# Income uncertainty, substitution effect and relative yield spreads.

The term structure of relative yield spreads between U.S. taxable and tax-exempt bonds is often discussed in economics and finance (Stiglitz, 1988; Van Horne, 1994). It is commonly recognized that the tax rate of the marginal investor affects the pricing of taxable and tax-exempt bonds. There is, however, little agreement on how the tax rate as one form of market frictions works, and why relative yield spreads of the two types of long-term bonds are generally much smaller than, but occasionally greater than that of their short-term counterparts.The theoretical attempts to study relative yield spreads of taxable and tax-exempt bonds started from Miller (1977) and Fama (1977). They noted that the corporate-capital-structure and bank tax-arbitrage could affect current relative yield spreads. Having noticed the inaccurate predictions and difficulties in reconciling the "implied tax rate" (the relative yield spread) with the actual average tax rate faced by nonbank corporations and banks, Trczinka (1982) tackled the issue by incorporating a time-varying default risk premium for prime-grade municipal bonds into his model, while Buser and Hess (1986) extended the Miller model by including financing costs. Skelton (1983), however, employed the historical restrictions of Regulation Q to explain the yield spreads.

The Tax Reform Act of 1986 had a profound impact on the tax-exempt bond market. Although relative yield spreads have become consistently narrower since then (see Table 1), the downward-sloping pattern of the term structure appeared [TABULAR DATA FOR TABLE 1 OMITTED] unchanged in general. In the 1980s, the municipal (tax-exempt) bonds became increasingly popular investment instruments among individual investors and mutual fund management firms.

Since then the research focus gradually shifted to the term structure of relative yield spreads. The relative yield spread of taxable and tax-exempt bonds for a particular term-to-maturity is defined as the ratio of the yield spread of the two types of bonds of that term-to-maturity to the yield of the corresponding taxable bond. Kochin and Park (1988) identified the downward-sloping pattern and suggested that it provides arbitrage opportunities (see Table 1). Other researchers, such as Piros (1987), Mitchell and McDade (1992), Green (1993), and Kryzanowski, Xu, and Zhang (1995), attempted to explain the factors that may cause the downward-sloping term structure. Piros (1987) noted that the yields of tax-exempt bonds must be sufficiently high to attract risk-averse investors whose future income is uncertain. By examining the behavior of property and liability insurance companies in the tax-exempt bond market, Mitchell and McDade (1992) found that the downward-sloping term structure of relative yield spreads may result from the market segmentation. Green (1993) suggested that the downward-sloping pattern is caused by the investor's taxable bond trading strategy. Kryzanowski, Xu, and Zhang (1995) found that, in addition to the time-varying risk premium and forward tax rate, the tax timing option and expected future inflation could also cause the term structure downward-sloping.

Researchers who have studied taxable versus tax-exempt bond yields have focused rather narrowly on one or another factor that may account for the observed pattern of relative yield spreads. None of these papers has considered the yield spread effect of investors who choose taxable and tax-exempt bonds in a multi-period context that includes uncertainty about future income and tax rates. This paper attempts to do exactly that based on Lucas (1978), Piros (1987), and Sargent (1987). The paper is limited, however, in that it makes no attempt to incorporate the empirical findings of other researchers, and that it assumes that the investors are not constrained from rearranging their portfolios freely. Some investors, in particular those institutional investors in these markets often lack complete flexibility to rearrange their portfolios. Their presence in the taxable and tax-exempt bond markets could be an important force affecting the term structure.(1)

The rest of the paper is organized as follows. In Section I, the proposed model is introduced and discussed. Section II contains the theoretical analysis and predictions. Finally, some concluding remarks are offered in Section III.

I. MODEL

This section establishes a model that describes the risk-averse investor's dynamic portfolio choice in a nominal framework. It is different from the conventional asset pricing model (Lucas, 1978; Sargent, 1987) in that income, assets, returns, and tax payments are all nominal. Within the model framework, this paper shows how uncertain future income and tax rates, and income and substitution effects are related to the term structure of relative yield spreads.

In this model, the representative investor is assumed to be rational and risk-averse. The investor receives income, [y.sub.t], and makes a consumption decision, [c.sub.t], and an investment decision at time t. Exogenous uncertainty is characterized by a time-homogeneous Markov process {[X.sub.t]}, where [X.sub.t] is the state at time t. The income process {[[Gamma].sub.t]} can be expressed as a function of {[X.sub.t]}. Thus, the source of uncertainty is from future income and the uncertainty is resolved only when income becomes known.

