October 16, 2011

Bonds: Part 1 - The Theory

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For the last six years I have developed software for the bonds desks of Investment Banks (usually the European Government Bonds desk). Now it is time to move on into a different business area. So I have decided to write some notes on the business and software of bond trading as I have witnessed it – starting here with the basics.

A bond is basically a loan that can be traded. Imagine company X needs a billion dollars to buy another firm. The company issues 1000 bonds each with a face value of a million dollars (also known as nominal value), which totals to the required billion dollars. These bonds are then sold to banks and other large entities. The bonds have a maturity date 10 years in the future and an annual coupon rate of 5%. On the maturity date the issuer will pay the holder of the bond (whoever that is) the face value of the bond, a million dollars in this case. In addition every year on the anniversary of issuing the bonds, company X will pay the current holder of the bond a coupon payment – 5% of the value of the bond, which is $50,000. This is essentially the interest rate company X is paying to borrow the money. The people who get the coupon payments and the maturity payment do not need to be the same people. The bonds can be bought or sold at any time for any price. It is similar to a mortgage, if the bank could sell the debt to someone else and your repayments went to the new owner (this actually happens – see MBS, but that is a little out of my area). The first transfer of ownership, from the issuer to the purchaser, is called the “primary market”. Any subsequent sales occur in the “secondary market”.

The instrument described above is a standard bullet bond. It is called a bullet bond because before bond administration was done electronically, the physical bond paper had little tear-away tabs for each coupon payment, and these tabs were know as bullets. Military (and sports) metaphors are common in banking. However, there are numerous different categories and types of bonds. The biggest split is between Corporates/Credit and Rates. Company issued bonds are considered Corporates or Credit and government issued bonds are Rates. Inside all the banks I know about, the two are always separate from a business standpoint (different traders sitting as physically distinct desks) and sometimes so is the supporting technology, but not necessarily. As I have always worked officially with Rates, only incidentally being involved with Credit, the rest of this series will lean heavily towards the world of government debt.

Rates are often further split by business type into separate desks. Here by desks I just mean an organisationally distinct group of traders that have supposed responsibility for a type of security. Although that won’t stop a trader transacting in other securities, just that their desk defines their focus or specialty. The desks themselves can be further subdivided, but more on that in the next post.

US government bonds are called Treasuries, and they normally have their own desk of traders (usually based in New York). There is the Gilts desk for British government bonds, and the European Government Debt desk (also known as the EGB or Eurogovvy desk) for dealers of Eurozone bonds. Depending on the size of the bank there may also be desks of traders for emerging markets, supras/agencies (debt issued by non-government organisations that is guaranteed by governments – like the European Development Bank or Fannie Mae) or even sometimes a Scandi desk just for the Scandanavian countries. There is also often a desk of just a trader or two for linkers – inflation-linked bonds where the return is derived from the inflation rate in a country.

Derivatives based on rates bonds also have their own desks – futures, swaps, options, and more. I have had little to do with these desks other than where they interact with bonds desks. Credit also breaks itself into different desks, but I have never interacted with them.

Beyond just different issuers there is also an entire spectrum of different types. Bullet bonds (as used above in examples) is the most common type of bond I have encountered. However, there are many more. Each requires a different calculation to determine their price and have different levels of popularity. Below is a list of some I have encountered on rates desks. There are many more types than listed here.

  • Floating Rate Bonds or floaters – the coupon payments of these bonds is not set in advance, but instead floats. The amount paid is determined by some algorithm based on the financial environment at the time (ie. central bank rate + 1%).
  • Inflation Linked Bonds or linkers – A type of floating rate bond where the coupon payment is based on an inflation rate.
  • Zero Coupon Bonds – bonds which make no coupon payments.
  • Perpetual Bonds or perps – bonds that never payback the nominal value and just continue paying coupons forever. There are a few of these issued by the UK government hundreds of years ago and the holders still collect the coupon payments every year.

These are more common on credit desks:

  • Callable or Putable Bonds – bonds where the issuer (for callables) or the holder (for putables) can call in the debt under certain conditions – thus forcing payment of the nominal and ending coupon payments.
  • Asset Backed Bonds – bonds where the nominal and coupon payments are based on the returns on some asset, for instance pools of mortgages.
  • Convertible Bonds – bonds where the debt can be converted into company shares under certain conditions.

