Your Money: Know how bond prices and coupons are calculated
February 9, 2021 1:00 AM
Interest on a bond accrues in a non-linear way. However, the market uses a linear approximation to compute the accrued interest. The fractional coupon period that has elapsed is computed using the day-count convention that is used to price the bond.
Thus, for US Treasury bonds we use the Actual/Actual convention, whereas for Government bonds in India we use the 30/360 European method.
By Sunil K. Parameswaran
Plain vanilla bonds pay coupons every period. As we go from one coupon date to the next, with the passage of each day, one day’s worth of coupon gets accrued. This is termed as accrued interest. The way to perceive it is as follows. If a buyer buys a bond between two coupon dates, he will receive the entire coupon on the next coupon date. A pro-rata share of this coupon however belongs to the seller.
Accrued interest This accrued interest has to be paid upfront at the time of acquisition of the bond. In some markets, there is the concept of an ex-dividend rate. This is a misnomer for bonds pay coupons and not dividends. But the implication is the same as for equity shares. If a buyer buys the bond after the ex-dividend date, the entire next coupon will be paid to the seller, although the buyer is entitled to a pro-rata share. This will manifest itself as a negative accrued interest.
Interest on a bond accrues in a non-linear way. However, the market uses a linear approximation to compute the accrued interest. The fractional coupon period that has elapsed is computed using the day-count convention that is used to price the bond. Thus, for US Treasury bonds we use the Actual/Actual convention, whereas for Government bonds in India we use the 30/360 European method.
The total price that is payable for the bond is called the ‘Dirty Price’. However, the price that is quoted in the market is the difference between the dirty price and the accrued interest and is termed as the ‘Clean Price’ or the ‘Add-Interest’ price. The term dirty is used for the former since it is contaminated with accrued interest. The latter excludes the accrued interest, which has to be added to it, and hence the name.
Yield to maturity You may wonder why the market chooses to quote clean and not dirty prices. If the yield to maturity of a bond does not change, the price will still change from one day to the next, due to one additional day’s interest being accrued.
Thus, if an analyst were to peruse dirty price data, he will be unable to discern as to how much of the observed price change is due to the extra accrued interest, and how much has been caused by a change in yield. If however, clean price data is used, any perceived spikes in the short run are due to yield changes, a factor that every analyst is looking for. Hence all bond markets report clean and not dirty prices. However, a buyer has to pay the dirty price, which is obtained by adding the accrued interest to the former.
The ‘Duration’ of a bond is a measure of price sensitivity. This statistic is based on the dirty price and not the clean price. Accrued interest has a duration of zero, for it is payable upfront. Thus, the higher the accrued interest component of the dirty price, the lower is the duration. If the yield remains constant, on every coupon date the duration of the bond will jump up sharply, since the accrued interest component of the dirty price will drop out.