How Not To Do Mean-Variance Analysis

Ex post mean-variance analysis is a financial application of descriptive statistics. But descriptive statistics, where the sum of square deviations from the mean plays a fundamental role, has a strong geometric flavor. In this paper we emphasize the geometry of mean-variance analysis.

Geometry starts with points. The primary points in our geometric exposition are two biotechnology exchange-traded funds,
    FBT – First Trust NYSE Arca Biotechnology Indx Fund
and
    XBI  – SPDR S&P Biotech ETF.
The graphs of the 2013-12-31-normalized adjusted closing prices, $\mathbf{a}_{\scalebox{0.7}{{FBT}}}$ and $\mathbf{a}_{\scalebox{0.7}{{XBI}}}$, of the two funds are shown in Figure 2.1, as well as the graph of an unattended long portfolio, UIP, in the two funds. UIP’s normalized adjusted closing price vector,

is necessarily a convex combination of $\mathbf{a}_{\scalebox{0.7}{{FBT}}}$ and $\mathbf{a}_{\scalebox{0.7}{{XBI}}}$.

Normalized adjusted closing prices are the horses that drive our mean-variance cart. When the adjusted closing price vectors  $\mathbf{a}_{\scalebox{0.7}{{FBT}}},\mathbf{a}_{\scalebox{0.7}{{XBI}}}$, and $\mathbf{a}_{\scalebox{0.7}{{UIP}}}$  are replaced by their daily return vectors  $\mathbf{r}_{\scalebox{0.7}{{FBT}}},\mathbf{r}_{\scalebox{0.7}{{XBI}}}$, and $\mathbf{r}_{\scalebox{0.7}{{UIP}}}$,  you move into a higher dimensional space with all geometry left behind.

The pictures that follow tell the whole story. Thanks to the TikZ vector-graphics language these pictures are precise, numerical images—not just schematic drawings.

Figure 2.1 shows the 2014 history of an unattended investment portfolio, UIP, in two high-returning exchange traded funds, FBT and XBI. The 2013-12-31 closing composition of UIP was

but these closing proportions never reoccured in 2014. Indeed UIP closed with more than 25% in XBI (green higher than blue) for most of the first quarter, whereas XBI was less than 25% of UIP (blue higher than green) for much of the remaining three quarters. However Figure 2.1 does shows the geometry of the 3:1 proportions. On every vertical line, the brown UIP point is exactly 3/4 of the way from the green XBI point to the blue FBT point.

There were exactly 253 market days from 2013-12-31 to 2014-12-31 inclusive. Each of the four adjusted closing price graphs in Figure 2.1 represents the changing value of $100 invested at 2013-12-31 closing prices in the corresponding fund or portfolio over the next 252 market days. Each graph corresponds to a point $\mathbf{a}\in\mathds{R}^{253}$, and the three points, $\mathbf{a}_{\scalebox{0.7}{{FBT}}},\mathbf{a}_{\scalebox{0.7}{{XBI}}},~\text{and}~\mathbf{a}_{\scalebox{0.7}{{UIP}}}$ of (2.1), are on the line segment

in $\mathds{R}^{253}$—with $\mathbf{a}_{\scalebox{0.7}{{UIP}}}$ corresponding to $t=0.75$. (Appendix A describes how normalized adjusted closing prices can be computed.)

Portfolio UIP represents a completely passive, unattended investment. Think of an investor as having money in FBT and XBI at the close of 2013. Suppose that FBT represents exactly 75% of his total investment at that time and XBI 25%. UIP then simply tracks each $100 of his investment through 2014. The investor does absolutely nothing, and all dividends from the two funds are automatically reinvested.

However the continually reallocated portfolio CRP, mostly hidden by UIP, is an entirely different matter. Here the investor decides, a priori, that 75% FBT and 25% XBI are the right proportions for his investment. Accordingly, before each market day of 2014, he reinvests his money so as to start the day with exactly these proportions in the two funds.
Algorithm 2.1 computes the growth of $100 under this scenario.

Here are the adjusted closing prices of the four funds over the month of December 2014.

The geometric equation (2.1) holds on every line of Table 2.1, but the proportions of FBT and XBI in UIP,

are different on every line. The 2013-12-31, 3:1 proportions come closest to being realized on the 2014-12-31 line of Table 2.1, where UIP closes at 75.33% FBT : 24.67% XBI.

