The initial regression using all variables, except 5Year NAV Return and 5Year Market Return (since they both are represented in 5Year Discount), with the ThaiFund removed from the data, gave the following results.
The regression equation is
NAV Discount = 0.0283 0.000745 1Year Avg Price + 0.00043 Morn. Rating
+ 0.690 5yr discount + 0.752 1yr Avg NAV Disc + 0.0033 Bond =1
 0.0397 For=1
36 cases used 11 cases contain missing values
Predictor Coef StDev T P VIF
Constant 0.02833 0.02540 1.12 0.274
1Year A 0.0007451 0.0007835 0.95 0.349 1.2
Morn. R 0.000426 0.005394 0.08 0.938 1.2
5yr dis 0.6901 0.2821 2.45 0.021 2.0
1yr Avg 0.75227 0.09339 8.06 0.000 2.6
Bond =1 0.00334 0.01244 0.27 0.790 1.5
For=1 0.03966 0.01265 3.13 0.004 1.5
S = 0.03009 RSq = 91.6% RSq(adj) = 89.8%
Analysis of Variance
Source DF SS MS F P
Regression 6 0.285690 0.047615 52.58 0.000
Error 29 0.026259 0.000905
Total 35 0.311950
Source DF Seq SS
1Year A 1 0.017689
Morn. R 1 0.000001
5yr dis 1 0.151241
1yr Avg 1 0.106678
Bond =1 1 0.001184
For=1 1 0.008898
Unusual Observations
Obs 1Year A NAV Disc Fit StDev Fit Residual St Resid
19 12.1 0.00290 0.04748 0.01941 0.05038 2.19R
29 15.3 0.08870 0.00731 0.00803 0.09601 3.31R
41 45.6 0.06790 0.06260 0.02619 0.00530 0.36 X
R denotes an observation with a large standardized residual
X denotes an observation whose X value gives it large influence.
The Rsquared is very high at 91.6%. However some of the variables (Morningstar Rating and Bond/Equity) have very high pvalues. Thus they are not likely to be relevant to the model. Additionally, there is a data point (#29) with a very high standard residual (3.31). This represents the MuniYield Fund that invests in municipal bonds. The fund is unusual in that its 5year discount and 1year average discount are both negative, yet it is currently trading at a premium. There can be two explanations for this. Either i)something unusual has happened to the fund in recent weeks creating enough buyside interest that the discount has become a premium (this seems unlikely given the steady performance of most municipal bond funds), or ii) I made an error inputting the data. It is impossible to tell which is the cause without having access to Bloomberg, which I currently do not. Nevertheless, if either case is true, the fund is unrepresentative of most funds and should therefore be excluded from the analysis.
Therefore I reran the regression without the MuniYield Fund and without the Morningstar Rating and Bond/Equity variables. The results are listed below.
The regression equation is
NAV Discount = 0.0304 0.000957 1Year Avg Price + 0.714 5yr discount
+ 0.748 1yr Avg NAV Disc  0.0365 For=1
35 cases used 11 cases contain missing values
Predictor Coef StDev T P VIF
Constant 0.030421 0.008830 3.45 0.002
1Year A 0.0009567 0.0005743 1.67 0.106 1.0
5yr dis 0.7145 0.2097 3.41 0.002 1.8
1yr Avg 0.74772 0.06638 11.26 0.000 2.1
For=1 0.036467 0.009170 3.98 0.000 1.3
S = 0.02342 RSq = 94.4% RSq(adj) = 93.6%
Analysis of Variance
Source DF SS MS F P
Regression 4 0.276890 0.069222 126.18 0.000
Error 30 0.016457 0.000549
Total 34 0.293347
Source DF Seq SS
1Year A 1 0.019408
5yr dis 1 0.148079
1yr Avg 1 0.100726
For=1 1 0.008676
Unusual Observations
Obs 1Year A NAV Disc Fit StDev Fit Residual St Resid
19 12.1 0.00290 0.05043 0.01184 0.05333 2.64R
32 18.2 0.03460 0.08376 0.00824 0.04916 2.24R
40 45.6 0.06790 0.07211 0.02043 0.00421 0.37 X
R denotes an observation with a large standardized residual
X denotes an observation whose X value gives it large influence.
The RSquared is now even higher at 94.4% with at least 90% significance of all variables. The Fstatistic is extremely high at 126.18 providing an extremely strong regression. The VIF of all variables is relatively low and therefore collinearity is not a problem. Yet before concluding anything about the model, we must look at the error assumptions and regression diagnostics
Regression Diagnostics
I first looked at the standard residuals. There is one data point with a residual of 2.64. This is the India Growth Fund, a fund that invests primarily in Indian equities. The situation with this fund is similar to the MuniYield fund. That is, it is currently trading at a discount (.029%) despite a 1year average premium (5.59%) and a 5year premium (3.67%). However, the Indian market is a lot more volatile than the US munibond market and the current discount is very small. Therefore I cannot see a good reason to exclude this data.
