text stringlengths 128 2.05k |
|---|
The main limitation of the existing literature in analyzing the value in investing in multiple assets is that two fundamentally different questions are often entangled: i) how much incremental value can an investment manager derive from trading a given additional asset, and ii) how should an investment manager go about... |
In the literature, the former question is seldom investigated, whereas attempts to address the latter abound . It is typically assumed that the universe of assets to trade is given, and the emphasis is placed on optimizing capital allocation across these assets. In the Mean-Variance framework and related approaches, th... |
. Naturally, if one is able to figure out how to extract value out of trading a given additional asset, then it means that the additional asset does indeed add intrinsic value to the existing pool. However, attempting to determine optimal allocation within a large pool of related assets can pose serious numerical chall... |
Interestingly, judging by the proliferation of fund-of-funds, practitioners have long grasped the importance of mitigating risk concentration across risk factors such as fund managers, asset classes, strategy styles, geographic locations etc., on top of or as constraints to the optimization process. However, their appr... |
Definition 1.1 We denote an asset as any investment resulting in a periodic stream of returns, realized or marked-to-market. Throughout the rest of this paper we identify an asset by its time series of returns, and we consider that two assets having identical time series of returns are identical for all investment purp... |
The rest of the paper is structured as follows. In Section we provide an intuitive answer to what makes an asset useful. We argue that the usefulness of an asset can only be considered relative to a reference pool of assets, and we argue that for an asset to be useful, it needs to sufficiently add to the diversificatio... |
WHAT EXACTLY MAKES AN ASSET USEFUL? So what exactly makes an asset useful to an investment manager? Intuitively, the answer ought to depend on what assets the investment manager already has access to, and what risk factors and benchmarks, if any, he/she wants to avoid exposure to. Indeed, no matter how high an asset’s ... |
Prior to any technical discussion, let us review four intuitive features we would expect an asset to exhibit to consider it incrementally useful relative to a reference pool of assets. Each of the four features corresponds to a motivation an investment manager might have for widening the universe of assets he/she trade... |
2.1 Incremental Diversification Perhaps the most fundamental reason why an investment manager might want to consider broadening the pool of assets he/she trades is diversification. To make matters precise, we provide the following definition for diversification. |
Definition 2.1 Throughout this paper we denote diversification as the act through which one aims at reducing the level of risk of a portfolio, for the same level of expected return, by adding one or more assets. |
We note however that not all new assets have the same potential for reducing the risk of a portfolio; some provide more diversification potential than others. Intuitively, adding shares of a U.S. bank to a universe of U.S. financial stocks might not present the same diversification benefits as adding a commodity future... |
2.2 Predictability of Returns Central to most popular portfolio optimization approaches, is the need to estimate expected values of asset returns, which practitioners typically do by forecasting future asset returns |
. However, if a time series of returns is pure noise, any attempt to forecast it would be vain, and most portfolio optimization processes would fail to make use of the new asset. Therefore, for a new asset to be useful, its time series of returns needs to be sufficiently predictable. |
2.3 Reduced Tail Risk Another reason that can motivate investment managers to add a new asset to their trading universe is to mitigate the likelihood that their portfolio can undergo a significant idiosyncratic move, thereby possibly causing their investors to panic and withdraw assets. Unlike the incremental diversifi... |
The rationale for widening the trading universe as a way of mitigating tail events is that, doing so could reduce the proportion of total exposure that is concentrated in a single asset, thereby reducing the sensitivity of the overall portfolio to idiosyncratic shocks. However, trading more assets doesn’t always result... |
2.4 Suitability for Passive Investment Perhaps the most wide-spread expectation one can have of an asset is that it appreciates over time. We slightly relax this requirement, and instead require of an asset that it be suitable for passive investment in order to be considered useful. In other words, it should be possibl... |
2.5 Relative Importance of Usefulness Criteria Incremental diversification and predictability of returns are primary criteria of incremental usefulness in that, if either one is not met, no investment manager will find the new asset incrementally useful. A new asset whose returns can be perfectly replicated using exist... |
The other two criteria on the other hand are secondary in that a new asset could be useful to some investment managers, even if those criteria are not met. For instance, a new asset that exhibits strongly predictable returns can be exploited by an active investment manager, even if it isn’t suitable for passive investm... |
