I first mentioned Zscores soon after I started writing about closedend funds here about three years ago in an analysis of how to look at discounts and premiums in this article: Need 78% Yields In Retirement? Build Your Income Portfolio With ClosedEnd Funds (Part III: Discounts And Premiums) about three years ago It was a fairly new concept to me at the time. At that time I had not seen any other discussion or reference to Zscores by other authors on Seeking Alpha. Morningstar published a oneyear Zscore, but cefconnect had not yet started to do so. This was also about the time I discovered cefanalyzer, which is the best data base for CEFs that I know of, where 3, 6, and 12 month Zscores are tabulated.
In that article I went into considerable detail on what the Zscore is and how it can be useful. The best way to understand a Zscore is to realize that it represents the distance a fund’s current premium/discount is from its average premium/discount over the time period under consideration. The distance is measured in standard deviations and the sign indicates the direction away from the mean. So a 1year Zscore of 1.2 means the current premium/discount is 1.2 standard deviations lower (more discounted) than the mean premium/discount for the past year. For normally distributed populations we have a clear understanding of how such distributions will behave and one can estimate probabilities for a given Zscore with the probability decreasing as the absolute (unsigned) value of the Zscore increases. If the assumptions are valid mean reversion of the premium/discount should be highly likely.
In subsequent article I refined my concept of the utility of Zscores and began to emphasize how any probabilities the standard calculators associate with it are based on the assumption that it is normally distributed for a fund. This turns out to be an invalid assumption for nearly every case, so one must approach such numbers with extreme caution.
Current Premiums, Discounts and ZScores for CEFs
Readers who follow CEFs will be aware that valuations (as indicated by premium/discount metrics) are stretched. Cefanalyzer tabulates data on 529 CEFs. The median discount/premium is a 5.46% discount and 80.2% of the funds are discounted.
But the median Zscores are 0.73, 0.90 and 0.61 for 3, 6 and 12Months indicating that funds have been giving up discount over all periods.
I’ll be showing an analysis of all CEFs listed on cefanalyzer and three cohorts of those funds: equity, taxable fixedincome and taxexempt municipal bonds. Let’s start with a look at the top and bottom ten Zscores for funds in those three cohorts.
Equity Lowest Ten ZScores 

Fund 
ZScore 

Asa Gold & Precious Metals Ltd (ASA) 
1.30 

John Hancock TaxAdvantaged Global Shareholder Yield Fund (HTY) 
1.20 

Dividend & Income Fund (DNI) 
0.88 

Kayne Anderson Energy Development Co (KED) 
0.76 

Barings Corporate Investors (MCI) 
0.75 

John Hancock Financial Opportunities Fund (BTO) 
0.70 

Eaton Vance TaxManaged BuyWrite Income Fund (ETB) 
0.62 

Pimco Global Stocksplus & Income Fund (PGP) 
0.55 

Gdl Fund (GDL) 
0.52 

Herzfeld Caribbean Basin Fund Inc (CUBA) 
0.45 
Equity Highest Ten ZScores 

Fund 
ZScore 

China Fund Inc (CHN) 
4.25 

Aberdeen Singapore Fund Inc (SGF) 
4.02 

Liberty All Star Growth Fund Inc. (ASG) 
3.80 

First Trust/Aberdeen Emerging Opportunity Fund (FEO) 
3.69 

Central Securities Corp (CET) 
3.59 

Calamos Global Dynamic Income Fund (CHW) 
3.57 

Zweig Total Return Fund Inc (ZTR) 
3.50 

Blackrock Global Opportunities Equity Trust (BOE) 
3.40 

Aberdeen Greater China Fund, Inc. (GCH) 
3.38 

Aberdeen Australia Equity Fund Inc (IAF) 
3.31 
Taxable FixedIncome Lowest Ten ZScores 

