# Dirty vs clean price

I just want to verify an understanding, if a bond has a coupon of 5% and a yield of 5% and assume that required yield is not changed during the life of the bond. Even in this case the clean price of the bond will have a slight variation right? Due to the way AI is calculated and the fact that it is not discounted. AI does not fully isolate the effect of time passage on a bond…

Thanks,

The way you asked this questions is a little confusing but I’ll try to help.

Clean price = Price of the bound calculated by the discounted coupon payments and principal payment

Dirty price = Clean price + AI

AI = (Coupon rate * Face value of bond)(Days since last coupon/Days in coupon periods)

AI will increase linearly towards the amount of the coupon payment until it reaches the value of the coupon payment on the day the coupon is made. Then it drops to 0 when the coupon is made.

Remember that assuming a constant required yield the clean price converges to the face value of the bond as time goes on (discount bonds increase in value towards the face value premium bonds decrease in value towards the face value)

The dirty price of the bond will simply fluctuate due to changes of these two components as well as changes in price for any change in yield.

If I’m understanding your question right then no the clean price will not vary. If a bond has a coupon 5% and yield of 5% it’s priced to par and assuming a constant yield it will not change (it’s currently priced to par and price and yield change with each other, since yield isn’t changing price isn’t either). And clean price never changes due to AI remember only the dirty price includes AI.

“If I’m understanding your question right then no the clean price will not vary. If a bond has a coupon 5% and yield of 5% it’s priced to par and assuming a constant yield it will not change (it’s currently priced to par and price and yield change with each other, since yield isn’t changing price isn’t either). And clean price never changes due to AI remember only the dirty price includes AI.”

Consider a bond issued at par with a coupon of 5% and YTM of 5%, for simplicity call it a 1 year annual coupon bond. Also assume a flat interest rate structure.

6 months before maturity the full price of the bond would be = 105/((1.05)^0.5)=102.46 however accrued interest in this case would be 2.5 and thus the dirty price is not exactly 100. I recall reading somewhere that this has to do with the way accrued interest is calculated and the fact the it ignores discounting but I can not remember where I read it…

yes are correct.

It is that fact that AI is linear and discounting of coupons is not it will produce a difference.