QUANTIFYING INCREMENTAL DIVERSIFICATION In order to motivate our approach to quantifying incremental diversification, let us first make precise scenarios in which we would intuitively conclude an asset incrementally diversifies a reference pool, and cases where we would consider the new asset to be redundant. |
3.1 Intuition The guiding principle to determining whether a new asset adds incremental diversification to a reference pool of assets ought to be that, if it is easy to replicate the stream of returns of the new asset using assets and other factors in the reference pool, then the new asset doesn’t add diversification t... |
To affine our intuition, let us consider some concrete examples that will help derive stylized features that a suitable quantitative measure of incremental diversification should exhibit. |
We start by considering a fund [MATH] whose (one-period) returns we assume are independent across time, and drawn from the same random variable [MATH] with mean [MATH] and standard deviation [MATH] . We consider another asset A, whose (one-period) returns we assume are also independent across time, and drawn from a ran... |
[EQUATION] whose mean and standard deviation read [EQUATION] and [EQUATION] where the last inequality stems from the fact that [MATH] and [MATH] . It is worth noting that the inequality is always strict, unless either [MATH] , which corresponds to only investing in [MATH] or A, or [MATH] , that is when [MATH] and A are... |
We also note that, for a fixed [MATH] , the lower the correlation between [MATH] and A, the lower the standard deviation of the new portfolio, and consequently the more ‘diversification value’ we get out of adding A to the portfolio. In this toy example, portfolio [MATH] is regarded as a single asset, that is, we do no... |
Definition 3.1 Let P be a reference pool of assets with returns [MATH] , and let A be an asset not in P, with return [MATH] . We denote the portfolio that best replicates A using P as the portfolio of assets in P whose allocation, which we denote [MATH] , satisfies |
[EQUATION] where [MATH] denotes the risk-free interest rate, and [EQUATION] denotes the return of the portfolio with allocation [MATH] across assets in P and whose excess cash (resp. net leverage) earns (resp. is funded by borrowing at) the risk-free interest rate. |
Optimization Problem ( 3.1 ) is quadratic and is easily found to have solution [EQUATION] Denoting [MATH] the return of the best replicating portfolio, it follows that |
[EQUATION] Additionally, the residual return of the replication —which we also refer to as tracking error—, namely [MATH] , has variance that is found to be |
[EQUATION] Intuitively, a low variance of residual returns of replication indicates that we are able to replicate the stream of returns of the new asset using assets in the reference pool fairly well, which in turn implies, according to our foregoing guiding principle, a low potential for incremental diversification. I... |
Stylized Fact 1: A good quantitative measure of the incremental diversification an asset A adds to a reference pool of assets should never be high when the correlation between the new asset A and the portfolio of assets in the reference pool that best replicates A is high. |
What about the reverse? What scenarios can we intuitively conclude wouldn’t result in incremental diversification? The first that comes to mind from our previous discussion is when the correlation between the new asset and its best replicating portfolio of assets in the reference pool is 1, in particular when the retur... |
Stylized Fact 2: A good quantitative measure of the incremental diversification an asset A adds to a reference pool of assets should be the lowest when returns of the new asset A can be obtained as a linear combination of returns of assets in the existing pool. |
Remark 3.1 The alert reader might be wondering what to make of trading strategies such as pairs trading, that are only possible when some assets exhibit strong relationships. We note that these strategies are only profitable if the spread between the two assets forming the pair deviates enough from its equilibrium regi... |
The second stylized fact deals with assets whose returns are a linear combination of returns of assets in the reference pool, and consequently can easily be emulated using a portfolio of assets in the reference pool, the allocation of which doesn’t change over time and can be obtained using linear regression. Additiona... |
Stylized Fact 3: Let [MATH] be the time series of asset returns and factor values of a reference pool of [MATH] assets and factors, and |
[MATH] the times series of returns of a new asset A. A good quantitative measure of the incremental diversification asset A adds to the reference pool should be the lowest when returns of the new asset A can be obtained as a function of present and past values of returns and factors in the pool, that is |
[MATH] for some function [MATH] and memory [MATH] It is worth stressing that the time series characteristic of the reference pool, namely [MATH] , is discrete-time, and each unit of discrete-time corresponds to the same wall-clock time. Thus, Stylized Fact 3 aims at discarding new trading strategies or assets that expl... |
