I’m confused between these two ways to get the price of a bond.

Discount the bond’s future cashflows back to today to get the theoretically correct price.

Add accrued interest to the quoted clean price.

Where does the clean price come from? Using formula 1 should already give us the complete price of the bond at any point in time. Is it that traders calculate the theoretically correct price then take off accrued interest since last coupon date (i.e. the DCF price = the dirty price?).

If so,

a) why do they do this? Isn’t it easier to just quote the complete (dirty) price

b) Shouldn’t they also require that the accrued interest include the reinvestment income that could have been earned while waiting for the coupon (i.e. the interest earned on the accrued portion of the coupon)?

Suppose that you have a bond that pays semiannual coupons on January 15 and July 15, and that you purchase it on, say, April 2. If you were to discount the future cash flows as of April 2 (the first cash flow being 104 days away, or 104/182.5 of a half-year), then, in theory, you would get the dirty price of the bond. (I say “in theory” because accrued interest is usually calculated as simple interest (number of days held / number of days between coupons), not as compound interest; the DCF calculation would give you the dirty price if the accrued interest were compound interest.) From that dirty price you will subtract the accrued interest to get the clean price.

It might be easier, but it’s incomplete. As an investor, you want to know what you’re paying for the bond itself (e.g., to know what value to place on your balance sheet) and what you’re paying for accrued interest (e.g., to calculate interest income on your income statement).

Interestingly, you have this one backward. The seller wouldn’t normally get their coupon payment (cash flow) until July 15; the buyer is giving it to the seller early (April 2). If you want to include an adjustment for reinvestment income, the buyer should pay the seller only the present value of the accrued interest, which the seller could then invest until July 15 to get the full value of the accrued interest. (You don’t earn reinvestment income until you have cash to reinvest; accrued interest plus $5.50 will buy you a cup of Starbuck’s.) In practice, nobody worries about such nuances (at least in part because no buyer and seller could ever agree on the reinvestment rate to use).