Trend Breaks and the Persistence of Closed-End Fund Discounts

Nazif Durmaz; Hyeongwoo Kim; Hyejin Lee; Yanfei Sun

doi:10.1515/snde-2024-0123

Article

Trend Breaks and the Persistence of Closed-End Fund Discounts

Nazif Durmaz , Hyeongwoo Kim , Hyejin Lee and Yanfei Sun

Published/Copyright: September 5, 2025

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From the journal Studies in Nonlinear Dynamics & Econometrics

Abstract

Closed-end fund (CEF) prices often exhibit large and persistent deviations from their associated net asset values (NAVs). This occurrence is puzzling, given that NAVs are openly accessible to the public for CEFs, which essentially consist of repackaged financial assets. The persistence of these deviations is particularly notable when using linear models, suggesting the need for nonlinear models to comprehend this phenomenon known as the CEF discount puzzle. To unravel this puzzle, we employ the RALS-LM framework, enabling the identification of multiple endogenously chosen trend-breaks, and conduct an analysis utilizing data from 31 CEF discounts. Our findings reveal that CEF prices tend to fluctuate around time-varying trends, which aligns with the characteristics of regime switching models. Additionally, we demonstrate that incorporating non-normal errors through moment conditions enhances efficiency at the margin. Moreover, we establish that nonlinearity solely in the form of level shifts falls short in explaining the persistent nature of CEF discounts.

Keywords: closed-end fund; CEF discount puzzle; residual augmented least squares; non-normal error; trend breaks

JEL Classification: C22; G12; G15

Corresponding author: Hyejin Lee, Department of Accounting, Economics & Finance, Tuskegee University, 1200 W Montgomery Rd, Tuskegee, AL 36088, USA, E-mail: hlee@tuskegee.edu

Appendix A: Derivation of h t in RALS(t _ν) test

Let y 1 , y 2 , ⋯ , y T be i.i.d. observations from a generalized Student’s t-distribution with the following pdf:

f y t ; ν , μ , σ = Γ ν + 1 2 σ Γ ν 2 ν π 1 + y t − μ 2 ν σ 2 − 1 2 ν + 1 ,

where μ is a location parameter, σ is a scale parameter, and ν denotes degrees of freedom. The log-likelihood function is the following.

ln ⁡ L T ν , μ , σ = 1 T ∑ t = 1 T ln ⁡ f y t ; ν , μ , σ

= 1 T ∑ t = 1 T ln ⁡ Γ ν + 1 2 − ln ⁡ σ − ln ⁡ Γ ν 2 − 1 2 ln ⁡ ν π − ν + 1 2 ln 1 + y t − μ 2 ν σ 2

= 1 T · T ln ⁡ Γ ν + 1 2 − ln ⁡ σ − ln ⁡ Γ ν 2 − 1 2 ln ⁡ ν π − 1 T ∑ t = 1 T ν + 1 2 ln ⁡ ν σ 2 + y t − μ 2 ν σ 2

Differentiating ln ⁡ L T · with respect to μ yields,

∂ ln ⁡ L T ∂ μ = − 1 T ∑ t = 1 T ν + 1 2 ν σ 2 ν σ 2 + y t − μ 2 2 y t − μ − 1 ν σ 2

= 1 T ∑ t = 1 T ν + 1 y t − μ ν σ 2 + y t − μ 2 = 1 T ∑ t = 1 T h y t

The first order condition is 1 T ∑ t = 1 T h y t = 0 . Replacing y t with e t ∼ μ = 0 , σ 2 = 1 , we obtain,

h e t = ν + 1 e t ν + e t 2

where ν is degrees of freedom. □

Appendix B: Additional tables

See Tables B1–B3

Table B1:

RALS(t _ν) test results with alternative degrees of freedom.