Stylized Fact 3 is however not to say that, if two fund managers trade the same universe of assets, their funds do not provide diversification to the universe of assets they trade. Diversification doesn’t arise solely as a result of what a fund manager trades, but also, and perhaps more importantly, as a result of how ... |
Stylized Fact 4: A good quantitative measure of incremental diversification should allow for manager diversification. That is, two funds with identical constituents but different time-varying allocations driven by different (random) signals, should be able to diversify each other, despite their identical constituents. |
Asset returns can be scaled up and down through leverage. Thus, the scale of a time series of returns should intuitively bear no relevance on whether the corresponding asset incrementally diversifies a reference pool, or more generally is incrementally useful. This observation gives rise to our fifth and last Stylized ... |
Stylized Fact 5: A good quantitative measure of incremental diversification should be scale-invariant. Equivalently, a good quantitative measure of incremental diversification should neither depend on the standard deviations of returns time series of assets in the reference pool, nor should it depend on the standard de... |
3.2 Differential Mutual Information Timescale as a Measure of Incremental Diversification To motivate our measure of incremental diversification, we start with the simple case of two assets [MATH] and A. |
Case 1: Two assets, i.i.d. Gaussians Returns of [MATH] (resp. A) are assumed to be independent draws across time from the same distribution [MATH] (resp. [MATH] ), with mean [MATH] and variance [MATH] . We assume that [MATH] is jointly Gaussian and the correlation between [MATH] and [MATH] is [MATH] . As previously dis... |
[EQUATION] When [MATH] [MATH] (Equation ( 3.1 )) increases with [MATH] , whereas [MATH] decreases with [MATH] , and the two requirements for diversification, namely high expected return per unit of risk and unrelated returns, are consistent. However, when [MATH] [MATH] still increases as a function of [MATH] but [MATH]... |
[EQUATION] still provides valuable insight on shared information between A and [MATH] , namely that knowing [MATH] never increases the uncertainty about [MATH] ; the uncertainty is preserved when the two assets are decorrelated ( [MATH] ), and decreases otherwise. A natural measure of the diversification A adds to [MAT... |
[EQUATION] Remark 3.2 Expected returns being equal, Equation ( ) as a measure of incremental diversification penalizes equally new assets with [MATH] and new assets with [MATH] , which could be perceived as a limitation, as the former can be used to construct portfolios with much higher return per unit of risk than the... |
Case 2: n assets, i.i.d. Gaussians This intuitive measure of incremental diversification easily extends to the multi-assets case. If we consider a pool of [MATH] assets and factors P, with corresponding returns and factor values drawn independently (across time) from a Gaussian random vector [MATH] that is also assumed... |
[EQUATION] It immediately follows from Gaussian identities that [EQUATION] where we assume that [MATH] is non-degenerate, from which we recover Equation ( ) in the two assets special case. |
Case 3: Beyond Gaussianity In the non-Gaussian case, the conditional variance [MATH] might very well be a function of [MATH] , so that a more suitable candidate to quantify incremental diversification is obtained by taking the expectation with respect to [MATH] |
[EQUATION] Expected conditional variance as measure of incremental diversification only captures the first two moments of the joint-distribution. This is sufficient for Gaussian distributions as they are fully determined by their first two moments. However, non-Gaussian distributions typical exhibit tail behaviors that... |
Proposition 3.1 Let [MATH] and [MATH] be two squared-integrable random variables. Then [EQUATION] and the inequality is an equality if and only if |
[EQUATION] or equivalently, if and only if [MATH] for any [MATH] Proof. Hint: This follows from the law of total variance and Hilbert’s projection theorem. |
The canonical measure of the amount of information in a random variable with probability measure [MATH] and admitting pmf or pdf [MATH] , is the notion of entropy (expressed in bits) defined as |
[EQUATION] Unless stated otherwise, throughout the rest of this paper we assume [MATH] admits a pdf. When we need both cases, we will use the expression continuous entropy or differential entropy to emphasize that [MATH] admits a pdf, and discrete entropy or Shanon entropy when [MATH] admits a pmf, in which case we wil... |
A related notion is that of conditional entropy, which can be defined as [EQUATION] when [MATH] and [MATH] exist, and that measures the amount of information contained in random variable [MATH] that is not already contained in random variable [MATH] . In the multi-assets Gaussian case, this measure of incremental diver... |
[EQUATION] In other words, in the Gaussian case, conditional entropy and conditional variance are equivalent measures of incremental diversification as one is fully determined by the other and is an increasing function of the other. In general however, conditional entropy is a more general measure of incremental divers... |
