3.1 Geometric Intuition for the CAPM
3.1.4 Mathematical Analysis of the Minimum-Variance Opportunity Set
In the following we make the arguments rigorous that led to the definition of the tangent portfolio. The mathematically less inclined reader can skip this subsection.
It is sometimes said that the minimum-variance opportunity set is convex (as it is depicted in Fig.3.3and mathematically defined in Sect.A.1). This is, however, not always the case: the mean-variance opportunity set does not need to be convex, as we can already see in the case of two assets where the opportunity set isonlyconvex if their correlation isC1(compare Fig.3.2). However, we don’t need this convexity to prove the existence of a tangent portfolio, but before we can obtain any existence result, we need first to distinguish whether we allow for short-selling or not.
This decision has two sides: a modeling one and a mathematical one. First, it is not so clear whether allowing for short-sales is appropriate or not for our model.
We could argue that in most developed markets short-selling is possible and hence our model should include it. On the other hand, there are markets where it is not possible (it might be banned or infeasible due to a lack of liquidity) and even on the most developed markets there are many market participants (small private investors) who do not have the chance to short-sell assets, at least not without steep costs.
The mathematical side of the story is even more difficult: we will see that without short-selling we can find a rigorous proof for the existence of a tangent portfolio
3We will see that economically spoken, this portfolio is such that the marginal rate of substitution between the investor’s preferences for risk and return equals the marginal rate of transformation offered by the minimum variance opportunity set.
3.1 Geometric Intuition for the CAPM 99 under quite natural assumptions, but when we allow for short-selling then existence might fail if we do not impose rigid assumptions. Later, however, when we derive the capital asset pricing model, we will need to allow short-selling. This inherent problem of the geometric and “intuitive” approach presented in this chapter can only be fixed by studying the more rigorous no-arbitrage approach that we will follow in Chap.4.
Let us now first consider the existence of the tangent portfolio when we exclude short-selling.
The main property of the opportunity set that we need is in this case that it is closed (compare Sect.A.3 for a definition). Moreover we need certain minor properties that we summarize later.
Lemma 3.1 If we have finitely many assets, the minimum-variance opportunity set is closed and connected.
Proof We give two proofs, the first based on the Bolzano-Weierstrass Theorem the second based on a property of continuous functions:
By construction it is clear that the opportunity set is connected. To see that the opportunity set is closed if we have finitely many assets is easy: letKdenote the number of assets and let us consider a sequence of points xn D .n; n/(n D 1; 2; : : :) in the opportunity set withxn !xD.; /. Each of thexncorresponds to a portfolio characterized by asset weightsn1; : : : ; nK withnk 0 for allk D 1; : : : ;K andPK
kD1nk D 1. Therefore the vector n WD .n1; : : : ; nK/ is for all n 2 Nin a compact set.4 According to the Bolzano-Weierstrass TheoremA.3we can select a converging subsequence of then. Let us denote its limit by, then defines a portfolio with meanand variance2, since mean and variance depend continuously on the asset weights. Thus any limit of points in the opportunity set is again in the opportunity set, in other words we have proved that the opportunity set is closed.
The second proof uses the function f that assigns mean and variance to a portfolio:
f W SWD (
2RKCC 1
ˇˇˇˇXK
kD0
kD1 )
! (
.; /ˇˇ ˇˇD
XK kD0
kk; 2 Dkcovjkj;2S )
:
Now, sinceSis obviously closed and connected and sincef is continuous, we can deduce that f.S/, i.e. the opportunity set, also is closed and connected, compare
Sect.A.3. ut
4We use in this chapter bold face characters for vectors to increase readability.
What about if we have infinitely many assets? In this case the opportunity set does not have to be closed. As a simple example think about perfectly correlated assets with k D 1 1=k and k D 1. The opportunity set is given by f.; 1/j2Œ0; 1/gand is obviously not closed. In this case we see also why we need closedness: the efficient frontier in the example does not exist, since any portfolio with meanand variance2in the opportunity set can be improved. (The potential
“best” portfolio with D D 1is not contained in the opportunity set.) We see that we better stick to the case of finitely many assets.
Since the opportunity set is closed, we can in fact construct the efficient frontier.
To construct a tangent portfolio, however, we need to know a little bit more about the geometric structure of the efficient frontier:
Lemma 3.2 If we have finitely many assets, the efficient frontier can be described as the graph of a function fWŒa;b, where0 a b <1. Moreover there exists a point c2Œa;bsuch that f is concave and increasing onŒa;cand decreasing on Œc;b.
Proof By construction it is clear that the functionfexists. It is also clear thatb<1, since there are no points in the minimum-variance set with > maxkD1;:::;Kk, compare (3.1). Suppose now thatfis increasing and strictly convex on some interval Œs1;s2, wheres2 >s1. Then we can combine the portfoliosA, with meanf.s1/and variances21, andB, with meanf.s2/and variances22. Using again the formula (3.1) we can find a2.0; 1/such that the new portfolioAC.1/Bhas the variance .s1Cs2/2=4. The mean of this portfolio depends on the correlation betweenAand B, but can be estimated from below by.f.s1/Cf.s2//=2, as a small computation shows. Given the strict convexity off, however,f..s1Cs2/=2/ < .f.s1/Cf.s2//=2, thus we have found a portfolio that is “better” than the efficient frontier (i.e., its variance is the same, but its mean larger). This is a contradiction, thusf has to be concave when it is increasing.
