Efficient Frontier without Risk-Free Asset

Now we have the solution to the mean-variance optimization problem, for every target expected return $E[\tilde r_p]$, we could find the optimal weights $\boldsymbol W^*$, and the corresponding variance $\sigma^2_p$ and expected return $E[\tilde r_p]$. We could explore:

• The weight vector $\boldsymbol W^*$ for different target expected returns $E[\tilde r_p]$. (Two Fund Theorem)
• The relationship between the expected return $E[\tilde r_p]$ and the variance $\sigma^2_p$. (Efficient Frontier)
• Covariance (covariance with minimum variance portfolio (MVP), zero covariance portfolio, etc.)

These three are the core features of the mean-variance optimization. We will explore them one by one.

Two Fund Theorem

From previous section, we have the optimal weight vector $\boldsymbol W^*$ for a given target expected return $E[\tilde r_p]$:

\[\boldsymbol W^* = \boldsymbol g + \boldsymbol h \cdot E[\tilde r_p],\]

where:

• $\boldsymbol g = \frac{1}{D} \left( B \cdot \boldsymbol V^{-1} \boldsymbol 1 - A \cdot \boldsymbol V^{-1} \boldsymbol \mu \right)$, $N \times 1$ vector
• $\boldsymbol h = \frac{1}{D} \left( C \cdot \boldsymbol V^{-1} \boldsymbol \mu - A \cdot \boldsymbol V^{-1} \boldsymbol 1 \right)$, $N \times 1$ vector

Clearly, all the optimal weights are a linear function of the target expected return $E[\tilde r_p]$. With a different target expected return, we could find a different optimal weight vector. All the optimal portfolios weights are on the same line in the $\mathbb{R}^N$ weight space . Since they are on a straight line, we could use any two optimal portfolios to construct the other optimal portfolios. This is the Two Fund Theorem.

Claim:

For any two optimal portfolios, the linear combination of them is also an optimal portfolio:

\[\boldsymbol W^* = \alpha \boldsymbol W^*_1 + (1 - \alpha) \boldsymbol W^*_2, \quad \forall \alpha \in [0, 1]\]

Proof:

Since they are optimal portfolios, we could write them as:

\[\begin{align*} \boldsymbol W^*_1 = \boldsymbol g_1 + \boldsymbol h_1 \cdot E[\tilde r_1], \\ \boldsymbol W^*_2 = \boldsymbol g_2 + \boldsymbol h_2 \cdot E[\tilde r_2], \end{align*}\]

where $\boldsymbol g_1, \boldsymbol h_1$ are the optimal weights for portfolio 1, and $\boldsymbol g_2, \boldsymbol h_2$ are the optimal weights for portfolio 2. We could rewrite the linear combination of them as:

\[\begin{align*} \boldsymbol W^* &= \alpha \boldsymbol W^*_1 + (1 - \alpha) \boldsymbol W^*_2 \\ &= \alpha (\boldsymbol g_1 + \boldsymbol h_1 \cdot E[\tilde r_1]) + (1 - \alpha) (\boldsymbol g_2 + \boldsymbol h_2 \cdot E[\tilde r_2]) \\ &= \alpha \boldsymbol g_1 + (1 - \alpha) \boldsymbol g_2 + \alpha \boldsymbol h_1 \cdot E[\tilde r_1] + (1 - \alpha) \boldsymbol h_2 \cdot E[\tilde r_2] \\ &= \boldsymbol g + \boldsymbol h \cdot E[\tilde r_p] \end{align*}\]

where:

• $\boldsymbol g = \alpha \boldsymbol g_1 + (1 - \alpha) \boldsymbol g_2$
• $\boldsymbol h = \alpha \boldsymbol h_1 + (1 - \alpha) \boldsymbol h_2$
• $E[\tilde r_p] = \alpha E[\tilde r_1] + (1 - \alpha) E[\tilde r_2]$
• $\boldsymbol W^*$ is the optimal portfolio for target expected return $E[\tilde r_p]$.

Thus, we have proved that the linear combination of any two optimal portfolios is also an optimal portfolio. This is the Two Fund Theorem.

Note: The optimal portfolios are also called frontier portfolios, see below Efficient Frontier section.

Frontier without Risk-Free Asset

Now we know weights $\boldsymbol W^*$ are a straight line in the weight space $\mathbb{R}^N$.

How about in the mean-variance space ($\mu, \sigma^2$)? or mean-standard deviation space ($\mu, \sigma$)?

We know that the optimal weights $\boldsymbol W^*$ are a linear function of the target expected return $E[\tilde r_p]$, the variance of portfolio $\sigma^2_p$ is a function of the weights ($\sigma^2_p = W^T_p V W_p$), where V is the covariance matrix of the asset returns. Clearly, we could find the relationship between the expected return $E[\tilde r_p]$ and the variance $\sigma^2_p$.

