As we have seen in Section (2.1.2), being humble about the knowledge of asset moments might pay off in terms of out-of-sample utility. Let’s for the moment exhaust this idea to the fullest and assume that we do not have any information about assets at all: no prior knowledge and no data. In reality, of course, there is plenty of data available for financial markets which generally could be used to draw conclusions about the behavior of individual assets. In particular, it is highly unlikely that all assets do have perfectly equal moments, and hence it might be worthwhile to try to use data in order to detect assets with particularly nice properties. Still, what would be the optimal investment strategy in case that we do not know anything about asset moments at all?
Even in that case we still might know that assets in reality are not equal. But as we do not have any data or knowledge about them, we can not distinguish between individual assets in any way. So basically each asset will appear the same, with equal expected value, equal volatility and the same correlation between all assets. We refer to this situation as “fully uninformed”, and from this perspective the market can be modeled as follows:
Definition 2.2.1: [Fully uninformed market view]
An investment situation shall be called fully uninformed market view when all assets are completely indistinguishable:
• all assets have the same unknown expected return µ¯
• all assets have the same unknown variance σ¯2
• all pairwise correlations have the same value σ¯ij The fully uninformed covariance matrix hence is given by:
¯ΣR=
¯
σ2 σ¯ij . . . σ¯ij
¯
σij σ¯2 . . . σ¯ij
... ... ... ...
¯
σij σ¯ij . . . ¯σ2
As all assets have the same expected value ¯µ, the expected portfolio return does not at all contribute to the portfolio decision: all portfolios have the same expectation ¯µanyways. The portfolio variance, however, does depend on the actual portfolio weights that are chosen (at least when assets are not perfectly correlated). For example, investment into a single asset only will give a portfolio variance of ¯σ2, while the variance of an equally weighted portfolio is
σEW2 =w0EWΣRwEW
=Xd
i=1
w2EW,iσ2i +Xd
i=1 d
X
j=1,j6=i
wEW,iwEW,jCov(Ri, Rj)
= 1 d2
d
X
i=1
σ2i + 1 d2
d
X
i=1 d
X
j=1,j6=i
Cov(Ri, Rj)
= 1 d2d 1
d
d
X
i=1
σi2
!
+ 1
d2d(d−1)
1 d(d−1)
d
X
i=1 d
X
j=1,j6=i
Cov(Ri, Rj)
= 1
d2dσ¯2+ 1
d2d(d−1)¯σij
= 1
dσ¯2+(d−1)
d ¯σij (2.1)
Based on this formula we can derive two optimal portfolio allocation rules.
Theorem 2.2.2: [Optimality under fully uninformed market views]
Given that an investor has a fully uninformed market view, the following two investment strategies can be shown to optimize the portfolio selection:
1. for any given subset of assets the equal weights portfolio is a variance minimizing strategy 2. expanding the equal weights portfolio to as many assets as possible further reduces portfolio
variance
In other words, when we do not have any knowledge about individual assets, we should invest in as many assets as possible, with wealth equally distributed between assets.
The second part of Theorem (2.2.2) can be seen by looking at the limit of Equation (2.1) ind. Holding ¯σ2 and ¯σij fix, the portfolio variance is decreasing in d and converges to the average covariance ¯σij:
d→∞lim σ2EW(d) = ¯σij
This limit can be interpreted as systematic risk that is common to all assets and can not be diversified away. The proof of Theorem (2.2.2), part 1, is a little bit more elaborate, and we need the inverse of the fully uninformed covariance matrix first.
Lemma 2.2.3: [Inverse of fully uninformed covariance]
Let ΣR be a fully uninformed covariance matrix with variances equal to σ¯ and covariances equal to σ¯ij. Then the inverse matrix of ΣR is given by
Σ−1R =
¯
a2 a¯ij . . . ¯aij
¯
aij ¯a2 . . . ¯aij ... ... . .. ...
¯
aij a¯ij . . . ¯a2
with elements
¯
a2 = 1−(d−1)¯σij
−¯σ4+(d−2)¯σij¯σijσ¯2−¯σij2(d−1)
¯ σ2
¯
aij =− σ¯ij
¯
σ4+ (d−2)¯σijσ¯2−σ¯ij2(d−1) Proof. [Lemma (2.2.3)]
To be the inverse of ΣR, matrix Σ−1R must fulfill the following general equation:
Σ−1R ΣR=Id
Due to the special nature of both matrices, all matrix elements of the product can be derived from one of two different equations:
¯
σ2¯a2 + (d−1)¯σij¯aij = 1
¯
σ2¯aij + ¯σij¯a2 + (d−2)¯σij¯aij = 0 Solving the first equation for ¯a2 gives:
¯
σ2¯a2+ (d−1)¯σij¯aij = 1
¯
σ2¯a2 = 1−(d−1)¯σij¯aij
¯
a2 = 1−(d−1)¯σij¯aij
¯
σ2 (?) Plugging this into the second equation we can now solve for ¯aij:
¯
σ2¯aij + ¯σij¯a2+ (d−2)¯σij¯aij = 0
¯
σ2¯aij + ¯σij 1−(d−1)¯σij¯aij
¯ σ2
!
