Deactivation corresponds to the increase in the value of an instrument updated as time passes and the expiry date approaches. The value of the instrument will increase (increase) at the interest rate implied by the discounted issue price, the value at maturity and the duration to maturity.

### Break down the increase in discount

A bond can be purchased at par, at premium or at a discount. Regardless of the purchase price of the bond, all bonds mature at face value. The face value is the amount of money that a bond investor will repay at maturity. A bond purchased at a premium has a value greater than par. As the bond nears maturity, the value of the bond decreases until it is at par on the maturity date. The decrease in value over time is called amortization of the premium.

A bond issued at a discount has a value lower than the nominal value. As the bond approaches its redemption date, its value will increase until it converges with the nominal value at maturity. This increase in value over time is called an increase in the discount. For example, a three-year bond with a face value of $ 1,000 is issued at $ 975. Between issue and maturity, the value of the bond will increase until it reaches its total face value of $ 1,000, which is the amount that will be paid to the bond holder at maturity.

The increase can be accounted for using a linear method, whereby the increase is evenly distributed over the duration. Using this portfolio accounting method, the increase in the discount can be thought of as a linear accumulation of capital gains on a discount bond in anticipation of receiving the peer at maturity. The increase can also be accounted for using a constant return, with the increase being the closest to maturity. The constant return method is the method required by the Internal Revenue Service (IRS) to calculate the cost base adjusted from the amount of the purchase to the expected amount of repayment. This method distributes the gain over the remaining life of the bond, instead of recognizing the gain in the year of repayment of the bond.

To calculate the amount of accretion, use the formula:

Discount amount = Basis of purchase x (YTM / Regularization periods per year) – Coupon interest

The first step in the constant return method is to determine the return to maturity (YTM), which is the return that will be earned on a bond held to maturity. The yield to maturity depends on the frequency with which the yield is compounded. The IRS allows the taxpayer some flexibility in determining the accrual period to use to calculate the return. For example, a bond with a nominal value of $ 100 and a coupon rate of 2% is issued for $ 75 with a maturity of 10 years. Suppose it is composed every year for the sake of simplicity. The YTM can therefore be calculated as:

$ 100 face value = $ 75 x (1 + r)^{ten}

$ 100 / $ 75 = (1 + r)^{ten}

1.3333 = (1 + r)^{ten}

r = 2.92%

The coupon interest on the bond is 2% x $ 100 par value = $ 2. Therefore,

Accumulation_{period1} = ($ 75 x 2.92%) – Coupon interest

Accumulation_{ period1} = $ 2.19 – $ 2

Accumulation_{period1} = $ 0.19

The purchase price of $ 75 represents the basis of the bond on issue. However, in subsequent periods, the base becomes the purchase price plus accrued interest. For example, after year 2, the accumulation can be calculated as follows:

Accumulation_{period.2} = [($75 + $0.19) x 2.92%] – $ 2

Accumulation_{period.2} = $ 0.20

Using this example, it can be seen that a discount bond has positive accumulation; in other words, the base increases, increasing over time by $ 0.19, $ 0.20, etc. Periods 3 to 10 can be calculated in the same way, using accrual accounting from the previous period to calculate the base for the current period.