The time-t traded short- and long-term securities are default-risk free, taxable one- and two-period bonds, denoted as [Mathematical Expression Omitted] and [Mathematical Expression Omitted], respectively. Their tax-exempt counterparts are denoted as [L.sub.1t] and [L.sub.2t], respectively.(2) These securities are denominated in units of time-t income. The after-tax gross returns on these securities, [Mathematical Expression Omitted] and [R.sub.it] (i = 1, 2), are the extra resources available to the investor when the bonds mature. The pre-tax net returns on taxable and tax-exempt bonds, [Mathematical Expression Omitted] and [r.sub.it] (i = 1, 2), are known when the bonds are purchased at time t. The tax rates imposed on pre-tax net returns at time t + 1 and t + 2 are denoted as [[Tau].sub.t+1] and [[Tau].sub.t+2], respectively. The after-tax gross returns and pre-tax net returns are related by [Mathematical Expression Omitted] and [R.sub.it] = 1 + [r.sub.it] for i = 1, 2.

The government collects taxes from both income and yields. The tax rate imposed on income at time t, [[Tau].sub.t], is a monotonic increasing function of the level of income and predictable with respect to the information set at time t, i.e., [[Tau].sub.t] = f([y.sub.t]([X.sub.t])), where [[Tau].sub.t] [element of] (0, 1). The same tax rate, [[Tau].sub.t], is also the one imposed on the yield of one-period taxable bonds purchased at time t - 1 because the taxable yield is paid at time t. The tax rate, imposed as of time t + 1 on the yield of two-period bonds purchased at time t - 1, is uncertain because the taxable yield is paid at time t+1, when the state is still unknown. This tax rate is denoted as [[Tau].sub.t+1] = f([y.sub.t+1]([X.sub.t+1])), where [[Tau].sub.t+1] [element of] (0, 1). Clearly, the uncertainty of the future tax rate implies the possibility of changes in the tax rate applicable to the investor due to possible changes in future income. Thus, the focus will be on the investor's expectations of uncertain future income and tax rates. Since uncertainty is characterized by a time-homogeneous Markov process, we specify [E.sub.t]([[Tau].sub.t+i]) = [[Tau].sub.t], i = 1, 2; that is, the expectations of the future tax rate based on the information set available at the current time is simply the current tax rate. The investor's optimization problem is constrained by

[Mathematical Expression Omitted] (1)

where t = 0, 1,...; [y.sub.0] and [L.sub.0] are initial values of income and assets, respectively. Since at time t only [[Tau].sub.t] applies to the budget constraint, [[Tau].sub.t] does not introduce any uncertainty at time t. This budget constraint suggests that the sum of consumption and new investment in bonds should be less than or equal to the sum of after-tax income and assets available for reinvestment.

The investor has a time-separable, strictly concave, monotonic and twice differentiable utility function, u = u(c). The discount rate is defined as [Beta] which is less than one and greater than zero. The set of choice variables at time t is defined by [Mathematical Expression Omitted]. Then the individual investor solves the following dynamic portfolio choice problem

[Mathematical Expression Omitted] (2)

subject to Equation 1. At time t, the variables [y.sub.t], [[Tau].sub.t], [d.sub.t], [R.sub.1t], [Mathematical Expression Omitted], [R.sub.2t], and [Mathematical Expression Omitted] are known, conditional on the information set available at time t.

II. THEORETICAL PREDICTIONS

This section shows how the term structure of relative yield spreads and uncertainty introduced by the exogenously-given income process are related.

Solve the problem by forming the Lagrangian function,

[Mathematical Expression Omitted], (3)

where {[[Lambda].sub.t]} is a sequence of random Lagrange multipliers. Given that the solution exists for this Markov model,(3) some of the first-order conditions are

u[prime]([c.sub.t]) - [[Lambda].sub.t] = 0, (4)

-[[Lambda].sub.t] + [Beta][E.sub.t]([[Lambda].sub.t+1][R.sub.1t]) = 0, (5)

[Mathematical Expression Omitted], (6)

[[Lambda].sub.t] + [[Beta].sub.2][E.sub.t]([[Lambda].sub.t+2][R.sub.2t]) = 0, (7)

and

[Mathematical Expression Omitted]. (8)

Rearranging these conditions yields

u[prime]([c.sub.t]) = [Beta][E.sub.t] [u[prime]([c.sub.t+1])[R.sub.1t]], (9)

[Mathematical Expression Omitted], (10)

u[prime]([c.sub.t]) = [[Beta].sup.2][E.sub.t][u[prime]([c.sub.t+2])[R.sub.2t]], (11)

and

[Mathematical Expression Omitted]. (12)

