THE INTEREST RATE SENSITIVITY OF TAX-EXEMPT BONDS UNDER TAX-NEUTRAL VALUATION

JOIM Journal Of Investment Management, Vol. 12, No. 1, (2014), pp. 62 68 JOIM 2014 www.joim.com THE INTEREST RATE SENSITIVITY OF TAX-EXEMPT BONDS UNDER TAX-NEUTRAL VALUATION Andrew Kalotay a We explore the effect of taxes on the prices of municipal bonds. Although interest is taxexempt, the gain resulting from purchasing a muni at a deep discount below the so-called de minimis threshold is subject to severe tax treatment. The gain is taxed as ordinary income at maturity; currently for a typical investor the applicable rate is roughly 40%. Thus, purchasing a bond at 80 would trigger an 8-point tax liability. The paper develops a rigorous approach to the pricing of munis by incorporating taxes into the industry-standard OAS-based valuation framework. The key concept is tax-neutral value, which is simply the fair value that takes into account potential tax payments. Taxneutral valuation allows us to explore how muni prices respond to changing interest rates. The basic insight is that due to the interaction of the purchase price and the related tax payment, discount tax-exempt bonds are significantly more sensitive to interest rates than taxable bonds. For example, currently the duration of a 10-year taxable bond is roughly 8.5 years, while that of a 10-year muni can exceed 13 years. Tax-neutral valuation provides the foundation for accurately projecting the prices of munis under various interest rate scenarios. The primary application of this approach is risk management, including hedging. It is also essential for determining the optimum time to recognize a loss in order to maximize after-tax performance. 1 Introduction While interest payments on municipal bonds ( munis ) are exempt from federal income taxes, a President, Andrew KalotayAssociates, Inc., 61 Broadway, Suite 1400, New York, NY 10006, USA. Tel.: (212) 482 0900; E-mail: andy@kalotay.com capital gains and losses are subject to complex tax treatment. Taxes affect investors after-tax performance. For example, because the gain on a bond purchased in the secondary market at a discount and held to maturity is subject to taxes, the after-tax yield of the investment will be lower than the pretax yield. The prices of discount taxexempt bonds are routinely converted to so-called 62 First Quarter 2014

The Interest Rate Sensitivity of Tax-Exempt Bonds under Tax-Neutral Valuation 63 after-tax cashflow yields to maturity. Converting price to an after-tax yield is straightforward, and it allows investors to compare alternative investments on an-apples-to-apples basis. The prices of tax-exempt bonds reflect these complex tax considerations. This phenomenon is well recognized by practitioners but their attempts to deal with the effects analytically have been limited to using yields and modified durations, and not contemporary fixed income analytics (e.g. see Leibowitz, 1981; Merrill Lynch, 2007). In this paper we will extend the conventional arbitragefree method of bond valuation (the so-called option-adjusted spread approach) to incorporate tax effects. We first determine the tax-neutral ( fair ) value of a bond assuming a buy-and-hold policy. This fair value provides the basis for rigorous risk analysis. 1 As we shall see, the interest rate sensitivity of a muni can be significantly greater than that of a like taxable bond. The implications of this observation are farreaching. At the present, standard commercially available analytical systems do not take taxes into account. This is particularly troublesome in the case of exchange traded funds and mutual funds that attempt to replicate the performance of a large index which may consist of over 10,000 bonds with a few hundred securities. Matching durations on a pretax basis does not assure that the same relationship holds when the effect of taxes is properly accounted for. In light of this, the large tracking errors of these index-matching portfolios do not come as a surprise. 2 Relevant tax treatment A thorough discussion of the tax treatment of munis, including original issue discount bonds (OIDs) and original issue premium bonds (OIPs), is provided by Ang et al. (2010). For illustrative purposes, we will assume that the bonds under consideration were originally sold at par. Investors who purchase a bond in the secondary market at a discount and hold it to maturity or call are taxed on the gain. At a modest discount to par (a so-called de minimis discount, defined as less than 0.25 times the number of years remaining to maturity) the applicable rate is the relatively low capital gains rate (at the time of writing, 20% if long-term). If the discount exceeds the de minimis threshold, the entire gain is taxed at the higher ordinary income rate (around 40% at the time of writing). We also note that the loss on a bond purchased at a premium and held to maturity has no tax effect. The tax treatment is more complicated if the bond is sold prior to maturity or call. But because the values in the current discussion are based on buyand-hold, we ignore taxation related to sales. 3 Market prices of munis The obvious first question to explore is the reasonableness of the assumption that market prices can be imputed from a buy-and-hold strategy. On the one hand, Constantinides and Ingersoll (1984) argue that active tax management can produce superior return over buy-and-hold. This would imply that market prices should be higher than that indicated by buy-and-hold. On the other hand, there is scant empirical evidence that such is the case. In fact, according to Ang et al. (2010), the market prices of deep discount munis are significantly lower than would be implied by buyand-hold, and they provide a possible explanation for this phenomenon. In any case, for risk management purposes it is irrelevant whether the actual prices are marginally higher or lower than that indicated by buy-andhold. The relevant fact is that prices of munis do reflect the presence of taxes, and our approach could be readily adapted to pricing models more sophisticated than buy-and-hold, should such become available. First Quarter 2014 Journal Of Investment Management

