf{y}$$ be any two minimum variance portfolios with different target expected returns $$\mathbf{x}^{\prime}\mu=\mu_{p,0}\neq\mathbf{y}^{\prime}\mu=\mu_{p,1}$$. That is, portfolio $$\mathbf{x}$$ solves: \begin{align*} \min_{\mathbf{x}}~\sigma_{p,x}^{2}&=\mathbf{x}^{\prime}\Sigma \mathbf{x}\\ \textrm{ s.t. }\mathbf{x}^{\prime}\mu&=\mu_{p,0}\textrm{ and }\mathbf{x}^{\prime}\mathbf{1}=1, \end{align*} and portfolio $$\mathbf{y}$$ solves, \begin{align*} \min_{\mathbf{y}}\sigma_{p,y}^{2}&=\mathbf{y}^{\prime}\Sigma \mathbf{y}\\ \textrm{ s.t. }\mathbf{y}^{\prime}\mu&=\mu_{p,1}\textrm{ and }\mathbf{y}^{\prime}\mathbf{1}=1. \end{align*} Let $$\alpha$$ be any constant and define the portfolio $$\mathbf{z}$$ as a linear combination of portfolios $$\mathbf{x}$$ and $$\mathbf{y}$$: \begin{align} \mathbf{z} & =\alpha\cdot\mathbf{x}+(1-\alpha)\cdot\mathbf{y}\tag{12.22}\\ & =\left(\begin{array}{c} \alpha x_{1}+(1-\alpha)y_{1}\\ \vdots\\ \alpha x_{N}+(1-\alpha)y_{N} \end{array}\right).\nonumber \end{align} Then the following results hold:

1. The portfolio $$\mathbf{z}$$ is a minimum variance portfolio with expected return and variance given by: \begin{align} \mu_{p,z} & =\mathbf{z}^{\prime}\mu=\alpha\cdot\mu_{p,x}+(1-\alpha)\cdot\mu_{p,y},\tag{12.23}\\ \sigma_{p,z}^{2} & =\mathbf{z}^{\prime}\mathbf{\Sigma z=\alpha}^{2}\sigma_{p,x}^{2}+(1-\alpha)^{2}\sigma_{p,y}^{2}+2\alpha(1-\alpha)\sigma_{xy},\tag{12.24} \end{align} where $$\sigma_{p,x}^{2}=\mathbf{x}^{\prime}\Sigma \mathbf{x},\sigma_{p,y}^{2}=\mathbf{y}^{\prime}\Sigma \mathbf{y},\sigma_{xy}=\mathbf{x}^{\prime}\Sigma \mathbf{y}.$$
2. If $$\mu_{p,z}\geq\mu_{p,m}$$, where $$\mu_{p,m}$$ is the expected return on the global minimum variance portfolio, then portfolio $$\mathbf{z}$$ is an efficient frontier portfolio. Otherwise, $$\mathbf{z}$$ is an inefficient frontier portfolio.

The proof of 1. follows directly from applying (12.21) to portfolios $$\mathbf{x}$$ and $$\mathbf{y}$$: \begin{align*} \mathbf{x} & =\Sigma^{-1}\mathbf{MB}^{-1}\tilde{\mu}_{x},\\ \mathbf{y} & =\Sigma^{-1}\mathbf{MB}^{-1}\tilde{\mu}_{y}, \end{align*} where $$\tilde{\mu}_{x}=(\mu_{p,x},1)^{\prime}$$ and $$\tilde{\mu}_{y}=(\mu_{p,y},1)^{\prime}$$. Then, for portfolio $$\mathbf{z}$$: \begin{align*} \mathbf{z} & =\alpha\cdot\mathbf{x}+(1-\alpha)\cdot\mathbf{y}\\ & =\alpha\cdot\Sigma^{-1}\mathbf{MB}^{-1}\tilde{\mu}_{x}+(1-\alpha)\cdot\Sigma^{-1}\mathbf{MB}^{-1}\tilde{\mu}_{y}\\ & =\Sigma^{-1}\mathbf{MB}^{-1}(\alpha\cdot\tilde{\mu}_{x}+(1-\alpha)\cdot\tilde{\mu}_{y})\\ & =\Sigma^{-1}\mathbf{MB}^{-1}\tilde{\mu}_{z}, \end{align*} where $$\tilde{\mu}_{z}=\alpha\cdot\tilde{\mu}_{x}+(1-\alpha)\cdot\tilde{\mu}_{y}=(\mu_{p,z},1)^{\prime}$$. Result 2. follows from the definition of an efficient portfolio.

