|Eclipse Software, Inc.||Bond Pricing in the Market|
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The pricing (valuation, more precisely) of bonds, and fixed income investments more generally, is a very large and complicated subject. Our focus here is on topics of the most importance to the intended Architecture. We are specifically interested in how pricing plays out in the marketplace. We are concerned with theory only to the extent that it supports that interest.
Our topics include:
This page builds on material developed on other pages on this site:
|Inventory and Trading P&L||Buying and selling an investment (of any type), position keeping, and calculating realized and unrealized profit and loss.|
|Coupon Interest||How coupon interest is figured on trades, and how interest income and expense is calculated.|
|Day Count Conventions||These determine the details associated with coupon interest.|
This page can also be read stand-alone. Nevertheless, it is true that to have an accurate understanding of bond pricing it is necessary to understand the big picture. For instance, the day count convention used on a bond impacts not only the interest bought/sold, but also the price and realized P&L. This is true even for constant cash flows, which may not be at all intuitive. Likewise, certain folk wisdom (e.g., how certain conventions favor lenders) also turn out to be untrue in the wider context.
There are a number of topics which we neither discuss nor consider, such as:
We also give very little attention to fees, commissions, and the like. Our focus is on principal and interest.
If you have any questions or comments on this material, we encourage you to e-mail us at BondPrc@eclipsesoftware.biz.
Before getting into the details of bond pricing, there are some background points that are useful to keep in mind.
Investing is bottom-line oriented: cash out vs. cash in. When you buy or sell a bond, the cash is generally split among various categories. For instance, when you sell a bond, you normally receive cash for the accrued interest since the last coupon date (see Trade Interest Bought/Sold). You may also pay a commission or fees. But on a cash-flow basis, you only care how much you're out of pocket.
That's why we are being careful to talk about "value", not "price" or even "market value" (we'll get to these later). We are only concerned with how much you pay or receive.
Another aspect of this is that the relevant trade field for evaluating investments is always settlement date. Thus interest bought/sold on a trade is based on settlement date rather than trade date. Likewise interest income is always figured on settlement date positions, even though inventory and trading P&L are calculated on trade date positions (for the reasons for using trade date for inventory, see Date Basis).
Risk affects value. In the Consumer Purchase Analogy, the less trustworthy vendor will not be able to get the same amount for the same goods.
More generally, the higher the risk associated with a transaction, the less you will be willing to pay. This is another way of saying you require a higher return.
This is a fancy name for "you can't get something for nothing".
Putting it more technically, equivalent cash flows (e.g., coupon payments plus the return of principal) with the equivalent risk must have the same value.
If two equivalent investments have different values, you can make a risk-free return with no investment: go short the higher value investment and long the lower value.
The bond HI-VAL has a value of $98. The bond LO-VAL, with the same risk (e.g., the same issuer and terms) and set of coupon and principal payments, has a value of $96.
To make your risk-free return, with no investment on your part, you sell the bond HI-VAL, receiving $98 in cash. You buy LO-VAL, paying $96 from the $98 you received for the short sale.
You now have $2, with no investment on your own. When you receive the coupon payments for the bond you own (LO-VAL), you transfer the (equal) payments to the party you shorted HI-VAL to. The same applies to the principal payment(s).
It is important not to forget the clause "equivalent risk". Otherwise equal cash flows won't have the same value. Conversely, if equal cash flows have the same value, they basically have the same risk.
Like so much of the securities market and processing, there are a number of conventions that you have to be aware of and simply get used to. It's part of the process. You may have a PhD in math, but if you ignore the conventions you will be out of step with the market. Which means you will lose money as well as being confused.
We will discuss the conventions as they come up. These are some of the ones we'll address:
The present value (we're still not ready to talk about a "price" – see price in the Glossary) of a sequence of future cash flows is equal to the sum of the discounted values of those cash flows. This is really just another way of stating the No-Arbitrage Principle. If the present value were anything else, you'd have a risk-free way to make money without investing anything up front.
The problem, of course, is determining the appropriate discount rate to use.
For our purposes, we take the rate set by the market. After all, that determines the cash you will have to invest.
If you think the market discount rate is too high (and thus the amount to pay is lower than what you figure is appropriate), then you will buy. Since a higher discount rate is associated with greater risk, another way of saying this is that you think the risk is less (i.e., usually that the issuer's credit worthiness is higher than what the market thinks).
If you think the issuer is more likely to default than the market thinks, you won't buy.
