6.2 Portfolio constraints
We now briefly describe the most commonly used constraints in the portfolio design or optimization. Some of the constraints are imposed by the regulators or brokers (like shorting constraints, leverage constraints, and margin requirements), while others are optional depending on the investor’s views (such as being market neutral or controlling the portfolio sparsity level or even enforcing diversity). For simplicity of notation, in the following we omit the time dependency and express a variety of constraints in terms of the (normalized) portfolio \(\w\) defined in (6.5). It is important to recognize whether the constraints are convex, as that means that they can be efficiently handled later in the optimization process (see Appendix A).
6.2.1 Long-only or no-shorting constraint
In financial markets, an investor or trader typically buys stocks taking what is called long positions. Interestingly, in some financial markets, a broker may allow the investor to short sell or short some stocks (i.e., to borrow some shares and sell them, with the commitment of buying them back later) for a corresponding borrowing fee. This means that the corresponding elements of \(\w\) can be negative.
borrows shares of a stock from a broker and sells them in the open market,
If shorting is not allowed, the constraint is \[\w \geq \bm{0},\] which is linear and, hence, convex.
6.2.2 Capital budget constraint
Assuming no shorting and no other kind of leverage, the portfolio must satisfy the budget constraint \[\bm{1}^\T\w + c = 1,\] which is linear and, hence, convex.
Alternatively, it can be rewritten without the cash variable as \[\bm{1}^\T\w \leq 1.\] If, instead, equality is imposed, the implication is that the cash is zero so that the portfolio is fully invested in the risky assets.
In practice, this constraint needs to be used in conjunction with some other constraint such as no-shorting constraint or leverage constraint. Otherwise the positions may be unbounded resulting in unrealistic large positions.
6.2.3 Holding constraints
Practitioners always set limits on maximum positions to avoid overexposure and ensure diversification, i.e., upper bounds \(\bm{u}\) on the portfolio elements. On occasions, they may also have minimum positions is they are sure they want to hold certain assets, i.e., lower bounds \(\bm{l}\).
Thus, holding constraints are imposed via lower and upper bounds: \[\bm{l}\leq\w\leq\bm{u},\] which are linear and, hence, convex.
6.2.4 Cardinality constraint
While the universe of assets may be large (say 500 stocks), an investor typically wants to limit the number of active positions (i.e., positions with nonzero allocations) to simplify the logistics of the operation and reduce the rebalancing.
Limiting the number of active positions to \(K\) is mathematically equivalent to placing an upper bound on the cardinality of the portfolio vector: \[\|\w\|_0\leq K,\] where \(\|\cdot\|_0\) is the cardinality operator or \(\ell_0\)-pseudo-norm (which counts the number of nonzero elements). This constraint is nonconvex and, therefore, difficult to handle.
6.2.5 Turnover constraint
As previously discussed in Section 6.1.2, transaction costs are approximately proportional to the turnover. Therefore, it makes sense to control the turnover when designing the portfolio.
Suppose the currently held portfolio is \(\w_0\). Then the turnover constraint is of the form \[\|\w - \w_0\|_1\leq u,\] where \(u\) denotes the maximum turnover allowed. This constraint is convex because norms are convex functions (see Appendix A).
6.2.6 Market-neutral constraint
According to factor modeling with the market as the single factor, the returns can be expressed as \(\bm{r}_t = \bm{\alpha} + \bm{\beta}\cdot r_t^\textm{mkt} + \bm{\epsilon}_t\) (see Chapter 3 for details).
Portfolio managers typically want to avoid exposure to the market, which means that they want the portfolio orthogonal to the “beta”: \[\bm{\beta}^\T\w = \bm{0},\] which is linear and, hence, convex.
6.2.7 Dollar-neutral constraint
When shorting is allowed, one may want to balance the long and short positions in what is called dollar-neutral position: \[\bm{1}^\T\w = \bm{0}.\]
In practice, the dollar-neutral constraint should be used in conjunction with some leverage constraint. Some academic papers and capital asset pricing models assume, in effect, that one can sell a security short without limit and use the proceeds to buy securities long. This is a mathematically convenient assumption for hypothetical models of the economy, but it is unrealistic (Jacobs et al., 2005).
6.2.8 Diversification constraint
Some portfolio designs tend to concentrate the allocation in few assets. The quantity \(\|\w\|_2^2=\w^\T\w =\sum_{i=1}^N w_i^2\) can be used as a diversification measure (DeMiguel, Garlappi, Nogales, et al., 2009; Goetzmann and Kumar, 2008). In fact, the quantity \(\|\w\|_2^2\) is the Herfindahl index of the weights of the portfolio. A lower value is indicative of a higher level of diversification; it is lower-bounded by \(1/N\) and upper-bounded by \(1\). Therefore, one can promote diversification with the constraint: \[\|\w\|_2^2\leq D,\] where \(1/N \leq D<1\). This constraint is convex because norms are convex functions (see Appendix A).
6.2.9 Leverage constraint
Regulations in countries typically limit the amount that can be borrowed for long and short positions. For example, in the U.S., the Federal Reserve System (FRS) established the Regulation T (Reg T) that limits the borrowing to be no greater than 50% of the securities purchase price. The 50% requirement is called the initial margin, but certain brokers may have stricter requirements.
The total amount of long and short positions can be measured with the \(\ell_1\)-norm \(\|\w\|_1=\sum_{i=1}^N |w_i|\) and the leverage constraint can be mathematically written as \[\|\w\|_1\leq 2,\] where the factor of 2 indicates that 50% is being borrowed.
6.2.10 Margin requirements
When borrowing is involved, apart from the leverage constraints imposed by regulators such as the Reg T, brokers will impose margin requirements to reserve collateral34 and ensure that later the borrower can pay back the loan.
The capital budget constraint \(\bm{1}^\T\w \leq 1\) cannot be used in isolation when borrowing is involved since it would imply that an investor can use all the cash from short-selling to buy stocks, which is not the case in real life (Jacobs et al., 2005).
The broker will impose margin requirements in both long and short positions. For a long position, the broker may allow to borrow, say, 50% of the value of the position. For short positions, by definition the investor is borrowing stocks; the cash from short-selling is controlled by the broker as cash collateral but one is also required to have some equity as the “initial margin” to establish the positions, usually the overall collateral is about 105%.
It is convenient to separate the positions of the portfolio into longs and shorts: \[\w = \w^+ - \w^-,\] where \(\w^+ \ge \bm{0}\) denotes the longs and \(\w^- \ge \bm{0}\) the shorts. The margin requirement can then be expressed as \[ m^\textm{long}\times\bm{1}^\T\w^+ + m^\textm{short}\times\bm{1}^\T\w^- \leq 1, \] where \(m^\textm{long}\) and \(m^\textm{short}\) control how much margin is required for longs and shorts, respectively. For example, a broker may impose \(m^\textm{long}=0.5\) and \(m^\textm{short}=0.05\).
References
The term collateral refers to an asset that a lender accepts as security for a loan. The collateral acts as a form of protection for the lender. That is, if the borrower defaults on their loan payments, the lender can seize the collateral and sell it to recoup some or all of its losses.↩︎