# How to calculate a Corporate Bond Transaction Price (Bond returns?)?

I am struggling with the concepts and variables of corporate bonds returns.

Bai, Bali and Wen (2019) define monthly corporate bond returns as:

$r_{i,t}&space;=\frac{P_{i,t}+&space;AI_{i,t}&space;+&space;C_{i,t}}{P_{i,t-1}+&space;AI_{i,t-1}}&space;-1$

Where where $P_{i,t}$ is transaction price, $AI_{i,t}$, is accrued interest, and $C_{i,t}$, t is the coupon payment, if any, of bond i in month t.

$R_{i,t}$ denote the return in excess of the risk-free rate (proxied by the one-month Treasury bill rate).

So far I have gathered data for accrued interest, coupon rate (coupon amount/frequency), Bid and Ask prices. But I cannot find a variable called transaction price.

How should I define the transaction price of a bond? and how to calculate it?

• Refs:

The paper cited: https://faculty.georgetown.edu/qw50/RiskFactor.pdf

Here is a a previous a similar question: Calculating historical Bond returns

The prices of most coupon-paying bonds (not only corporates, but treasuries, munis, etc) are quoted "clean" (without the accrued coupon) rather than "dirty" (with the accrued coupon).

If you agree to buy a bond for a (clean) price $$P$$, then on the settlement date (generally 1-3 business days after your trade date, depending on the kind of the bond), you pay $$P$$ plus the interest accrued until the settlement date - what your paper denotes $$AI$$, times the notional. $$P+AI$$ is called the dirty price.

To mark to market your bond position on any day, you'd multiply the notional of your position by the that day's dirty price - i.e. by the sum of the clean price at which you could sell it (we'll ignore the bid-ask spread for this discussion) plus the interest accrued until the settlement date if you sold it that day.

So your formula just divides the dirty price that you could get at the end of the period plus and coupon payments by the dirty price at the beginning of the period. This is similar to how you'd calculate the reuturn on a stock over a period of time: divide the stock price at the end plus the dividends during the period by the stock price at the beginning.

There are two things I don't like about this formula. One is - it assumes that once you receive a coupon (or a dividend), you don't invest it somewhere where you'd earn interest on it. It's not a huge difference, but it implies that you're indifferent whether you receive a coupon sooner rather than later. The second thing that I don't like is that if you buy a corporate bond with money borrowed from your corporate treasury, then your corporate treasury will charge your financing, which will be much higher for a corporate bond (because it's credit-risky and illiquid) than for a treasury bond. Ignoring the cost of financing when you are trying to define the return on your position is not good.

The accrued interest is easy to compute, but how do you "compute" the bond price? Generally, you don't. It's something you observe or estimate, rather than compute, just like a stock price or a commodity price. If the bond is quoted as yield, rather than as price, then you compute the price from the yield by projecting its cash flows and discounting with this yield.

Observing a price of a corporate bond is harder than than of a stock, because a lot of bonds that don't trade frequently. If you find that this bond last traded several days ago at some price, this would not be a great prediction of the price today, unlike a stock that trades throughout the day. There are databases like FINRA's (most USD corporate bonds) and MiFID Liquid on Bloomberg Terminal (lots of EUR bonds) where you can find the last traded price. But you should not use stale prices for your marks or P&L.

Another approach is to use some kind of a simple model to estimate what a bond price should be by observing other proxy instruments (for example, other bonds from the same issuer that trade) and applying some kind of interpolation. Source BVAL on Bloomberg Terminal is a well-known example of model price.

Another approach is to collect quotes contributed by people who want to buy or sell this bond or mark their positions, and compute some consensus price (typiclaly, drop the outliers and compute the median). Sources BGN and CBBT on Bloomberg Terminal are examples. Other sources of consensus prices for corporate are IHS Markit (they are being bought by S&P, who'll probably rename it) and Refinitiv (former Reuters).

• Effectively, your argument on illiquidity might add to the complexity of calculating monthly bond returns, no? Very good read, thanks from my side. – Kermittfrog Feb 6 at 16:20
• Thank you for such a complete answer. It is really useful. Indeed, it is true that bond databases are less complete than stock databases and more difficult to build. I have access to the Refinitiv and EURO Corporate Markit EUR IBOXX constituents. I hope it has the necessary information to replicate the document. For the liquidity problem, when trying to implement a backtesting strategy the paper calculates the covariance between $P_t$ and $P_{t-1}$. Maybe not a best approximation, but at least it takes into account the concern. – pinpss Feb 7 at 12:31

"How should I define the transaction price of a bond? and how to calculate it?"

"The Trade Reporting and Compliance Engine is the FINRA-developed vehicle that facilitates the mandatory reporting of over-the-counter transactions in eligible fixed income securities" (my emphasis) - https://www.finra.org/filing-reporting/trace.

Please correct me if I am wrong.

In the US market, there are pricing differences between TRACE trade prices versus Lehman quotes data.

From one side, TRACE prices equal quotation midpoints ($\frac{Bid&space;Price&space;-&space;Ask&space;Price}{2}$), where most academics would use the midpoint of the spread as the fair price.

From the other side, Lehman quotes bid prices, where bids reveal the preferences from the highest willingness to pay from investors at the time of the quote.

Adding up, since illiquidity is a major pricing factor (with low trading volume in the day < \$1 million). On average lower liquidity tends to go further away from the true mid-point in an institutional market (hence, bigger Bid-Ask Spreads).

So in conclusion, both data measures contain similar information regarding bond valuations prices. But with a different angle, Right?

The original pricing formula can be found in from Bessembinder (2008) where they control for additional credit factors. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=650883