It's as simple as that. But the rate is set by the market.
Valuing an investment basically looks at four factors: cash in, cash out, time, and risk. We are taking the discount rate as established by the market (though we will discuss it again briefly in Determining the Discount Rate), leaving three factors.
There are a number of analyses of interest, defined by the factor we are solving for.
All three ways are mathematically equivalent. We will discuss them in turn.
If you invest $100,000 in a security and get back $107,000 in one year, this is a 7% annual return, of course. If you'd invested $200,000, you would have received $214,000. For this reason, the future value after one year is normally written as:
Future_value (at one year) = Amount_invested * (1 + Rate)
If your $100,000 investment has a constant 7% return and you hold it for a second year, how much will it be worth at the end of the second year?
The above formula still holds. For the second year the Amount_invested is $107,000. Doing the math you will have $114,490.00 at the end of the second year. That it is not simply another $7,000 reflects the compounding that is going on.
This can be written more generally as
(1)Future_value (after Num_years) = Amount_invested (at Time_0) * (1 + Rate)Num_years
where "Time_0" refers to your initial investment.
The term of most general interest is the one on the right:
(2)Future_value_of_$1 (after Num_years) = (1 + Rate)Num_years
For the value at the end of the second year, Num_years is 2 and (1 + Rate)Num_years is (1 + 0.07)2, or 1.144900.
This equation applies to any value of Num_years, not just integers. The same 7% return for a half year is (1 + 0.07)0.5, or 1.034408. For one day we have (1 + 0.07)(1/365), or 1.000185.
The above example highlights four amounts of interest:
|PV||Present value. In our example this was $100,000, in (2) it was $1.|
|FV||Future value. After one year $1 had a future value of $1.0700 and after two years a value of $1.1449.|
|R||This is the return on an annual basis, as described in the sidebar Rate, Yield, and Return.
Note that it is not the coupon rate. We haven't reached coupons yet.
|T||Time, expressed in terms of years or fractions of a year. For simplicity we take 365 days in every year.|
Knowing any three of these values allows us to calculate the remaining one. In the above we solved for FV as a function of the others, thus FV = FV(PV, R, T).
We can rewrite eq. 1 as
(3)FV(PV, R, T) = PV * (1 + R)T
The bond pricing problem is different. We are given a set of future cash flows (multiple FVs) at specific times, from the bond prospectus. The market determines the discount rate. We want to determine the value as of today.
We start by determining the present value of a single payment, for example a coupon payment (FV) to be received a certain number of days from now (giving us T). The rate (R) comes from the market (we continue to assume 7%). We simply re-arrange the terms of eq. 3 to get:
(4)PV(FV, R, T) = FV/(1 + R)T
If we are to receive $35 in 90 days, we would have:
PV(FV, R, T) = $35/(1.07)(90/365)
PV(FV, R, T) = $34.42
If you have more than one future cash flow, you simply add the present values of each (FV(1) is the first cash flow, FV(2) the second, etc.):
(5)PV(FVs, R, T) = Σi [FV(i)/(1 + R)T(i)]
What if we have a present value and a future value (at some given time in the future)? We want to determine the implied interest rate (more commonly called "yield" in this context).
We can again rearrange the terms of eq. 3:
(6)R(PV, FV, T) = (FV/PV)(1/T) - 1
For instance if we are being asked to pay $34.42 in order to receive $35 in 90 days, what is the yield? Using eq. 6 we obtain the expected 7%.
In practice this is the most important, as it enables you to compare different cash flows on a consistent basis. And yield is what an investor is interested in: am I getting more or less for what I invest?
For instance, if someone (with the same risk profile, of course) offers to sell you an investment that returns $40 in 101 days and wants to charge you $39.36, is that a better or a worse deal than the one above? You calculate the yield, which is 6%, and decide the first deal is better.
Computationally, the problem is determining the yield of a series of cash flows (eq. 5). Unfortunately there is no closed-form solution (i.e., something like eq. 6). The only way to solve the problem is by a series of trials and errors, converging on the yield. See Determining Yield Given Price below.
Fortunately for this discussion, we will take the rate determined by the market.
Now that we have the math out of the way, it's time to see how bonds are priced in practice and what this means.
First, forget the math. The present value of a bond is determined by people participating in the market. If someone thinks that, given the current price, they will be able to get a greater return than they can from equivalent investements, they will buy the bond. Otherwise they won't, or, if they own the bond, they may sell it.