Categories	Fund	τ RALS t 3	ρ ̂ t 3 2	τ RALS t 7	ρ ̂ t 7 2	k ̂
Core funds	ADX	−3.470^a	0.753	−3.476^a	0.792	8
	CET	−2.211	0.612	−2.009	0.646	5
	CLM	−1.878	0.865	−1.684	0.852	10
	CRF	−1.959	0.793	−1.825	0.681	0
	FUND	−2.426^c	0.789	−2.688^c	0.731	2
	GAB	−3.367^b	0.852	−3.274^b	0.817	0
	GAM	−3.395^a	0.451	−3.220^a	0.492	9
	GRF	−1.696	0.913	−1.804	0.820	10
	RMT	−1.713	0.819	−1.769	0.825	10
	RVT	−1.740	0.828	−1.846	0.811	7
	SOR	−1.601	0.585	−1.471	0.631	3
	SPE	−4.190^a	0.615	−3.835^a	0.681	1
	TY	−3.851^a	0.385	−3.669^a	0.410	6
	USA	−1.861	0.486	−1.914	0.530	1
Corp debt BBB	ICB	−4.419^a	0.630	−4.354^a	0.661	7
	INSI	−3.701^a	0.780	−3.904^a	0.756	5
	MGF	−3.027^b	0.630	−3.050^b	0.622	8
	MIN	−0.878	0.794	−0.766	0.802	5
	PAI	−3.356^b	0.829	−3.266^b	0.820	8
	VBF	−3.141^b	0.847	−3.104^b	0.842	3
General bond	DUC	−0.736	0.753	−0.830	0.752	1
	JHI	−2.145	0.874	−2.138	0.876	1
	KMM	−3.187^b	0.676	−3.289^a	0.681	7
	KST	−2.933^b	0.631	−2.936^b	0.624	10
	MCI	−2.848^b	0.979	−2.817^b	0.945	2
	MCR	−2.735^c	0.831	−2.706^c	0.825	1
	MMT	−3.372^a	0.837	−3.486^a	0.847	1
	MPV	−2.809^b	0.777	−2.818^b	0.722	8
	PCM	−1.749	0.956	−1.409	0.889	4
	PIM	−3.130^b	0.594	−3.306^a	0.587	8
	PPT	−1.496	0.684	−1.495	0.672	10

Note: (a) τ RALS t v are the test statistics for the RALS( t v ) test. (b) ρ ̂ t v 2 indicates the ratio of the estimated error variances for the RALS( t v ). (c) We chose the optimal number of lags ( k ̂ ) based on the general-to-specific rule with a maximum 10 lags. (d) ^a, ^b and ^cdenote a 1 %, 5 % and 10 % rejection, respectively. (e) The critical values of RALS tests are dependent on ρ ̂ 2 and were obtained from Hansen (1995).

Table B2:

LM and RALS-LM test results with one level shift.

Categories	Fund	τ L M	τ RALS‐LM 2 & 3	ρ ^ 2 & 3 2	τ RALS‐LM t 5	ρ ^ t 5 2	τ RALS‐LM t 3	ρ ^ t 3 2	τ RALS‐LM t 7	ρ ^ t 7 2	T ^ B	k ^
Core funds	ADX	−1.952	−1.787	0.827	−1.741	0.723	−1.911	0.695	−1.649	0.740	2002:07	8
	CET	−3.531^b	−2.671	0.843	−2.806^c	0.714	−3.012^b	0.702	−2.874^c	0.725	2008:10	8
	CLM	−2.357	−1.906	0.869	−2.831^c	0.805	−1.954	0.850	−1.848	0.817	2008:08	0
	CRF	−1.803	−0.886	0.793	−2.123	0.712	−2.066	0.789	−2.124	0.679	2008:08	0