Fund 
ZScore 

Nuveen High Income 2020 Target Term Fund (JHY) 
1.43 

Western Asset Investment Grade Defined Opportunity Trust Inc. (IGI) 
1.32 

Blackrock Enhanced Government Fund, Inc. (EGF) 
1.28 

Federated Premier Municipal Income Fund (FMN) 
1.18 

Nuveen High Income December 2018 Target Term Fund (JHA) 
1.14 

Templeton Emerging Markets Income Fund (TEI) 
0.96 

First Trust Mortgage Income Fund (FMY) 
0.85 

Nuveen Build America Bond Fund (NBB) 
0.69 

Blackrock Income Trust, Inc. (BKT) 
0.62 
Taxable FixedIncome Highest Ten ZScores 

Fund 
ZScore 
Western Asset Mortgage Defined Opportunity Fund Inc. (DMO) 
2.84 
First Trust Strategic High Income Fund Ii (FHY) 
2.76 
Ares Dynamic Credit Allocation Fund, Inc. (ARDC) 
2.68 
Legg Mason Bw Global Income Opportunities Fund Inc. (BWG) 
2.54 
Apollo Tactical Income Fund Inc. (AIF) 
2.53 
Nuveen MultiMarket Income Fund (JMM) 
2.51 
Ivy High Income Opportunities Fund (IVH) 
2.49 
Morgan Stanley Emerging Markets Debt Fund Inc (MSD) 
2.39 
Pimco Income Strategy Fund Ii (PFN) 
2.35 
KKR Income Opportunities Fund (KIO) 
2.30 
TaxExempt Municipal Bonds Lowest Ten ZScores 

Fund 
ZScore 
Pimco Municipal Income Fund (PMF) 
1.92 
Neuberger Berman Intermediate Municipal Fund Inc (NBH) 
1.59 
Pimco Municipal Income Fund III (PMX) 
1.21 
Blackrock Investment Quality Municipal Trust Inc. (BKN) 
1.19 
Blackrock Municipal Income Trust (BFK) 
1.13 
Pioneer Municipal High Income Trust (MHI) 
1.13 
Blackrock Munivest Fund Ii, Inc. (MVT) 
1.07 
Nuveen Enhanced Municipal Value Fund (NEV) 
0.91 
Nuveen Quality Municipal Income Fund (NAD) 
0.90 
Western Asset Managed Municipals Fund Inc. (MMU) 
0.81 
TaxExempt Municipal Bonds Highest Ten ZScores 

