Figure 4.1 shows the Bloomberg Yield and Spread Analysis page for the 3.85% Apple Inc. (AAPL) bond that matures on May 4, 2043. Its flat price was 87.24 (percent of par value) for settlement on March 5, 2014. Its street convention yield to maturity was 4.653675%. This bond was issued on May 3,2013, as a 30-year noncallable offering at a de minimis OID price of 99.418 (percent of par value). This information is shown on the Bloomberg Descriptive page; see the lower right side of Figure 4.2. The bond's price

FIGURE 4.1 Bloomberg Yield and Spread Analysis Page, AAPL 3.85% Bond Due 5/04/2043, Assuming 43.40% Ordinary Income Tax Rate and 23.80% Capital Gains Rate

has fallen because long-term benchmark Treasury bond yields rose in the 10 months after issuance.

First, let's confirm the pretax street convention yield to maturity of 4.653675%. This entails substituting into equation 3.11 from Chapter 3, repeated here as equation 4.1.

(4.1)

Using the 30/360 day-count convention, 121 days have elapsed since the last coupon on November 4 and there are 59 semiannual periods between November 2013 and May 2043. So, Flat = 87.24, t/T = 121/180, PMT = 1.925 [= 3.85/2], and N = 59.

Solving by arduous trial-and-error search (or, better, setting up the equation in an Excel spreadsheet and using Solver) obtains the result that y = 0.023268374. Annualized to a semiannual bond basis and rounded to six decimals, the yield to maturity is 4.653675%. A more direct way of confirming the Bloomberg street convention yield is to use the YIELD financial function in Excel.

The inputs are the settlement date, maturity date, annual coupon rate as a decimal, flat price, par value, periodicity, and the code for the 30/360 day- count convention.

The reported after-tax rate of return of 2.749418% for this bond is troubling for two reasons. First, there is a small error in the calculation, as I demonstrate. But more important, this to me is bad bond math because it is very misleading, at least to a U.S. investor. The problem is that this is a global bond, as indicated in Figure 4.2 for the market of issue. For a reason unknown to me, Bloomberg does not use U.S. tax rules on global bonds (or on at least some private placements). Instead, the Bloomberg calculation assumes that the “gain” from buying at 87.24 and redeeming the bond at 100 (assuming no default) will be taxed at the assumed capital gains tax rate of 23.80%. Bloomberg only uses U.S. tax rules on U.S. domestic issuances. Clearly, you need to know how to do this calculation yourself.

Let's assume a U.S. investor elects to not bring the accrued market discount into income each year. That's the much easier assumption to deal with analytically. The investor's projected after-tax yield on a market discount bond is the solution for aty in this general expression.

The left side is the purchase price, including accrued interest (AI). The first term on the right side is important to understanding this calculation. When the investor receives that first coupon payment of PMT, the accrued interest account is closed out. Only (PMT – AI) is taxable interest income. The ordinary income tax rate is denoted tax and is assumed to be constant. The remaining coupon payments are all reduced by the ordinary income tax payments. At redemption, tax is owed on the “gain” from buying the bond at a market discount, the difference between the par value and the flat price. Importantly, the same ordinary income tax rate on that “gain” is used because that is U.S. tax law and has been so since 1984. There is no difference to a U.S. taxpayer if the bond is designated global or domestic.

Now we can collect terms, multiply the numerator and denominator by (1 + aty)t/T, and do the usual sum-of-a-finite-geometric-series reduction, as shown in equation 4.3.

(4.3)

Note that AI = t/T * PMT = 121/180 * 1.925 = 1.294028.

Solving again by tedious trial-and-error search (or by using an Excel spreadsheet and Solver) obtains the result that aty = 0.013432181. Annualized, the projected after-tax rate of return on this bond is 2.686436%, not 2.749418% as reported on the Bloomberg Yield and Spread Analysis page.

How does Bloomberg get its after-tax rate on a market discount bond? I suspect some adjustments are made to a program equivalent to the YIELD financial function in Excel. To see this, recall the entries used above to get the before-tax, street convention yield to maturity of 4.653675%.

Now multiply the third item, the annual coupon rate, by one minus the ordinary tax rate of 43.40%. Then subtract the tax on the deferred market discount from the fifth entry, the redemption amount, using the assumed capital gains tax rate of 23.80%.

This produces an after-tax yield of 2.749418%, the same as reported on the Bloomberg Yield Analysis page.

I suggest that U.S. users of Bloomberg who would like to see a projected after-tax rate of return set the capital gains rate to whatever assumption is used for ordinary interest income. This will work for all bonds – global, domestic, private placements – because the same tax law applies to each type. Then the output will be the same as if it is done with the “tax-adjusted” YIELD function in Excel. For example, use the ordinary income tax rate of 43.40% in the fifth entry.

This gives an after-tax yield of 2.686572%. This is virtually the same as the result calculated above, 2.686436%. But still, why is there a small difference? The (admittedly minor) problem with the “tax-adjusted” YIELD approach is that the accrued interest needed in equation 4.2 is calculated using the after-tax coupon rate. That reduces the left side of the equation. Also, the after-tax first cash flow is not corrected by the accrued interest. That reduces the right side of the equation. These two errors offset. This is no doubt a concern to only those of us who are fastidious about bond math.

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