). Proposition 3.2 Let [MATH] and [MATH] be two random variables having finite entropies [MATH] and [MATH] . Then [EQUATION] and the inequality is an equality if and only if [MATH] and [MATH] are independent. |
Unlike expected conditional variance that cannot differentiate decorrelation from independence, conditional entropy, as a measure of incremental diversification, is informative about the full distribution tails, and is maximized (for a given entropy [MATH] ) when [MATH] is independent from [MATH] , which we recall impl... |
Case 4: Beyond temporal independence Conditional entropy as a measure of incremental diversification satisfies both Stylized Facts 1 and 2. To see why, we note that, in the Gaussian case, [MATH] is also the variance of the residual return of the best replicating portfolio, and using Equation ( ) we obtain |
[EQUATION] which confirms that [MATH] decreases with the correlation between A and its best replicating portfolio (Stylized Fact 1), and is lowest when [MATH] is a linear combination of [MATH] —i.e. where [MATH] (Stylized Fact 2). In the general case, using the fact that Gaussian random variables are maximum-entropy am... |
[EQUATION] where [MATH] is a Gaussian distribution with equal covariance matrix to that of [MATH] . Hence, [MATH] can be made arbitrarily small by increasing [MATH] or equivalently by jointly increasing [MATH] and [MATH] , which is consistent with Stylized Facts 1 and 2. |
However, conditional entropy as a measure of incremental diversification does not satisfy Stylized Fact 3, as we have been ignoring the temporal aspect of our time series because of our i.i.d. assumption (across time). To see why, we consider [MATH] , and note that, under our memoryless assumption on the reference pool... |
The notion of entropy of random variables is extended to discrete-time stochastic processes by the notion of entropy rate which is defined as |
[EQUATION] when the limit exists. The notion of conditional entropy is then extended to define conditional entropy rate as [EQUATION] |
when [MATH] and [MATH] exist. Similarly to the random variable case, the conditional entropy rate measures the amount of information per unit of time contained in stochastic process [MATH] that is not already reflected in [MATH] . Moreover, the conditional entropy rate fully captures dependencies between time series ac... |
Proposition 3.3 Let [MATH] and [MATH] be two discrete-time stationary stochastic processes having finite entropy rates [MATH] and [MATH] . Then |
[EQUATION] If we further assume that [MATH] and [MATH] have bounded memory in the sense that there exists [MATH] and [MATH] such that |
[EQUATION] where [EQUATION] then the inequality in Equation ( 21 ) is an equality if and only if [MATH] and [MATH] are independent. |
Proof. The inequality in Equation ( 21 ) is a direct consequence of Proposition 3.2 . Moreover if [MATH] and [MATH] are independent, then it is easy to see that [MATH] |
To prove the reverse, we note that [EQUATION] Hence, if [MATH] then [MATH] . As [MATH] for every [MATH] , it follows that [MATH] for every [MATH] , which implies that [MATH] and [MATH] are independent for every [MATH] or, equivalently, [MATH] and [MATH] are independent. |
Remark 3.3 By definition, [MATH] if and only if the mutual information between [MATH] and [MATH] grows too slowly with [MATH] , specifically in [MATH] . Such slow growth can only be attributed to excessive cross-sectional and/or temporal coupling as [MATH] increases. Moreover, it is easy to see that when [MATH] has no ... |
The following corollary is a direct consequence of Proposition 3.3 Corollary 3.1 Let [MATH] and [MATH] be two discrete-time stationary stochastic processes such that |
[EQUATION] then for any function [MATH] , time [MATH] , memory [MATH] , and lag [MATH] . The random variables [MATH] and [MATH] are independent. |
Another perspective on Corollary 3.1 is that, using conditional entropy rate as measure of incremental diversification, the best case scenario corresponds to assets whose current and future returns are independent from (and therefore cannot be predicted using) past returns of assets and values of factors in the referen... |
The flip side of the foregoing observation is provided in Proposition 3.4 Proposition 3.4 Let [MATH] and [MATH] be two discrete-time stochastic processes that admit finite entropy rates. If there exist a function [MATH] , and [MATH] such that |
[EQUATION] then [EQUATION] Proof. This proposition is a direct consequence of [EQUATION] which follows from Equation ( 22 ).∎ Proposition 3.4 shows that conditional entropy rate, as a measure of incremental diversification an asset A adds to a reference pool of assets and factors P, satisfies both Stylized Facts 2 and ... |
Let’s study the consistency of conditional entropy rate with Stylized Fact 1. Entropy rates do not always exist in general, nor are there generic analytic formulae to compute them when they exist. For stationary stochastic processes however, entropy rates are guaranteed to exist. |
The notion of best replicating portfolio in the mean-squared sense (Definition 3.1 ) is easily extended to the non-i.i.d. case as the portfolio that has dynamic allocation that is solution to the Optimization Problem |
[EQUATION] whose solution is found to read [EQUATION] and the covariance between the return of the new asset [MATH] and the return of the best replicating portfolio [MATH] reads |