	FUND	−1.480	−1.204	0.861	−0.992	0.776	−0.717	0.829	−0.776	0.785	2007:10	3
	GAB	−2.312	−1.222	0.878	−1.562	0.830	−2.179	0.864	−2.075	0.856	2009:04	1
	GAM	−2.854^c	−2.580	0.720	−2.378	0.632	−2.371	0.635	−2.373	0.635	2008:10	9
	GRF	−2.315	−3.467^b	0.827	−2.221	0.985	−2.144	0.994	−2.052	0.947	2003:06	9
	RMT	−1.273	−1.672	0.920	−1.401	0.844	−1.322	0.848	−1.463	0.843	2005:12	10
	RVT	−1.505	−1.699	0.914	−1.423	0.814	−1.336	0.832	−1.479	0.809	2009:04	7
	SOR	−2.567	−2.016	0.856	−2.481	0.615	−2.859^b	0.589	−2.525	0.654	2008:10	6
	SPE	−1.911	−0.614	0.306	−1.211	0.200	−1.025	0.210	−0.843	0.201	2001:02	0
	TY	−3.671^a	−3.230^b	0.617	−2.896^b	0.433	−2.849^b	0.415	−2.755^b	0.444	2008:11	6
	USA	−2.564	−1.995	0.773	−2.180	0.590	−1.997	0.575	−1.992	0.605	2008:10	5
Corp debt	ICB	−2.769^c	−2.745^c	0.843	−2.935^b	0.741	−2.578	0.730	−2.478	0.755	2000:11	1
BBB	INSI	−2.481	−2.060	0.914	−1.698	0.876	−2.143	0.905	−2.128	0.876	2008:08	10
	MGF	−2.135	−2.660^c	0.790	−3.094^b	0.663	−3.004^b	0.669	−3.128^b	0.663	2008:08	8
	MIN	−2.004	−1.570	0.975	−1.466	0.916	−1.487	0.896	−1.462	0.927	2013:04	5
	PAI	−1.936	−1.888	0.922	−1.948	0.878	−1.681	0.875	−1.724	0.892	2009:09	9
	VBF	−3.210^b	−3.739^a	0.893	−3.433^b	0.832	−3.399^b	0.833	−3.457^b	0.835	2003:06	3
General	DUC	−2.202	−1.597	0.914	−1.211	0.781	−1.597	0.757	−1.683	0.778	2008:12	10
Bond	JHI	−2.178	−2.531	0.951	−2.429	0.873	−2.443	0.854	−2.434	0.881	2012:09	1
	KMM	−2.647	−2.503	0.906	−2.406	0.676	−2.383	0.661	−2.419	0.689	2008:10	7
	KST	−2.035	−1.746	0.713	−1.520	0.726	−1.419	0.777	−1.606	0.701	2000:12	10
	MCI	−2.546	−2.884^c	0.927	−2.426	0.978	−2.448	0.977	−2.409	0.976	2009:04	2
	MCR	−2.836^c	−2.646	0.879	−2.258	0.847	−2.676	0.860	−2.612	0.839	2014:11	10
	MMT	−2.794^c	−2.541	0.950	−2.280	0.856	−2.247	0.836	−2.306	0.867	2008:08	1
	MPV	−2.734	−2.822^c	0.824	−1.990	0.772	−2.125	0.834	−1.910	0.733	2006:12	8
	PCM	−3.078^b	−3.085^b	0.820	−2.450	0.905	−2.059	0.924	−1.933	0.888	2008:08	8
	PIM	−3.546^b	−3.566^a	0.794	−3.832^a	0.614	−3.797^a	0.617	−3.851^a	0.617	2007:09	8
	PPT	−3.143^b	−3.059^b	0.834	−3.335^b	0.582	−3.434^a	0.545	−3.274^b	0.605	2007:10	10

