Bond Accrued Interest, Clean Prices, and Coupon-Date Effects

# How to properly interpret accrued interest of bonds


# How to properly interpret accrued interest of bonds












Ever since I work in finance I was wondering what accrued interest (AI) are good for (see the wikipedia article for a short introduction). I think I have a clear picture in mind now and the usual explanations are misleading.

Clean prices (=quoted prices) are needed to show a smooth price evolution and they prevent the zig-zag that I get in dirty prices after the coupon payment- alright, I understand that.

When I sell a bond I get the cash (=dirty = full) price which is $$ \text{full price} = \text{clean price} + \text{AI}, $$ where AI is some defined fraction of the coupon that is zero on a coupon date.

Most explanation say something like "AI are the compensation if I sell the bond before the coupon payment". But isn't that wrong?

I get the full price for my bond - the discounted cashflows. So I also get the discounted value of the next coupon. It is the discounted coupon - but the whole discounted coupon - that's it. I get the full price on the first day after the preceding coupon payment all the way to the last day before the next payment. When there is a trade, discounted cashflows as a whole are traded i.e. not a fraction of any of them.

My summary

- dirty prices are the object of interest (they tell you the yield-to-maturity, they are traded but they have a drop after the coupon payment).

- clean prices are there for quotation and for graphing prices - no drop because of coupon payments.

- AI are one way to remove the drop at a coupon payment date. Of course there must be a convention for AI in order that every market participant can go from dirty to clean and back. But the usual explanation as a reward is misleading.

What do you think about this interpretation of these terms?

## Answer by Steven (score 1, accepted)

https://quant.stackexchange.com/a/71962

The core of your explanation is almost decent. Let me examine your summary point by point.

For the first one, dirty prices are the object of interest in pricing process, and will have a drop right after coupon date, but as far as Zvi Bodie is concerned, clean price is the one that tells us YTM (see Investments 9th, Zvi Bodie, et al., 14.3 Bond Yields).

For the second and the third one, I had the same problem as the questioner when I ran across bond princing at college, but now I figure it out, and will discuss them altogether. Let's begin with a general version of the pricing formula: Suppose that the full coupon period covers T days and that the bond is being priced and settled at date t ($t \in T$) into the period. Therefore, t/T is the fraction of the period that has gone by and 1 – t/T is the fraction that remains. Here is a general version of pricing, discounting the coupon payments (C) and principal redemption (P) over the remaining N payments at the yield to maturity per period (y). $$Clean+Accrued=\frac{C}{(1+y)^{1-\frac{t}{T}}}+\frac{C}{(1+y)^{2-\frac{t}{T}}}+...+\frac{C+P}{(1+y)^{N-\frac{t}{T}}}\tag{1}$$ Use the formula of the sum of geometric series, we can have $$Clean+Accrued=[\frac{C}{y}(1-\frac{1}{(1+y)^N})+\frac{P}{(1+y)^N}]*(1+y)^{t/T}\tag{2}$$ We can interpret (1) and (2) as discounting future cash flows to the last coupon date before settlement date, then compounding until settlement date t by multiplying $(1+y)^{t/T}$. For (2), we use some approximating technique to examine it more closely: if y is near coupon rate, then we have: $$Clean+Accrued\approx [P(1-\frac{1}{(1+y)^N})+\frac{P}{(1+y)^N}]*(1+y)^{t/T}=P(1+y)^{t/T}\tag{3}$$ Then we use Maclaurin series expansion for (3): $$Clean+Accrued\approx P(1+y)^{t/T}\approx P(1+\frac{t}{T}y)=P+P\frac{t}{T}y\tag{4}$$ So the price does flunctuate because of $P\frac{t}{T}y$. In order to smooth the cyclical peak, we can take the term out and call it Accrued Interest. P is principal and $P\frac{t}{T}y$ is linear, then $P\frac{t}{T}y$ is naturally considered as some kind of compensation for holding the bond. The most important is that the $\bbox[red]{\color{lime}{prerequisite}}$ on which we talk about compensation is that the compensation is included in dirty price or is decomposed out of dirty price, or it is the compensation regarding clean price.

One should not trap himself in the pitfall of "Why call AI compensation for the seller while everything is included in the dirty price which the buyer already pays", because we talk about AI under the precondition that we first have dirty price as a whole (where it's meaningless to create a concept of compensation), then we minus AI to have clean price (this is where compensation works).

## Answer by jeff m (score 0)

https://quant.stackexchange.com/a/3859

I'm not quite sure what the question is - are you asking if your explanations are correct? Are you wondering why the full discounted coupon cash flow is accrued up to the settlement date? It sounds like you are implying there may be some arbitrage opportunity by collecting the full discounted value of the next coupon payment immediately following receipt of a coupon. This wouldn't work out for a multitude of reasons, most notably transaction costs and reinvestment risk.

Also, it might be worth mentioning that not all countries trade cum-coupon, some country's bonds trade ex-coupon for a certain period until the next coupon payment date.

## Answer by Richi Wa (score 0)

https://quant.stackexchange.com/a/3884

I found the following reference http://www.financetrainer.com/fileadmin/inhalte/TOOLS_SKRIPTEN/0302_fie.pdf (page 18,19) which clearly states that with ISMA and Moosmüller method my explanation (hopefully clearly written) is correct. In a summary: There is no compensation for holding the bond between coupon dates by AI (at least in the German market). AI is just one way to make the jump at coupon dates disappear. The dirty price is the real thing, clean is just quoted.

## Answer by jordi (score -2)

https://quant.stackexchange.com/a/3860

Your interpretation is not correct. When you sell a bond you get the "clean price", not the dirty price. That's just how it is.

And if you think about it, that's also the only logical way to do it. Otherwise, no one would ever be able to buy or sell bonds except right on or near the coupon date which would be the only time that the price would be fair to both parties.

When you buy a coupon bond, you must (1) pay for bond itself (the dirty price) and (2) compensate ("prepay") the seller for the pro-rated part of the coupon that they are entitled to (but that you will be getting instead of them in the future). Part (2) is a very real cash flow to/from your broker and not some theoretical construction. All in all you will be paying the clean price.

So in other words, the "clean" price is the value you would get for the bond if you sold it today in the market. The dirty price is just the sum of clash flow (i.e. the value of the bond itself if you were to steal it or obtain it for free).