Static Markowitz mean-variance portfolio selection model with long-term bonds

Dennis Ikpe; Romeo Mawonike; Frederi Viens

doi:10.3934/naco.2022030

Article Contents

2023, Volume 13, Issue 3&4: 523-554. Doi: 10.3934/naco.2022030

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Static Markowitz mean-variance portfolio selection model with long-term bonds

1.
Michigan State University, USA
2.
Great Zimbabwe University, Zimbabwe
3.
Rice University, USA

^*Corresponding author: Frederi Viens
^*Corresponding author: Frederi Viens

Received: April 2022

Revised: September 2022

Early access: October 2022

Published: September 2023

The first author is supported by [National Research Foundation, South Africa]

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Abstract

We propose a static Markowitz mean-variance portfolio selection model suitable for long-term zero-coupon bonds. The model uses a multi-factor term structure model of Vasicek (Ornstein-Uhlenbeck) type to compute the portfolio's expected return and its variance in the model. German Government zero-coupon bonds with short to very long time to maturity are considered; the data spans August 2002 to December 2020. The main investment assumption is the re-investment of cash flows of zero-coupon bonds with maturities less than the planning horizon at the current spot interest rate. Solutions for the zero-coupon holding vector and the tangency portfolio are obtained in closed form. Model parameters are estimated under an assumption of modeling ambiguity which takes the form of Knightian uncertainties at the level of the latent factors, allowing the use of a Kalman filter. Different investment strategies are examined on various risk portfolios. Results show that one- and two-factor Vasicek models produce attractive out-of-sample portfolio predictions in terms of the Sharpe ratio especially on long-term investments. It is also noted that a small number of risky bonds can adequately produce very attractive portfolio risk-return profiles.

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Mathematics Subject Classification: Primary: 91G10, 91G30; Secondary: 60G35, 60H10.

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Figure 1. Portfolio frontier of two risky zero-coupon bonds; 24-year and 30-year under two-factor Vasicek model invested over 25-year investment period

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Figure 2. Vasicek: Expectation structure of r(T) over one-year investment horizon

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Figure 3. Vasicek: Volatility structure of r(T) over one-year investment horizon

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Figure 4. Vasicek: Zero coupon bond price over one year investment period

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Figure 5. Portfolio frontier of two risky zero coupon bonds; 4-year and 10-year, under one-factor Vasicek model invested over one-year period

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Figure 6. Portfolio frontier of five risky zero-coupon bonds; 4-year, 6-year, 7-year, 8-year and 10-year under two-factor Vasicek model invested over 5-year investment period

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Table 1. Vasicek: parameter estimation based on one- two- and three-factor models with bond maturities: $ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 $

Parameter	One-factor	two-factor	Three-factor
$ \mu $	0.0255	0.0292	0.0243
$ \xi_{1} $	0.3245	0.2554	0.2966
$ \xi_{2} $		0.3017	0.3288
$ \xi_{3} $			0.2860
$ c_{1} $	0.0256	0.0208	0.0211
$ c_{2} $		0.0228	0.0232
$ c_{3} $			0.0191
$ \lambda_{1} $	0.1172	0.1151	0.1124
$ \lambda_{2} $		0.1073	0.0963
$ \lambda_{3} $			0.0954
$ \rho_{1} $		0.9396	0.9531
$ \rho_{2} $			0.8077
$ \rho_{3} $			0.8247
$ L $	1033.4	1048.8	1047.0
$ \sigma_{\epsilon2} $	0.00267	0.00019	0.00018
$ \sigma_{\epsilon3} $	0.00268	0.00021	0.00019
$ \sigma_{\epsilon4} $	0.00266	0.00021	0.00018
$ \sigma_{\epsilon5} $	0.00263	0.00021	0.00020
$ \sigma_{\epsilon6} $	0.00259	0.00020	0.00019
$ \sigma_{\epsilon7} $	0.00092	0.00020	0.00019
$ \sigma_{\epsilon8} $	0.00251	0.00020	0.00018
$ \sigma_{\epsilon9} $	0.00248	0.00021	0.00018
$ \sigma_{\epsilon10} $	0.00246	0.00021	0.00019

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Table 2. Vasicek: Parameter estimation based on one-two-and three-factor models with bond maturities: $ 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30 $. Here the shorthand notation $ \sigma_{\epsilon i} $ represents the estimated standard deviation $ \sigma(\epsilon_{T_i}) $