With four predictors and 35 data cases, the relevant leverage factor is 2.5*(4+1)/35 = .3571. At this level there is one case with high leverage. The Source Capital Midcap Value Fund has a leverage value of .7611. This is likely due to the fact that the fund’s 1year average share price is 45.86 versus a mean for the data set of 13.76. This made me take another look at 1Year Average Share Price as a predictor in the model. Its coefficient in the model is 0.0009567. Despite a low pvalue, this coefficient is still pretty close to 0 and thus may not have much significance in the model. So I reran the regression with the three remaining predictors.
The regression equation is
NAV Discount = 0.0186 + 0.704 5yr discount + 0.766 1yr Avg NAV Disc
 0.0363 For=1
35 cases used 11 cases contain missing values
Predictor Coef StDev T P VIF
Constant 0.018622 0.005423 3.43 0.002
5yr dis 0.7040 0.2155 3.27 0.003 1.8
1yr Avg 0.76611 0.06731 11.38 0.000 2.1
For=1 0.036274 0.009428 3.85 0.001 1.3
S = 0.02408 RSq = 93.9% RSq(adj) = 93.3%
Analysis of Variance
Source DF SS MS F P
Regression 3 0.275367 0.091789 158.26 0.000
Error 31 0.017980 0.000580
Total 34 0.293347
Source DF Seq SS
5yr dis 1 0.158615
1yr Avg 1 0.108167
For=1 1 0.008586
Unusual Observations
Obs 5yr dis NAV Disc Fit StDev Fit Residual St Resid
19 0.0367 0.00290 0.05101 0.01217 0.05391 2.59R
32 0.0036 0.03460 0.08044 0.00822 0.04584 2.03R
R denotes an observation with a large standardized residual
Without 1year average share price, the Rsquared is still extremely strong at 93.9%. The remaining three variables now reject the Beta Null hypothesis at a 99% confidence level and the regression F statistic is still extremely high at 158.26. The Regression diagnostics still shows the India Growth Fund with a high standard residual, yet as explained before I am choosing to ignore this.
With three predictors now, the relevant leverage value is 2.5*(3+1)/35 = .2857. At this level there are two funds with significant leverage values. These are the First Financial Fund (#13) and the High Income Advantage Fund (#18). Their leverage values are 0.322191 and 0.288414 respectively. The High Income Advantage Fund falls has the highest 1Year Average Discount in the data set and the highest current NAV, thereby explaining its leverage. However, there is nothing that intrinsically makes this fund different from the others. The First Financial Fund has no characteristics that differentiate it from the data set. Therefore I can see no reason to exclude them. Furthermore, running the regression without them barely changed the results. Thus despite high leverage values, I can conclude that the model is stable with or without these points.
Finally, there are no data with cook values above 1. Thus there are no influence points in the data set.
Error/Residual Assumptions
The standard residuals vs. fits plot on the left shows no apparent relationship among residuals; thus the residuals are not correlated. The graph on the above left shows that the residuals are roughly normally distributed. The normality assumption is further reinforced by the histogram of the residuals below showing a roughly Gaussian distribution.
Therefore, our regression assumptions are valid and we can conclude that the model is significant.
CONCLUSION
The regression equation of the model is:
NAV Discount = 0.0186 + 0.704 5yr discount + 0.766 1yr Avg NAV Disc  0.0363 For=1
Thus with an intercept of .0186, we would expect a brand new domestic closed end fund (i.e. with no discount history) to trade at a premium of .0186. The Beta coefficients say that i)for every 1 percentage point increase in five year discount/premium, the current NAV discount/premium increases by .704 percentage points, ii)for every 1 percentage point increase in 1year average discount, the current NAV discount/premium increases by .766 percentage points, and iii)if a fund invests in foreign securities, its discount increases, or its premium decreases, by .0363 percentage points. In addition to the statistical significance of the model, the coefficients make practical sense. We would expect past discount performance to give a good indication of current performance, despite the many disclaimers in all fund prospectuses. Additionally, we would expect the higher risk in foreign funds and the lackluster performance of foreign markets, relative to the US bull market over the past five years, to detract from a fund’s discount/premium.
Now that we have a good predictive model it is interesting to test its accuracy on data not included in the study data set. I ran a prediction interval on the Colonial Intermediate High Income Fund, a domestic highyield bond fund. It’s 5year discount is .001, its one year average discount is.0294, and it doesn’t invest in foreign securities. The model predicted a fit of .04044 with a 95% prediction interval of (0.01037, 0.09125). The fit is very close to the actual premium of .0374 that the fund was trading at on October 24, 1997 and thus the model seems to be accurate with market data.
It is interesting to know that past performance is the best indicator of current performance. However, current performance tells us nothing about future performance. A model that could predict future discounts/premiums would be of strong practical value and might have saved me some money in my Asia Pacific Fund investment. Perhaps I already have a topic for next semester’s regression project.