64 Andrew Kalotay 4 Methodology Our initial goal is to determine the tax-neutral value ( fair value ) of tax-exempt munis using arbitrage-free analysis (see Kalotay et al. (1993)). In the absence of taxes and options, fair value is obtained by discounting prospective cash flows at the appropriate spot rates; if options are present, such discounting is performed on a lattice. The taxation of munis complicates the calculation because the cash flows depend on the purchase price; roughly speaking, the lower the purchase price the more taxes will be due when the bond is retired. 4.1 Practical considerations Arbitrage-free valuation requires an issuerspecific optionless (par) yield curve for discounting, and for valuing options at a specified interest rate volatility. In the case of tax-exempt bonds, optionless long-term rates are not readily available, because the industry-standard MMD and MMA yield curves assume that the underlying bonds are callable at par after 10 years. The approach of extracting optionless par curves from callable curves is described in Kalotay and Dorigan (2008). The numerical examples below assume that the optionless curve has been provided and that it evolves according to an industrystandard lognormal process. Our methodology, of course, is applicable to arbitrary interest rate processes. the case of a callable bond the fair value has to be determined iteratively, as the timing of the tax payment depends on the evolution of interest rates. The calculation can be simplified if the bond is optionless, as illustrated below. Assume that the bond has 10 years remaining to maturity, its pretax value is 80, the discount factor for a cash flow occurring 10 years from now is 0.45, and the tax rate applicable to the gain is 40%. Solving V = 80 0.45 0.40 (100 V) gives the fair value V = 75.610. Figure 1 displays the fair values of 10-year bonds with various coupons. For comparison purposes, we also show the values in the absence of taxes. The calculations are based on the yield curve displayed in Table 1. The assumed volatility for callable bonds (see Figure 3) is 20%. The longterm capital gains rate is 20%, and the tax rate applicable to ordinary income is 40%. Note that the pretax value of a discount muni exceeds its fair value by the present value of the taxes paid at the time the bond is redeemed. (The discount rate applied to the tax payment is 5 Tax-neutral value We now determine the value of a bond under the buy-and-hold strategy. We define the taxneutral (fair) value as the price which is equal to the present value of after-tax cashflows (i.e. interest and principal payments minus the taxes paid at the time of redemption). Simply put, the fair value is the pretax value adjusted for taxes. Because taxes depend on the purchase price, in Figure 1 Fair value of 10-year bonds. Table 1 Issuer s optionless par yield curve. Maturity (yrs) 1 2 5 10 15 20 30 Rate (%) 1.0 1.5 2.0 3.0 3.5 4.0 4.5 Journal Of Investment Management First Quarter 2014