A hugely important part of bond trading is knowing what the bonds are worth. In this respect bonds have an advantage over other security types like equities and currencies – the cash flows to the bond holder are known in advance. Thus the Discounted Cash Flow technique can be used to price the bond. From the bullet bond example above, we know that it returns its nominal value ($1M) in 10 years (on the maturity date) and that every year (because the coupon frequency is annual) it also returns a coupon payment of $50K (as the coupon rate is 5% of the nominal). Thus the cash flows of the bond after purchasing it on issue are:

Year 1 $50,000
Year 2 $50,000
Year 3 $50,000
Year 4 $50,000
Year 5 $50,000
Year 6 $50,000
Year 7 $50,000
Year 8 $50,000
Year 9 $50,000
Year 10 (Maturity Date) $1,050,000

Using DCF and setting the discount rate to the current US Federal Reserve Base Rate of 0.25% gives the present value of the bond as $1,468,534. Thus suggesting a buyer of the bond at issue would be willing to pay upto that amount for it. The interest rate a bond returns, known as its yield, is different to its coupon rate. In the example above, purchasing the bond at issue for $1,468,534 means a yield of 0.25%, despite a coupon rate of 5%. If $1,000,000 is paid for the bond at issue, then the yield will be 5%, the same as the coupon. So price and yield are correlated, but the coupon is unchanging. The only thing that matters are the cash flows associated with holding the bond and the appropriate yield curve (that is, the expected interest rates). Thus the price of a bond drops, its yield increases and vice versa (a common entry-level interview question in Fixed Income technology). Astute readers will immediately see two problems: the choice of discount rate; and the risk of bankruptcy.

The risk that the issuer of the bond may not repay the debt is known as the credit risk. The higher the credit risk the more likely the issuer won’t pay, thus the lower the price of the bond and the higher the rate the issuer has to offer to tempt buyers. Levels of credit risk are categorised by the grades produced by ratings agencies. So other things being equal, a AAA rated bond is worth more than a C rated bond. Countries are generally considered safer than companies, so they pay less of a credit risk premium. However, countries themselves can vary in perceived risk. In the Eurozone, the spread (or difference) over German 10-year bonds (also known as bunds) is the standard measure of perceived risk – as Germany is seen as very likely to repay its debt (it is rated AAA). At the moment German 10-year bonds yield 1.95% while Greek 10-year bonds yield 23.02% (unrated due to likelihood of default) and the French 10-year rate is 2.66% (AAA rated). The Greek spread is 21.07% or 2107 basis points (1%=100 basis points), while for France it is 71 basis points. As both are in the Eurozone, both have their bonds denominated in the same currency and thus their post-issue value is driven by the same ECB interest rate. So the price difference is largely a measure of risk. Unsurprisingly, Greece is seen as incredibly more likely to default than France. Credit Default Swaps are securities to insure against credit risk, but they do not seem to be used much by EGB traders.

The issue with the discount rate is that in reality it is unlikely to remain static for all but the shortest maturity bonds. Instead a changing interest rate needs to be used that accounts for predicted changes in rates. Luckily such a projection of future interest rates exists – the yield curve. The rates to use when discounting a government bond’s cash flows is determined by a yield curve for the bond’s currency. Below is an example yield curve. As can be seen, the rate changes with time, typically with an upward slope.

Yield Curve

The process of constructing a yield curve is known as bootstrapping. First find a set of instruments with maturities that correspond to desired tenors on the curve (a tenor is just a period of time, eg 1 day or 3 months or 15 years). These instruments tend to be the most liquid (effectively the instruments with the most volume available for buying or selling) at the desired tenors. This usually means derivatives of various flavours are used (why this the case is explained in the next post). The interest rate implied by the market price of the chosen instruments becomes a point on the curve. The actual curve is then interpolated from these points (a process known as bootstrapping).

