Portfolios with correlated assets

Following the discussion of optimal portfolios for uncorrelated assets in the last lecture, we study now portfolios of correlated assets.

Let again be $M$ = number of assets, $x_0$ = value of riskless asset, $x_i$ = value of risky asset $i$, $i\ne 0$, and $n_i$ = number of assets of kind $i$.

Investing in a portfolio with initial wealth

$$W = \sum_{i=0}^M n_i x_i^0 ,$$ (1)

at some later time $T$ the wealth of portfolio will be

$$S(T) = \sum_{i=0}^M n_i x_i(T) .$$ (2)

For convenience, we take again $W$ = 1, $x_i^0$ = 1, i.e., $p_i$ = $n_i$ so that the condition $\sum_{i=0}^M p_i$ = 1 must hold.

To be specific, we will now asssume the $\delta x_i$ to be Gaussian random variables

$$p(\delta x_1,\cdots,\delta x_M) = \frac{1}{\sqrt{(2\pi)^M \d... ...2}\sum_{ij}^M (\delta x_i-m_i)C^{-1}_{ij}(\delta x_j-m_i)} .$$ (3)

The real symmetric, positive definite covariance matrix $C$ can be diagonalized by an orthogonal transformation $O^T$ = $O^{-1}$. Thus, the asset combinations $e_i$ defined by

$$\delta x_i = m_i+\sum_j O_{ij}e_j ,$$ (4)

are uncorrelated.

In particular, we assume the expected returns of asset $i$

$$m_i = <\delta x_i> ,$$ (5)

to be known. Hence, the expected return of portfolio is the linear combination

$$m_p = \sum_{i=0}^M p_i m_i .$$ (6)

Furthermore, we also assume the covariance matrix $C_{ij}$ of assets to be known

$$C_{ij} = <(\delta x_i-m_j)(\delta x_j-m_j)>$$ (7)

$$= <\delta x_i\delta x_j> -<\delta x_i>m_j-m_i<\delta x_j>+m_im_j$$ (8)

$$= <\delta x_i\delta x_j> -m_im_j .$$ (9)

This is a nontrivial assumption, as guesses for covariances are difficult to obtain in practice. Similarly, we find for the covariance between asset $i$ and the portfolio $p$

$$C_{ip} = <(\delta x_i-m_j)(\delta S-m_p)>$$ (10)

$$= \sum_{i=1}^M p_j <(\delta x_i-m_j)(\delta x_j-m_j)>$$ (11)

$$= \sum_{i=1}^M C_{ij}p_j ,$$ (12)

and finally for the variance of portfolio $p$

$$D_p = C_{pp} = \sum_{i=1}^M p_i C_{ip} = \sum_{i,j=1}^M p_i... ...m_{i=1}^M p_i^2 C_{ii} +\sum_{i,j=1,j\ne i}^M p_i C_{ij} p_j .$$ (13)

The main idea of Markowitz was to find an optimal portfolio by minimizing its variance $D_p$ fixing its expected return $m_p$. Minimizing $D_p$ must be done under some constraints. Necessary constraints are

$$\mbox{\lq\lq normalization'':} \sum_{i=0}^M p_i = 1,$$ (14)

$$\mbox{expected return:} \sum_{i=0}^M p_i m_i= m_p .$$ (15)

Note, that some $p_i$ might be negative if short selling of assets is allowed. Other possible constraints, describing specific situations, can be, for example,

$$\mbox{dividends:} \sum_{i=0}^M p_i d_i= D,$$ (16)

$$\mbox{no short selling:} \sum_{i=0}^M \vert p_i\vert > 0,$$ (17)

$$\mbox{diversification:} \sum_{i=0}^M p_i^2 = C,$$ (18)

$$\mbox{margins:} \sum_{i=0}^M \vert p_i\vert x_i= F .$$ (19)