As for daily returns,

the 2014 return vector equation,

is valid in $\mathds{R}^{252}$ due to continual reallocation. This insures that the corresponding mean returns satisfy

as required by “The Standard Mean-Variance Portfolio Selection Model” of [2, pp. 3-4].

The ancillary folder that accompanies this article includes three files, FXUZ7.csv,
matlab/FXUC2014.mat, and matlab/FXZC2014.mat, that contain the normalized adjusted closing prices used for this article.

The $\text{MathWorks}^{\text{\tiny{\textregistered}}}$ Financial Toolbox with the MATLAB programming language is perhaps the most popular resource for doing mean-variance analysis. We have computed our mean-variance tables using MATLAB scripts in the matlab subdirectory of the ancillary folder that accompanies this article.

Our MATLAB script hn2mv1a.m illustrates the problem with mean-variance analysis as it is usually practiced. The script begins with the line
         load FXUC2014.mat; % A dates fundsA legendA
which loads the $253\times 4$ matrix of adjusted closing prices, $A=\left[~\mathbf{a}_{\scalebox{0.7}{{FBT}}},~\mathbf{a}_{\scalebox{0.7}{{XBI}}},~\mathbf{a}_{\scalebox{0.7}{{UIP}}},~\mathbf{a}_{\scalebox{0.7}{{CRP}}}\,\right],$
displayed in Figure 2.1. This matrix has rank 3 rather than 4, since $\mathbf{a}_{\scalebox{0.7}{{UIP}}}$ is a linear combination of $\mathbf{a}_{\scalebox{0.7}{{FBT}}}$ and $\mathbf{a}_{\scalebox{0.7}{{XBI}}}$ (1.1).

Next we remove $\mathbf{a}_{\scalebox{0.7}{{CRP}}}$ from $A$ and append the columns
       $\mathbf{a}_{\scalebox{0.7}{{UIP2}}}=0.50\cdot\mathbf{a}_{\scalebox{0.7}{{FBT}}}+0.50\cdot\mathbf{a}_{\scalebox{0.7}{{XBI}}}~~\text{and}~~\mathbf{a}_{\scalebox{0.7}{{UIP3}}}=0.25\cdot\mathbf{a}_{\scalebox{0.7}{{FBT}}}+0.75\cdot\mathbf{a}_{\scalebox{0.7}{{XBI}}}$
so that  $A=\left[~\mathbf{a}_{\scalebox{0.7}{{FBT}}},~\mathbf{a}_{\scalebox{0.7}{{XBI}}},~\mathbf{a}_{\scalebox{0.7}{{UIP}}},~\mathbf{a}_{\scalebox{0.7}{{UIP2}}},~\mathbf{a}_{\scalebox{0.7}{{UIP3}}}\,\right]$  becomes a $253\times 5$ matrix of rank 2. Geometrically, $A$ describes 5 points on a line in $\mathds{R}^{253}$, and $\verb!rank!(A)=2$ since the line does not pass through the origin.

The hn2mv1a.m script continues with the lines
⬇ %% get asset moments from adjusted closing prices ptf = Portfolio; ptf = estimateAssetMoments(ptf, A, ’dataformat’, ’prices’); [mn, cv] = getAssetMoments(ptf); Here mn and cv are the $5\times 1$ and $5\times 5$ mean daily return and covariance of daily return matrices corresponding to the $252\times 5$ daily return matrix
       $R=\left[~\mathbf{r}_{\scalebox{0.7}{{FBT}}},~\mathbf{r}_{\scalebox{0.7}{{XBI}}},~\mathbf{r}_{\scalebox{0.7}{{UIP}}},~\mathbf{r}_{\scalebox{0.7}{{UIP2}}},~\mathbf{r}_{\scalebox{0.7}{{UIP2}}}\,\right]$
derived from $A$ via (2.4). The annualized results are shown in Table 3.1.

One problem with Table 3.1 is immediately obvious. How can the mean returns of the long portfolios UIP and UIP2 be less than the mean return of either component fund? A dimensional problem is less obvious but just as troubling. We start with an adjusted closing price history, $A$, which corresponds to five points on a line segment (a one simplex) in $\mathds{R}^{253}$;  but, to do mean-variance analysis on $A$, we must jump to the four simplex in $\mathds{R}^{252}$ generated by the five linearly independent columns of the daily return matrix $R$ ($\verb!rank!(R)=\verb!rank!(V)=5$). It simply doesn’t make sense to us!

