## Abstract

In this article, we propose a generalized expected shortfall risk-budgeting investing framework, which offers a simple and flexible way to deal with various risks beyond volatility, namely valuation, asymmetry, tail, and illiquidity risks. We empirically illustrate the methodology by proposing a risk-based strategic allocation for a multi-asset portfolio made of traditional and alternative assets with different degrees of liquidity.

**TOPICS:** Tail risks, portfolio construction, real assets/alternative investments/private equity

Risk-based investment strategies have attracted considerable attention from the investor community and academia since the advent of the Great Financial Crisis (GFC). In most applications, risk is measured through volatility, i.e., the standard deviation of a portfolio’s returns. To many investors, associating risk with volatility is highly disputable.

Asness (2014) illustrated this debate through the opposition between two schools, with the “quant/geeks” using volatility as their preferred risk measure while some other investors argue that valuation is the only true risk.^{1} ,Asness (2014) suggested that the two approaches can be reconciled by incorporating valuation risk in quant models through a positive (negative) expected return hypothesis for the cheap (expensive) assets. More generally, risk is a multi-faceted concept. Corporate bonds are characterized by an intrinsic asymmetry of risk associated with default risk, while equities are characterized by tail risks with financial crashes that appeared more frequently than predicted by a normal distribution. Some prominent alternative assets such as private equity, real estate, and distressed securities are affected by structural liquidity risks.

The purpose of our article is to propose a simple and flexible analytical solution to deal with this diversity of risks in an asset allocation framework. We extend previous literature that in general has focused solely on the asset allocation implications of one specific dimension of risk, along two dominant research paths. On one side, there is abundant literature seeking to analyze the implications of deviations from Gaussian-normality on asset allocation (Favre and Galeano 2002, Jondeau and Rockinger 2006, and Hitaj et al. 2012). In parallel, academics and practitioners have analyzed the implicatons of illiquidity on asset moments and portfolio selection (Getmansky et al. 2004, Cao and Teiletche 2007, and Pedersen et al. 2014).

Our research integrates these distinct approaches in a unified framework. We start from developments in the literature that analyze how the various components of a portfolio contribute to extreme risk measures such as Value-at-Risk (Gourieroux et al. 2000, Hallerbach 2002, and Martellini et al. 2015) or Conditional Value-at-Risk/Expected Shortfall (Scaillet 2002, Boudt et al. 2008, and Darolles et al. 2015). We consider a simplified version of the Cornish-Fisher approximation (Zangari 1996) of the portfolio return quantile function that allows us to analytically break down the portfolio’s expected shortfall in terms of individual risk contributions. We also extend the existing literature on liquidity risk by deriving new analytical expressions for the scaling coefficients of the co-skewness and co-kurtosis matrixes, so that the impact of the liquidity of each asset on the portfolio risk can be assessed independently. One of the key benefits of our framework is the fact that we are able to derive closed-form expressions for the generalized risk contributions, which simplifies practical implementation.

We first review the definitions and general properties of the portfolio expected shortfall in a risk-budgeting context. We then detail our expected shortfall risk decomposition framework. Finally, we illustrate the methodology with a risk-based asset allocation simulation spanning traditional and alternative assets with different degrees of liquidity. A technical appendix contains mathematical details.

## EXPECTED SHORTFALL AS A RISK MEASURE

### Expected Shortfall: Definition and Properties

Our modelling framework is built around Expected Shortfall (ES), also known as Conditional Value at Risk. For a given confidence level (1 − α), ES corresponds to the expected loss of the portfolio during the worst α-proportion of times for a predefined investment horizon\protect.^{2} ES is a more sensible risk measure than volatility as it concentrates on losses and has superior mathematical properties than other tail loss estimators such as Value-at-Risk (VaR).

Formally, we consider a portfolio ** w** = (

*w*

_{1}, …,

*w*)

_{n}^{T}made of

*n*risky assets. Let

**= (**

*R**R*

_{1}, …,

*R*)

_{n}^{T}denoting the vector of individual asset returns with mean vector

**μ**= (μ

_{1}, …, μ

_{n})

^{T}and covariance matrix

**Ω**=

*E*[(

**−**

*R***μ**)(

**−**

*R***μ**)

^{T}]. The associated portfolio returns are then given by

*R*(

**) =**

*w*

*w*^{T}

**with mean μ(**

*R***) =**

*w*

*w*^{T}μ and volatility σ(

**) =(**

*w*

*w*^{T}

**Ω**

**)**

*w*^{1/2}. The Expected Shortfall (ES) of the portfolio returns then equals:

with:

where *Z* is a zero-mean unit-variance random variable and *G*^{−}^{1}(.) is the quantile function associated with the portfolio’s return distribution.

The measurement of (**1**) requires one to estimate the quantities *G* ^{−}^{1}(α) and *E*[*Z*|*Z* ≤ *G*^{−}^{1}(α)]. The simplest choice is to assume that *Z* follows a standard normal distribution, in which case (**1**) becomes:

with:

where φ(.), Φ(.) and *z*_{a} = Φ^{−}^{1}(α) are the standard Gaussian density, cumulative distribution, and quantile functions, respectively. is a positive constant depending only on α; it is higher for lower α, i.e., larger losses. While it has the merit of being simple, the approach might be insufficient as Gaussian-normality does not always describe financial return distributions well. Later in the text, we relax the Gaussian hypothesis.