It is true that they will make use of the techniques we discuss here for evaluating future cash flows. However, that is not enough, because there is uncertainty and risk in every bond (even Treasuries, which have the least). See More on Net Present Value for a discussion of these factors.
The bottom line is that the present value is set by buyers and sellers in the market. That is the starting point. Now back to the math.
We use the bond FORD 5.700% 20 Jan 2012, CUSIP 34539CXR8. This bond has a 5.7% nominal coupon rate (CR) and pays semi-annually, every January and July 20th. It matures on 20 Jan 2012.
To obtain the full set of bond and trade parameters, follow the instructions below at [FORD_2012].
We start with the date 8 Feb 2008. The price is 80.087 for settlement 8 Feb 08. Note that though prices are normally quoted on a trade date basis for normal settlement (e.g., T+1, T+3), everything on this page is settlement date basis.
The discount rate (yield) set by the market is 12.572%. (We note that the "yield" quoted in the market reflects a market convention and is not the same as what we will develop on this page. See Market Yield for a discussion and references.)
Let's see if we can use eq. 5 to verify the price. You can easily program this analysis in a spreadsheet program.
There are 8 coupon dates after 8 Feb 08, including the maturity date. We assume we are purchasing $100,000 in par value of the bonds. Each semiannual coupon payment is $2,850.00. On maturity date we receive the $100,000 par plus the final coupon payment of $2,850.00. (Throughout this example we ignore weekends and holidays.)
|20 Jul 08||20 Jan 09||20 Jul 09||20 Jan 10||20 Jul 10||20 Jan 11||20 Jul 11||20 Jan 12|
|FV (total cash)||2,850.00||2,850.00||2,850.00||2,850.00||2,850.00||2,850.00||2,850.00||102,850.00|
|T (in days)||163||347||528||712||893||1,077||1,258||1,442|
|T (in years)||0.446575||0.950684||1.446575||1.950684||2.446575||2.950684||3.446575||3.950684|
|(1 + R)T||1.054308||1.119164||1.186855||1.259866||1.336067||1.418256||1.504037||1.596559|
|FV/(1 + R)T||2,703.19||2,546.54||2,401.30||2,262.14||2,133.13||2,009.51||1,894.90||64,419.76|
These are the terms on the right side of eq. 5.
Eq. 5 has us add up the values in final row in the table. Thus,
PV = $80,370.47
This gives us a price of 80.37 for the $100,000 par bond.
But the actual price for the trade is 80.087, which is much lower than what we calculated. What's going on?
Interest bought/sold is what's going on. Here's how the market actually works:
The bond uses the 30U/360 day count convention. Between 20 Jan 08 (the preceding coupon date) and the settlement date (8 Feb 08) there are 18 days (this convention uses 30 days per month). With 180 days in each coupon period (a characteristic of this convention), there is one-tenth of the 20 Jul 08 coupon payment included with the trade. The coupon payment on $100,000 on bonds is $2,850.00. Thus the trade includes $285.00 in interest.
Remember that our present value is total cash. Since there is $285.00 in interest out of the $80,370.47 present value, that leaves $80,085.47 for the rest of the trade, which by definition is the principal. This equates to a price of 80.085, which is different from 80.087 because we are not using Market Yield in our calculations (nor would we want to).
Recapping, this is the way bonds are priced in the market:
It is not the case that the price is determined, the principal is calculated, and the interest is added in to get the net money. The math works out to the same result, and that is the way it is normally treated in IT and Ops, but it is putting things backwards from the point of view of the wider context. (This can be seen in [MSRB_RuleG33] , where the formulas in Sections (b)(i)(B)(1) and (b)(i)(B)(2) have the interest bought/sold as the last term on the right.)
That's really all there is to it. For the rest of this discussion we explore the implications.
The discussion above focused on a single day or a single week. In this section we look at what happens over longer periods, and what happens over a coupon payment date.
The following table looks at two coupon payment dates, 20 Jul 08 and 20 Jan 09. Both payments are for $2,850.00. Besides the coupon date, the day before and two days after are also shown.
The values are for trades settling on the given dates; these are not "trade dates". The top part of the table has the trade values, and the bottom has the daily differences in these values.
|19 Jul 08||20 Jul 08||21 Jul 08||22 Jul 08||. . .||19 Jan 09||20 Jan 09||21 Jan 09||22 Jan 09|
What can we learn from this data about what happens around coupon date? These observations apply to bonds trading at a discount. We will discuss the premium case later.
|Net Money||Net money drops on coupon date because we now have one less future discounted cash flow.