Parameter	One-factor	two-factor	Three-factor
$ \mu $	0.0353	0.0405	0.0369
$ \xi_{1} $	0.3245	0.2551	0.2523
$ \xi_{2} $		0.3011	0.3297
$ \xi_{3} $			0.2846
$ c_{1} $	0.0226	0.0184	0.0165
$ c_{2} $		0.0200	0.0203
$ c_{3} $			0.0167
$ \lambda_{1} $	0.1172	0.1152	0.1125
$ \lambda_{2} $		0.1072	0.0999
$ \lambda_{3} $			0.0941
$ \rho_{1} $		0.9396	0.9515
$ \rho_{2} $			0.8050
$ \rho_{3} $			0.8209
$ L $	1042	1057	1052
$ \sigma_{\epsilon21} $	0.00236	0.00019	0.00018
$ \sigma_{\epsilon22} $	0.00236	0.00019	0.00019
$ \sigma_{\epsilon23} $	0.00236	0.00018	0.00018
$ \sigma_{\epsilon24} $	0.00236	0.00018	0.00020
$ \sigma_{\epsilon25} $	0.00235	0.00019	0.00019
$ \sigma_{\epsilon26} $	0.00237	0.00019	0.00019
$ \sigma_{\epsilon27} $	0.00237	0.00018	0.00018
$ \sigma_{\epsilon28} $	0.00238	0.00019	0.00018
$ \sigma_{\epsilon29} $	0.00237	0.00019	0.00019
$ \sigma_{\epsilon30} $	0.00238	0.00019	0.00018

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Table 3. Vasicek: Spot prices of long-term maturity zero coupon bonds

1-factor
$ T_{i} $	20	21	22	23	24	25	26	27	28	29	30
$ P(0, T_{i}) $	0.2454	0.2281	0.2120	0.1970	0.1831	0.1702	0.1582	0.1471	0.1367	0.1270	0.1181
2-factor
$ T_{i} $	20	21	22	23	24	25	26	27	28	29	30
$ P(0, T_{i}) $	0.2106	0.1939	0.1785	0.1643	0.1512	0.1392	0.1281	0.1180	0.1086	0.100	0.0920
3-factor
$ T_{i} $	20	21	22	23	24	25	26	27	28	29	30
$ P(0, T_{i}) $	0.2477	0.2297	0.2130	0.1975	0.1832	0.1698	0.1575	0.1460	0.1354	0.1255	0.1164

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Table 4. Vasicek: Expected zero coupon bond prices and standard deviations over 25-year investment period

1-factor
$ T_{i} $	20	21	22	23	24	25	26	27	28	29	30
$ \mathbb{E}_{0} $[$ P(25, T_{i}) $]	1.3689	1.2749	1.1894	1.1136	1.0495	1.000	0.9566	0.9016	0.8441	0.7875	0.7334
std[$ P(25, T_{i}) $]	0.0038	0.0033	0.0028	0.0025	0.0022	0.000	0.0018	0.0016	0.0014	0.0012	0.0011
2-factor
$ T_{i} $	20	21	22	23	24	25	26	27	28	29	30
$ \mathbb{E}_{0} $[$ P(25, T_{i}) $]	1.4032	1.3004	1.2085	1.1277	1.0585	1.000	0.9488	0.8910	0.8319	0.7735	0, 7170
std[$ P(25, T_{i}) $]	0.0042	0.0030	0.0019	0.0010	0.0003	0.000	0.0002	0.0006	0.0009	0.0011	0.0011
3-factor
$ T_{i} $	20	21	22	23	24	25	26	27	28	29	30
$ \mathbb{E}_{0} $[$ P(25, T_{i}) $]	1.3587	1.2711	1.1918	1.1206	1.0573	1.000	0.9500	0.8972	0.8445	0.7926	0.7421
std[$ P(1, T_{i}) $]	0.0114	0.0087	0.0063	0.0043	0.0028	0.000	0.0022	0.0028	0.0032	0.0034	0.0034

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Table 5. Vasicek: Expected zero coupon bond holding period returns over 25-year investment horizon