### Contribution to Expected Shortfall and Risk-based Portfolios

The risk budgeting principle is based on the breakdown of a risk measure in contributions coming from the different components of a portfolio (Tasche 2004). A portfolio’s ES can be expressed as , where CES_{α(i)}(** w**) =

*w*∂ES

_{i}_{α}(

**)/∂**

*w**w*is the contribution of asset

_{i}*i*to the ES of the portfolio. The percentage risk contributions are given by %CES

_{α}

_{(}

_{i}

_{)}(

**) = CES**

*w*_{α}

_{(}

_{i}

_{)}(

**)/ES**

*w*_{α}(

**), with .**

*w*Risk-based investing is an application of risk budgeting principles, which does consist of determining portfolio weights such that the individual risk contributions match a set of prespecified risk budgets. We consider here the case of fully invested long-only portfolios.^{3} For the ES risk measure, a risk-based portfolio can be obtained by solving the following risk minimization program (Roncalli 2013):

where *b _{i}*> 0 correspond to strictly positive individual risk budgets, with ; and

*c*is an arbitrary constant. From the first-order conditions of the Lagrangian function of (

**3**), we verify that CES

_{α}

_{(}

_{i}

_{)}(

*w*^{*}) = λ

_{c}

*b*∀

_{i}*i*, where λ

_{c}is the Lagrange multiplier associated with the logarithmic constraint. Using the individual risk contribution and risk budget definitions, it is easy to show that the optimal solution verifies that . The risk-based portfolio solution is then deduced simply by rescaling the solution (e.g., by constraining the sum of weights to be equal to 100%). The ES minimization formulation (

**3**) demonstrates that a risk-based portfolio solution exists and is unique since it corresponds to the solution of the minimization of a convex function subject to convex constraints (Bertsimas et al. 2004, and Bruder and Roncalli 2012).

While the risk minimization program (**3**) has more theoretical appeal, in the empirical section we follow the standard approach in the industry that consists of solving the following optimization program:

A simple example is the risk parity/equal risk contribution strategy where one gives the same risk budget to each asset in the portfolio, i.e., *b _{i}*=

*n*

^{–1}(Maillard et al. 2010); a key difference is that risk is here measured by ES and not by volatility only.

One can remark that “traditional” risk-based solutions are obtained as a special case by assuming that all assets have zero expected returns and the distribution of return innovations is Gaussian.^{4} Departing from these hypotheses is what we propose in the next section.

## EXPANDING THE RISK MEASUREMENT FRAMEWORD

We now introduce our methodology to incorporate different types of risk within the ES framework.

### Volatility Risk

The usual risk-based investment strategies determine their capital allocations such that all components of the portfolio contribute equally to its volatility. In general terms, volatility mainly measures the amplitude of returns. A common criticism is that this measure is symmetric and hence does not differentiate between gains and losses, while the left-tail of the distribution (losses) is more critical than the right-tail (gains) from a risk perspective. Hence, it is often recommended to look at downside risk measures concentrated on losses, and in particular on extreme losses. One of the main purposes of the generalized ES measure that we develop below is to correct for this pitfall while keeping the role of volatility in sizing returns.

### Valuation Risk

Valuation is one of the major risks investors face. A natural way to include valuation risk within the ES framework is to consider that the expected returns **μ** are reflecting valuation misalignment (hence as a directional risk). This way of proceeding is notably suggested in Asness (2014) who mentions that valuation risk can be incorporated in a quant model through a positive (negative) expected return hypothesis for the cheap (expensive) assets. A similar approach is followed by Roncalli (2014) who suggests using a Gaussian ES to incorporate tactical views in a risk-budgeting framework. The author draws a parallel with the ltraditional Markowitz allocation problem, which does consist in the minimization of the quantity , where γ is the risk-aversion parameter. Comparing this expression with (**2**), we deduce the implied risk-aversion parameter in ES as being equal to . It is interesting to notice that the incorporation of expected returns into the ES is made at very specific conditions, and notably at very high level of risk aversion. For instance, if α = 5% and σ(** w**) = 10%, we have γ

^{imp}= 41.2542, which is extremely high, as typical risk-aversion estimates are below 10 (Mehra and Prescott 1985). Hence, even if we allow the incorporation of expected returns in a risk-based framework, the program remains fairly different from Markowitz’s in practice.

^{5}

To model expected returns, we recommend using carry. Carry corresponds to the expected return of an asset if its price does not move in the future (Koijen et al. 2018). As a first-order approximation, carry will typically be measured through bond or earnings yields. Carry is intrinsically related to valuation, since it is obtained as the ratio between the expected income from holding an asset and its current price. For the same income stream, carry is larger (smaller) when asset prices are lower (higher). All in all, our approach leads to the following mechanism: lower (higher) carry implies lower (higher) expected returns, which translates into higher (lower) ES through (**1**).

### Asymmetry and Tail Risks

It is widely acknowledged that financial return distributions frequently deviate from a Gaussian distribution, as they present both asymmetrical and fat-tailness characteristics. Asymmetry of risk describes a situation where potential gains and losses are uneven. This risk is typical of asset classes such as credit, where issuers are subject to default risk, implying a fundamentally negatively-skewed distribution of returns. Tail risk expresses the fact that extreme events occur more frequently than expected from a normal distribution, leading to a higher probability of big losses or big gains.