Because the coupon yield is less than the discount rate, the drop is by less than the coupon amount. This is another way of saying that the principal is below par (i.e., trading at a discount). The general trend has to be upwards in order to converge to par on maturity date. Note how much higher the Net Money is on coupon date 20 Jan 09 than on 20 Jul 08).
Thus, though Net Money drops on coupon date, the drop decreases as you get closer to maturity (e.g., the drop is less on 20 Jan 09 than on 20 Jul 08.
Net money increases at an increasing rate within a coupon period (the exponential behavior of eq. 4). You can see this in the period from 20 Jul 08 to 19 Jan 09.
|Accrued Interest||Other than on the coupon dates themselves, the increase in accrued interest is constant (except for rounding differences). This is built into the day count conventions.|
|Principal||Because the Net Money increase is always higher then the increase in Accrued Interest (for this particular bond), Principal will exhibit a pattern similar to Net Money.|
|Price||Price will mirror Principal, of course.
Most importantly, splitting coupon interest out separately insures a more even price. In our example, where the market discount rate never changes, we see that the price increases very gradually and smoothly. It is true that the increases changes just after coupon date (because of the difference between the coupon yield and the discount rate), but it is much smoother than it would be if we did not account for coupon interest separately.
The value of the bond from the 20 Jan 2008 to maturity on 20 Jan 2012 is given in the figure below. It assumes the market discount rate for the bond does not change.
The general upward trend is due to the fact that the bond is trading at a discount. The upper saw-tooth line is the net present value of the future cash flows. The slope within each coupon period corresponds to the market discount rate (12.572%).
The light area just below it is the accrued interest determined from the day count convention and coupon rate. It is actually a fairly constant sawtooth pattern (see Day Count Conventions). The interest is going towards $2,850 on 20 Jan 2012, and the principal is converging to $100,000.
The principal is the lower portion of the figure and is by definition the difference between the net present value and the accrued interest.
At this scale the fine structure given in the earlier table is difficult to pick up. It is also true that many of the effects we are discussing are quite small in terms of the total present value, and in the real world they can be swamped by changes in the market interest rates and issuer evaluation.
But what's happening to our original investment on coupon payment date? How is it doing? Are we actually losing money now?
On 20 Jul 08 we receive the $2,850 coupon payment. Adding this to the present value (i.e., Net Money) takes our total value to 84,735.24. This is an increase of $27.49 over the prior day, compared to an increase of $27.48 on the prior day.
If you assume you can reinvest the coupon payment at the same return (not coupon rate) as the bond, your total investment would show a smooth, exponentially increasing curve.
This reinvestment assumption also comes into play when evaluating the initial investment decision. We invest $80,370.47 at a yield of 12.572%, maturing in 1,442 days. We might expect at maturity to have about:
FV= $80,370.47 * (1 + .12572)1,442/365
But our total cash received is the sum of the 8 coupon payments and the principal:
Cash= 8 * $2,850.00 + $100,000.00
The difference between the calculated future value and the cash received is the reinvestment of the coupon payments. If we can (and do) reinvest them at a yield of 12.572%, we will end up with a total cash amount at maturity of $128,316.27. Obviously there is risk in this assumption.
It is interesting that traders in investment firms do not get credit for the reinvestment income on cash generated. A trader who held the security to maturity would be allocated $42,429.53 in income, not $47,945.80. The trader would also be charged a money market rate of interest expense on the amount invested over the full holding period, of course.
There is generally no difference between the final coupon period and the earlier ones (the exception being a long or short final coupon period). The accrued interest again goes toward $2,850, while principal goes towards par: $100,000. Bond holders will receive $102,850 on January 20th.
|Trade values||19 Jul 11||20 Jul 11||21 Jul 11||22 Jul 11||. . .||17 Jan 12||18 Jan 12||19 Jan 12|
What happens if Ford has also issued a discount security, i.e., one that has no periodic coupon payments, only a final payment at maturity (20 Jan 2012, the same as the bond)? We will call this Ford-DSC. It has all the other provision of the bond we discussed earlier, so it has the same risk profile.
We buy $102,850 par value of XYZ-DSC on 20 Jul 2011 (settlement date), the beginning of the final coupon period. How much should we expect to pay for it?