We should note that the MATLAB lines of hn2mv1a.m,
     eCRP = 252 * mean(rCRP); % eCRP = 0.4265 sigCRP = sqrt(252) * std(rCRP, 1); % sigCRP = 0.2783
produced the coordinates for the two CRP points of Figure 3.1. The rCRP in this code corresponds to the $\mathbf{r}_{\scalebox{0.7}{{CRP}}}$ vector of (2.5) or the $\mathbf{r}_{\scalebox{0.7}{{CRP}}}$ from $\mathbf{a}_{\scalebox{0.7}{{CRP}}}$ via (2.4). They are the same.

Figure 4.1 on the next page and the hn2domv3.m script below it illustrate a serious problem with mean periodic returns. This is a simple, artifical example, where a fund gains 50% in the first quarter of the year, loses 67% in the second quarter, gains 200% in the third quarter, and loses 17% in the forth quarter.

An investor in the fund realizes that his fund has returned 25% over the year, but the trip has been terribly rocky; he decides to bail out.

Not so fast, his investment adviser tells him. Just add up the quarterly returns:
        $50\%-67\%+200\%-17\%=166\%.$
You have avergaed averaged over 40% per quarter. The fund may seem a bit risky, but, in view of its history, you should expect to average 40% per quarter next year as well. It’s clear from the numbers.

Figure 4.1 tells the whole story. Mean periodic returns tend to accentuate the positive. Mean periodic discounts do just do just the opposite.

Definition 1 is a paraphrase of definitions in The Theory of Interest, [1].

The means of periodic changes in year-to-date return and periodic changes in date-to-end-of-year discount are the appropriate measures of effective performance of a security over a year. In the the example of Figure 4.1, the annualized mean of the changes in year-to date return is
        $e^{0}=50\%-00\%+100\%-25\%=25\%,$
and the annualized mean of changes in date-to-end-of-year discount is
        $e^{1}=40\%-80\%+80\%-20\%=20\%.$
These annualized means do satisfy Definition 1,  $(1+e^{0})(1-e^{1})=1$.

On the other hand, the MATLAB code shows that the mean return, $e^{r}$, and the mean discount, $e^{d}$, have opposite signs, and $(1+e^{r})(1-e^{d})=5.8667$. These computations show that mean returns and mean discounts are essentially incompatible with the theory of interest. The mean return problem has been noted, for example, in [3, pp. 104-105].

SEC Rule 156 below might apply to the situation illustrated by Figure 4.1 and the hn2mv3.m script which follows it.

     Rule 156: Investment Company Sales Literature

     Under the federal securities laws, including section 17(a) of the
     Securities Act of 1933 (15 U.S.C. 77q(a)) and section 10(b) of the
     Securities Exchange Act of 1934 (15 U.S.C. 78j(b)) and Rule 10b-5
     thereunder (17 CFR Part 240), it is unlawful for any person,
     directly or indirectly, by the use of any means or instrumentality
     of interstate commerce or of the mails, to use sales literature
     which is materially misleading in connection with the offer or sale
     of securities issued by an investment company. Under these
     provisions, sales literature is materially misleading if it:

       1. Contains an untrue statement of a material fact or
       2. omits to state a material fact necessary in order to make a
          statement made, in the light of the circumstances of its use,
          not misleading.

Securities Act of 1933: Rule 156

Rule 156 raises an interesting question. Is the use of mean-variance analysis, as it appears to be practiced today, “materially misleading” when an investment company tells a client to “expect” a 160% return over the next year based on a 25% total return over the past year? This sort of reasoning reminds us of Mark Twain’s analysis of the expected shortening of the lower Mississippi due to the rounding of its bends over time (Appendix B).

More realistically, consider the example of FBT. In Table 3.1 we have seen that the annualized mean daily return of FBT over 2014 was  $e_{\scalebox{0.7}{{FBT}}}=e^{r}_{\scalebox{0.7}{{FBT}}}=42.45\%$. An investor with a marginal knowledge of the theory of interest might ask his advisor what the corresponding annualized mean daily discount was. If the advisor were perplexed by this question, the investor could explain that to get the annualized mean discount you simply replace the daily return equation (2.4) by the daily discount equation

and sum the results. The advisor might then be mildly concerned by the annualized average discount,  $e^{d}_{\scalebox{0.7}{{FBT}}}=35.34\%$, if he were told that returns and discounts over the same period of time are supposed to satisfy the equation
         $(1+r)(1-d)=1$,
according to the theory of interest, but, in fact, $(1+e^{r}_{\scalebox{0.7}{{FBT}}})(1-e^{d}_{\scalebox{0.7}{{FBT}}})=0.9211$.