From Equations 9 and 10 with the time subscripts shifted backward once and the definition of conditional covariance, it can be shown

[Mathematical Expression Omitted]. (13)

Since [E.sub.t]([[Tau].sub.t]) = [[Tau].sub.t] and there is no uncertainty in both income and consumption at time t, the right-hand side of Equation 13 is zero, yielding the (indifference) tax rate

[Mathematical Expression Omitted]. (14)

Now consider the future relative yield spreads. From Equations 9, 10, 11, and 12, it can be shown that

[Mathematical Expression Omitted], (15)

and

[Mathematical Expression Omitted]. (16)

Equations 15 and 16 can be rearranged, respectively, as

[Mathematical Expression Omitted], (17)

where [[Delta].sub.1] = [Cov.sub.t] [u[prime]([c.sub.t+1]), [Mathematical Expression Omitted], and

[Mathematical Expression Omitted], (18)

where [Mathematical Expression Omitted].

Equations 17 and 18 indicate that there are two major factors affecting the future relative yield spreads. The first factor is the expected value of the future tax rate. It is noted that [[Tau].sub.t+1] and [[Tau].sub.t+2] are unknown at time t. Given the setup of the model, [E.sub.t]([[Tau].sub.t+i]) = [[Tau].sub.t] for i = 1, 2. This simply says that when the information about the future tax rate is not in the information set, using the (indifference) tax rate to form expectations is the optimal choice.

The second factor is the scaled covariance between the future marginal utility derived from the future consumption and the future tax rate,

[Mathematical Expression Omitted], (19)

where i = 1, 2. Because [E.sub.t]u[prime]([c.sub.t+i]) for i = 1, 2 is a positive term based on the monotonic assumption of u(c), the sign of [[Delta].sub.i] is dependent on the sign of the covariance. It is known that a higher (lower) future tax rate [[Tau].sub.t+i], which is dependent upon a higher (lower) future income, leads to a lower (higher) future after-tax yield, [Mathematical Expression Omitted], which may affect the future consumption in the following two ways.(4)

CASE 1

Case 1 (the income effect) is the case where a lower (higher) future after-tax yield makes the future consumption lower (higher) and hence the marginal utility derived from it becomes higher (lower). In this case, the terms [[Delta].sub.i] (i = 1, 2) are less than zero. [E.sub.t]([[Tau].sub.t+i]) = [[Tau].sub.t] for i = 1, 2, and Equations 14, 17, and 18 imply that the future relative yield spreads are higher than the current relative yield spread:

[Mathematical Expression Omitted], (20)

and

[Mathematical Expression Omitted]. (21)

Based on the above inequalities, it is clear that the term structure would be downward-sloping if

[Mathematical Expression Omitted].

As term-to-maturity increases, relative yield spreads would approach the (indifference) tax rate from above. Hence, we have [Mathematical Expression Omitted]. Since [Mathematical Expression Omitted] usually exceeds [Mathematical Expression Omitted], i.e., the yield curve of taxable bonds is normally upward-sloping, [Mathematical Expression Omitted] reflects the fact that the income effect is greater in a more distant future because of greater future uncertainty.

CASE 2

Case 2 (the substitution effect) is the case where a lower (higher) future after-tax yield leads more (less) investment in other possibilities (such as bonds with shorter terms-to-maturity or that are tax-exempt). Such portfolio rebalancing makes the future consumption higher (lower) and the marginal utility derived form it becomes lower (higher). In this case, the terms [[Delta].sub.i] (i = 1, 2) are greater than zero. [E.sub.t]([[Tau].sub.t+i]) = [[Tau].sub.t] for i = 1, 2, and Equations 14, 17, and 18 imply that the future relative yield spreads are lower than the current relative yield spread:

[Mathematical Expression Omitted], (22)

and

[Mathematical Expression Omitted]. (23)

As can be derived from the above inequalities, the term structure would be downward-sloping if

[Mathematical Expression Omitted].

As term-to-maturity increases, relative yield spreads would fall way from the (indifference) tax rate. Hence, we have [Mathematical Expression Omitted]. Since [Mathematical Expression Omitted] usually exceeds [Mathematical Expression Omitted], [Mathematical Expression Omitted] reflects the fact that the substitution effect is greater in a more distant future because of greater future uncertainty.