Given that it has the exact same cash flows as the bond at that point, and that they have the same risk profile, the No-Arbitrage Principle tells us they have to have the same value. So, we would pay $96,889.73 (see the table for The Last Coupon Period).
This is also what we would pay for the bond in total, of course.
What happens to the end-of-day P&L accruals for the two securities? Because they will have the same total value the next day, we would expect to have the same P&L. And we do, but it looks a little different:
If both securities are held to maturity, they will both have $5,960.27 in income: $102,850.00 - $96,889.73. But the bond will show $2,850 in interest income and $3,110.27 in realized P&L, while the discount note will show it all in realized P&L.
Given that cash is king, what's the justification for the different treatments? We invest the same amount up front and get the same amount out at the end of the period, after all.
To confuse things further, if there were other bonds in the final period but with different stated coupon rates, we could buy an amount of bonds that would return us $102,850 at maturity. We would then have different splits between interest income and trading P&L as well.
On the off chance we're not confused enough, couldn't we argue that we should recognize our 12.572% annual return as interest, with a straight return of the principal we invested? That's how a bank account would be handled, with no realized P&L at all.
The short answer is that this is the convention. This situation is well understood, so we all benefit from a common foundation.
Note that this situation exists only when we are in a single payment period.
The discussion above has been for a bond being discounted more than the coupon yield (5.7812%). This results in the generally increasing curve seen above.
The picture is somewhat different if the discount rate is below the coupon return. If the market were discounting the bond at 3.00%, the bond would be worth more than par. The bond converges to the same values on 20 Jan 2012, but the path is different as shown below.
The net present value is still increasing at the discount rate (3.00% in this case). Because the coupon yield is greater than this, on coupon date the net present value drops by more than the amount of the coupon payment. This results in the downward trend in the principal.
What happens if the bond had a different convention, say Act/Act (ICMA)? This convention has the same coupon amounts as above, but treats every day in a coupon period equally; no days are skipped. There are 182 days between 20 Jan 08 and 20 Jul 08, and 19 days between 20 Jan 08 and 8 Feb 08. Thus,
Trade interest = (19/182)*$,2850.00
Trade interest = $297.53
Doing the same math as before, this results in principal on the trade of $80,072.94, and a price of 80.073.
It may seem counter-intuitive that two bonds with the exact same cash flows and risk profiles could have different prices. But this just reinforces the fact that it is cash that is king, and that price is backed into. If this weren't true, you would have a risk-free arbitrage opportunity.
Likewise, the two bonds may reflect different realized trading P&L for transactions involving the exact same cash flows.
If the discount rate for the bonds doesn't change, and they are sold to settle one week later, the present value of the cash flows goes up by $182.74, to $80,553.21. This $182.74 is indeed our profit, but how it is accounted for is different for the two day count conventions.
|8 Feb 08||15 Feb 08|
|Trade Interest||285.00||395.83||110.83||Interest Income|
|Net Money||80,370.47||80,553.21||182.74||Same for both DCCs|
|Trade Interest||297.53||407.14||109.61||Interest Income|
|Net Money||80,370.47||80,553.21||182.74||Same for both DCCs|
|Price||80.0729||80.1461||0.0732||Reflects differences in principal, and thus realized P&L|
Given that the values of the two bonds are the same and the interest is different, something else has to change. The only possibilities are realized and unrealized P&L.
Thus it is generally not helpful to describe one day count convention as "better" than another, or that one benefits the lender or the borrower. Just because a 30/360 bond may "lose" a day's interest in July, it does not mean that the market allows it to get lost in a total return sense. This is one reason why fixed income investments are normally reported on an equivalent yield basis.
On 28 Feb 08 our bond ([FORD_2012]) has a value of $80,370.47. We then compute the interest based on the day count convention and back into a price for the remainder: 80.0855.
Given that what we really care about is the money going out (or coming in), why not just skip the interest step and quote a price of 80.3705 in total? Stocks pay dividends, and that's the way they're priced.
We could go that route. A trade done with interest figured separately is said to be done on a clean price basis. The 80.0855 above is the clean price. A trade done for the total amount is said to be done on a dirty price basis. 80.3705 is the dirty price. Both trades will settle for the same amount. One is not cheaper (or more expensive) than the other.
For this example we have:
To a large extent this is a matter of following the market convention, which is to use clean prices (we look at the reasons behind this convention in detail below). Conventions do matter.
The price used also depends on the context. It is rare to use a dirty price on a trade or in accounting. In various analytic applications, however, the dirty price is the norm. Measures such as duration and convexity are always done on a dirty price basis. We will return to these areas after discussing the reasons behind the clean price convention.
