U.S. tax authorities were very busy in the 1980s. After fixing the taxation of market discount bonds originated after 1984, the next task was bonds purchased at a premium. In principle, the premium over par is the present value of the “excessive” coupon payments because the coupon rate is greater than the yield to maturity. You are getting more than you need (in order to pay par value). So, for tax purposes, the investor naturally would want to amortize the premium over the lifetime of the bond. That means reducing the amount of taxable interest income each year.

According to IRS Publication 550, for bonds acquired before 1985, “you can amortize the premium using any reasonable method,” including straight-line amortization and something called Revenue Ruling 82-10, which the IRS does not even bother to explain. The big change was that for bonds acquired after 1985, “you must amortize the bond premium using a constant-yield method on the basis of the bond's yield to maturity.” For some fun, read the torturous way that Publication 550 tries to explain to the public how to calculate the constant-yield price trajectory. It's really just basic bond math.

FIGURE 4.3 Bloomberg Yield and Spread Analysis Page, IBM 8.375% Bond Due 11/01/2019, Assuming 43.40% Ordinary Income Tax Rate and 23.80% Capital Gains Rate

In Chapter 3,1 use the 8 3/8% IBM bond due November 1, 2019, to illustrate the various calculations on its Bloomberg Yield and Spread Analysis page; it is shown again here as Figure 4.3. This bond was priced at 132.209 flat to yield 2.322082% for settlement on February 14, 2014, Valentine's Day. Its after-tax rate is reported to be 1.314666%, assuming an ordinary income tax rate of 43.40% and a capital gains rate of 23.80%. Table 4.1 displays the spreadsheet that I used to confirm that calculation.

The first column shows the dates for the cash flows, assuming street convention for the timing of the payments. Solving for an after-tax true yield would be a lot more work. The second column is the before-tax cash flows. The full price of the bond at settlement is 134.605181, the flat price of 132.209 plus accrued interest of 2.396181 [= (103/180) * (8.375/2)]. The third column is the all-important projected after-tax cash flows. But first we need the constant-yield price trajectory shown in the fourth column. I get this using the PRICE financial function in Excel. For instance, the value of 108.8725568 for May 1, 2018, comes from:

TABLE 4.1 After-Tax Rate Calculation on the 8 3/8% IBM Bond Due 11/01/2019

Date

Cash Flow

After-Tax Cash Flows

Constant Yield Price

Change in Price

1-t/T

Present

Value

2/14/14

134.605181

134.605181

132.2090000

5/1/14

4.1875

3.898312

131.0840133

1.1249867

0.4278

3.887405

11/1/14

4.1875

3.526979

128.4184522

2.6655611

1.4278

3.494150

5/1/15

4.1875

3.540410

125.7219429

2.6965093

2.4278

3.484559

11/1/15

4.1875

3.553998

122.9941260

2.7278169

3.4278

3.475097

5/1/16

4.1875

3.567743

120.2346380

2.7594880

4.4278

3.465764

11/1/16

4.1875

3.581648

117.4431112

2.7915268

5.4278

3.456558

5/1/17

4.1875

3.595714

114.6191737

2.8239375

6.4278

3.447479

11/1/17

4.1875

3.609943

111.7624491

2.8567246

7.4278

3.438527

5/1/18

4.1875

3.624338

108.8725568

2.8898923

8.4278

3.429702

11/1/18

4.1875

3.638900

105.9491116

2.9234452

9.4278

3.421002

5/1/19

4.1875

3.653631

102.9917240

2.9573876

10.4278

3.412428

11/1/19

104.1875

103.668533

100.0000000

2.9917240

11.4278

96.192509

32.2090000

134.605181

The fourth entry item is the yield to maturity, calculated using the YIELD function as in Chapter 3.

The fifth column reports the change in the constant-yield price from period to period. The sum of the column is 32.209, the premium over par value at purchase. The value of 120.2346380 on May 1, 2016, is 2.7594880 lower than the price on November 1, 2015. Now we can see the after-tax cash flow calculation for that date: 3.567743 = 4.1875 – (4.1875 – 2.7594880) * 0.4340. The amount of tax for each period is the interest payment less the amortization of the premium times the assumed ordinary tax rate. The cash flow that requires special attention is the first one on May 1, 2014, because we have to subtract the accrued interest from the interest payment: 3.898312 = 4.1875 – (4.1875 – 2.396181 – 1.1249867) * 0.4340.

The after-tax rate is the internal rate of return on the series of aftertax cash flows in column three. The sixth and seventh columns are used in getting the internal rate of return using Solver. Each after-tax cash flow is discounted at the rate per semiannual period for the time until the receipt of payment. For example, the payment on November 1, 2014, is discounted back for 1.4278 [= 2 – (103/180)] periods. The discount rate that makes the sum of the present values equal the total price of 134.605181 turns out to be 0.6571075%. Annualizing this by multiplying the two and rounding to six digits gives the after-tax rate of 1.314215%. I have no explanation for the insignificant difference between this result and that reported on the Bloomberg page, 1.314666%.

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