LITMIR.BIZ

Скачать книгу

costs and fees have not been taken into account. Therefore, it is reasonable to assume that the mean return would have been closer to 9.92 % if costs had been taken into account. However, the risk reduction would not be impacted by these costs.

      2.2 Extensions to the Mean-Variance Approach

      This section examines some extensions to the mean-variance approach that have been developed to address some of the issues raised in Chapter 1. First, it discusses how illiquidity can be incorporated into the mean-variance model. Using a measure of illiquidity, we can adjust the model to penalize those assets that are illiquid. That is, everything else being equal, portfolio managers would prefer to invest in liquid assets; therefore, illiquid assets will be considered only if they provide some additional benefit. Second, we discuss how limits on factor exposures can be incorporated into a mean-variance approach. Finally, we address how estimation risk can be taken into account. Estimation risk refers to the risk that the estimated parameters that are used as inputs in the mean-variance approach could be different from the true values of those parameters. For example, in Chapter 1 of this book, expected returns on asset classes were shown to be quite inaccurate if a long series of data was not available. A method for incorporating the risk of misestimating the mean return of an asset class will also be discussed.

      2.2.1 Adjustment of the Mean-Variance Approach for Illiquidity

      Investors, especially after the 2008–9 global financial crisis, have become more concerned with illiquidity risk and its costs. Before discussing liquidity risk and its incorporation into the mean-variance framework, we need to define liquidity more precisely. There are two types of illiquidity risks: market liquidity risk and funding liquidity risk (Hibbert et al. 2009). Market liquidity risk arises when an event forces an investor to sell an asset that is not actively traded and there are a limited number of active market participants. Under such a circumstance, the price of the asset may have to be reduced significantly in order to bring a bid from market participants. The discount offered by the seller will have to be significantly higher during periods of financial stress. It is important to note that some apparently liquid assets could become illiquid during periods of economic stress. Certain mortgage-backed securities that appeared to be quite liquid became highly illiquid during the 2008–9 financial crisis. Two factors especially contributed to increased illiquidity of certain assets during the period of financial stress: lack of transparency in some products and the presence of asymmetric information between sellers and buyers. For example, during the 2008–9 financial crisis, investors (buyers) were reluctant to purchase complex mortgage-backed securities because (1) buyers could not value the pools of mortgages that acted as collateral for those securities, and (2) buyers assumed that sellers must know that the pools contained many nonperforming mortgage loans, which was why they were willing to sell them at such deep discounts. Many alternative investments – such as private equity, some real assets, and hedge funds with long lockups – could expose an investor to market liquidity risk.

      Funding liquidity risk arises when the investor is unable to obtain financing, cannot roll over currently available debt, or lacks liquidity to meet capital commitments. Funding liquidity risk can arise in the context of private equity as well, but this time it is related to the inability of the investor to meet the general partner's capital calls. The two risks are related in the sense that by having a substantial amount of cash, one would be able to reduce both types of liquidity risk. An unlevered portfolio of publicly traded securities has relatively low market liquidity risk because the securities can be sold on the exchange at relatively low cost. Also, funding liquidity risk is absent because the portfolio is not levered, and therefore funding cannot be withdrawn by lenders.

      In this section, the focus is on market liquidity risk and its incorporation into the mean-variance framework. The following presentation is kept simple to highlight the impact of liquidity risk on optimal allocations. In particular, it presents a model that modifies the mean-variance optimization framework by incorporating a liquidity penalty function, the purpose of which is to allow for an explicit, easily communicated, natural specification of liquidity preferences that works in conjunction with the standard mean-variance approach that was discussed in Chapter 1.13 The liquidity penalty function reflects the cost of illiquidity and the preference for liquidity. By incorporating this penalty function into the traditional mean-variance optimization model, we can construct a framework for asset allocation involving illiquid assets.

      In Chapter 1, the general framework of the mean-variance optimization was presented as:

(2.4)

      Here,

is the expected rate of return of the portfolio, w_i is the weight of asset i in the portfolio, λ is the investor's measure of risk aversion, and σ²P is the variance of the rate of return on the portfolio.

      To account for illiquidity, we begin by assigning each asset class i an illiquidity level, denoted by L_i, which takes values between 0 and 1. Perfectly liquid assets are assigned 0 and highly illiquid assets are assigned 1. The liquidity level of the portfolio is measured by

. The objective function of the mean-variance approach is now adjusted to reflect a penalty for the extent that each feasible portfolio displays illiquidity when the investor has a preference for liquidity:

      (2.5)

Here, Φ is a positive number and it represents the investor's preference for liquidity (i.e., aversion to illiquidity). If all assets are highly liquid, then L_P = 0 and the liquidity preference will have no impact. However, when some assets are illiquid, L_P > 0 and the investor's aversion to illiquidity will reduce the attractiveness of portfolios with illiquidity. The impact of the liquidity penalty will be to reduce the value of the objective function by subtracting the illiquidity penalty from the expected returns on illiquid assets. In other words, solving this problem would be similar to solving the original optimization problem of Equation 2.4, in which the expected return on each asset is reduced by ΦL_i. For the most illiquid assets, L_i = 1, which reduces the expected returns on such assets by Φ. This insight can help the portfolio manager select a reasonable value for Φ.

      For example, a portfolio manager has assigned a liquidity level of 0.5 to the private equity asset class. The expected annual mean return on this asset class is estimated to be 18 %. The asset owner is an endowment and therefore does not have strong preference for liquidity; this has led the portfolio manager to set Φ = 0.10. The adjustment to the mean return of the private equity asset class is:

      The manager of a family office portfolio is considering the same asset class and assigns a liquidity level of 0.5 as well. However, liquidity is more important to this family office, and therefore the manager has set Φ = 0.20. The adjustment to the estimated expected return of the private equity asset class for the family office inside the model should be:

      Obviously, everything else being equal, the family office will make a smaller allocation to the private equity class.

      2.2.2 Adjustment of the Mean-Variance Approach for Factor Exposure

      In the process of asset allocation, whether strategic or tactical, the investor may wish to limit the exposure of the portfolio to certain sources of risk (and potentially some returns) other than or in addition to the risk of the overall market. For example, the investor may wish to cap the exposure of the portfolio to changes in the price of oil. As long as an observable factor representing the source of risk exists, the constraints on factor exposures can be incorporated into the mean-variance approach with little difficulty.

      The first step is to estimate the factor exposures of the

Скачать книгу