Finally, "dirty" is a value-neutral term. It simply means interest is not broken out separately (this is more accurate than saying the dirty price includes interest; interest as a separate component is an outcome of the clean price convention, not anything inherent in the bond). You could call it the "full value" price just as well. "Full price" is sometimes used as another term for dirty price.
Stocks (often) make periodic dividend payments, just like bonds make periodic coupon payments. Stocks are traded (and reported and analyzed) on a dirty price basis. Why not bonds?
A better question is, why aren't stocks traded on a clean price basis? The reason is, they can't. Coupon payments are a contractual feature of a bond, included in the prospectus. The issuer does not have the right to suspend them. The issuer may be forced to default, for reasons of business survival, but this is a dire step.
On the other hand, dividends are elective on the part of the issuer, even for preferred stock. The timing and amount (if any) is at the discretion of the issuer. Stocks are thus always traded on a dirty price basis.
With bonds, then, we have the choice of using clean or dirty prices. Why would we want to use clean prices? After all, we start with the dirty price and back into the clean price. If the convention was to use the dirty price, there would be no need for day count conventions.
There are several very good reasons to prefer a clean price:
Splitting out the interest component results in the remainder of the bond value (the principal, by definition) being much more closely aligned to the changes in the yield. In our example we kept the yield fixed, and the clean price is really quite smooth. The dirty price (and associated principal) would be very uneven, reflecting the jumps on coupon date.
Stock dividends are usually for a smaller percentage than coupons, so the drop is smaller. Stocks are also much more volatile. If you look at the price history for a stock, it is usually difficult to pick up the drop in price on ex-date.
The clean price approach thus leads to improvements in the quality of the data on both the balance sheet and the income statement. If you need to work with the dirty price, you can always create it from the clean price and interest bought/sold.
The decision to use the clean price convention results in the following effects:
Given that the market adopted clean prices as the convention for trading, dirty prices have to be used with caution. We look at the situation in a number of different areas.
In certain cases trades are done with a dirty price: no separate interest bought/sold is incorporated into the trade figuration. This is sometimes referred to as trading on a net money basis. The trade settles for the full cash amount, just as any other trade. It is not true that any interest is "lost"; the total value of the trade will not change.
The vast majority of bond trades are figured on a clean price basis.
As long as both sides agree, a trade can clear and settle (see clearance). However, Ops is accustomed to handling trades on a clean price basis. Anything that introduces non-standard features increases both cost and the risk of operational error.
Dirty trades present a number of problems for the Trade Accounting and Risk functions.
This impacts accounting (as we saw before), compliance, and risk.
For all these reasons, in every instance where we have encountered a dirty-priced trade the firm has treated it on a clean price basis on its books and records.
Here the situation is more variable, because of the number of different audiences and the different interests. Dirty-pricing can refer to reporting on both trades and positions.
For the reasons given in the preceding discussion, reports for accounting, auditing, risk, and compliance are almost always done on a clean-price basis. The pricing choice affects P&L, both interest and trading, so you have to be consistent. Reporting interest income/expense and trading P&L implies a clean-priced approach to principal (and interest bought/sold) as well.
If you are interested primarily in the value of your bond (and this is of course a tremendously important issue), you can certainly report it as the sum of the principal and accrued interest. Many analyses are based on the full value of the investment, so they use the dirty price approach. Examples are duration and convexity.
The real issue is one of terminology. In the industry "price" explicitly means the clean price. Similarly, the convention is that "market value" refers to quantity times the (clean) price.
There are no standard terms for the dirty-priced values. Sometimes the market value based on principal alone is called the "clean market value" or the "flat market value". Likewise the combination of interest and principal (at market price) may be referred to as the "full market value" (and less frequently the "dirty market value").
If you are in an area where both full and flat market values are often reported, the paramount consideration is to be explicit and consistent in how the values are labeled and referred to.
Once you have the cash flows and the discount rate to apply, calculating the net present value of a bond (and, knowing the applicable day count convention, the accrued interest, and then the price and principal) is straightforward. This was discussed above in Valuing a Sequence of Cash Flows.
Though not the focus of this page, those "givens" require a little further discussion.
In our [FORD_2012] example, we took the discount rate of 12.572% as a given. Where did it come from? The market, certainly. But how do we decide if we think this is the appropriate value to use, and thus whether the bond is a good investment or not?