The Interest Rate Sensitivity of Tax-Exempt Bonds under Tax-Neutral Valuation 65 the spot rate derived from the issuer s yield curve.) Here the de minimis threshold is 97.50% of par (100 10 0.25). In the absence of taxes, a bond with a 2.72% coupon would be valued at 97.50 (blue line), but due to the tax on the gain the value would be less. Conspicuous in Figure 1 is how the fair value (red line) falls off the cliff at the de minimis level. The critical coupon (discussed below) is 2.76%. If the coupon is 2.75%, the fair value declines by 0.60% to about 96.90. The reason is that above 97.50 the gain is taxed at the 20% capital gains rate, but below 97.50 the gain is taxed at 40%. The de minimis threshold with 30 years to maturity is 92.50%. The critical coupon for a bullet is 4.10%; at 4.09% the fair value drops by 0.36% to 92.14 (red line). The effect is less pronounced than in Figure 1, because the maturity is farther away. In the absence of taxes, a coupon of 4.07% would give a 92.50% value (blue line). As discussed above, it is more difficult to determine the fair value of a callable bond than that of a bullet, because the redemption date is uncertain and in turn so is the present value of the tax payment. We also note that the redemption date depends on the interest rate volatility (assumed to be 20% in the illustration below). Thus in this case the fair value must be calculated by lattice-based recursion. Figure 2 Fair value of 30-year bullets. Figure 3 Fair value of 30-year NC10 s. In comparison to Figure 2, the presence of the call option reduces both the pretax and after-tax values. In the case of lower-coupon bonds the results are similar to those in Figure 2, because the effect of the call option is relatively insignificant. In the case of higher-coupon bonds the tax effect is relatively insignificant to begin with, because the price is closer to par. 5.1 Critical coupon level As we saw in the examples above, for a given yield curve and bond structure (i.e. maturity and optionality) there is a theoretical coupon level where the fair value falls off the cliff. While in reality the price decline is not as abrupt as indicated by our model, this critical coupon level is still of practical interest. If a bond is purchased at a price slightly above the de minimis threshold, its price could decline significantly even if rates rise only modestly. Because the market anticipates this possibility, the price experiences downward pressure when the coupon is slightly above the critical level. Similarly, the price can be higher than that predicted by our model if the coupon is slightly lower than the critical level. Price behavior near the de minimis level is of independent interest and could be explored as a separate study. We also observe that while market prices are extremely important to investors whose portfolios First Quarter 2014 Journal Of Investment Management

66 Andrew Kalotay must be marked to market (such as mutual funds and ETFs), they are less significant for buy-andhold investors. The point is that the value of the remaining cashflows to an existing investor can be higher or lower than to a marginal buyer. This is discussed in Kalotay and Howard (forthcoming). For a given yield curve, the critical coupon of an optionless can be determined easily (see below). For a callable bond the calculation is obviously more complicated. Illustration: Determining the Critical Coupon C of an Optionless 10-Year Bond Assume the present value of a 10-year $1 annuity is $8.50, the discount factor for a cash flow occurring 10 years from now is 0.70, and the capital gains rate is 20%. Solving 8.50 C + 0.70 (100 0.20 2.50) = 97.50, results in C = 3.28%. Note that the fair value of a bond whose coupon is slightly below 3.28% is 97.50 0.70 (0.40 0.20) 2.50 = 97.15, which accounts for the incremental tax bill arising from the applicability of the ordinary income tax rate (40%) over the tax bill arising from applying the short-term gain rate (20%). 6 The interest rate sensitivity of tax-exempt bonds In the preceding sections of this paper we developed a tax-neutral valuation model for tax-exempt bonds assuming buy-and-hold, and explored the behavior of this model for various bond structures. Given the above foundation, we are ready to investigate the interest rate sensitivity of tax-exempt bonds. The general approach to determining interest rate risk is as follows: (1) Determine the OAS of the bond at the given price relative to the benchmark yield curve. (2) Shock the yield curve. (3) Reprice the bond at the OAS obtained in (1). (4) Calculate risk measures. In the case of tax-exempt bonds, it is imperative to recognize that the prices may be depressed by potential tax payments; otherwise the interest rate sensitivity can be severely underestimated. The intuition is clear: higher rates depress the price, and a lower price increases taxes. The OAS should be an indication of only credit risk or mispricing, but it should not reflect tax effects. The appropriate spread measure for munis is tax-neutral OAS, that is, the OAS that equates the price to the tax-neutral value. Naturally the higher the applicable tax rate the greater is the above effect, so it is most pronounced when the price is below the de minimis threshold. At the de minimis threshold the price is discontinuous, and therefore interest rate sensitivity is not defined. Whenever the tax treatment is discontinuous it is desirable to distinguish between up and down durations. The exhibits below display the durations of various tax-exempt bond structures using tax-neutral prices (based on the benchmark curve). Figure 4 displays the durations of 10-year optionless bonds. As we saw in Figure 1, the critical coupon in this case is 2.76%. The durations of bonds with coupon below 2.76% exceeds 12 years, which is 2 years longer than the bonds maturity! Duration slightly exceeds 10 years even in the de minimis region (coupons larger than 2.76% but less than 3.00%). Since the pretax duration of a bond cannot exceed the bond s maturity, any calculator that disregards Journal Of Investment Management First Quarter 2014

