# 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.

• You might want to clarify what exactly your question is. As you say, the convention is to report clean prices since they are easier to interpret and graph. I find YTM is easier to interpret than prices, but the distribution of (dirty) prices would be an important input in optimization (when different bonds have different maturities).
– John
Jul 26, 2012 at 18:41
• Here's a worthwhile book on bonds amazon.com/The-Treasury-Bond-Basis-Arbitrageurs/dp/1557384797 Jul 28, 2012 at 14:45

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).

• Hi @Steven and welcome! Thank you for this answer. Do I see it correctly that the algebra that you present works to a large extent as the bond is priced at par - $y \approx C/P$? Aug 21 at 10:18
• @RichiW Yes, the assumption under which I derive (3) is that y is near coupon rate, namely, the bond is priced at par ($y\approx C/P$). It is not always the case in reality, though. I made this assumption because it's easier to derive the term $P\frac{t}{T}y$ and clarify the intuition behind accrued interest. Aug 22 at 3:15
• thank you. It would be interesting to set the general case though ... Maybe I find time .. Or you :) Aug 23 at 16:54
• @RichiW You're welcome. A little bit challenging, but worth a try. ^_^ Aug 24 at 13:51

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.

• Yes, the question is whether this framwork is correct. As I see from yours and user1628's answer it seems to be wrong. I know this argument that there is no arbitrage due to costs and market risk. Do you know a website where all these things are well explained especially for the US and some European markets? Thank you! Jul 27, 2012 at 7:04

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.

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).

• You have clean and dirty mixed up. Jul 27, 2012 at 4:47
• @user1628 I agree - there seems to be a mix up with clean and dirty. And as we start talking about clean and dirty. What is the genuine thing? Thinking in zero yield curves, for me, it must be the dirty price as sum of discounted cash flows. Then I subtract AI and get clean. Can I somehow start form clean? Of course it is quoted - so traders may start from there. But even when I think in terms yield-to-maturity, I use the dirty price ... do I make myself clear? Jul 27, 2012 at 7:07
• @Richard - You can start from clean by discounting back to the settlement date. Jul 27, 2012 at 12:28
• Also, as @JL344 stated, this answer has the two mixed up from the US perspective. According to this explanation you are paying AI twice. The buyer of a bond is out of pocket the AI and receives the full coupon at the following coupon date. Think of the coupon as an amount split proportionately at the settlement date, where the seller is paid by the buyer the proportional coupon(the AI) from the prior coupon to the settlement date, and the buyer will receive the full coupon at the coupon date following settlement. Jul 27, 2012 at 12:46
• @jeffm: thanks for the explanation. This is what I thought in the beginning. Putting the whole thing in a framework where I have a spot yield curve of zeros. Let's say the bond is 3 months to maturity (semi-annual coupon payments), there is payment of a coupon C and 100 at maturity. Can I calculate the clean price as: 100*discountfactor and forget about the coupon and just add the AI for the dirty price? So the follwoing coupon is just excluded for the clean price and is just added as AI to the dirty price? Is this correct? Thanks! Jul 28, 2012 at 12:01