Determining the discount rate is a huge topic, so we will only point out that it is based on a number of factors, including:
A difficulty is that a bond typically has a sequence of cash flows, not just one at the end, as we have seen. One approach is to weight the cash flows by the period until they are received and use that as a measure. This is one of the ideas behind the calculation of duration.
Firms have credit analysts whose explicit function is determining the credit worthiness of bond issuers, which factors directly into determination of the discount rate.
Determining the cash flows of a bond can be more difficult than determining the discount rate, because it requires assumptions about future events.
Some of these events are features contained in the bond's prospectus and typically include an element of choice. These are called embedded options and are sometimes referred to collectively as a bond's optionality.
Our Ford bond ([FORD_2012]) has call provisions. This enables Ford to buy back the bonds at certain times if they choose to. Typically they will want to do this if interest rates fall to the extent that they can re-issue the bonds at a lower interest rate (and thus lower cost to themselves). Or, less often, they may be awash in cash.
This makes the calculation of net present value problematic, as you no longer know the cash flows. It certainly makes applying eq. 5 difficult.
You do know up front that the call provision makes the investment riskier. You can't even do a buy-and-hold strategy counting on the stated coupon and principal payments. This tells you that the bond must have a higher return than it would without the call provision.
Another source of uncertainty comes with amortizing securities, such as pools of mortgages, where the underlying mortgage holders have the option of paying off the principal faster than called for in the mortgage documents. The future cash flows are even more variable and difficult to predict than with corporate bonds. Most asset-backed securities have such a feature.
Mortgage paydowns are also interest-rate sensitive, of course. If rates fall, the person who took out the mortgage will be motivated to refinance, i.e., pay the existing mortgage off and obtain a new one at a lower rate. If you hold such a security, it also means that your reinvestment assumption will probably not hold.
But there may be other factors as well. A community's economic well-being can influence mortgage paydown rates.
Another complicating factor of these securities is that the underlying borrower may default.
Each type of securitized asset has different repayment characteristics and various approaches to modeling them. One of the more common is the PSA prepayment model for mortgage-backed securities.
Another embedded option is a sinking fund provision, where the issuer is required to call a certain number of the outstanding bonds according to a fixed schedule.
The bond holder may also have an option, called a put option, that enables the holder to receive payment of the outstanding principal from the issuer in advance of its stated maturity. The holder may wish to do so if interest rates rise or the issuer is deemed to be less credit-worthy.
Being in many ways the opposite of a call option, this feature reduces the risk to the bond holder, which increase the value.
Variable rate securities pose additional complications. These bonds typically tie their coupon rates to a benchmark such as LIBOR or the 30-year Treasury. This means that the coupon payments can change from period to period, and they are not predictable.
Many of the features we've been discussing in this section, though not all, are sensitive to future (and unknown) interest rate movements. These movements impact both the discount rate (R) and the cash flows (FVs).
Some of the features, such as the call and put options, are binary in nature. If the interest rate at the time when the embedded option can be exercised is below (for a call) or above (for a put) a given level, the bond effectively matures at that point.
Other features, such as prepayments of principal on a mortgage-backed security, are more involved. If the interest rate comes down, prepayments will go up. This not only impacts the cash flow at the time of prepayment, but into the future as well. The remaining interest payments will go down because of the reduced principal outstanding. The future principal payments will also decrease, of course.
Interest rates can vary over time, so the prepayments must adjust on an on-going basis.
How can these features be handled? Closed form mathematical solutions are generally impossible for them. A common approach is to devise a number of future interest rate scenarios and then model them, advancing the time period-by-period and see what happens to the future cash flows (e.g., a lower rate may result in a bond being called). Each scenario can result in a different present value. The scenarios are assigned probabilities and the results combined.
Of course this leaves you with the problem of assigning priorities to each of the future interest rate scenarios....
This section has a selection of sub-topics that amplify the material presented above.
In Valuing a Sequence of Cash Flows we derived the basic formulas relating present and future values, discount rate (yield), and time, as summarized in the Equation Summary. This is what a mathematician would consider yield and is what you get if you use standard spreadsheet functions. There are a number of other yield measures in use in the market, and it's helpful to be aware of them.
It is important to remember (see the recap at the end of Bond Prices in the Market) that the yield we have been discussing was backed into from the present value asssigned by the market. It is an after-the-fact measure.