With a similar construction we can prove that iff is decreasing at some points then it cannot be increasing at any point larger thans. Putting everything together,
we have proved the lemma. ut
We mention that it is possible that the efficient frontier is a decreasing and strictly convex function and that the efficient frontier does not have to be continuous, see Fig.3.5for an example.
Using the above lemmas we can now prove the existence of a tangent portfolio:
Proposition 3.3 If we have finitely many assets, and at least one asset has a mean which is not lower than the return Rf of the risk-free asset, then a tangent portfolio exists.
Proof Given the above conditions, an efficient frontier exists according to Lemma 3.1. Using Lemma 3.2, we know that there are points a, b, such that the efficient frontier is the graph of a function f on Œa;b, which is concave and
3.1 Geometric Intuition for the CAPM 101
Fig. 3.5 An example for a discontinuous, partially decreasing and strictly convex efficient frontier
c
a a c a c
Rf
Rf
Rf
Fig. 3.6 The three cases for the construction of the tangent portfolio
increasing onŒa;c. We denote this part of the graph byF. Using the condition on the asset returns, we see thatf.c/Rf.
Now, we have to distinguish three cases: If there is a tangent onF through the point.Rf; 0/, i.e., the risk-free asset, we have found a tangent portfolio by taking a tangent point inFon this line. Otherwise, if a line from.Rf; 0/to.f.c/;c/lies nowhere belowF, the point.f.c/;c/is the tangent point. If both is not the case, then the tangent portfolio is given by the point.f.a/;a/. Compare Fig.3.6for an illustration of the three cases.
In all three cases, the constructed line cannot lie below other points of the efficient frontier, sincef is decreasing for values larger thanc, but the tangent line is increasing (or at least horizontal), sincef.c/Rf. ut What changes in this argument if we allow for short-selling? In a nutshell:
everything. – In fact, the existence is not guaranteed anymore! Take as simple example two assets.1; 1/ D .1; 1/and.2; 2/ D .1:1; 1/with a correlation ofC1. Then we have for a portfolio of2 Runits of asset 1 and1units of asset 2 that.; /D.C1:1.1/; 1/D.1:10:1; 1/. Thus we can construct portfolios with arbitrarily large returns and a variance of1by choosingnegative enough. It is now easy to see that we are not able to construct any tangent portfolio in this case.
The example can be modified such that1 ¤ 2: even then we will usually not be able to define a tangent portfolio. A similar construction is possible for correlation1.
Assuming that no pair of assets has a correlation of1orC1does not fix this problem either, since a combination of two such assets might have a correlation of C1or1with another asset.
If we exclude this possibility as well, then we finally get an existence result:
Theorem 3.4 Let X be a mean-variance opportunity set (with short-selling) and assume that for any two portfolios Xj;Xk 2 X with j ¤ k the correlation between their returns is in.1;C1/, i.e. neither1 nor C1, then there exists a tangent portfolio for X.
Proof LetD f2RKj PK
kD1kD1g. SubstitutingkD1PK1
kD1k, we can transformtoRK1. Now we consider the compactificationCK1. We define this in two steps: first transformRK1toDVK1via
.x1; : : : ;xK1/7!
2
arctan.x1/; : : : ; 2
arctan.xn/
:
Now addSK1 D @DVK1and use the standard topology ofDK1:Now consider a sequencenthat maximizes..n/Rn/ f. A subsequence ofnconverges inCK1, since CK1is compact. Now consider the following two cases:
Case 1: The limit is in DVK1, then also k is finite. Now let limn!1.n/Rf
.n/ <1, since otherwise we have a finite portfolio which is riskless and this could only happen if it is composed of two risky portfolios with correlationC1or1– a contradiction.
Case 2: The limit is inSK1, then alsokis infinite. DefineA1 as assetK,A2as all other assets in the relative weights specified by the limit inSK1. Then we can find a sequence Qn with the same limit, but being composed only of two portfolios A1, A2 with Qn1A1 C Qn2A2 where Qn1 ! C1 and Qn2! 1.
By assumption, corr.A1;A2/2.1;C1/, thus
.Qn1A1C Qn2A2/; .Qn1A1C Qn2A2/
n2N
is a curve with slope going to zero (computation in Chap.3). Therefore limn!1.Qn/Rf
.Qn/ D0, but sinceandare contained in the portfolio weights, the original sequencencould not have been maximizing which contradicts our initial
assumption. ut
This result, however, is not as useful to determine the existence of a tangent portfolio: we would have to check all (infinitely many) portfolios of our assets and their correlations with each other to verify the condition of the theorem. We will see
3.1 Geometric Intuition for the CAPM 103 in the next chapter how the no-arbitrage condition can help us to avoid this problem and secure the existence of a tangent portfolio under a more reasonable condition.
Up to then, we will tacitly assume the existence of a tangent portfolio, although we know now that this is not a trivial matter.
By the way: whether we allow for short-selling or not, the tangent portfolio does not have to be unique. Non-uniqueness, however, occurs only in very specific situations and is not important for practical applications where we are usually happy with finding an optimal portfolio and do not care that much about whether there would have been another equally good portfolio. . .