Why Par Bonds Don't Sell at Par

On the morning of October 30, the current thirty year Treasury that matures August 15, 2039 was priced to yield 4.283%. This Treasury pays a coupon at the rate of 4.5%. Suppose the bond were yielding the same rate as the coupon rate. What is the price (specfically the flat price) of the bond? Well, that's an easy one. In an introductory finance class, we learned that when the yield equals the coupon rate, the bond price sells at par. The question is so simple the obvious answer of par must be wrong and it is. When the current 30-year is priced at 4.5%, the flat price is 99.99387 tantalizingly close to par but not exactly. What the hell happened? Simply put, accrued interest is the culprit. When a bond is traded between two coupon payments attention must be given to how the next coupon payment should be divided between buyer and seller. Coupon interest accrues linearly during the period. Accordingly, if the bond is sold halfway through the period, the seller is entitled to half the next coupon payment. Note that the seller is paid coupon interest for his/her without a discount even though the coupon payment is still three months away. If the seller keep the bond rather than selling it their half of the coupon payment would worth its present value discounted back three months. The accrued interest grows linearly on a simple interest basis while the present value of the next coupon payment follows a slightly curved path due to the impact of compounding. The market convention (i.e., linear accrual) introduces a slight wrinkle into bond valuation. If the yield and the coupon rate are the same, the bond's flat price will sell at slight discount to par between coupon payment dates. A par bond will sell at par only on coupon payment dates when accrued interest equals zero.