Par, Discount, and Premium

Par

A bond is issued at an initial price of $1,000. As explained above, the terms of a bond include time to maturity, level of the coupon, guarantees, etc.

Fluctuations in the value of a bond over time are not necessarily a result of supply and demand. During the year, a bond coupon that is due for payment at the end of the year accumulates. The value of index-linked bonds also changes according to variations in the index.

In order to calculate the adjusted value of a bond, the coupon accumulated to date must be calculated as follows:

PAR = [B + C x ^days/₃₆₅] = The adjusted value of the bond in terms of dollars.

• B – the original price in dollars when the bond was issued.
• C – the annual coupon (i.e., the annual amount of interest in dollars).
• Days – the number of days that have passed since the coupon’s distribution.

Calculating the Adjusted Value for Linked Bonds

As stated above, the change in the index should be added to the adjusted value of linked bonds, as follows:

PAR = [B + C x ^days/₃₆₅] x ^Mt/_Mo = The adjusted value of the bond in terms of dollars.

B – the original price in dollars when the bond was issued.

C – the annual coupon.

Days – the number of days that have passed since the  coupon’s distribution.

M_t – the current index.

M_o – the base index.

If the bond is traded at $1,000 (or at the adjusted price), then the yield to maturity of the bond will be identical to the level of its coupon. In this situation, the bond is said to be “traded at par,” i.e., at its face value (or its adjusted value).

 

Discount

This refers to a situation when a bond is traded at a lower price than its face value (or, in the case of an index-linked bond, its adjusted value). In this situation, the investor‘s yield is higher than the coupon since both investor’s yield and capital gain contributes to the total yield.

Example: a bond bearing 4% annual interest is issued for four years at $1,000. After one year, the bond is traded at $990. In this situation, the yield to maturity from this bond when purchased for $990 can be roughly calculated as follows:

(⁴⁰/₉₉₀ x 100) = 4.04%

In addition, if the bond is held until maturity (i.e., for three additional years), then the investor will also benefit from a capital gain since the bond will be redeemed for $1,000, which will yield a further 1% profit (10/1,000) over three years, i.e., ¹/₃% per year.

It can therefore be roughly estimated that the yield on the purchase of the bond at a discount is 4.37% per year. This means that the fact that the bond has been purchased at a discount contributed 0.37% to the level of the coupon paid by the bond.

In practice, in order to achieve an accurate calculation of the exact yield to maturity, a financial calculator should be used. The exact yield to maturity is 4.363%. A rough calculation indicates 4.37%; this calculation method provides a fairly good approximation.

Premium

A premium reflects the opposite situation, i.e., when a bond is purchased at a price that is higher than when it was issued, or higher than its adjusted value.

Using the bond data from the preceding example, let us assume that the price after a year is $1,010, and that a bond is purchased at this price. For the coupon, the resulting yield is calculated as follows:

(⁴⁰/_1,010 x 100) = 3.96%

Since, however, the bond was purchased at a price higher than the adjusted price, the bond purchaser loses yield since the bond, which was purchased for $1,010, will be redeemed for $1,000 thereby resulting in a loss of yield amounting to 1% (¹⁰/_1,010) over three years, i.e., ¹/₃% per year.

According to a rough calculation, it can be stated that the yield on the purchase of the bond at a premium is 3.63% per year, which means that the fact that the bond was purchased at a premium resulted in a capital loss of 0.37% (4% – 3.63%).

A bond is issued at an initial price of $1,000. As previously explained, the terms of a bond include time to maturity, level of the coupon, guarantees, etc. Fluctuations in the bond value over time are not necessarily a result of supply and demand.

During the year, a bond coupon that is due for payment at the end of the year accumulates. The value of index-linked bonds also changes according to variations in the index. In order to calculate the adjusted bond value, the coupon accumulated to date must be calculated as follows:

PAR = [B + C x days/365] = the adjusted bond value in terms of dollars.

• B – the original price in dollars when the bond was issued.

• C – the annual coupon (i.e., the annual amount of interest in dollars).

• Days – the number of days that have passed since the coupon’s distribution.