Following Zangari (1996), Cornish-Fisher (CF) approximations have gained in popularity thanks to their convenience and flexibility. The advantage of this approach in the present context is to offer the opportunity of obtaining closed-form expressions for the risk contributions. Let Σ = *E*[(** R** −

**μ**)(

**−**

*R***μ**)

**⊗ (**

*T***−**

*R***μ**)

^{T}] be the coskewness matrix and

**Γ**=

*E*[(

**−**

*R***μ**) ⊗ (

**−**

*R***μ**)

^{T}⊗ (

**−**

*R***μ**)

**] be the cokurtosis matrix, where ⊗ stands for the Kronecker product operator. Denote**

*T**m*

_{3}(

**) =**

*w*

*w*^{T}Σ

**(**

**⊗**

*w*

*w***)**and

*m*

_{4}(

**) =**

*w*

*w*^{T}

**Γ**(

**⊗**

*w***⊗**

*w***) as the third- and fourth-order portfolio centered moment, respectively. The skewness and kurtosis of the portfolio are then given by**

*w**s*(

**) =**

*w**m*

_{3}(

**)/σ**

*w*^{3}(

**) and**

*w**k*(

**) =**

*w**m*

_{4}(

**)/σ**

*w*^{4}(

**). Under a second-order CF approximation, with**

*w**s*

^{2}(

**) = 0, ES is equal to (see Appendix A):**

*w*where is the adjustment to ES for the non-Gaussian features of the distribution of returns. Like for the Gaussian-normal case in (**2**), ES decreases with expected returns μ(** w**) and increases with volatility σ(

**). But contrary to the Gaussian case, ES is now affected by higher moments as is linearly decreasing in skewness**

*w**s*(

**) and linearly increasing in kurtosis**

*w**k*(

**).**

*w*^{6}These are desirable features because they imply that an asset with negative skewness or high kurtosis will tend to receive a lower weight in a risk-based portfolio based on ES, all other things being equal.

To determine the exact risk-based portfolio composition, the next step is to get individual ES contributions as inputs to program (**4**), which, under a CF hypothesis, can be shown to be equal to (see Appendix A):

with

where CMEAN_{i}(** w**), CVOL

_{i}(

**), CSKEW**

*w*_{i}(

**), and CKURT**

*w*_{i}(

**) are asset**

*w**i*mean, volatility, skewness, and kurtosis contributions to portfolio ES; (

**Ω**

**)**

*w*_{i}, (Σ(

**⊗**

*w***))**

*w*_{i}, and (

**Γ**(

**⊗**

*w***⊗**

*w***))**

*w*_{i}correspond respectively to the i-th entry of the vectors

**Ω**

**, Σ(**

*w***⊗**

*w***) and**

*w***Γ**(

**⊗**

*w***⊗**

*w***).**

*w**c*

_{1},

*c*

_{2},

*c*

_{3}, and

*c*

_{4}are constants with

*c*

_{1}= −1, , , and . Hence, individual contributions of each asset to ES can be broken down into their risk contributions per moment.

Note that, in the same way as the volatility of the portfolio depends on covariance, the skewness and kurtosis individual risk contributions depend on codependencies between assets. For instance, the individual contributions to skewness CSKEW_{i}(** w**) depend on the covariance and co-skewness matrixes while the individual contributions to kurtosis CKURT

_{i}(

**) depend on the covariance and co-kurtosis matrixes.**

*w*^{7}As we will illustrate in the empirical section, this decomposition facilitates the identification of the various sources of risk at the individual asset level, but also at the aggregated portfolio level, as summing over all ES contributions (

**6**) provides the total ES risk itself, i.e., .

### Illiquidity Risk

Illiquidity implies that transactions cannot be observed frequently, leading to some form of stale pricing and smoothing return patterns. A common way to correct for the implied downward bias in the estimation of (co)variance is to assume that current asset returns form a moving average of true returns and to unsmooth them (Geltner 1993 and Getmansky et al. 2004). In Appendix B, we show that illiquidity also affects higher-order (co)moments. To correct for this, we follow this literature and assume that for each asset, the actual returns *R _{i}*

_{,}

_{t}cannot be observed directly due to illiquidity, and that the reported returns of the illiquid asset, denoted by , are governed by a MA(

*K*) process

where ∈_{i}_{,}_{k} ⊗ [0,1] for *k* = (0, …, *K*) and . Once these MA models are estimated for each asset, we can correct the covariance, co-skewness and co-kurtosis empirical estimates as shown in Appendix B (see equation **20**). On that basis, we finally derive a new expression for individual contributions to ES, isolating illiquidity risk as follows:

where and are defined as in equation 6 and obtained through the application of illiquidity-corrected and raw moments respectively. ILLIQUID_{i(w) represents the contribution of asset i to portfolio ES coming from illiquidity, with ILLIQUIDi(w}) = 0 for liquid assets.

## EMPIRICAL ILLUSTRATION

In this section, we empirically illustrate the ES decomposition methodology by determining a risk-based strategic allocation for a multi-asset portfolio across a diversified range of traditional and alternative assets.