Our MATLAB script hn2mv1L.m is a linear variant of the hn2mv1a.m script of Section 3. We again remove $\mathbf{a}_{\scalebox{0.7}{{CRP}}}$ from $A$ and append the unattended portfolio vectors $\mathbf{a}_{\scalebox{0.7}{{UIP2}}}$ and $\mathbf{a}_{\scalebox{0.7}{{UIP3}}}$ to the result. Then we add the unattended, long-short portfolio ZNS, with normalized adjusted closing price vector

to $A$, so that $A$ becomes the $253\times 6$ matrix
            $A=\left[\,\mathbf{a}_{\scalebox{0.7}{{FBT}}},\,\mathbf{a}_{\scalebox{0.7}{{XBI}}},\,\mathbf{a}_{\scalebox{0.7}{{UIP}}},\,\mathbf{a}_{\scalebox{0.7}{{UIP2}}},\,\mathbf{a}_{\scalebox{0.7}{{UIP3}}},\,\mathbf{a}_{\scalebox{0.7}{{ZNS}}}\,\right]$
of rank 2. Finally, we put $\mathbf{a}_{\scalebox{0.7}{{CRP}}}$ back into $A$,

and, since $\mathbf{a}_{\scalebox{0.7}{{CRP}}}$ is independent of the other six columns of $A$, the rank of $A$ increases to 3.

When the lines
   ⬇ %% get asset moments from adjusted closing prices ptf = Portfolio; ptf = estimateAssetMoments(ptf, A, ’dataformat’, ’prices’); [mn, cv] = getAssetMoments(ptf);
of hn2mv1a.m are replaced with the lines
   ⬇ %% get daily changes in year to date return R_0 = diff(A / 100); % divide by 100 so that A(1, :) == 1 %% get (annualized) asset moments E_0 = sum(R_0) % total return Sig_0 = sqrt(252) * std(R_0, 1); % standard deviation of return V_0 = 252 * cov(R_0, 1); % covariance of return
in hn2mv1L.m, we arrive at the mean-variance results

The covariance matrix $V^{0}$ has rank three, but the upper-left $6\times 6$ block has rank only two, since the four unattended portfolios UIP through ZNS are affine combinations of the two funds FBT and XBI. Moreover, the corresponding portion of the total return matrix $E^{0}$ mirrors these affine combinations (in contrast to the confusing order of the five mean return values in the $E$ of Table 3.1).

Figure 5.1 shows the obtainable $(e^{0},\sigma^{0})$ corresponding to Table 5.1. This nonlinear triangle is the $(e^{0},\sigma^{0})$-image of the triangle in $\mathds{R}^{252}$ with vertices $\mathbf{r}^{0}_{\scalebox{0.7}{{XBI}}},\mathbf{r}^{0}_{\scalebox{0.7}{{ZNS}}}$, and $\mathbf{r}^{0}_{\scalebox{0.7}{{CRP}}}$.

Thus $e^{0}$ and $e^{1}$ conform to the return-discount requirement of the theory of interest,

but a corresponding summand pair only conforms by accident.

Let us close this article with a revised version of Figure 2.1. The revision, Figure 6.1, shows the same adjusted-closing-price history of the exchange traded funds FBT and XBI and the continually reallocated portfolio CRP, but now the unattended long-short portfolio, ZNS, has replaced UIP. Of the four funds and portfolios, ZNS (purple) had the highest 2014 return with the least volatility.

These normalized adjusted closing prices were generated from UEZ2014.mat by the MATLAB script hn2mv1L3.m. They are recorded, to 5-decimal places (along with the prices of UIP, UIP2, and UIP3), in the comma-separated-value file FXUZC7.csv.

Here the continually-reallocated black-dotted portfolio points are exactly 1/2 and 3/4 of the $e$-way from FBT to XBI on the red, continually-reallocated, FBT-to-XBI path.

The growth in value of an unattended investment portfolio P over a given interval of time can be completely described by a normalized adjusted closing price equation

where the $p_{j}$ are the proportions of the securities in the portfolio P at the close of the day of normalization. The corresponding mean periodic return equation,

does not follow when $e=\verb!mean!(\mathbf{r})$ and periodic return vectors $\mathbf{r}$ are defined by

The mean periodic return equation (7.2) does hold with (7.3) when one restricts his attention to continually reallocated portfolios. Unfortunately continually reallocated investment portfolios are more numerical artifact than financial reality.