Example 2.16 (Create an arbitrary efficient frontier portfolio from two efficient minimum variance portfolios)

Consider the data in Table 1 and the previously computed minimum variance portfolios that have the same expected return as Microsoft and Starbucks, respectively, and let $$\alpha=0.5$$. From (12.22), the frontier portfolio $$\mathbf{z}$$ is constructed using: \begin{align*} \mathbf{z} & =\alpha\cdot\mathbf{x}+(1-\alpha)\cdot\mathbf{y}\\ & =0.5\cdot\left(\begin{array}{c} 0.8275\\ -0.0907\\ 0.2633 \end{array}\right)+0.5\cdot\left(\begin{array}{c} 0.519\\ 0.273\\ 0.207 \end{array}\right)\\ & =\left(\begin{array}{c} (0.5)(0.8275)\\ (0.5)(-0.0907)\\ (0.5)(0.2633) \end{array}\right)+\left(\begin{array}{c} (0.5)(0.519)\\ (0.5)(0.273)\\ (0.5)(0.207) \end{array}\right)\\ & =\left(\begin{array}{c} 0.6734\\ 0.0912\\ 0.2354 \end{array}\right)=\left(\begin{array}{c} z_{A}\\ z_{B}\\ z_{C} \end{array}\right). \end{align*} In R, the new frontier portfolio is computed using:

a = 0.5
z.vec = a*x.vec + (1-a)*y.vec
z.vec
##   MSFT   NORD   SBUX
## 0.6734 0.0912 0.2354

Using $$\mu_{p,z}=\mathbf{z}^{\prime}\mu$$ and $$\sigma_{p,z}^{2}=\mathbf{z}^{\prime}\mathbf{\Sigma z}$$, the expected return, variance and standard deviation of this portfolio are:

mu.pz = as.numeric(crossprod(z.vec, mu.vec))
sig2.pz = as.numeric(t(z.vec)%*%sigma.mat%*%z.vec)
sig.pz = sqrt(sig2.pz)
c(mu.pz, sig.pz)
## [1] 0.0356 0.0801

Equivalently, using $$\mu_{p,z}=\alpha\mu_{p,x}+(1-\alpha)\mu_{p,y}$$ and $$\sigma_{p,z}^{2}=\alpha^{2}\sigma_{p,x}^{2}+(1-\alpha)^{2}\sigma_{p,y}^{2}+2\alpha(1-\alpha)\sigma_{xy}$$ the expected return, variance and standard deviation of this portfolio are:

mu.pz = a*mu.px + (1-a)*mu.py
sig.xy = as.numeric(t(x.vec)%*%sigma.mat%*%y.vec)
sig2.pz = a^2 * sig2.px + (1-a)^2 * sig2.py + 2*a*(1-a)*sig.xy
sig.pz = sqrt(sig2.pz)
c(mu.pz, sig.pz)
## [1] 0.0356 0.0801

Because $$\mu_{p,z}=0.0356>\mu_{p,m}=0.0249$$ the frontier portfolio $$\mathbf{z}$$ is an efficient frontier portfolio. The three efficient frontier portfolios $$\mathbf{x},\,\mathbf{y}$$ and $$\mathbf{z}$$ are illustrated in Figure 12.6 and are labeled “E1” , “E2” and “E3” , respectively.

$$\blacksquare$$

Example 2.18 (Create an efficient frontier portfolio with a given expected return from two efficient minimum variance portfolios)