The Interest Rate Sensitivity of Tax-Exempt Bonds under Tax-Neutral Valuation 67 again when it falls below 92.50 (the de minimis threshold price occurring at around a 4.33% coupon). There is an obvious extension to key rate durations (not discussed here). Figure 4 Duration of 10-year optionless bonds. taxes will severely underestimate the true duration of discount bonds. But the error is significant even if the price is close to par for example, the pretax duration of a bond in the de minimis region is 8.85 years, considerably shorter than the true duration. It is well known that taxes can have a drastic effect on interest rate sensitivity. For example, as pointed out by Kalotay (1984), the after-tax duration of an original issue discount bond issued by a taxable corporation can exceed the bond s maturity. Figure 5 below shows similar information for 30- year bonds callable in 10 years. As long as the price is below par, the duration is longer than it would be in the absence of taxes (latter not shown). The duration is not defined when the price is discontinuous in this case when it declines below par (slightly below a 5.00% coupon) and 6.1 Observations about pretax risk measures As described at the beginning of this section, interest rate risk measures are calculated using a fixed OAS relative to a benchmark yield curve. Note that OAS depends on whether or not tax effects are incorporated. Example: Solve for OAS, Calculate Duration Suppose that the price of an optionless 10-year 2.5% bond is 84.15; calculate its OAS relative to the benchmark curve given earlier. Pretax (incorrect): OAS = 147 bps, duration = 8.87 years Tax-neutral: OAS = 90 bps, duration = 12.14 years Correct calculation of interest rate risk requires an explicit adjustment for taxes. In the absence of such, the risk of tax-exempt bonds is underestimated. 7 Conclusion Figure 5 Duration of 30NC10 bonds. It is generally recognized that taxes on capital gains depress the prices of tax-exempt bonds. We have presented a rigorous approach to incorporating this effect in the valuation of tax-exempt bonds. Specifically, we extended the conventional OAS framework to after-tax analysis, including tax-neutral value and tax-neutral OAS. Using tax-neutral values as a foundation, we have shown that the interest rate sensitivity of taxexempt bonds can be significantly greater than First Quarter 2014 Journal Of Investment Management

68 Andrew Kalotay that indicated by pretax calculation, which unfortunately is the current standard in the industry. The difference is most pronounced for shorter-term bonds selling below the de minimis level, whose duration can exceed their maturity by several years. In light of the fact that under current practice the interest rate sensitivity of tax-exempt bonds is miscalculated, the large tracking errors of index-matched ETFs and mutual funds are not surprising. Tax-adjusted analytics are essential for proper management of tax-exempt bond portfolios. Note 1 Tax-neutral values, durations, and other values in this paper were calculated using Kalotay Analytics MuniOAS TM library (patent pending). References Ang, A., Bhansali, V., and Xing, Y. (2010). Taxes on Tax- Exempt Bonds, Journal of Finance 65(2), 565 601. Constantinides, G. M. and Ingersoll, J. E. (1984). Optimal Bond Trading with Personal Taxes, Journal of Financial Economics 13(3), 299 335. Kalotay, A. (1984). An Analysis of Original Issue Discount Bonds, Financial Management (Autumn), 29 38. Kalotay, A. and Dorigan, M. (2008). What Makes the Municipal Yield Curve Rise, Journal of Fixed Income (Winter), 65 71. Kalotay, A. and Howard, C. D. The Tax Option in Municipal Bonds, Journal of Portfolio Management. forthcoming. Kalotay, A., Williams, G., and Fabozzi, F. (1993). A Model for Valuing Bonds and Embedded Options, Financial Analysts Journal (May/June), 34 46. Leibowitz, M. (1981). Volatility in Tax-Exempt Bonds: A Theoretical Model, Financial Analysts Journal (November/December). Merrill Lynch (not an individual) (2007). Dealing with Deeper Discounts, Muni & Derivatives Commentary (June 18), 1 2. Keywords: Municipal bonds; tax-neutral value; taxes; duration; de minimis rule; bond valuation; after-tax; interest rate sensitivity; risk management; tax-neutral OAS; option-adjusted spread; critical coupon Journal Of Investment Management First Quarter 2014