The variance of the portfolio is given by:

\[\begin{align*} \sigma^2_p &= \boldsymbol W^T \boldsymbol V \boldsymbol W \\ &= \left( \boldsymbol g + \boldsymbol h \cdot E[\tilde r_p] \right)^T \boldsymbol V \left( \boldsymbol g + \boldsymbol h \cdot E[\tilde r_p] \right) \\ &= \frac{C}{D} [E(\tilde r_p)] - \frac{A}{C}]^2 + \frac{1}{C} \\ \end{align*}\]

The variance ($\sigma^2_p$) is a quadratic function of the expected return ($E[\tilde r_p]$), if we plot the expected return and variance on a 2D graph, we could find the relationship between them is a hyperbola. The minimum variance portfolio (MVP) is the point on the hyperbola with the minimum variance.

Minimum Variance Portfolio (MVP)

The minimum variance portfolio (mvp) is the point on the hyperbola with the minimum variance. Clearly, it has the expected return $E[\tilde r_{mvp}] = \frac{A}{C}$ and variance $\sigma^2_{mvp} = \frac{1}{C}$.

Efficient Frontier

Recall the utility theory, if we assume only the mean and variance of the wealth are relevant for decision-making, any rational investor would choose the portfolio with highest expected return at given level of variance. (Top left corner of the mean-variance space). Therefore, the lower half of the hyperbola is not desirable, and the upper half of the hyperbola is the efficient frontier. In summary:

• Frontier Portfolios
  • All hyperbolas are frontier portfolios
  • All frontier portfolios is the lowest variance portfolio for a given expected return
  • The frontier is the edge of all feasible portfolios
  • Any two frontier portfolios could be combined to form another frontier portfolio
• Minimum Variance Portfolio (MVP)
  • The pivot point of the hyperbola
  • The MVP is the lowest variance portfolio of all portfolios
• Efficient Frontier
  • The upper half of the hyperbola
  • Higher than the MVP ($E(\tilde r_p > E(\tilde r_{mvp}) = \frac{A}{C}$)
• Inefficient Portfolios
  • The lower half of the hyperbola
  • Higher than the MVP ($E(\tilde r_p < E(\tilde r_{mvp}) = \frac{A}{C}$)
  • Not desirable

Mean-Standard Deviation space

Asymptotes:

\[E[\tilde r_p] = \frac{A}{C} + \sqrt{\frac{D}{C}} \cdot \sigma_p\]

Explore Covariance

Covariance

The covariance between any portfolios could be calculated using the covariance matrix $V$:

\[Cov(\tilde r_p, \tilde r_q) = \boldsymbol W^T_p \boldsymbol V \boldsymbol W_q\]

where $\boldsymbol W_p$ and $\boldsymbol W_q$ are the weight vectors of the two portfolios.

Covariance of Frontier Portfolios

Recall the variance of the frontier portfolio is given by:

\[\sigma^2_p = \frac{C}{D} [E(\tilde r_p)] - \frac{A}{C}]^2 + \frac{1}{C} \\\]

With the same logic, we could find the covariance between any two frontier portfolios is given by:

\[Cov(\tilde r_p, \tilde r_q) = \frac{C}{D} [E(\tilde r_p)] - \frac{A}{C}] \cdot [E(\tilde r_q)] - \frac{A}{C}] + \frac{1}{C} \\\]

Covariance with Minimum Variance Portfolio (MVP)

Claim:

The covariance between any portfolio (not necessarily frontier portfolio) and the minimum variance portfolio (MVP) is equal to variance of the minimum variance portfolio (MVP):

\[Cov(\tilde r_p, \tilde r_{mvp}) = \sigma^2_{mvp}\]

Proof:

Consider the following portfolio: $\alpha \tilde r_p + (1 - \alpha) \tilde r_{mvp}$, where $\alpha$ is a constant. The variance of this portfolio is given by:

\[\begin{align*} Var(\alpha \tilde r_p + (1 - \alpha) \tilde r_{mvp}) &= \alpha^2 Var(\tilde r_p) + (1 - \alpha)^2 Var(\tilde r_{mvp}) + 2\alpha(1 - \alpha) Cov(\tilde r_p, \tilde r_{mvp}) \\ &= \alpha^2 \sigma^2_p + (1 - \alpha)^2 \sigma^2_{mvp} + 2\alpha(1 - \alpha) Cov(\tilde r_p, \tilde r_{mvp}) \\ &= \alpha^2 \sigma^2_p + (1 - \alpha)^2 \sigma^2_{mvp} + 2\alpha(1 - \alpha) \sigma^2_{mvp} \\ \end{align*}\]