+ (d−2)¯σij¯aij = 0
¯
aijσ¯2+ (d−2)¯σij+σ¯ij
¯
σ2 − σ¯2ij(d−1)¯aij
¯
σ2 = 0
¯
aij σ¯2+ (d−2)¯σij − σ¯ij2(d−1)
¯ σ2
!
=−σ¯ij
¯ σ2
¯ aij
¯
σ4+ (d−2)¯σijσ¯2−σ¯ij2(d−1)
¯ σ2
!
=−σ¯ij
¯ σ2
¯
aij =− σ¯ij
¯
σ4+ (d−2)¯σijσ¯2−σ¯2ij(d−1) This, together with equation (?) gives the final expression for ¯a2.
Now we are ready to proof the variance minimizing property of the equal weights portfolio.
Proof. [Theorem (2.2.2), part 1]
From (2.1.1) we know that the global minimum variance portfolio is given by
wGMV= Σ−1R 1d 10dΣ−1R 1d and its portfolio variance can be computed by
σGMV2 = 1 10dΣ−1R 1d
We will now prove that the variance of the equal weights portfolio is equal to the variance of the global minimum variance portfolio under fully uninformed market views. The variance of the equal weights portfolio is already known from Equation (2.1):
σ2EW=wEW0 ΣRwEW
= 1
d¯σ2+ d−1 d σ¯ij
= σ¯2+ (d−1)¯σij d
Similarly, using the representation of the inverse covariance matrix for a fully uninformed market view given in Lemma (2.2.3), we get:
wEW0 Σ−1R wEW= ¯a2+ (d−1)¯aij d
Now the variance of the global minimum variance portfolio in a fully uninformed market view can be expressed in terms of ¯a2 and ¯aij:
σGMV2 = 1 10dΣ−1R 1d
= 1
(dwEW)0Σ−1R (dwEW)
= 1
d2 ¯a2+(d−1)¯d aij
= 1
d(¯a2+ (d−1)¯aij)
=
1 (¯a2+(d−1)¯aij)
d
The remaining step to be done is to re-express the variance of the global minimum variance portfolio in terms of ¯σ2 and ¯σij, in order to verify equation
¯
σ2+ (d−1)¯σij = 1
(¯a2+ (d−1)¯aij) This is shown in Appendix (6.0.2), so that altogether we get:
σGMV2 =
1 (¯a2+(d−1)¯aij)
d
= σ¯2+ (d−1)¯σij
d
=σ2EW
To sum up, we now have several optimal asset allocation strategies, depending on the amount of information that we have for asset moments. If we really have no information at all, we will have to treat all assets as equal, and the optimal portfolio then is the equal weights portfolio with as many assets included as possible. Diversification pays off, and the portfolio variance will be below the individual asset variances. Next, if we have information about the covariance matrix only, without any knowledge about expected asset returns, then the optimal portfolio allocation is the global minimum variance portfolio. This way, lower risk assets will be preferred, and the portfolio will usually consist of several assets in order to also benefit from diversification effects. Theory suggests, however, that risk and return are related, so that low-risk assets generally also should be assets with lower expected returns. Given that
this relation holds, the global minimum variance portfolio might not be the best way to bring knowledge of the covariance matrix to use anymore. Depending on the degree of risk aversion, higher risk assets might be preferable. This applies even more in a situation where expected asset returns are fully known. In such a case, Markowitz’s portfolio theory will guide us to the optimal asset allocation, resulting in portfolios with optimal risk-return tradeoffs.
In reality, however, the situation that we face should generally be somewhere in-between.
Clearly, we do not have perfect information about true asset moments, but they only can be estimated with uncertainty. Still, with data on historic price trajectories of assets we should at least get some idea about return distributions such that we could improve on naive equal-weights diversification. This is exactly what we will try to achieve in the following chapter: we want to design a portfolio selection approach that can explicitly be adjusted to reflect the actual level of uncertainty of asset moments. In the limiting case of perfect information about asset moments this selection approach should converge to Markowitz portfolio selection, while it should simply pick an equal weights portfolio for the other extreme case of full ignorance about asset moments. In other words, we want to both enjoy the benefits of diversification and still make informed bets on assets with supposedly good risk-return profiles.