### Data

Our dataset covers 10 traditional and alternative asset classes, over the period spanning from the first quarter of 1990 to the second quarter of 2017. Due to limitations related to Private Equity and Real Estate, we use quarterly data. Data are downloaded from Bloomberg, with the exception of Private Equity, which is taken from the Cambridge Associates website. In Exhibit 1, we give the list of asset classes and the associated indexes. We also present how we measure carry for the various asset classes. Our measures are inspired by Ilmanen (2011) and Koijen et al. (2018), but we also extend the approach of these authors for some additional alternative assets.

For public equities (S&P 500 for large caps, Russell 2000 for small caps), we use earnings yields rather than dividend yields, since the latter forecasting abilities have notoriously decreased as companies have increasingly used share buy-backs or accumulated cash rather than distributing it. As earnings yields constitute a real variable, we add inflation expectations as measured by the University of Michigan Survey of Consumers one-year ahead expectations. For fixed income securities, we use Bloomberg Barclays U.S. Treasury, U.S. Credit Investment Grade and U.S. High Yield indexes. We compute carry as the sum of current yield and the roll-down. The yield measure is the yield to worst. Roll-down is inferred from the comparison of yield/maturity couples for intermediate and all maturities indices.

Commodity carry is based on the one-year roll-yield of 13 primary commodities future curves: Sugar, Coffee, Cotton, Cocoa, Live Cattle, Lean Hogs, Gold, Silver, Copper, WTI Crude Oil, Brent, Heating Oil, and Gasoil. The basket is equally-weighted and rebalanced every end of quarter. Carry for commodities is finally the sum of the average roll yield and the cash returns earned by investing every three months at USD Libor three months.

Hedge fund returns are measured through the Hedge Fund Research (HFR) fund-of-hedge funds index. We select funds of hedge funds rather than single hedge funds as the associated performance indexes are much less subject to the typical biases affecting single fund indexes, such as survivorship and backfill biases. While no direct measure of carry for hedge funds exists, we rely on academic research that has shown that a significant portion of their returns can be linked to exposures to other asset classes either contemporaneously or with a lag (Hasanhodzic and Lo 2007). On that basis, we estimate the implied carry for hedge funds through a regression-based aggregation of other individual asset classes’ carry.

More precisely, we follow a two-step process. We start by a regression of hedge funds excess returns on contemporaneous and lagged excess returns of equities large and small caps, fixed income Treasuries, and high yield bonds and commodities. We propose up to four quarterly lags and retain significant lags through a stepwise regression at the 10% significance level over the full sample. After the model has been selected, we compute carry for hedge funds as the beta-weighted average of individual carry for the retained factors. We use the same methodology for private equity.

Contrary to other alternative assets, sensible measures for real estate carry are readily available. In particular, real estate returns can be divided into two components: an income return coming from gross rental income minus operating expenses, and a capital return coming from the change in market value of the property. We retain the former as our measure of carry, as it is the return captured by a real estate investor if the price does not move. In practice, income returns computed by NCREIF have been historically fairly stable around 2% per quarter^{8} and this is the estimate we retain in our empirical section.

Finally, we replicate variance risk premiums (VRP) returns through the comparison of VIX index (implied volatility) and the realized volatility in the underlying equity market. More specifically, the VRP returns are based on the difference, during a given month, between the squared level of VIX at the end of the previous month and the realized variance of the S&P 500 daily observations. This measure of performance approximately replicates the performance of a short position on one-month maturity variance swap and its use has been advocated since Bollerslev et al. (2009). The VRP carry is based on the lagged implied-realized spread.

### Asset Returns Analysis

Exhibit 2 displays descriptive statistics for the asset returns. All assets have delivered substantial premiums over the risk-free rate (one-month USD LIBOR) which was 3.3% p.a. over the same time period. Sharpe ratios gravitate around 0.5 for most assets with the noticeable difference of private equity and real estate. However, the latter asset classes are characterized by substantial illiquidity, which puts downward bias on the volatility measure and leads to an overestimation of the Sharpe ratio, as we detail below. Deviations from Gaussian-normality are substantial and frequently significant as shown by Jarque-Bera statistics. Most asset classes are indeed characterized by a negative skewness and an excess kurtosis of the return distribution, real estate, and variance risk premiums being the two most striking examples. Incorporating this dimension of the asset returns is thus critical.

Carry estimates vary from a small amount for commodities to close to 10% p.a. for high yield. For assets such as fixed income or real estate, the carry constitutes the main source of return in the long run. At the opposite, for small caps and commodities, the portion due to carry is low. Large cap public equities, private equity, variance risk premiums, and hedge funds fall between these extremes. As far as correlation is concerned, we can isolate two groups of assets. On the one hand, public equities, private equity, high yield bonds, hedge funds, and variance risk premiums all have average correlations with the rest of the universe of assets, which are roughly 0.35 on average. On the other hand, investment grade bonds, commodities, real estate, and even more spectacularly government bonds appear as the most diversifying assets, as their correlation with the rest of the universe is close to zero or even negative for government bonds.