Given the two minimum variance portfolios with target expected returns equal to the expected returns on Microsoft and Starbucks, respectively, consider creating a frontier portfolio with target expected return equal to $$0.05$$. To determine the value of $$\alpha$$ that corresponds with this portfolio we use the equation $\mu_{p,z}=\alpha_{.05}\mu_{p,x}+(1-\alpha_{.05})\mu_{p,y}=0.05.$ We can then solve for $$\alpha_{.05}$$: $\alpha_{.05}=\frac{0.05-\mu_{p,y}}{\mu_{p,x}-\mu_{p,y}}=\frac{0.05-0.0285}{0.0427-0.0285}=1.51.$ Given $$\alpha_{.05}$$ we can solve for the portfolio weights using: $\begin{eqnarray*} \mathbf{z}_{.05} & = & \alpha_{.05}\mathbf{x}+(1-\alpha_{.05})\mathbf{y}\\ & = & 1.51\times\left(\begin{array}{c} 0.8275\\ -0.0907\\ 0.2633 \end{array}\right)-0.514\times\left(\begin{array}{c} 0.519\\ 0.273\\ 0.207 \end{array}\right)\\ & = & \left(\begin{array}{c} 0.986\\ -0.278\\ 0.292 \end{array}\right). \end{eqnarray*}$ We can then compute the mean and variance of this portfolio using $$\mu_{p,z_{.05}}=\mathbf{z}_{.05}^{\prime}\mu$$ and $$\sigma_{p,z_{.05}}^{2}=\mathbf{z}_{.05}^{\prime}\mathbf{\Sigma z}_{.05}.$$ Using R, the calculations are:

# compute alpha
a.05 = (0.05 - mu.py)/(mu.px - mu.py)
a.05
## [1] 1.51
# compute weights
z.05 = a.05*x.vec + (1 - a.05)*y.vec
z.05
##   MSFT   NORD   SBUX
##  0.986 -0.278  0.292
# compute mean and volatility
mu.pz.05 = as.numeric(crossprod(z.05,mu.vec))
sig.pz.05 = as.numeric(sqrt(t(z.05)%*%sigma.mat%*%z.05))
c(mu.pz.05,sig.pz.05)
## [1] 0.050 0.107

This portfolio is labeled “E4” in Figure 12.6.

Example 2.20 (Create an inefficient frontier portfolio with a given expected return from two minimum variance portfolios)

Given the two minimum variance portfolios with target expected returns equal to the expected returns on Microsoft and Starbucks, respectively, consider creating a frontier portfolio with target expected return equal to the expected return on Nordstrom. Then, $\mu_{p,z}=\alpha_{nord}\mu_{p,x}+(1-\alpha_{nord})\mu_{p,y}=\mu_{\textrm{nord}}=0.0015,$ and we can solve for $$\alpha_{nord}$$ using: $\alpha_{nord}=\frac{\mu_{\textrm{nord}}-\mu_{p,y}}{\mu_{p,x}-\mu_{p,y}}=\frac{0.0015-0.0285}{0.0427-0.0285}=-1.901.$ The portfolio weights are: $\begin{eqnarray*} \mathbf{z}_{nord} & = & \alpha_{nord}\mathbf{x}+(1-\alpha_{nord})\mathbf{y}\\ & = & -1.901\times\left(\begin{array}{c} 0.8275\\ -0.0907\\ 0.2633 \end{array}\right)+2.9\times\left(\begin{array}{c} 0.519\\ 0.273\\ 0.207 \end{array}\right)\\ & = & \left(\begin{array}{c} -0.0064\\ 0.9651\\ 0.1013 \end{array}\right). \end{eqnarray*}$ Using R, the calculations are:

# compute alpha
a.nord = (mu.vec["NORD"] - mu.py)/(mu.px - mu.py)
a.nord 
## NORD
## -1.9
# compute weights
z.nord = a.nord*x.vec + (1 - a.nord)*y.vec
z.nord 
##    MSFT    NORD    SBUX
## -0.0664  0.9651  0.1013
# compute mean and volatility
mu.pz.nord = as.numeric(crossprod(z.nord,mu.vec))
sig.pz.nord = as.numeric(sqrt(t(z.nord)%*%sigma.mat%*%z.nord))
c(mu.pz.nord,sig.pz.nord)
## [1] 0.0015 0.1033

Because $$\mu_{p,z}=0.0015<\mu_{p,m}=0.02489$$ the frontier portfolio $$\mathbf{z}$$ is an inefficient frontier portfolio. This portfolio is labeled “IE1” in Figure 12.6.

$$\blacksquare$$

### 12.4.1 Algorithm for computing efficient frontier

The efficient frontier of portfolios, i.e., those frontier portfolios with expected return greater than the expected return on the global minimum variance portfolio, can be conveniently created using (12.22) with two specific efficient portfolios. The first efficient portfolio is the global minimum variance portfolio (12.2). The second efficient portfolio is the efficient portfolio whose target expected return is equal to the highest expected return among all of the assets under consideration. The choice of these two efficient portfolios makes it easy to produce a nice plot of the efficient frontier. The steps for constructing the efficient frontier are:

1. Compute the global minimum variance portfolio $$\mathbf{m}$$ by solving (12.2), and compute $$\mu_{p,m}=\mathbf{m}^{\prime}\mu$$ and $$\sigma_{p,m}^{2}=\mathbf{m}^{\prime}\Sigma \mathbf{m}$$.
2. Compute the efficient portfolio $$\mathbf{x}$$ with target expected return equal to the maximum expected return of the assets under consideration. That is, solve (12.10) with $$\mu_{0}=\max\{\mu_{1},\ldots,\mu_{N}\}$$, and compute $$\mu_{p,x}=\mathbf{x}^{\prime}\mu$$ and $$\sigma_{p,x}^{2}=\mathbf{x}^{\prime}\Sigma \mathbf{x}$$.
3. Compute $$\mathrm{cov}(R_{p,m},R_{p,x})=\sigma_{mx}=\mathbf{m}^{\prime}\Sigma \mathbf{x}$$.
4. Create an initial grid of $$\alpha$$ values $$\{0,0.1,\ldots0.9,1\}$$, and compute the frontier portfolios $$\mathbf{z}$$ using $\mathbf{z}=\alpha\times\mathbf{x}+(1-\alpha)\times\mathbf{m},$ and compute their expected returns and variances using (12.22), (12.23) and (12.24), respectively.
5. Plot $$\mu_{p,z}$$ against $$\sigma_{p,z}$$ and adjust the grid of $$\alpha$$ values appropriately to create a nice plot. Negative values of $$\alpha$$ will give inefficient frontier portfolios. Values of $$\alpha$$ greater than one will give efficient frontier portfolios with expected returns greater than $$\mu_{0}.$$
Example 2.22 (Compute and plot the efficient frontier of risky assets (Markowitz bullet))

To compute the efficient frontier from the three risky assets in Table 12.1 in R use:

a = seq(from=0, to=1, by=0.1)
n.a = length(a)
z.mat = matrix(0, n.a, 3)
colnames(z.mat) = names(mu.vec)
mu.z = rep(0, n.a)
sig2.z = rep(0, n.a)
sig.mx = t(m.vec)%*%sigma.mat%*%x.vec
for (i in 1:n.a) {
z.mat[i, ] = a[i]*x.vec + (1-a[i])*m.vec
mu.z[i] = a[i]*mu.px + (1-a[i])*mu.gmin
sig2.z[i] = a[i]^2 * sig2.px + (1-a[i])^2 * sig2.gmin +
2*a[i]*(1-a[i])*sig.mx
}

The variables z.mat, mu.z and sig2.z contain the weights, expected returns and variances, respectively, of the efficient frontier portfolios for a grid of $$\alpha$$ values between 0 and 1. The resulting efficient frontier is illustrated in Figure 12.7 created with:

plot(sqrt(sig2.z), mu.z, type="b", ylim=c(0, 0.06), xlim=c(0, 0.16),
pch=16, col="green", cex = cex.val, ylab=expression(mu[p]),
xlab=expression(sigma[p]),cex.lab=1.5)
points(sd.vec, mu.vec, pch=16, cex=2, lwd=2, col="blue")
points(sig.gmin, mu.gmin, pch=16, col="green", cex=2)
points(sig.px, mu.px, pch=16, col="green", cex=2)
text(sig.gmin, mu.gmin, labels="GLOBAL MIN", pos=2, cex = cex.val)
text(sd.vec, mu.vec, labels=asset.names, pos=4, cex = cex.val)
text(sig.px, mu.px, labels="E1", pos=2, cex = cex.val)

Each point on the efficient frontier is a portfolio of Microsoft, Nordstrom and Starbucks. It is instructive to visualize the weights in these portfolios as we move along the frontier from the global minimum variance portfolio to the efficient portfolio with expected return equal to the expected return on Microsoft. We can do this easily using the PerformanceAnalytics function chart.StackedBar():

chart.StackedBar(z.mat, xaxis.labels=round(sqrt(sig2.z),digits=3),
legend.loc="upper",
colorset=rainbow8equal[1:3], ylab="Weights")

The resulting plot is shown in Figure 12.8. As we move along the frontier, the allocation to Microsoft increases and the allocation to Nordstrom decreases whereas the allocation to Starbucks stays about the same.

$$\blacksquare$$