In particular, this yield is applied uniformly to all future cash flows, regardless of tenor. But this is not the way the market actually operates. Future cash flows are valued based on the yield curve. Generally, the yield is higher the longer the investment is held.
This means that in the Equation Summary the term R, the discount rate, should be a function of the tenor, R = R(tenor).
"The yield curve" itself is a bit of a misnomer. Every investment has an implicit yield curve. The ones most quoted are those based on Treasury instruments with various tenors and one based on LIBOR instruments.
Adapting these curves for use with an arbitrary investment is not straightforward.
This assumes that the shape of the curve matches that of the Treasuries. There is also the fact that the spread (e.g., 1.5%) over the Treasuries will usually change over time.
The term "yield" in the bond market is defined slightly differently. The market convention is based on coupon periods, not straight time to the future cash flow. This is how trades are quoted and is the value that will be printed on confirms. This is the value we saw quoted for the Ford bond.
The conventions can be found in [MSRB_RuleG33]. The formula in Section (b)(i)(B)(2) corresponds to our eq. 4. All bonds are assumed to have a coupon frequency of 2, which gives rise to the "Y/2" terms. The reliance on coupon periods is clear.
Further, there is are different function for securities in their final coupon period. Section (b)(i)(B)(1) has that formula.
Though different from our variable R, the values are quite close.
Note that this yield is sometimes referred to as the "yield price".
The web page for [FORD_2012] has a value for "Semiannual Yield" of 12.200. The "Annual Yield" is 12.572.
We saw this concept in Coupon Amounts. Using eq. 3, $1.00 invested in this bond would have the following value in half a year:
FV = (1 + R)T
FV = (1 + 0.12572)0.5
FV = 1.061
What quoted bond coupon rate corresponds to such a return? Remembering that coupon payments are done on a simple interest basis, we simply double the 0.061 to get 0.122, or a coupon rate of 12.200%.
This is "Semiannual Yield".
In Determining Yield we noted that there is no closed form solution to the problem of determining yield if we are given the present value and a series of future cash flows.
If you build a spreadsheet with the future cash flows, you parameterize the present value function with a value for yield. You sum up the present value of each cash flow, and that sum gives you the present value (and thus the price) for the entered yield.
If you were given the price and asked to find the yield, however, you would probably proceed by fiddling with the yield parameter until the price you calculate equals the one you were given. In this way you've determined the yield.
This trial and error approach is what is used in practice. The most common method encountered is Newton's Method. It is used because of its convergence characteristics.
To use Newton's Method, the "zero" you are searching for is the expression "price_in_the_market - price_you_calculate".
If you buy a bond that will pay $1,000 at maturity for a price of 98.5, you pay $985.00 in principal, not $98,500. (If you buy 1,000 shares of stock at a price of 98.5, you would pay $98,500.)
This is simply the bond market convention. It is sometimes referred to as "the price per $100 par value".
In Coupon Amounts we saw that receiving a semiannual coupon at half the nominal coupon rate (CR) results in a higher annual yield (R): (1 + CR/2)2 - 1.
R continues to increase as we increase the compounding frequency (2 in the example) and converges in the case of continuous compounding to:
1 + R= eCR (continuous compounding)
and for a time T:
(1 + R)T= eCR*T (continuous compounding)
People sometimes use eCR*T in place of the correct equation (eq. 3), (1 + R)T, when computing interest for a given annual return (i.e., there is no compounding involved).
If you want to take advantage of the exponential formulation and still be accurate you need to adjust the exponent. We use "r" to refer to the value we'll use in the exponential formula ("ln" is the natural logarithm).
er= (1 + CR/Fr)Fr
r= Fr*ln(1 + CR/Fr)
erT= (1 + R)T
The topics of bond pricing, yield calculations, coupon interest, and day count conventions can be very challenging, at least as much for the mechanics exhibited in the market as for the inherent complexity. These Review Questions have been developed to help you to think about the topics more broadly.
If you would like to discuss these Review Questions, please feel free to e-mail us at RQ-BondPricing@eclipsesoftware.biz.
We discussed under A Second Security some of the implications of the market convention that equates interest income with the amount of coupon interest earned.
Consider a coupon-bearing bond that we are holding (thus we don't have to consider realized P&L arising from a sale). The value goes up (or down) as time passes, the market discount rate fluctuates, and cash is received.
It is a natural reaction to these topics to feel that there must be a better way for the markets to operate. The second Review Question is:
Be complete, yet concise.
This will give you all the data we discuss on this page.
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