Exhibit 3 summarizes the estimations performed to correct asset co-moments for illiquidity. We follow the steps described in Appendix B. We first estimate moving-average models and then infer corrections for the different co-moment matrixes. In Panel A, we first report Bayesian information criterion (BIC) for the different moving-average models. We highlight in bold the lag that we retain for each asset as it leads to minimize the BIC statistic. With the exception of high yield, traditional assets and commodities are all characterized by a zero-lag, meaning they are deemed liquid through this criterion. Alternative assets are characterized by estimated poor liquidity, particularly for private equity and even more for real estate.

Panel B displays the associated smoothing coefficients, while in Panel C we report the implied scaling coefficients for the moments.^{9} Cases where θ_{i}_{,0} = 1 indicate situations where there is no estimated illiquidity-bias and no need for scaling up the moments. On the contrary, cases where θ_{i}_{,0} < 1 indicate examples where some illiquidity bias is estimated. Correcting for this leads to an inflation of moments. For instance, the volatility of real estate is estimated to be understated by a factor of more than two due to the illiquid nature of the asset. This notably implies that the impressive Sharpe ratios of Exhibit 2 for the less liquid assets should be questioned.

To shed additional light on those issues, we report in Exhibit 4 different risk-adjusted performance measures. On top of the traditional Sharpe ratios, we report illiquidity-corrected Sharpe ratios where volatility is incorporating the illiquidity correction displayed in Exhibit 3. We also report so-called Modified Sharpe ratios, which are obtained as the ratio between asset excess returns and the 95% ES, where ES is estimated alternatively through three different models of increased completeness: a Gaussian approximation (see equation **2**) a skewness-kurtosis expanded version (see equation **5**) or the most complete model incorporating expansions for non-normal behavior and illiquidity-bias. The risk measure can have a significant implication on the asset ranks, notably when one considers Modified (ES-based) Sharpe ratios involving the different risk dimensions. Most spectacularly, while real estate is the second best asset as far as Sharpe ratio is concerned, it becomes the worst (jointly with variance risk premiums) when one incorporates corrections for both non-normality and illiquidity. These differences will also appear in the risk-based allocations as we will see now.

### Risk-Based Asset Allocation Results

Suppose an investor wants to build a fully-invested portfolio where each asset will contribute to the same extent to the total risk of the portfolio, i.e., she wants to follow a risk parity/equal risk contribution approach (Maillard et al. 2010).

Our purpose is to illustrate how the portfolio composition will be different when the investor uses different risk models. More specifically, we consider five different models of increasing sophistication:

• Naive Risk Parity (NRP): risk measure is volatility, ignoring correlation;

• Risk Parity (RP): risk measure is volatility, incorporating correlation;

• Gaussian Expected Shortfall (GES): risk measure is ES, assuming a Gaussian distribution;

• Modified Expected Shortfall (MES): risk measure is ES, where ES is modified to incorporate skewness and kurtosis;

• Liquidity Adjusted Modified Expected Shortfall (LAMES): risk measure is ES, where ES is modified to incorporate skewness and kurtosis, and co-moment matrices are corrected for illiquidity.

Associated portfolio weights are given in Exhibit 5. We observe a substantial variation across models. NRP naturally overweights assets with low volatility such as government bonds, investment grade bonds, hedge funds, and real estate, and underweights high-volatility assets such as equities, commodities, and variance risk premiums. RP introduces correlation and modifies the NRP allocation by increasing the allocation to low-correlation assets such as government bonds, commodities, or real estate, and by reducing the allocation to the other ones.

The next column marks the entry into ES risk measures with firstly the Gaussian approximation. Relative to RP, the key change in GES is to introduce carry. The allocation is increased to assets presenting a high carry to volatility ratio, and most notably real estate. Portfolio weights change significantly when one corrects ES for non-normal behavior, with notably a significant drop in real estate weight, an asset characterized by high kurtosis and fairly negative skewness, to the benefit of equities and bonds. An interesting result is the fact that the allocation to high yield bonds barely changes as its high kurtosis is compensated by a modest positive skewness. Finally, the last column presents the portfolio weights associated with the LAMES risk model, i.e., based on ES incorporating both non-normal and illiquidity adjustments. Assets characterized by poor liquidity, such as real estate, private equity, or variance risk premiums see their weights reduced further. The illiquidity adjustment also leads to an overall increase in weight for the bond bucket.

In Exhibit 6, we provide more details on the Equal Risk Contribution portfolios, by contrasting the two extreme portfolios, NRP and LAMES. For each of them, we give a breakdown of the portfolio ES along two axes, splitting contributions by asset class and by moments, where ES is estimated through the complete—LAMES—risk model. The last column gives the individual contributions to total ES. LAMES portfolio is represented in Panel A. As an example, government bonds contribute to 1.15% out of the ES of the portfolio (3.27%) through volatility but only at 0.33% in total, as they reduce the risk through average return, skewness, kurtosis, or illiquidity. At the opposite, real estate has an almost zero-contribution through volatility but contributes more positively to ES through higher moments and illiquidity. More generally, for liquid asset classes, the largest contributions come from volatility.

In Panel B, we represent the risk contributions of the NRP portfolio, which displays a much higher estimated ES (16.9%). First of all, we observe that the bulk of the additional ES is coming from higher moments and even more from illiquidity underestimation. In terms of assets, the risks of real estate and variance risk premiums are largely underestimated, leading to a very unbalanced portfolio in terms of risks when appropriately corrected for non-normality and illiquidity risks.