1-factor
$ T_{i} $	20	21	22	23	24	25	26	27	28	29	30
ExpReturn(%)	457.8	458.9	461.0	465.3	473.2	487.5	504.7	512.9	517.5	520.0	521.0
2-factor
$ T_{i} $	20	21	22	23	24	25	26	27	28	29	30
ExpReturn(%)	566.3	570.7	577.0	586.4	600.0	618.4	640.7	655.1	666.0	673.5	679.3
3-factor
$ T_{i} $	20	21	22	23	24	25	26	27	28	29	30
ExpReturn(%)	448.5	453.4	459.5	467.4	477.1	488.9	503.2	514.5	523.7	531.6	537.5

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Table 6. Expected returns, Expected terminal wealth, Sharpe Ratios, and risk, over $ 25 $-year investment period

Risky bonds(yrs)	Exp Returns (%)	$ E[\hbar_{25}] $	Sharpe Ratio	Risk (%)
		1-factor
27	500.2	5.894	2.23	5.7
24;30	497.1	5.882	1.68	5.7
24;27;30	497.3	5.892	1.72	5.7
20;21;22;23;24;26;27;28;29;30	489.2	5.868	0.26	6.5
		2-factor
27	655.1	7.206	4.89	7.5
24;30	639.7	7.198	3.80	6.0
24;27;30	638.4	7.208	3.28	6.1
20;21;22;23;24;26;27;28;29;30	621.5	7.172	0.46	6.7
		3-factor
27	514.5	5.908	4.83	5.3
24;30	507.3	5.907	3.23	5.7
24;27;30	504.6	5.913	2.34	6.7
20;21;22;23;24;26;27;28;29;30	491.6	5.880	0.40	6.8

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Table 7. Vasicek: Expected zero coupon bond prices and standard deviations over one year investment period

1-factor
$ T_{i} $	1	2	3	4	5	6	7	8	9	10
$ \mathbb{E}_{0} $[$ P(1, T_{i}) $]	1	0.9694	0.9280	0.8840	0.8400	0.7971	0.7559	0.7165	0.6791	0.6436
std[$ P(1, T_{i}) $]	0.0000	0.0755	0.0721	0.0686	0.0656	0.0616	0.0592	0.0556	0.0529	0.0500
2-factor
$ T_{i} $	1	2	3	4	5	6	7	8	9	10
$ \mathbb{E}_{0} $[$ P(1, T_{i}) $]	1	0.9563	0.9066	0.8555	0.8046	0.7550	0.7073	0.6619	0.6188	0.5783
std[$ P(1, T_{i}) $]	0.0000	0.0693	0.0787	0.0848	0.0883	0.0894	0.0883	0.0860	0.0831	0.0800
3-factor
$ T_{i} $	1	2	3	4	5	6	7	8	9	10
$ \mathbb{E}_{0} $[$ P(1, T_{i}) $]	1	0.9626	0.9231	0.8838	0.8449	0.8068	0.7696	0.7336	0.6988	0.6653
std[$ P(1, T_{i}) $]	0.0000	0.0748	0.0927	0.1063	0.1140	0.1183	0.1191	0.1140	0.1170	0.1127

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Table 8. Vasicek: Expected zero coupon bond holding period returns over one - year investment horizon

1-factor
$ T_{i} $	1	2	3	4	5	6	7	8	9	10
ExpReturn(%)	3.1660	4.4612	4.9773	5.2381	5.3820	5.4505	5.4990	5.5072	5.5158	5.5428
2-factor
$ T_{i} $	1	2	3	4	5	6	7	8	9	10
ExpReturn(%)	4.5697	5.4820	5.9731	6.3261	6.5695	6.7440	6.8590	6.9651	7.0032	7.0728
3-factor
$ T_{i} $	1	2	3	4	5	6	7	8	9	10
ExpReturn(%)	3.8853	4.2791	4.4467	4.6041	4.7224	4.8337	4.9073	4.9800	5.0353	5.0861

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Table 9. Expected returns, Expected terminal wealth and Sharpe Ratios of risky zero-coupon bond portfolios over one-year investment period