[EQUATION] Remark 3.4 When [MATH] is jointly stationary, it is easy to see from Equations ( 24 ) and ( 25 ) that the best replicating portfolio is in fact a static portfolio (i.e. its target allocation is constant over time), [MATH] is stationary, and the correlation between the new asset and its best replicating portf... |
Using the following property of stationary processes [EQUATION] and using the maximum-entropy property of Gaussian random variables to upper bound [MATH] , we obtain a generalization of Equation ( 3.2 ) to the non-i.i.d. case: |
[EQUATION] where [MATH] a Gaussian with the same covariance matrix as [MATH] . This confirms that, in the stationary case, using conditional differential entropy rate as measure of incremental diversification is consistent with Stylized Fact 1 in that the correlation between an asset and its best replicating portfolio ... |
Finally, conditional differential entropy rate also satisfies Stylized Fact 4. To see why, we consider two funds [MATH] and [MATH] whose constituents are the same and have returns [MATH] . Let’s denote |
[EQUATION] the funds’ respective allocations, each driven by a different time series of signals [MATH] or [MATH] . The funds’ respective time series of returns read |
[EQUATION] It is clear that the amount of diversification [MATH] adds to [MATH] [MATH] depends on the joint law of [MATH] , and is certainly not always [MATH] . Thus, the two funds can diversify each other; the lower the stochastic similarity between their signal processes, the more they can diversify each other. |
Remark 3.5 We stress that, to ensure manager diversification, conditional differential entropy rate does not require knowing what underlying assets or asset classes the fund manager is trading, what his/her trading thesis is, or what types of data (alternative or otherwise) drive his/her trading decisions. Our approach... |
Conditional differential entropy rate, however, is scale-sensitive, and consequently does not satisfy Stylized Fact 5. Indeed, for any scalar [MATH] and vector [MATH] |
[EQUATION] where [MATH] denotes the Hadamard product. Another limitation of its use as incremental diversification is that it can be negative. Both drawbacks are related to the difference between differential and Shanon/discrete entropies. Strictly speaking, unlike their discrete counterparts that quantify information ... |
[EQUATION] is an absolute measure of the amount of information per unit of time contained in [MATH] that is also contained in [MATH] . It is always non-negative and invariant by any smooth change of variable (a.k.a. diffeomorphism), including linear rescaling. A large differential mutual information rate corresponds to... |
[EQUATION] is a candidate measure of incremental diversification. Definition 3.2 Let P be a reference pool of assets and factors, whose time series of returns and factor values we denote [MATH] . Let A be an asset not in P, whose time series of returns we denote [MATH] . Let us further assume that entropy rates of [MAT... |
[EQUATION] where we use the convention [MATH] and [MATH] [MATH] represents the amount of time required to see [MATH] bit of mutual/shared information between returns of the new asset A and returns of the reference pool P. Intuitively, it is always non-negative, takes its lowest value [MATH] when A is fully determined b... |
Theorem 3.1 For any reference pool P and new asset A satisfying the conditions of Definition 3.2 (a) [MATH] (b) When [MATH] [MATH] if and only if [MATH] |
(c) [MATH] if and only if [MATH] and [MATH] are independent. (d) For any continuously differentiable bijections [MATH] and [MATH] |
[EQUATION] Proof. Let us generically denote [MATH] the probability density function of random variable [MATH] . We note that, [EQUATION] |
where [EQUATION] and [EQUATION] and [MATH] denotes the Kullback-Leibler divergence . (a) Follows from the non-negativity of KL-divergence (also from Proposition 3.3 ), and (d) follows from the invariance of the KL-divergence by smooth bijections. As for (b), [MATH] when [MATH] , which given [MATH] , is equivalent to [M... |
Proposition 3.5 The measure of incremental diversification [MATH] satisfies Stylized Facts 1-5. Proof. As previously discussed [MATH] satisfies Stylized Fact 1 (see Equation ( 3.2 )). Moreover, for every finite [MATH] , the function |
[EQUATION] is a strictly increasing function, whence [MATH] also satisfies Stylized Fact 1. This also implies that [MATH] is lowest if and only if [MATH] is lowest. Hence, the fact that conditional entropy rate satisfies Stylized Facts 2 and 3 extends to [MATH] . In a similar reasoning, the fact that conditional entrop... |
3.3 Differential Entropy Rates From Discrete Entropy Rates In the previous Section, we discussed two major differences between differential (continuous) and Shanon (discrete) entropies, namely that, unlike discrete entropy, differential entropy is neither non-negative nor invariant by rescaling. Another oddity of the d... |
Theorem 3.2 Let [MATH] be a random variable taking values in [MATH] and that admits differential entropy [MATH] , and let [MATH] be the random vector taking values in [MATH] and satisfying |
[EQUATION] if and only if [EQUATION] and that admits discrete entropy [MATH] Denoting [MATH] the probability density function of [MATH] , if the following properties are met, |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.