How do these different portfolios fare from a risk-return perspective? Exhibit 7 answers this question from different angles. In Panel A, we report the first four moments of the distribution of returns. Average returns are in general lower for more complete risk models, reflecting the higher allocation to low-return assets such as fixed income and the lower allocation to high-return assets such as private equity and variance risk premiums. Volatility does not show a consistent pattern, contrary to skewness and kurtosis that are greatly reduced for more complete risk models.

In Panel B we investigate various tail-risk measures. While normal-Gaussian ES estimate barely changes, risk estimated through LAMES sharply reduces when one corrects for non-normality and illiquidity. Those results are also valid for most extreme events, as shown by maximum drawdowns and the returns during the Great Financial Crisis. In Panel C, we finally compare different risk-adjusted performance ratios. While the simple Sharpe ratio does not advocate the usefulness of more complete risk models, Modified Sharpe (i.e., based on Expected Shortfall) and Calmar (the ratio between average returns and maximum drawdowns) convey a totally different message. All in all, despite the fact that the objective set in the program is to diversify risk and not to minimize it, the use of more sophisticated risk models seems to lead to a significant reduction of risks and particularly of extreme risks.^{10} Notice that the incorporation of illiquidity risk (i.e., the transition from MES to LAMES) does not seem to trigger significant changes in the various risk-return metrics here. This is linked to the fact that the assets that are the less liquid in our sample are also the ones that display the strongest departures from normality,^{11} the most spectacular examples being real estate and variance risk premiums.

As a conclusion of this empirical section, note that our risk-based allocation results are horizon-dependent, as the relative importance of the individual moment contributions to portfolio ES changes with the return period considered.^{12} Indeed, under the assumption that all individual prices follow a strongly stationary martingale process, one can show that the high frequency return (co)-variances increase with time horizon while high-frequency return (co)-skewness and (co)-kurtosis attenuate with horizon (with the square root of time and time respectively). Despite these temporal aggregation differences, the asymmetry and tail risks will generally not disappear as the horizon increases due to the presence of individual leverage and GARCH effects in the return distributions (see Neuberger and Payne 2018). So the proposed moment and liquidity adjustments still matter for long-term investors.

## CONCLUSION

Risk-based investing strategies are becoming increasingly popular among investors. Yet, in most applications, risk is usually measured through volatility, while other risk characteristics are also important, especially when considering investing in alternative and/or illiquid asset classes.

In this article, we extend the existing literature that addresses the implication of higher moments or illiquidity on portfolio selection. We propose a generalized risk-based investing framework based on a liquidity-adjusted modified expected shortfall risk measure. This allows investors to deal with various sources of risk beyond volatility, such as valuation, asymmetry, tail, and illiquidity risks.

We obtain closed-form expressions for the moment contributions of each asset to the portfolio expected shortfall, i.e., mean, volatility, skewness, and kurtosis risk contributions. We also show how to isolate the individual illiquidity risk contributions through an adjustment of the asset co-moments. This decomposition facilitates the identification of the various sources of risks both at the individual and aggregate portfolio levels.

We illustrate the usefulness of our methodology with a strategic asset allocation problem applied to a diversified range of traditional and alternative assets. We show how the non-normality of returns and the illiquidity biases can affect significantly the capital allocation across asset classes. We also illustrate that the breakdown of the individual risk contributions can be very different across assets, with the largest contributions coming from volatility (respectively, non-normality/liquidity) for traditional (respectively, alternative) asset classes.

A natural extension of our research would consist of comparing the out-of-sample performance of our generalized risk-based approach with competing asset allocation methodologies,^{13} or applying the framework to other asset universes such as equity sectors or fixed income buckets. As previously discussed, the analysis of the sensitivity of the resulting asset allocation at different investment horizons is also an interesting venue for future research as different moments are not affected to the same extent by time-aggregation.

## ADDITIONAL READING

**The Impact of Illiquidity and Higher Moments of Hedge Fund Returns on Their Risk-Adjusted Performance and Diversification Potential**

Laurent Cavenaile, Alain Coën, and Georges Hübner

*The Journal of Alternative Investments*

**https://jai.pm-research.com/content/13/4/9**

**ABSTRACT:** * This article studies the joint impact of smoothing and fat tails on the risk–return properties of hedge fund strategies. First, the authors adjust risk and performance measures for illiquidity and the non-Gaussian distribution of hedge funds returns. They use two risk metrics: the Modified Value-at-Risk and a preference-based measure retrieved from the linear exponential utility function. Second, they revisit the hedge fund diversification effect with these adjustments for illiquidity. Their results report similar fund performance rankings and optimal hedge fund strategy allocations for both adjusted metrics. They also show that the benefits of hedge funds in portfolio diversification persist but tend to weaken after adjustments for illiquidity are made.*

