Portfolio Optimization

Portfolio opportunity set

The collection of all feasible investment possibilities set, or portfolio opportunity set, in the case of two assets is simply all possible portfolios that can be formed by varying the portfolio weights such that the weights sum to one.

We summarize the expected return-risk (mean-volatility) properties of the feasible portfolios in a plot with portfolio expected return, \(\mu_p\), on the vertical axis and portfolio standard deviation, \(\sigma_p\), on the horizontal axis.

Recall we have the expected return and variance of the portfolio return as follows:

\[ \begin{aligned} E[r_{p,t}] &= \sum_{i=1}^k w_{i,t} E[r_{i,t}] \\ \text{Var}[r_{p,t}] &= \sum_{i=1}^k\sum_{j=1}^k w_{i,t}\,w_{j,t}\,\text{Cov}(r_{i,t}, r_{j,t}) \end{aligned} \]

Given the following set up for two assets A and B:

	Asset A	Asset B
\(\mu\)	0.175	0.055
\(\sigma\)	0.258	0.115
\(\rho_{AB}\)	-0.164

where \(\mu\) is the expected return, \(\sigma\) is the volatility (standard deviation) of returns, \(\rho\) is the correlation coefficient between assets A and B.

Equally weighted

First we consider an equally weighted portfolio \(p_1\).

# initialize parameters
mu.A = 0.175
sig.A = 0.258
sig2.A = sig.A^2

mu.B = 0.055
sig.B = 0.115
sig2.B = sig.B^2

rho.AB = -0.164
sig.AB = rho.AB*sig.A*sig.B

# equally weighted
x.A = 0.5
x.B = 0.5
mu.p1 = x.A*mu.A + x.B*mu.B
sig2.p1 = x.A^2 * sig2.A + x.B^2 * sig2.B + 2*x.A*x.B*sig.AB
sig.p1 = sqrt(sig2.p1)
data.tbl = t(c(mu.p1=mu.p1, sig.p1=sig.p1, sig.avg=0.5*sig.A+0.5*sig.B))
data.tbl

##      mu.p1    sig.p1 sig.avg
## [1,] 0.115 0.1323416  0.1865

This portfolio has expected return half-way between the expected returns on assets A and B, but the portfolio standard deviation is less than half-way between the asset standard deviations. This reflects risk reduction via diversification.

The equally weighted portfolio has expected return half way between the expected returns on assets A and B, but has standard deviation (volatility) that is less than half way between the standard deviations of the two assets.

For long-only portfolios, we can show that this is a general result as long as the correlation between the two assets is not perfectly positive.

Long-short portfolio

Now we consider a second portfolio \(p_2\) which allows for short selling.

To short an asset one borrows the asset, usually from a broker, and then sells it. The proceeds from the short sale are usually kept on account with a broker and there may be restrictions on the use of these funds for the purchase of other assets. The short position is closed out when the asset is repurchased and then returned to original owner. If the asset drops in value then a gain is made on the short sale and if the asset increases in value a loss is made.

# long-short portfolio
x.A = 1.5
x.B = -0.5
mu.p2 = x.A*mu.A + x.B*mu.B
sig2.p2 = x.A^2 * sig2.A + x.B^2 * sig2.B + 2*x.A*x.B*sig.AB
sig.p2 = sqrt(sig2.p2)
data.tbl = t(c(mu.p2=mu.p2, sig.p2=sig.p2, sig.avg=1.5*sig.A-0.5*sig.B))
data.tbl

##      mu.p2    sig.p2 sig.avg
## [1,] 0.235 0.4004673  0.3295

This portfolio has both a higher expected return and standard deviation than asset A. The high standard deviation is due to the short sale, which is a type of leverage, and the negative correlation between assets A and B.

Leverage refers to an investment that is financed through borrowing. Here, the short sale of asset B produces borrowed funds used to purchase more of asset A. This increases the risk (portfolio standard deviation) of the investment.

