15-portfolio-estimation.Rmd

# Portfolio Estimation

```{r, message = FALSE, echo = FALSE}
library(CVXR)
library(ggplot2)
library(RColorBrewer)
library(tidyr)
```

## Goals

- Demonstrate `CVXR` on a financial example

## Markowitz Portfolio Problem

The Markowitz portfolio problem [@Markowitz:1952; @Roy:1952;
@LoboFazelBoyd:2007] is well known in finance. We will solve this
problem under various constraints.

We have $n$ assets or stocks in our portfolio and must determine the
amount of money to invest in each. Let $w_i$ denote the fraction of
our budget invested in asset $i = 1,\ldots,m$, and let $r_i$ be the
returns (\ie, fractional change in price) over the period of
interest. We model returns as a random vector $r \in {\mathbf R}^n$ with
known mean $\mathbf{E}[r] = \mu$ and covariance $\mathbf{Var}(r) = \Sigma$. Thus,
given a portfolio $w \in {\mathbf R}^n$, the overall return is $R = r^Tw$.

Portfolio optimization involves a trade-off between the expected
return $\mathbf{E}[R] = \mu^Tw$ and the associated risk, which we take to be the
return variance $\mathbf{Var}(R) = w^T\Sigma w$. Initially, we consider only
long portfolios, so our problem is
$$
\begin{array}{ll} 
\underset{w}{\mbox{maximize}} & \mu^Tw - \gamma w^T\Sigma w \\
\mbox{subject to} & w \geq 0, \quad \sum_{i=1}^n w = 1,
\end{array}
$$
where the objective is the risk-adjusted return and $\gamma > 0$ is a
risk aversion parameter.

## Example

We construct the risk-return trade-off curve for $n = 10$ assets and
$\mu$ and $\Sigma^{1/2}$ drawn from a standard normal
distribution.

```{r}
## Problem data
set.seed(10)
n <- 10
SAMPLES <- 100
mu <- matrix(abs(rnorm(n)), nrow = n)
Sigma <- matrix(rnorm(n^2), nrow = n, ncol = n)
Sigma <- t(Sigma) %*% Sigma

## Form problem
w <- Variable(n)
ret <- t(mu) %*% w
risk <- quad_form(w, Sigma)
constraints <- list(w >= 0, sum(w) == 1)

## Risk aversion parameters
gammas <- 10^seq(-2, 3, length.out = SAMPLES)
ret_data <- rep(0, SAMPLES)
risk_data <- rep(0, SAMPLES)
w_data <- matrix(0, nrow = SAMPLES, ncol = n)

## Compute trade-off curve
for(i in seq_along(gammas)) {
    gamma <- gammas[i]
    objective <- ret - gamma * risk
    prob <- Problem(Maximize(objective), constraints)
    result <- solve(prob)
    
    ## Evaluate risk/return for current solution
    risk_data[i] <- result$getValue(sqrt(risk))
    ret_data[i] <- result$getValue(ret)
    w_data[i,] <- result$getValue(w)
}
```

Note how we obtain the risk and return by _directly evaluating_
the value of the separate expressions:

```{r, eval = FALSE}
result$getValue(risk)
result$getValue(ret)
```

The trade-off curve is shown below. The $x$-axis represents the standard
deviation of the return. Red points indicate the result from investing
the entire budget in a single asset. As $\gamma$ increases, our
portfolio becomes more diverse, reducing risk but also yielding a
lower return.

```{r}
cbPalette <- brewer.pal(n = 10, name = "Paired")
p1 <- ggplot() +
    geom_line(mapping = aes(x = risk_data, y = ret_data), color = "blue") +
    geom_point(mapping = aes(x = sqrt(diag(Sigma)), y = mu), color = "red")

markers_on <- c(10, 20, 30, 40)
nstr <- sprintf("gamma == %.2f", gammas[markers_on])
df <- data.frame(markers =  markers_on, x = risk_data[markers_on],
                 y = ret_data[markers_on], labels = nstr)

p1 + geom_point(data = df, mapping = aes(x = x, y = y), color = "black") +
    annotate("text", x = df$x + 0.2, y = df$y - 0.05, label = df$labels, parse = TRUE) +
    labs(x = "Risk (Standard Deviation)", y = "Return")
```

We can also plot the fraction of budget invested in each asset.

```{r}
w_df <- data.frame(paste0("grp", seq_len(ncol(w_data))),
                   t(w_data[markers_on,]))
names(w_df) <- c("grp", sprintf("gamma == %.2f", gammas[markers_on]))
tidyW <- gather(w_df, key = "gamma", value = "fraction", names(w_df)[-1], factor_key = TRUE)
ggplot(data = tidyW, mapping = aes(x = gamma, y = fraction)) +
    geom_bar(mapping = aes(fill = grp), stat = "identity") +
    scale_x_discrete(labels = parse(text = levels(tidyW$gamma))) +
    scale_fill_manual(values = cbPalette) +
    guides(fill = FALSE) +
    labs(x = "Risk Aversion", y = "Fraction of Budget")
```

## Discussion

Many variations on the classical portfolio problem exist. For
instance, we could allow long and short positions, but impose a
leverage limit $\|w\|_1 \leq L^{max}$ by changing
```{r, eval = FALSE}
constr <- list(p_norm(w,1) <= Lmax, sum(w) == 1)
```

An alternative is to set a lower bound on the return and minimize just
the risk. To account for transaction costs, we could add a term to the
objective that penalizes deviations of $w$ from the previous
portfolio. These extensions and more are described in
@BoydBusseti:2017. The key takeaway is that all of these convex
problems can be easily solved in `CVXR` with just a few alterations
to the code above.

## References