# Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

## [PGP-I FM] Bond pricing

Dirty Price: Price of a bond that the buyer must pay / seller gets = PV of all future cash flows from the bond

Accrued Interest: Portion of the coupon accrued to the seller if the bond is sold after the issue date and before a coupon payment date =

$\displaystyle \frac{\mbox{Number of days since the previous coupon payment date } (d)}{\mbox{Number of days in the year } (Y) } \times \mbox{ Annual Coupon } (C)$

$\displaystyle \frac{d}{Y} \times C$

where $d$ and $Y$ would depend on the day-count convention followed.

Clean PriceDirty Price – Accrued Interest

The fair price of a bond is given by its Dirty Price and the Clean Price is just the quoting convention that avoids ‘spikes’ due to Accrued Interest.

Even though it’s merely a convention so not saddled with having to justify its logic, but one can try and make sense of it in a certain way. (For those who do not want to bother with the whys and hows of it, may safely ignore what follows.)

Clean Price (Wonkish)

Now we show that the definition of the Clean Price as the Dirty Price net of the Accrued Interest is equivalent to the PV of Coupons (as if) “deserved” by the buyer.

All we need to do is explain the equivalence for the first coupon, since after the first coupon payment there are no complications.

To the buyer what matters is the coupon received on the coupon payment date. While a part of that coupon, the accrued interest, can be said to “deservedly” belong to the seller, that amount in our discussion is deemed paid on the day of the transaction and not on the coupon payment date.

Another way to think about this is to say that we net the coupon amount due to the buyer by the accrued Interest brought forward to the coupon payment date. Basically what we are doing is bringing both payments on the same date, i.e. the coupon payment date.

Net first coupon “deserved” by the buyer is:

\begin{aligned} \displaystyle \mbox{Clean First Coupon} &=\mbox{Coupon} - \mbox{Future Value of Accrued Interest} \\&= \frac{C}{2} - \frac{d}{Y} \times C \times \big(1 + \frac{r}{2} \big)^{2 \times d/Y} \end{aligned}

Since Clean Price is the present value of the coupons “deserved” by the buyer, we have:

\begin{aligned} \displaystyle \mbox{Clean Price} &= PV \Big(\mbox{Clean First Coupon}\Big) \\&= PV \Big(\frac{C}{2} -\frac{d}{Y} \times C \times \big(1 + \frac{r}{2} \big)^{2d/Y}\Big) \\&= PV\Big(\frac{C}{2} \Big) - PV\Big(\frac{d}{Y} \times C \times\big(1 + \frac{r}{2} \big)^{2d/Y}\Big) \\&= PV\Big(\frac{C}{2} \Big) - \big(1 + \frac{r}{2} \big)^{-2d/Y}\Big(\frac{d}{Y} \times C \times\big(1 + \frac{r}{2} \big)^{2d/Y}\Big) \\&= PV\Big(\frac{C}{2} \Big) -\frac{d}{Y} \times C \\& \equiv \mbox{Dirty Price - Accrued Interest}\end{aligned}