Fund 
ZScore 
Eaton Vance Municipal Income 2028 Term Trust (ETX) 
1.78 
Mfs Investment Grade Municipal Trust (CXH) 
1.60 
Nuveen Enhanced Municipal Credit Opportunities Fund (NZF) 
1.28 
Mainstay Definedterm Municipal Opportunities Fund (MMD) 
1.24 
Blackrock Muniyield Investment Fund (MYF) 
1.07 
Nuveen Municipal 2021 Target Term Fund (NHA) 
1.06 
Blackrock Municipal 2030 Target Term Trust (BTT) 
0.98 
Nuveen Municipal Income Fund Inc (NMI) 
0.96 
Blackrock Muniassets Fund, Inc. (MUA) 
0.91 
Blackrock Municipal 2020 Term Trust (BKK) 
0.91 
ZScore Distributions
What I want to do now is look at the distributions of those Zscores. I’ll be referencing an expected normal distribution. If Zscores are distributed normally among the population of funds, this sample is large enough that we should expect little deviation from the expected normal distribution. In this next chart, the expected normal is the background distribution shaded in gray. The actual distribution of Zscores is shown in red.
Clearly, the shift is strongly away from the expected normal distribution toward higher Zscores.
Another way to look at these data is to calculate the ratio of the actual and expected distributions. For normally distributed population this value would equal 1 across the distribution. The next chart should the actual/expected normal frequency for Zscores of all CEFs.
Note that the frequency ratio is plotted on a log axis. The single occurrence of that deeply negative Zscore would be expected 0.000287% of the time, so the ratio to the expected value of is enormous. But let’s ignore that outlier and look at the other 528 Zscores. The skew to higher than normally expected Zscores is much more evident in this chart. At the highest bin, the actual distribution is 1000 times the expected normal, and at the negative Zscores distributions are a fraction of their normally expected values.
The situation is much the same for 6 month and 3 month Zscores
To look at this phenomenon in a bit more detail I split the full set of CEFs into three broad categories based on cefanalyzer’s classification scheme: Equity, Taxable FixedIncome and TaxExempt Municipal Bonds.
Equity CEFs
For equity CEFs (n=201)n the skew to the left is extreme. There are no funds in this category with 1year Zscores below 1.5.
Taxable FixedIncome CEFs
Taxable fixedincome funds (n= 113) have similarly skewed Zscores although the right tail (highest Zscores) is less extended than it is for the equity cohort.
TaxExempt Muni Bond CEFs
By comparision, the taxexempt municipal bond CEFs (n119) are nearly normally distributed with regard to 1 year Zscores although even for this category the distribution is shifted to high Zscores
How Do ZScores Correlate with NAV and Market Returns?
One might expect Zscores to be positively correlated with returns, when funds are showing gains, investors will be inclined to bid them up and discount points will be lost. The next two charts show the relationships between Zscores and return at NAV and at market price for the entire set of CEFs.
The correlation with NAV return is positive, but with an r^{2} of 0.17 the correlation is week. The correlation with market price is, unsurprisingly, stronger, r^{2 }= 0.43. The higher correlation to market price than with NAV should be obvious as movement in market price is what moves a fund’s discount/premium status.
For equity there is no correlation with NAV return. There is a positive correlation to return at market but the correlation coeficient is trivial: r^{2 is less than 0.1 and the slope of the regression line is nearly half of the full data set.}
Taxable fixedincome CEFs are interesting if for no other reason than there has not been a single NAV loser for the past year. The positive correlation between NAV return and Zscore has an r^{2} less than 0.1. The positive correlation with price return (0.39) is close to that for the full set.
For the year, the taxexempt munibond funds show an interesting pattern. The correlation to NAV return is slightly positive but below 0.05, so essentially meaningless. But the correlation with price return is very strong, r^{2} = 0.71 and, as shown in the table below, marked by a much steep slope, nearly three times that of the full data set.
Writers, myself included, sometimes make much of Zscores when considering valuation of CEFs. But it is important to realize that most statistical assumptions assume normal distributions, and when such an assumptions is at the base of a statistical argument it must be met or the argument is invalid. I have shown here that, at the present time, the distribution of Zscores among all CEFs and at least two cohorts (Equity and Taxable FixedIncome) this is clearly not the case.
Furthermore, there is a correlation between Zscores and market price. In one sense this correlation is trivial. Market price movements affect premiums and discount; this should be obvious. But in another sense it is not. It is also telling us that investors tend to drive discounts and premiums beyond what an efficient market (as seen in NAV returns) would suggest. If this were not the case, the correlation should be flat. The strongly positive slopes on the market price regressions, especially on the muni bonds, tell us that investors are willing to pay handsome premiums for those funds that outperform the market.
What I’ve done here is look at the distributions of current Zscores across the range of CEFs. The results provide an informative snapshot of the state of the CEF marketplace at this time. For Zscores to be useful as a signaling tool it would be more valuable to have an analysis of historical distributions and changes in Zscores. In addition, we have seen that in today’s market Zscores for equity and fixedincome CEFs are strongly skewed from a normal distribution toward the higher values. This leads to an intuitive conclusion that mean reversions will lead to a categorywide move toward lower Zscores. I’ll be following up on this examination of Zscores in future articles with the intention of examining those two issues: How are Zscores distributed on an historical basis and what how well is the intuitive conclusion of mean reversion supported by past evidence?
Disclosure: I/we have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours.
I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.