Over the short term bonds tend to trade at a set spread to their yield curve. That is its yield is not exactly on the curve, but at a small distance from it, the size of this distance is the spread. This spread represents the credit risk and any other issues (eg. liquidity) that affect the value of the bond beyond interest rate expectations.

The value of a bond also depends on when the coupon payment is due. A bond is worth more the day before a coupon payment (as the next day the purchaser will get the payment) than the day after (when the purchaser won’t get the payment). The amount the coupon payments change the price is known as accrued interest. Traders don’t like to calculate how much of an affect the coupon payments have on a price. Thus for the purposes of trading the price is stripped of accrued interest. The price with the coupon effect included is called the dirty price and with it excluded is the clean price. The rough diagram below shows how these concepts are related.

Clean Price vs Dirty Price

The price calculated may be the actual value of the bond, but it is not where banks want to trade. They want to buy low and sell high. The price at which they will buy is the bid, and the price at which they will sell is the ask (or offer) price. The calculated price is known as the mid price as it is normally halfway between the bid and ask. If it isn’t halfway then the actual halfway point is known as the skewed mid and the difference between that and the calculated mid is known as the skew. Skews are not often applied in my experience and only if the trader is particularly keen to do a certain trade. The difference between the bid and ask is called the spread (not to be confused with the spread off the yield curve – also known as the spread). The wider the spread, the worse the prices as the further they are from the calculated value of a bond. The bid and the ask prices are sent out to a market by the bank as the prices at which they are willing to trade – the mid is kept hidden. The process of publishing the prices is known as quoting. When quoting, the total nominal value of the bonds willing to be bought or sold as part of the quote is known as the volume.

For example, if a bond has a calculated price of $100, this is the mid. If the trader uses a spread of $1, then the prices quoted are 99.50/100.50. If the mid then moves to $100.50, then the quote is 100/101. If a skew of -0.5 is applied then the quote becomes 99.5/100.5 again (but the mid is still 100.5). If the spread is then widened to 2, the the quote is 99/101.

When working out whether or not a bond trade has made money the term PnL is used (short for “Profit and Loss”). There are two types: realized and unrealized. Realized PnL is the actual cash flow coming from a trade. If you have some bonds and sell all of them, then the price difference between the buy and sell multiplied by the number of bonds is the realized PnL. Thus if you have 1000 bonds bought at $100 and sell 500 at $99 then your realized PnL is -$500 (-$1 per bond for 500 bonds), so you made a loss. Unrealized PnL is the amount of realized PnL that would be made if your holdings are sold at current market rates. Thus if you have 1000 bonds bought at $100 and now the market price is $101, although your realized PnL is 0 (as they haven’t been sold), your unrealized PnL is $1000. Calculating the unrealized PnL by comparing the price paid against the market price is known as “marking to market”, and is easy in the liquid government bond market. A trader’s PnL is considered the sum of their realized and unrealized and is usually quoted on the basis of the difference day-by-day. So if a trader says their PnL is down $100K, it usually means so far they are down $100K that day.

After determining a bond’s price, measuring its risk is the next most important concept. The traders need to have some idea of how the value of their positions (that is the bonds in which they are long or short) will change when the market moves. The most common measure of this is DV01, which describes the amount a bond’s price will change given a 1 basis point change in yield. DV01 is similar to duration and gives an indication of a bond’s sensivity to changes in the yield curve. DV01 is technically only for US dollar bonds. PV01 is the same concept for other currencies (or even other security types), but on EGB desks the terms are used interchangeably (same with pvbp). To get the risk of a position, just multiply the net position by DV01. So if you own $10,000,000 of a bond with a DV01 of 0.02578, then you can expect the value of your bonds to go down approximation $25,780 for each basis point rise in yield (and vice-versa). It should be noted the above is a simplification, as the calculation only works for small changes in yield. Larger changes need to take into account convexity, but that is beyond this discussion (and I’ve rarely needed that information).

There are numerous other theoretical concepts in fixed income. There are large books devoted to understanding the myriad of derivatives based on bonds. However, the above are the fundamentals as I experienced them. Other concepts or maths came and went as required, but from a technology standpoint I often needed to understand the different types of bond, risk and valuation.

Continue to Part 2a – The Reality

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