Risky bonds(yrs)	Exp Returns (%)	$ E[\hbar_{1}] $	Sharpe Ratio	Risk (%)
		1-factor
7	5.4990	1.039	0.78	3.0
4;10	5.3900	1.042	0.57	3.9
4;7;10	5.3720	1.043	0.50	4.4
3;7;8;10	5.3820	1.043	0.47	4.7
4;6;7;8;10	5.4475	1.045	0.46	5.0
2;3;...; 10	5.1885	1.047	0.47	6.2
		2-factor
7	6.8590	1.054	0.37	6.2
4;10	6.6990	1.055	0.39	5.5
4;7;10	6.6851	1.057	0.34	6.3
3;7;8;10	6.7270	1.057	0.32	6.7
4;6;7;8;10	6.7934	1.059	0.30	7.3
2;3;...; 10	6.4894	1.060	0.24	8.2
		3-factor
7	4.9073	1.043	0.09	11.9
4;10	4.8451	1.044	0.08	7.3
4;7;10	4.8172	1.045	0.11	8.5
3;7;8;10	4.8550	1.045	0.11	8.9
4;6;7;8;10	4.8822	1.046	0.11	9.8
2;3;...; 10	4.7184	1.046	0.09	10.7

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Table 10. Vasicek: Expected zero-coupon bond prices and standard deviations over 5-year investment period

1-factor
$ T_{i} $	1	2	3	4	5	6	7	8	9	10
$ \mathbb{E}_{0} $[$ P(5, T_{i}) $]	1.1976	1.1380	1.0841	1.0378	1.000	0.9694	0.9280	0.8840	0.8400	0.7971
std[$ P(5, T_{i}) $]	0.0066	0.0062	0.0059	0.0057	0.000	0.0053	0.0051	0.0049	0.0046	0.0044
2-factor
$ T_{i} $	1	2	3	4	5	6	7	8	9	10
$ \mathbb{E}_{0} $[$ P(5, T_{i}) $]	1.2500	1.1751	1.1083	1.0502	1.000	0.9563	0.9066	0.8555	0.8046	0.7550
std[$ P(5, T_{i}) $]	0.0052	0.0041	0.0030	0.0017	0.000	0.0016	0.0025	0.0031	0.0033	0.0035
3-factor
$ T_{i} $	1	2	3	4	5	6	7	8	9	10
$ \mathbb{E}_{0} $[$ P(5, T_{i}) $]	1.1926	1.1390	1.0893	1.0435	1.000	0.9626	0.9231	0.8838	0.8449	0.8068
std[$ P(5, T_{i}) $]	0.0089	0.0077	0.0064	0.0052	0.000	0.0048	0.0055	0.0060	0.0063	0.0065

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Table 11. Vasicek: Expected zero coupon bond holding period returns over five - year investment horizon

1-factor
$ T_{i} $	1	2	3	4	5	6	7	8	9	10
ExpReturn(%)	23.5	22.6	22.6	23.5	25.5	28.2	29.5	30.2	30.5	30.7
2-factor
$ T_{i} $	1	2	3	4	5	6	7	8	9	10
ExpReturn(%)	30.7	29.6	29.5	30.5	32.4	35.2	37.0	38.3	39.1	40.0
3-factor
$ T_{i} $	1	2	3	4	5	6	7	8	9	10
ExpReturn(%)	23.8	23.4	23.3	23.5	23.9	25.1	25.8	26.5	27.0	27.9

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Table 12. Expected returns, Expected terminal wealth and Sharpe Ratios over $ 5 $-year investment period

Risky bonds(yrs)	Exp Returns (%)	$ E[\hbar_{5}] $	Sharpe Ratio	Risk (%)
		1-factor
7	29.5	1.271	0.58	6.9
4;10	27.1	1.263	0.23	7.1
4;7;10	27.6	1.268	0.30	7.0
3;7;8;10	28.25	1.271	0.38	7.2
4;6;7;8;10	28.42	1.273	0.41	7.1
1;2;3;...; 10	26.54	1.261	0.14	75
		2-factor
7	37.0	1.339	0.62	7.7
4;10	35.25	1.337	0.51	5.6
4;7;10	35.48	1.343	0.50	6.2
3;7;8;10	36.20	1.345	0.56	6.6
4;6;7;8;10	36.20	1.347	0.56	6.8
1;2;3;...; 10	34.09	1.333	0.26	6.2
		3-factor
7	25.8	1.246	0.26	7.4
4;10	25.70	1.250	0.31	5.9
4;7;10	25.48	1.251	0.22	7.1
3;7;8;10	25.88	1.252	0.26	7.5
4;6;7;8;10	25.76	1.252	0.24	7.9
1;2;3;...; 10	24.89	1.247	0.16	6.2

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