**Mean-Modified Value-at-Risk Optimization with Hedge Funds**

Laurent Favre and José-Antonio Galeano

*The Journal of Alternative Investments*

**https://jai.pm-research.com/content/5/2/21**

**ABSTRACT:** * Based on the normal value-at-risk, we develop a new value-at-risk method called modified value-at-risk. This modified value-at-risk has the property to adjust the risk, measured by volatility alone, with the skewness and the kurtosis of the distribution of returns. The modified value-at-risk allows us to measure the risk of a portfolio with non-normally distributed assets like hedge funds or technology stocks and to solve for optimal portfolio by minimizing the modified value-at-risk at a given confidence level.*

**Optimal Hedge Fund Allocation with Improved Estimates for Coskewness and Cokurtosis Parameters**

Asmerilda Hitaj, Lionel Martellini, and Giovanni Zambruno

*The Journal of Alternative Investments*

**https://jai.pm-research.com/content/14/3/6**

**ABSTRACT:** * Since hedge fund returns are not normally distributed, mean–variance optimization techniques are not appropriate and should be replaced by optimization procedures incorporating higher-order moments of portfolio returns. In this context, optimal portfolio decisions involving hedge funds require not only estimates for covariance parameters but also estimates for coskewness and cokurtosis parameters. This is a formidable challenge that severely exacerbates the dimensionality problem already present with mean–variance analysis. This article presents an application of the improved estimators for higher-order co-moment parameters, in the context of hedge fund portfolio optimization. The authors find that the use of these enhanced estimates generates a significant improvement for investors in hedge funds. The authors also find that it is only when improved estimators are used and the sample size is sufficiently large that portfolio selection with higher-order moments consistently dominates mean–variance analysis from an out-of-sample perspective. Their results have important potential implications for hedge fund investors and hedge fund of funds managers who routinely use portfolio optimization procedures incorporating higher moments.*

## ACKNOWLEDGMENTS

For their helpful comments, we thank participants in the QMI Risk-Based Investing workshop at Imperial College. We thank the anonymous referee for his or her thorough review and suggestions, which significantly contributed to improving the quality of the article. The first author gratefully acknowledges the financial support of the chair QuantValley/Risk Foundation. This manuscript solely reflects the views of the authors, not necessarily those of their respective employers. We remain solely responsible for any mistake or error.

## APPENDIX A

### CORNISH-FISHER EXPECTED SHORTFALL APPROXIMATION AND RISK CONTRIBUTIONS

We consider a portfolio ** w** = (

*w*_{1}, …,

*w*_{n)T made of n risky assets. Let R = (R1, …, Rn)T denote the vector of individual asset returns with mean vector μ = (μ1, …, μn)T, covariance matrix Ω = E[(R − μ)(R − μ)T], coskewness matrix Σ = E[(R − μ)(R − μ)T ⊗ (R − μ)T], and cokurtosis matrix Γ = E[(R − μ)(R − μ)T ⊗ (R − μ)T ⊗ (R − μ)T], where ⊗ stands for the Kronecker product operator. The portfolio returns are given by R(w) = wTR with mean μ(w) = wTμ, volatility σ(w) = (wTΩw)1/2, third-order centered moment m3(w) = wTΣ(w ⊗ w), and fourth-order centered moment m4(w) = wTΓ(w ⊗ w ⊗ w). The skewness and kurtosis of the portfolio are defined as s(w) = m3(w)/σ3(w) and k(w) = m4(w)/σ}

^{4}(

**), respectively.**

*w*The intuition of the Cornish-Fisher (CF) expansion is to approximate the quantile function of a standardized non-Gaussian random variable *Z* by the quantile of a standard normal variable *z*_{a} augmented by terms capturing the non-normal characteristics through the direct introduction of skewness and kurtosis (Zangari 1996). Following Maillard (2012), and assuming *s*^{2}(** w**) = 0, the CF quantile function

*G*(α) for confidence level (1 − α) can be expressed as:

_{CF}where , , , and . The ES of the standardized return *Z* can then be obtained by integrating the approximate CF quantile function (**9**):

where (.) is given in (9) and ϕ(.) is the standard normal density function. Using the fact that and , for *k* > 0, we deduce:

where . Substituting (**11**) in the ES portfolio formula **1** yields:

where is the adjustment to ES for the non-Gaussian features of the distribution of returns.

The estimation of a risk-based portfolio necessitates to obtain the individual contributions to ES, CES_{α}_{(}_{i}_{)}(** w**). Computing the first derivative of (

**12**) relatively to

*w*

_{i}, we get:

where ∂_{i} denotes the partial derivative relatively to *w*_{i}. From the centered moments derivatives (Jondeau and Rockinger 2006), we infer:

where (**Ω**** w**)

_{i}, (Σ(

**⊗**

*w***))**

*w*_{i}, and (

**Γ(**

**⊗**

*w***⊗**

*w*

*w***)**)

_{i}denote respectively the

*i*-th entry of the vectors

**Ω**

**, Σ(**

*w***⊗**

*w***) and**

*w***Γ(**

**⊗**

*w***⊗**

*w*

*w***)**. Inserting into (

**13**) and rearranging terms leads to:

Premultiplying by the portfolio’s weights *w _{i}* yields the contribution of each asset to the total ES, i.e., , which can finally be rewritten as a weighted sum of contributions to the first four moments of the portfolio as follows:

with

where CMEAN_{i}(** w**), CVOL

_{i}(

**), CSKEW**

*w*_{i}(

**), and CKURT**

*w*_{i}(

**) denote the contribution from the**

*w**i*-th asset mean, volatility, skewness, and kurtosis to the portfolio risk.

