Log in / Register
Home arrow Economics arrow Bond Math
< Prev   CONTENTS   Next >


Now consider another 4-year, annual payment corporate bond that also is priced to yield 4.182% on a pretax basis. This one has a low 1% coupon rate and trades at a much deeper discount than the 4% bond. Its price is 88.499 (percent of par value).

Assume that this 1% bond originally was issued after 1984 at par value (or with de minimis OID). This is an example of buying a seasoned bond at a market discount. Its yield has risen and its price has fallen since issuance, perhaps due to an increase in the credit risk of the issuer or perhaps due to a higher rate of inflation.

Assuming an ordinary income tax rate of 25% and a capital gains rate of 15%, this 1% bond will have a projected after-tax yield of 3.171%. That is the internal rate of return on its after-tax cash flows.

The numerators in the first four terms are the interest payment multiplied by one minus the ordinary income tax rate. But notice that in the fifth term the gain from buying at a discount is taxed at the ordinary rate. Back in the olden days it would have been taxed at the capital gains rate.

These post-1984 tax rules make sense to me because the amount of the premium or discount is the present value of the “excess” or “deficiency” in the coupon rate compared to the yield to maturity. Remember that the coupon rate is what you are promised to get; the yield is what you need (in order to pay par value). For example, the 1% bond has a “deficient” coupon in the amount of 3.182 (percent of par value) per year because its yield to maturity is 4.182%. The present value of the deficiency equals the amount of the discount: 11.501 = 100 – 88.499.

So, the discount at the time of purchase is compensation for low coupon interest income. A good portion of the investor's total return on the 1% bond is in the “capital gain.” That “gain” really is just deferred interest income, and it is taxed appropriately at the ordinary income rate. Only a discount deemed to be de minimis OID at purchase generates a capital gain for tax purposes when the bond is held to maturity.

Another wrinkle in the post-1984 U.S. tax rules is that the buyer of this market discount bond can elect to report the “capital gain” as income year by year. The investor calculates the accrued market discount either “ratably” (i.e., using straight-line amortization), or using the constant-yield price trajectory. For example, for the 1% bond the accrual each year is 2.87525 per year [= 11.501/4] using the ratable method. That amount would be taxed annually at the ordinary income tax rate.

Usually we assume the investor would prefer to defer the tax obligation to the time of sale or redemption. Suppose the investor sells the 1% bond at 96 (percent of par value) after two years. Then the total gain of 7.501 [= 96 – 88.499] is broken down into 5.7505 taxable at the time of sale as ordinary income [=2*2.87525] and the remainder of 1.7505 as a capital gain [=7.501- 5.7505]. In general, only gains above the accrued market discount benefit from the lower capital gains rate. If the investor holds the bond to maturity, the full accrued market discount is taxed as ordinary income at that time.

The after-tax rates of return on these two 4-year corporate bonds illustrate the usual tax advantage to discount bonds due to the timing of tax payments. The projected after-tax rate of return of 3.171% on the 1% bond is a bit higher than 3.154% on the 4% bond. That's because the tax obligation is deferred until maturity. This assumes a constant ordinary income tax rate over time, an assumption commonly made in practice. If you anticipate higher tax rates in the future, perhaps because of unending government budget deficits, you might want to amend that assumption. Notice that it would be easy to introduce an entire term structure of tax rates into the after-tax rate of return calculations.

Found a mistake? Please highlight the word and press Shift + Enter  
< Prev   CONTENTS   Next >
Business & Finance
Computer Science
Language & Literature
Political science