The solution of:

\[\min_{\alpha} Var(\alpha \tilde r_p + (1 - \alpha) \tilde r_{mvp}),\]

should be $\alpha = 0$, since the mvp is the lowest variance portfolio. The First Order Condition (FOC) is given by:

\[\begin{align*} \frac{\partial Var(\alpha \tilde r_p + (1 - \alpha) \tilde r_{mvp})}{\partial \alpha} &= 2\alpha \sigma^2_p + 2(1 - \alpha) \sigma^2_{mvp} + 2\alpha Cov(\tilde r_p, \tilde r_{mvp}) - 2(1 - \alpha) Cov(\tilde r_p, \tilde r_{mvp}) = 0 \\ \end{align*}\]

plugging in $\alpha = 0$, we have:

\[\begin{align*} 2(1 - 0) \sigma^2_{mvp} + 2(0) Cov(\tilde r_p, \tilde r_{mvp}) - 2(1 - 0) Cov(\tilde r_p, \tilde r_{mvp}) &= 0 \\ Cov(\tilde r_p, \tilde r_{mvp}) &= \sigma^2_{mvp} \\ \end{align*}\]

Thus, we have proved that the covariance between any portfolio (not necessarily frontier portfolio) and the minimum variance portfolio (MVP) is equal to variance of the minimum variance portfolio (MVP).

Zero Covariance Portfolio

Claim:

For any frontier portfolio $p$, except for the minimum variance portfolio (MVP), there exists a unique frontier portfolio $zc(p)$ such that the covariance between $p$ and $zc(p)$ is zero.

Proof:

We set

\[\begin{align*} Cov(\tilde r_p, \tilde r_q) &= W^T_p \boldsymbol V W_q \\ &= (\lambda \boldsymbol V^{-1} \boldsymbol \mu + \gamma \boldsymbol V^{-1} \boldsymbol 1)^T \boldsymbol V W_q \\ &= \lambda \mu^T V^{-1} V W_q + \gamma \boldsymbol 1^T V^{-1}V W_q \\ &= \lambda \mu^T W_q + \gamma \boldsymbol 1^T W_q \\ &= \lambda E[\tilde r_q] + \gamma. \end{align*}\]

Using the definition of $\lambda$ and $\gamma$:

\[\begin{align*} \lambda &= \frac{C \cdot E[\tilde r_p] - A}{D}, \\ \gamma &= \frac{B - A \cdot E[\tilde r_p]}{D}, \end{align*}\]

we could obtain:

\[\begin{align*} E[\tilde r_q] &= E[\tilde r_{zc(p)}] + \frac{cov(\tilde r_p, \tilde r_q)}{\sigma^2_{p}} \cdot \{E[\tilde r_p] - E[\tilde r_{zc(p)}]\} \\ &= E[\tilde r_{zc(p)}] + \beta_{qp} \cdot \{E[\tilde r_p] - E[\tilde r_{zc(p)}]\}. \end{align*}\]

Where $\beta_{qp} = \frac{cov(\tilde r_p, \tilde r_q)}{\sigma^2_{p}}$ is the beta of portfolio $q$ with respect to portfolio $p$.

Interpretation:

Any portfolio $q$ that is not the minimum variance portfolio (MVP) could be expressed or explained or constructed by the a frontier portfolio $p$ and its zero covariance portfolio $zc(p)$.

This give us a new perspective of how to explain the expected return of a portfolio. In the mean-variance setup, we explain the expected return by the variance (measure the relationship between the mean and variance). Now we could explain the expected return by the $\beta_{qp}$, higher the beta, higher the expected return.

Later, we will introduce the Market Portfolio, which is a special case of the equation above, substituting the market portfolio for $p$ will give us the a line called Security Market Line (SML). And it will further lead us to the Capital Asset Pricing Model (CAPM). We will discuss this in the next section.

Conclusion

Key takeaways from this section:

• The optimal weights $\boldsymbol W^*$ are a linear function of the target expected return $E[\tilde r_p]$.
• The variance of the portfolio $\sigma^2_p$ is a quadratic function of the target expected return $E[\tilde r_p]$.
• The minimum variance portfolio (MVP) is the point on the hyperbola with the minimum variance.
• The efficient frontier is the upper half of the hyperbola, and the lower half is the inefficient portfolios.
• The covariance between any portfolio (not necessarily frontier portfolio) and the minimum variance portfolio (MVP) is equal to variance of the minimum variance portfolio (MVP).
• The covariance between any two frontier portfolios is given by the covariance matrix $V$.
• For any frontier portfolio $p$, except for the minimum variance portfolio (MVP), there exists a unique frontier portfolio $zc(p)$ such that the covariance between $p$ and $zc(p)$ is zero.
• Any portfolio $q$ that is not the minimum variance portfolio (MVP) could be expressed or explained or constructed by the a frontier portfolio $p$ and its zero covariance portfolio $zc(p)$.