Now we calculate the portfolio opportunity set. Here the portfolio weight on asset A, \(x_A\), is varied from \(-0.4\) to \(1.4\) at a step of 0.1 and, since \(x_B=1−x_A\), the weight on asset B then varies from \(1.4\) to \(-0.4\).

x.A = seq(from=-0.4, to=1.4, by=0.1)
x.B = 1 - x.A
mu.p = x.A*mu.A + x.B*mu.B
sig2.p = x.A^2 * sig2.A + x.B^2 * sig2.B + 2*x.A*x.B*sig.AB
sig.p = sqrt(sig2.p)
port.names = paste("portfolio", 1:length(x.A), sep=" ")
data.tbl = as.data.frame(cbind(x.A, x.B, mu.p, sig.p))
rownames(data.tbl) = port.names
col.names = c("$x_{A}$","$x_{B}$", "$\\mu_{p}$", "$\\sigma_{p}$" )
kbl(data.tbl, col.names=col.names) %>%
   kable_styling(full_width=FALSE)

	\(x_{A}\)	\(x_{B}\)	\(\mu_{p}\)	\(\sigma_{p}\)
portfolio 1	-0.4	1.4	0.007	0.2049903
portfolio 2	-0.3	1.3	0.019	0.1792663
portfolio 3	-0.2	1.2	0.031	0.1550554
portfolio 4	-0.1	1.1	0.043	0.1331855
portfolio 5	0.0	1.0	0.055	0.1150000
portfolio 6	0.1	0.9	0.067	0.1024794
portfolio 7	0.2	0.8	0.079	0.0978237
portfolio 8	0.3	0.7	0.091	0.1021143
portfolio 9	0.4	0.6	0.103	0.1143487
portfolio 10	0.5	0.5	0.115	0.1323416
portfolio 11	0.6	0.4	0.127	0.1540890
portfolio 12	0.7	0.3	0.139	0.1782216
portfolio 13	0.8	0.2	0.151	0.2038943
portfolio 14	0.9	0.1	0.163	0.2305932
portfolio 15	1.0	0.0	0.175	0.2580000
portfolio 16	1.1	-0.1	0.187	0.2859111
portfolio 17	1.2	-0.2	0.199	0.3141923
portfolio 18	1.3	-0.3	0.211	0.3427518
portfolio 19	1.4	-0.4	0.223	0.3715255

Portfolios 1-4 and 16-19 are the long-short portfolios and portfolios 5-15 are the long-only portfolios.

The risk-return properties of this set of feasible portfolios can be visualized using:

# mu.p ~ sig.p scatter
cex.val = 1.5
col_vec = c(rep("blue", 4), rep("black", 2),
            "green", rep("black", 8), rep("blue", 4))
plot(sig.p, mu.p, type="b", pch=16, cex=cex.val,
     ylim=c(0, max(mu.p)), xlim=c(0, max(sig.p)), 
     xlab=expression(sigma[p]), ylab=expression(mu[p]),
     cex.lab=1.5, col=col_vec)
text(x=sig.A, y=mu.A, labels="Asset A", pos=4)
text(x=sig.B, y=mu.B, labels="Asset B", pos=4)

The black dots plus the green dot represent the long-only portfolios;
- The black dot labeled “Asset A” is the portfolio which invests \(100\%\) in asset A, i.e., (\(x_A=1, x_B=0\));
- The black dot labeled “Asset B” is the portfolio which invests \(100\%\) in asset B, i.e., (\(x_A=0, x_B=1\));
The blue dots represent long-short portfolios;
The green dot represents the global minimum variance portfolio (GMV). This portfolio has the property that it has the smallest variance/volatility among all feasible portfolios.

Global minimum variance portfolio

Computing the global minimum variance portfolio is a constrained optimization problem:

\[ \begin{align*} \underset{x_{A},x_{B}}{\min}\sigma_{p}^{2} & =x_{A}^{2}\sigma_{A}^{2}+x_{B}^{2}\sigma_{B}^{2}+2x_{A}x_{B}\sigma_{AB}\\ ~s.t.~x_{A}+x_{B} & =1. \end{align*} \]

This constrained optimization problem can be solved using two methods analytically.

