The Interest Rate Sensitivity of Tax-Exempt Bonds under Tax-Neutral Valuation

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1 The Interest Rate Sensitivity of Tax-Exempt Bonds under Tax-Neutral Valuation Andrew Kalotay President, Andrew Kalotay Associates, Inc. 61 Broadway, Suite 1400, New York, NY , While interest payments on municipal bonds ( munis ) are exempt from federal income taxes, 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 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 ad hoc, using yields and modified durations for example, and not contemporary fixed income analytics (e.g., see Merrill Lynch [2007]). In this paper we will extend the conventional arbitrage-free method of bond valuation (the so-called OAS 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 far-reaching. At the present, standard commercially available analytical systems do not take taxes into account. This is particularly troublesome in the case of ETF s 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. 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 1 Tax-neutral values, durations, and other values in this paper were calculated using Kalotay Analytics MuniOAS library (patent pending). 1

2 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, 15% if longterm). If the discount exceeds the de minimis threshold, the entire gain is taxed at the higher ordinary income rate (35% 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 our values are based on buy-and-hold, sales are irrelevant for the current discussion. 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 buyand-hold. This would imply that market prices should be higher than indicated by buyand-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 buy-and-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 indicated by buy-and-hold. The 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. Methodology Our initial goal is to determine the tax-neutral value ( fair value ) of tax-exempt munis using arbitrage-free analysis (see Kalotay, Williams, and Fabozzi [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. Practical considerations Arbitrage-free valuation requires an issuer-specific 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 it evolves as an industry- 2

3 standard lognormal process. Our methodology, of course, is applicable to arbitrary interest rate processes. Tax-Neutral Values We now determine the value of a bond under the buy-and-hold strategy. We define the tax-neutral (fair) value as the price which is equal to the present value of future 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 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 Illustration: Determine the Fair Value V of an Optionless Bond Assume the bond has10 years remaining to maturity, its pretax value is 80, the discount factor for a cash flow occurring 10 years from now is 0.40, and the tax rate applicable to the gain is 35%. Solving V=80-0.4*0.35*(100-V) gives the fair value V= Examples Next, we determine the fair values of various structures. 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 below. The assumed base case volatility is 20%. The longterm capital gains rate is 15%, and the tax rate applicable to ordinary income rate is 35%. Table 1. Issuer s Optionless Par Yield Curve Maturity (yrs) Rate (%)

4 Exhibit1. Fair Value of 10-Year Bonds Pre-Tax Fair Value Value (% Par) Coupon (%) First, 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. 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, but the tax on the gain reduces the value. Standing out in Exhibit 1 is how the fair value falls off the cliff at this level. The critical coupon (discussed below) is 2.75%. But if the coupon is 2.74%, the fair value declines by 0.60% to The obvious reason is that above the gain is taxed at the 15% capital gains rate, but below the gain would be taxed at 35%. 4

5 Exhibit 2. Fair Value of 30-Year Bullets Pre-Tax Fair Value Value (% Par) Coupon (%) The de minimis threshold with 30 years to maturity is 92.5%. The critical coupon is 4.09%; at 4.08% the fair value drops by 0.46%% to The effect is less pronounced than in Exhibit 1, because the maturity is farther away. In the absence of taxes, a coupon of 4.07% would give a 92.50% value. 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 should be calculated by lattice-based recursion. 5

6 Exhibit 3. Fair Value of 30-Year NC10 s Pre-Tax Fair Value Value (% Par) Coupon (%) In comparison to Exhibit 2, the presence of the call option reduces the pretax and aftertax values. In the case of lower-coupon bonds the results are similar to those in Exhibit 2, because the effect of the call option is relatively insignificant. In the case of highercoupon bonds the tax effect is relatively insignificant to begin with. Critical Coupon Level As we saw in the examples above, for the given yield curve and bond structure (i.e. maturity and optionality) there is a theoretical coupon level where the fair value declines discontinuously (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 take a large hit if rates rise modestly. Because the market anticipates this possibility, the price experiences downward pressure even though the bond has a coupon above the critical level. In a like manner, the prices can be higher than predicted by our model if the coupon is slightly lower than the critical level. (This price behavior will be explored in another paper). For a given yield curve and optionless bond, the critical coupon can be determined in a straightforward manner (see below). In case of a callable bond the calculation is obviously more complicated. 6

7 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 15%. Solving 8.50*C * ( *2.5) = results in C=3.27%. Note that the fair value of a bond whose coupon is slightly below 3.27% is *( )*2.50, or The Interest Rate Sensitivity of Tax-Exempt Bonds In the preceding sections of this paper we proposed a tax-neutral valuation model for taxexempt 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 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 Calculate risk measures In the case of tax-exempt bonds, it is imperative to recognize that the prices incorporate potential tax effects; otherwise the interest rate sensitivity can be severely underestimated. The intuition is clear: higher rates depress the price, and a lower price increases taxes. This, in turn, puts additional pressure on the price, etc., until the iteration converges to the fair 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 based on tax-neutral prices (at a 0 bps OAS to the benchmark curve). 7

8 Exhibit 4. Duration of 10-Year Optionless Bonds Duration (yrs) Coupon (%) Exhibit 4 displays the durations of 10-year optionless bonds. Recall that the critical coupon in this case is 2.75% (see Exhibit 1). The durations of bonds with coupon below 2.75% exceeds 12 years, which is 2 years longer than the bonds maturity! And durations exceed 10 years slightly even when the price is above the de minimis region (coupons larger than 2.75%).. Since the pretax duration of a bond cannot exceed the bond s maturity, any calculator that disregards 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 3.00% bond is 8.85 years, considerably shorter than 10 years. 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. Exhibit 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 again when it falls below (the de minimis threshold price occurring at around a 4.33% coupon). 8

9 Exhibit 5. Duration of 30NC10 Bonds Duration (yrs) Coupon (%) 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. (A naïve and erroneous approach is to use a fixed YTM or YTC spread to a given benchmark maturity.) It should be recognized that the OAS depends on whether or not the given price is assumed to reflect tax effects. 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: OAS=148 bps, duration=8.87 years Tax-adjusted: OAS=100 bps, duration=12.15 years The important observation is that 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. 9

10 Conclusion It is generally recognized that taxes on capital gains impact the prices of tax-exempt bonds. We have presented a methodology to incorporate this effect in the tax-neutral valuation of tax-exempt bonds. Using tax-neutral values as a foundation, we have shown that the interest rate sensitivity of tax-exempt bonds can be significantly greater than indicated by pretax calculation, which has been the 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 misspecified, the large tracking errors of index-matched ETF s and mutual funds are not surprising. Tax-adjusted analytics are essential for proper management of tax-exempt bond portfolios. 10

11 References Ang, A., V. Bhansali, and Y. Xing, 2010, "Taxes on Tax-Exempt Bonds," Journal of Finance 65(No. 2), Constantinides, G. M. and J. E. Ingersoll, 1984, "Optimal Bond Trading with Personal Taxes," Journal of Financial Economics, 13(No.3), Kalotay, A., 1984, An Analysis of Original Issue Discount Bonds, Financial Management (Autumn) Kalotay, A., G. Williams, F. Fabozzi, 1993, A Model for Valuing Bonds and Embedded Options, Financial Analysts Journal (May/June), Kalotay, A., and M. Dorigan, 2008, What Makes the Municipal Yield Curve Rise, Journal of Fixed Income, (Winter). Merrill Lynch, 2007, Dealing With Deeper Discounts, Muni & Derivatives Commentary (June 18),