Note: (a) τ L M , τ RALS ‐ LM 2 & 3 , τ RALS ‐ LM t v are the test statistics for the LM, RALS-LM(2&3), and RALS-LM( t v ) tests, where v = 5, 3, 7, respectively. (b) ρ ̂ 2 & 3 2 and ρ ̂ t v 2 indicate the ratio of the estimated error variances for RALS-LM(2&3) and RALS-LM( t v ) tests, respectively. (c) k ̂ and T ̂ B denote the optimal number of lags and the estimated break point, respectively. Since the number of lags and the break date determined using max F statistic are used in both LM and RALS-LM unit root tests, we report them one time. (d) The critical values for LM and RALS-LM tests are reported in Table 11.1 of Meng et al. (2014). (e) ^a, ^b, and ^cdenote a 1 %, 5 %, and 10 % rejection, respectively.

Table B3:

LM and RALS-LM test results with two level shifts.

Categories	Fund	τ L M	τ RALS‐LM 2 & 3	ρ ^ 2 & 3 2	τ RALS‐LM t 5	ρ ^ t 5 2	τ RALS‐LM t 3	ρ ^ t 3 2	τ RALS‐LM t 7	ρ ^ t 7 2	T ^ B		k ^
Core	ADX	−1.780	−1.356	0.875	−1.499	0.772	−1.668	0.747	−1.410	0.787	2002:07	2007:11	8
Funds	CET	−3.180^b	−2.887^c	0.839	−3.361^b	0.684	−3.386^b	0.668	−3.328^b	0.696	2001:01	2008:10	5
	CLM	−2.163	−1.981	0.871	−2.598	0.825	−1.901	0.858	−1.764	0.842	2008:08	2010:01	0
	CRF	−2.118	−2.172	0.880	−2.788^c	0.801	−2.682^c	0.835	−2.840^c	0.782	2002:09	2008:08	0
	FUND	−1.523	−1.081	0.893	−0.910	0.773	−0.342	0.814	−0.487	0.819	2007:10	2009:02	4
	GAB	−2.453	−1.654	0.857	−2.213	0.861	−3.021^c	0.971	−2.923^c	0.917	2008:11	2009:04	7
	GAM	−2.658	−3.289^b	0.752	−3.341^b	0.620	−3.402^b	0.603	−3.291^b	0.633	2008:10	2008:12	9
	GRF	−3.474^b	−3.897^a	0.812	−3.895^a	0.698	−2.732^c	0.714	−2.807^c	0.669	2003:05	2004:03	1
	RMT	−1.194	−0.699	0.949	−0.997	0.898	−0.938	0.882	−1.027	0.903	2008:12	2009:04	10
	RVT	−1.903	−2.112	0.932	−1.850	0.828	−1.767	0.834	−1.897	0.830	2008:08	2009:04	7
	SOR	−2.127	−1.960	0.858	−2.067	0.675	−2.210	0.670	−2.140	0.702	2008:10	2008:12	6
	SPE	−2.010	−0.851	0.289	−1.478	0.198	−1.331	0.206	−1.074	0.200	2001:02	2009:12	0
	TY	−2.024	−1.489	0.654	−0.297	0.464	−0.897	0.454	−1.272	0.480	2008:11	2009:02	10
	USA	−2.579	−2.367	0.813	−1.773	0.630	−1.291	0.610	−1.455	0.635	2004:04	2008:10	0
Corp debt	ICB	−2.982^c	−2.911^c	0.858	−3.233^b	0.736	−2.790^c	0.720	−2.707^c	0.746	2000:11	2008:11	1
BBB	INSI	−1.856	−1.384	0.937	−0.929	0.811	−0.985	0.812	−1.106	0.834	2008:08	2009:01	1
	MGF	−2.342	−2.826^c	0.803	−3.508^a	0.677	−3.465^a	0.676	−3.523^a	0.681	2008:08	2008:10	8
	MIN	−1.412	−1.033	0.981	−1.039	0.907	−1.072	0.889	−1.023	0.919	2010:11	2013:04	5
	PAI	−2.600	−2.880^c	0.959	−2.909^c	0.926	−2.417	0.955	−2.504	0.945	2008:12	2009:09	9
	VBF	−3.406^b	−3.665^a	0.916	−3.719^a	0.835	−3.199^b	0.855	−3.257^b	0.858	2003:06	2008:12	6
General	DUC	−3.173^b	−3.018^b	0.905	−1.976	0.815	−1.811	0.791	−2.083	0.823	2006:02	2008:12	1
Bond	JHI	−2.309	−2.467	0.977	−2.286	0.918	−2.288	0.907	−2.293	0.923	2011:07	2012:09	1
	KMM	−2.540	−2.499	0.931	−2.647^c	0.631	−2.742^c	0.585	−2.595^c	0.663	2000:11	2008:10	7
	KST	−2.289	−2.042	0.761	−2.110	0.791	−1.997	0.843	−2.189	0.762	2000:12	2001:07	10
	MCI	−2.673	−3.066^b	0.939	−2.661	0.991	−2.654	0.997	−2.668	0.986	2008:09	2009:04	2
	MCR	−2.341	−1.723	0.886	−1.513	0.859	−1.752	0.871	−1.675	0.849	2008:08	2014:11	10
	MMT	−3.126^b	−3.239^b	0.956	−3.078^b	0.874	−3.034^b	0.859	−3.101^b	0.884	2008:08	2008:12	1
	MPV	−2.202	−1.971	0.872	−1.692	0.819	−1.948	0.834	−1.817	0.804	2003:11	2006:12	10
	PCM	−2.078	−2.044	0.888	−2.101	0.895	−2.050	0.928	−2.133	0.875	2007:02	2008:08	4
	PIM	−3.389^b	−3.222^b	0.787	−3.455^a	0.605	−3.408^b	0.603	−3.484^a	0.613	2007:09	2008:09	8
	PPT	−2.759	−2.275	0.827	−2.098	0.563	−2.176	0.510	−2.072	0.592	2007:10	2008:09	10

Note: (a) τ L M , τ RALS ‐ LM 2 & 3 , τ RALS ‐ LM t v are the test statistics for the LM, RALS-LM (2&3), and RALS-LM ( t v ) tests, where v = 5, 3, 7, respectively. (b) ρ ̂ 2 & 3 2 and ρ ̂ t v 2 indicate the ratio of the estimated error variances for RALS-LM (2&3) and RALS-LM ( t v ) tests, respectively. (c) k ̂ and T ̂ B denote the optimal number of lags and the estimated break points, respectively. Since the number of lags and the break dates determined using max F statistic are used in both LM and RALS-LM unit root tests, we report them one time. (d) The critical values for LM and RALS-LM tests are reported in Table 11.1 of Meng et al. (2014). (e) ^a, ^b, and ^cdenote a 1 %, 5 %, and 10 % rejection, respectively.

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Received: 2024-11-08

Accepted: 2025-08-08

Published Online: 2025-09-05

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https://doi.org/10.1515/snde-2024-0123

Keywords for this article

closed-end fund; CEF discount puzzle; residual augmented least squares; non-normal error; trend breaks

Trend Breaks and the Persistence of Closed-End Fund Discounts

Article

Abstract

Appendix A: Derivation of h t in RALS(t ν ) test

Appendix B: Additional tables

References

Appendix A: Derivation of h t in RALS(t _ν) test