The first method, called the method of substitution, uses the constraint to substitute out one of the variables to transform the constrained optimization problem in two variables into an unconstrained optimization problem in one variable. \[ \min_{x_{A}}\sigma_{p}^{2}=x_{A}^{2}\sigma_{A}^{2}+(1-x_{A})^{2}\sigma_{B}^{2}+2x_{A}(1-x_{A})\sigma_{AB}. \]

Apply the first order condition to solve the unconstrained optimization problem.

\[ 0=\frac{d\sigma_{p}^{2}}{dx_{A}}=2x_{A}^{\min}\sigma_{A}^{2}-2(1-x_{A}^{\min})\sigma_{B}^{2}+2\sigma_{AB}(1-2x_{A}^{\min}). \]

We have the asset allocation for GMV portfolio:

\[ \begin{align} x_{A}^{\min} &= \frac{\sigma_{B}^{2}-\sigma_{AB}}{\sigma_{A}^{2}+\sigma_{B}^{2}-2\sigma_{AB}}, \label{eq:gmv} \tag{1} \\ x_{B}^{\min} &= 1-x_{A}^{\min}. \end{align} \]
The second method, called the method of Lagrange multipliers, introduces an auxiliary variable called the Lagrange multiplier and transforms the constrained optimization problem in two variables into an unconstrained optimization problem in three variables.

The Lagrangian function is formed by adding to \(\sigma_p^2\) the homogenous constraint multiplied by an auxiliary variable \(\lambda\) (the Lagrange multiplier) giving:

\[ L(x_{A},x_{B},\lambda)=x_{A}^{2}\sigma_{A}^{2}+x_{B}^{2}\sigma_{B}^{2}+2x_{A}x_{B}\sigma_{AB}+\lambda(x_{A}+x_{B}-1). \]

The shape of the investment possibilities set is very sensitive to the correlation between assets \(A\) and \(B\) given the other parameters. The following figure shows the portfolio frontiers for \(\rho_{AB}=-0.9, -0.5, 0, 0.5, 0.9\).

The closer \(\rho_{AB}\) is to \(-1\) the more curved is the frontier toward the y-axis and the higher is the possible diversification benefit.
There is noticeable curvature even for positive values of \(\rho_{AB}\).

If \(\rho_{AB}=1\), then the portfolio frontier approaches a straight line connecting assets A and B.
If \(\rho_{AB}=-1\), there exists a portfolio of A and B that has positive expected return and zero variance. Meaning arbitrage oppotunity.

Efficient portfolios are the feasible portfolios that have the highest expected return for a given level of risk as measured by portfolio standard deviation.
Inefficient portfolios are the portfolios such that there is another feasible portfolio that has the same risk but with a lower expected return.

Tangent portfolio

Tangent/optimal portfolio represents the portfolio which maximizes Sharpe ratio.

We can determine the proportions of each asset in the tangency portfolio by finding the values of \(x_A\) and \(x_B\) that maximize the Sharpe ratio of a portfolio.

Formally, we solve the constrained maximization problem:

\[ \begin{align*} \underset{x_{A},x_{B}}{\max}~\mathrm{SR}_{p} &= \frac{\mu_{p}-r_{f}}{\sigma_{p}}~s.t.\\ \mu_{p} & =x_{A}\mu_{A}+x_{B}\mu_{B},\\ \sigma_{p}^{2} & =x_{A}^{2}\sigma_{A}^{2}+x_{B}^{2}\sigma_{B}^{2}+2x_{A}x_{B}\sigma_{AB},\\ 1 & =x_{A}+x_{B}. \end{align*} \] The solution to this optimization problem is:

\[ \begin{align} x_{A}^{tan} & =\frac{(\mu_{A}-r_{f})\sigma_{B}^{2}-(\mu_{B}-r_{f})\sigma_{AB}}{(\mu_{A}-r_{f})\sigma_{B}^{2}+(\mu_{B}-r_{f})\sigma_{A}^{2}-(\mu_{A}-r_{f}+\mu_{B}-r_{f})\sigma_{AB}},\label{eq:tangency-port} \tag{2}\\ x_{B}^{tan} & =1-x_{A}^{tan}.\nonumber \end{align} \]