Example 1: Minimize portfolio variance $D_p$ subject to (i) $\sum_{i=0}^M p_i$ = $1$ and (ii) $\sum_{i=0}^M p_i m_i$ = $m_p$. Implementing the first constraint explicitly and introducing a Lagrange multiplier $\zeta$ for the second results in

$$E(p_0,p_1,\cdots,p_M) = \sum_{i,j=1}^M p_i C_{ij}p_j-\zeta \sum_{i=0}p_i m_i$$ (20)

$$= \sum_{i,j=1}^M p_i C_{ij}p_j-\zeta \sum_{i=1}p_i (m_i-m_0)-\zeta m_0 .$$ (21)

It follows from $\partial E/\partial p_i$ = 0 for $i\ne 0$

$$2 \sum_{i=1}^M C_{ij} p_j = \zeta (m_i-m_0) \quad \Big(=2 C_{ip} \Big) ,$$ (22)

and thus, inverting the covariance $C_{ij}$

$$p_i = \frac{\zeta}{2} \sum_{j=1}^M C^{-1}_{ij}(m_i-m_0) .$$ (23)

The Lagrange multiplier $\zeta$ is obtained from

$$m_p = \sum_{i=0}^M p_i m_i$$ (24)

$$= m_0+\sum_{i=1}^M p_i (m_i-m_0)$$ (25)

$$= m_0+\frac{\zeta}{2} \sum_{i,j=1}^M (m_i-m_0)C^{-1}_{ij}(m_j-m_0) ,$$ (26)

i.e.,

$$\zeta = 2 \frac{m_p-m_0}{\sum_{i,j=1}^M (m_i-m_0)C^{-1}_{ij}(m_j-m_0)} .$$ (27)

For $D_p$ we find

$$D_p = \sum_{i=1}^M p_i C_{ij}p_j$$ (28)

$$= \frac{\zeta^2}{4}\sum_{i,j}^M (m_i-m_0)C^{-1}_{ij}(m_j-m_0)$$ (29)

$$= \frac{(m_p-m_0)^2}{\sum_{i,j}^M (m_i-m_0)C^{-1}_{ij}(m_j-m_0)} ,$$ (30)

hence

$$\zeta = 2\frac{D_p}{m_p-m_0}$$ (31)

Example 2: Minimize portfolio variance $D_p$ without riskless asset subject to (i) $\sum_{i=0}^M p_i$ = $1$ and (ii) $\sum_{i=0}^M p_i m_i$ = $m_p$. Implementing the both constraints by introducing Lagrange multiplier $\zeta$ and $\mu$ results in

$$E(p_1,\cdots,p_M) = \sum_{i,j=1}^M p_i C_{ij}p_j-\zeta \sum_{i=1}p_i m_i-\mu \sum_{i=1}p_i .$$ (32)

It follows from $\partial E/\partial p_i$ = 0 for $i\ne 0$

$$p_i = \sum_{j=1}^M C^{-1}_{ij}(\frac{\zeta}{2} m_i+\frac{\mu}{2}) .$$ (33)

The point with the lowest risk has $\xi$ = 0, so that

$$p_i = \frac{\mu}{2}\sum_{j=1}^M C^{-1}_{ij} ,$$ (34)

and for $\mu$ follows

$$1 = \sum_i p_i = \frac{\mu}{2} \sum_{ij} C^{-1}_{ij} ,$$ (35)

i.e.,

$$\mu = \frac{2}{\sum_{ij} C^{-1}_{ij}} .$$ (36)

All optimal portfolios must have more risk. For the general case the Lagrange multipliers $ßxi$ and $\mu$ can be obtained similarly as in Example 1.

Fig. 1 shows the dependency of a portfolio of two risky assets from their correlation coefficient. The correlation coefficient, defined as $\rho$ = $\frac{C_{12}}{\sqrt{C_{11} C_{22}}}$. = $\frac{C_{12}}{\sigma_1\sigma_2}$, can only take values between $-1$ (perfect anti-correlation) and $+1$ (perfect correlation). In the extreme case of perfect anti-correlation, i.e., $\rho$ = $-1$, the two risky assets can be combined to a riskfree portfolio. If one of the two assets is riskfree, all the straight lines in the figure coincide (see example 1).

Figure 1: Portfolios of two risky correlated assets without short selling in the $m_p$-$\sigma _p$ plane.