In summary, the theoretical conjecture suggests that the persistent pattern of the downward-sloping term structure could potentially result from income and substitution effects. When the income effect dominates, relative yield spreads would approach the (indifferent) tax rate from above. When the substitution effect dominates, the spreads would fall from the tax rate. The income uncertainty and hence tax rate uncertainty in a more distant future lead to greater income and substitution effects. Since relative yield spreads tend to fall away from the (indifference) tax rate as term-to-maturity increases, it can be inferred that this may reflect mainly the current and future substitution effects, and the investors' efforts to rebalance their portfolio in the face of future income uncertainty. The substitution effect in the model may also be reflected in the various degrees of market segmentation observed in real life.(5)

III. CONCLUSION

This paper extends the analysis of the impact of income uncertainty on relative yield spreads between taxable and tax-exempt bonds based on the investor's dynamic portfolio choice model. The analysis indicates that the slope of the term structure of relative yield spreads could be related to the investors' decisions and corresponding income and substitution effects. The persistent pattern of the term structure may be largely due to the substitution effect since relative yield spreads tend to fall away from the (indifference) tax rate as term-to-maturity increases.

As is noted, the downward-sloping term structure started to flatten from the late 1980s and this continued gradually into the 1990s (see Table 1). Within the proposed dynamic portfolio choice model, this tendency may be due to the decreasing importance of substitution effects. It is also possible that other factors such as institutional changes may have come into play. This relatively recent phenomenon obviously represents a new challenge for future research.

Acknowledgment: Research support from the Concordia Professional Development Fund, Dalhousie Research Development Fund, and Canadian SSHRC research grant is gratefully acknowledged. I would like to thank L. Mazany and anonymous referees for their helpful and constructive comments. The usual caveats apply.

NOTES

* Direct all correspondence to: Kuan Xu, Department of Economics, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5 <kxu@is.dal.ca>.

1. The representative agent model is also limited in that it is not a suitable framework for studying heterogeneous behaviors among taxable and tax-exempt bond investors.

2. As can be seen later, the selection of one- and two-period coupon bonds is made for simplicity and the analysis in this model could be fairly general.

3. Duffie (1988) provides a general proof of the existence of a unique solution of a more general Markovian model.

4. The author would like to thank one referee for very stimulating comments on the two cases.

5. See Fama (1977), and Mitchell and McDade (1992), among others.

REFERENCES

Buser, Stephen A. and Patrick J. Hess. 1986. "Empirical Determinants of the Relative Yields on Taxable and Tax-exempt Securities," Journal of Financial Economics, 17: 335-355.

Duffie, Darrell. 1988. Security Markets: Stochastic Models. New York: Academic Press.

Green, Richard C. 1993. "A Simple Model of the Taxable and Tax-Exempt Yield Curves," The of Financial Studies, 2: 233-264.

Fama, Eugene F. 1977. "A Pricing Model for the Municipal Bond Market," Mimeo, University of Chicago.

Kochin, Levis A. and Richard W. Parks. 1988. "Was the Tax-Exempt Bond Market Inefficient or Were Future Expected Tax Rates Negative?," Journal of Finance, 43: 913-931.

Kryzanowski, Lawrence, Kuan Xu, and Hua Zhang. 1995. "Determinants of the Decreasing Term Structure of Relative Yield Spreads for Taxable and Tax-Exempt Bonds," Applied Economics, 27: 583-590.

Lucas, Robert E. 1978. "Asset Prices in an Exchange Economy," Econometrica, 46: 1429-1445.

Miller, Merton H. 1977. "Debt and Taxes,"Journal of Finance, 32: 261-275.

Mitchell, Karlyn and Michael D. McDade. 1992. "Preferred Habitat, Taxable/Tax-Exempt Yield Spreads, and Cycles in Property/Liability Insurance," Journal of Money, Credit, and Banking, 24: 528-552.

Piros, Christopher D. 1987. "Taxable vs. Tax-Exempt Bonds: A Note on the Effect of Uncertain Taxable Income," Journal of Finance, 42: 447-451.

Sargent, Thomas J. 1987. Dynamic Macroeconomic Theory. Cambridge, MA: Harvard University Press.

Skelton, Jeffrey L. 1983. "Banks, Firms and the Relative Pricing of Tax Exempt and Taxable Bonds," Journal of Financial Economics, 12: 343-355.

Stiglitz, Joseph E. 1988, Economics of the Public Sector, 2nd ed. New York: W.W. Norton.

Trzcinka, Charles. 1982. "The Pricing of Tax-Exempt Bonds and the Miller Hypothesis," Journal of Finance, 37: 907-923.

Van Horne, James C. 1994. Financial Market Rates & Flows, 4th ed. Englewood Cliffs, NJ: Prentice Hall.

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Author: | Xu, Kuan |
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Publication: | Quarterly Review of Economics and Finance |

Date: | Jun 22, 1998 |

Words: | 3038 |

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