*c*

_{1},

*c*

_{2},

*c*

_{3,}and

*c*

_{4}are constants with

*c*

_{1}= − 1, , , and . Summing across all assets, we retrieve the portfolio’s ES as the sum of individual contributions .

## APPENDIX B

### MODELING ILLIQUIDITY THROUGH SMOOTHING

Following Getmansky et al. (2004), we assume that, due to illiquidity, the actual returns *R _{i}*

_{,}

_{t}cannot be observed directly and that the reported returns of the illiquid asset, denoted by , are governed by a MA(

*K*) process:

where **θ**_{i}_{,}_{k} ∈ [0, 1] for *k* = (0, …, *K*) and . Getmansky et al. (2004) and Cavenaille et al. (2011) have shown that the smoothing mechanism (**17**) does not affect individual mean returns but leads to a downward bias in observed individual volatility, skewness, and kurtosis as follows:

Following Cao and Teiletche (2007), we expand these results to co-moments:

19where *s _{ijl}* =

*E*[(

*R*

_{i}_{,}_{t}− μ

_{i})(

*R*

_{j}_{,}

_{t}− μ

_{j})(

*R*

_{l}_{,}

_{t}− μ

_{l})] and

*k*=

_{ijlm}*E*[(

*R*

_{i}_{,}

_{t}− μ

_{i})(

*R*

_{j}_{,}

_{t}− μ

_{j})(

*R*

_{l}_{,}

_{t}− μ

_{l})(

*R*

_{m}_{,}

_{t}− μ

_{m})] denote the co-skewness and co-kurtosis elements. In all cases, we also see that the smoothing process (

**17**) leads to an underestimation (in absolute terms) of co-moments.

To cope with those biases, we correct the covariance, co-skewness, and co-kurtosis matrixes before applying the risk contribution calculations (**6**) and the program (**3**). More specifically, we correct the co-moments in the following way:

where the terms are inferred from the estimation of moving-average models. We first estimate the MA(*K*) process, , and then simply deduce the smoothing parameters as and for *k* > 1.

For each asset *i*, the implication of the illiquidity can then be judged by contrasting the portfolio’s risk allocation on the basis of the corrected statistics versus the ones based on the raw statistics with:

where and correspond to the unsmoothed and smoothed individual ES risk contributions respectively. It is straightforward to see that ILLIQUID_{i}(** w**) is equal to 0 for liquid assets as

**θ**

_{i,k}= 1 for

*k*= 1 and

**θ**

_{i,k}= 0 for

*k*> 1.

## ENDNOTES

↵

^{1}As Montier (2014) wrote:*“Putting volatility at the heart of your investment approach seems very odd to me as, for example, it would have had you increasing exposure in 2007 as volatility was low, and decreasing exposure in 2009 since volatility was high—the exact opposite of the value-driven approach.”*↵

^{2}In practice, the risk level α is typically chosen as being low, e.g., α = 5%.↵

^{3}As shown by Bai et al. (2016), the existence of short-selling positions can lead to multiple solutions for risk-based portfolios.↵

^{4}From the Gaussian ES specification (**2**), one immediately deduces that . It is straightforward to see that total ES contributions becomes equivalent to total risk contributions to volatility when μ_{i = 0 ∀}*i*.↵

^{5}See Jurczenko and Teiletche (2018) for an analytical framework allowing the combination of active views with a risk-based portfolio.↵

^{6}In practice, one needs to consider that under a CF approximation, the set of skewness-kurtosis pairs must be restricted in order to yield monotonic quantile functions, with the skewness varying between −3 and 3 and the excess kurtosis ranging between 0 and 8 (Maillard 2012).↵

^{7}From this perspective, one of the challenges of our framework is to estimate these parameters as they rapidly increase in number, making estimation more cumbersome. For application to large cross-section of assets, we suggest using either factor models (Jondeau et al. 2018 and Boudt et al. 2015) or shrinkage estimators (Martellini and Ziemann 2015 and Boudt et al. 2017).↵

^{8}See e.g., https://epitest.ncreif.org/documents/event_docs/NCREIF_Academy/NCREIF-Database-Query-Tools.pdf, page 38.↵

^{9}Co-moments are also corrected through appropriate products of the smoothing coefficients of the involved assets. See Appendix B for more details.↵

^{10}It is possible to target specifically for a ES minimization by setting*c*equal to −∞ in program (**3**).↵

^{11}Indeed, we find striking similarities between the risk-based asset allocations where ES is corrected for illiquidity only and the ones where ES is modified to incorporate non-normality (MES).↵

^{12}On the contrary, the traditional risk parity approaches are not affected by the investment horizon since individual variances and pairwise covariances are affected in the same way by the return measurement horizon, so that the relative risk structure is left unaffected. We thank the referee for emphasizing this difference to us.↵

^{13}In the current application, such out-of-sample investigations are limited due to the low-frequency of observations of illiquid asset classes. For instance, with nearly 30 years of quarterly data, there are 110 observations available to estimate all the comoment matrixes. A natural solution to circumvent this problem would be to use a factor model or shrinkage estimators as discussed in endnote 7 above. We leave this to future research.

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