Bond Math

In this section, we give some basics on “bond math”. The reader should be warned that the practical details are messy.

Discount Factors

Discount factors are essential to understand because they describe, once a cash flow is determined in the future, how much the cash flow is worth today. The set of zero-coupon bonds

{

^P,(:’> are the set of discount factors. The zero-coupon bond

Pzc

^(TI equals today’s value of a future cash flow of amount $1 to be paid by issuer ABC at a time

T

in the future. Conversely, l/P!E) tells you how much $1 invested today will be worth at time

T

if invested in an interest-bearing account with rate typical of the ABC coupon. Theoretically,

{

^P$] arise from stripping coupon bonds of different maturities. Since in general there are only a discrete (and sometimes small) number of bonds for a given issuer, aggregation is used to get discount factors for a given credit rating in a given sector. Interpolation schemes must be adopted in order to calculate the discount factors for arbitrary maturities between the known ZC bond maturity dates.

Yields

Bond prices can be recast equivalently as “yields”. The yield

y

is a common rate to be used in all discount factors for all coupons in order to produce the given price of the bond”. A quick mnemonic is “yield up, price down“. What this means is that if rates go up and the coupons of newly-issued bonds increase, investors will pay less for an already-issued bond because these pay a lower coupon.

Take a coupon bond paying f coupons per year, with the

j r h

coupon to be paid at date

q.,

a time interval Zj years from today. There are “day-count

l o Other Yield Conventions: There is also a “nominal yield”. This is the annual coupon

and a “current yield”, which is the annual coupon divided by the price in decimal. If the bond is callable, the coupons up to the first call date are employed to give the “yield to call” YTC. The “yield to maturity” YTM is the definition in the text. The “yield to worst”

YTW is the minimum of YTM and YTC. For a premium bond called at par, YTW = YTC and for a discount bond YTW = YTM. For munis, by regulation the YTW must be the yield quoted to clients.

Bonds: An Overview 119

conventions" for what constitutes a year]'. The yield has the attribute of the frequency fcompounding, and the discount factor for that cash flow is

pzc

(Ti) = [ 1 + ~ / f ] - ~ "

.

Note as

f +

⁰⁰ we get continuous compounding, i.e.

Pzi

( T .

1 +

e x p ( - y z , )

.

^{The price}

BgL

^{of the} ABC bond of maturity

T

, a time interval Z from today, with annual coupon

c A B C ,

is the sum of all the discounted cash flows. i.e.

N

BjlTB)c ( y ) =

cF[l+

y/f]-"'

+

100[1+ y / f I p f r (9.1)

j =O

The j = 0 accrued interest term is omitted for the clean price, and is present for the "dirty price" with the discount factor omitted12. The j = 1 term has the first full coupon. The last

N"'

coupon is paid at maturity. In real calculations, there are a myriad of details13.

Duration and Convexity

The duration D and convexity C of a bond with price B are defined as:

(9.2)

The duration, defined with the minus sign, is positive. Note that the duration, up to a factor, is the weighted average time of

payment^'^.

To second order, we get the relation

' I 30/360 Day Count: This assumes that there are 30 days per month and 360 days per

year, as described in the preceding chapter.

Accrued Interest: Accrued interest is calculated from and including the last interest payment date, up to but not including the settlement date of the trade. Settlement (regular way) is t

+

3 days for corporates and t

+

1 day for US governments, where t is the trade date. Accrued interest is paid to the seller of the bond.

Bond Conventions: These can get very messy and depend on the contract details of the bonds. In particular, bonds have a variety of conventions determining the actual interest payments. T o get a complete description of the complexities, consult The Bloomberg (hit the GOVT button and then type DES and HELP). You will get around 45 pages listing 600 conventions for calculating interest for government bonds in different countries. This is cool.

13

l 4 Other Durations: Macaulay duration is defined to include a factor (I+y/f).

The DVOl is the change in the bond price SB for a one bp/yr change in yield, viz 6 y ⁼ 104/yr. This includes all changes in the bond price, including any changes due to embedded options in the bond if they exist. The DVOl is generally defined numerically, since call features in bonds and other complexities cannot be described analytically. The conventions for the actual number of basis points moved vary. If the move Sy is too small, the numerical algorithms can become unstable and/or produce unphysical numerical jumpsI5. If the move is too big, large second order convexity effects enter. For example we can move S y = 5 0 b p / y r . Then the change SB is scaled back down by S y to get the DVO 1.

Spreads

Universally, the world of bonds is described by their spreads. Differences between a bond’s yield and the yield at the same maturity for a given benchmark (e.g. government or Libor) define that bond’s spread. Typically, spreads are quoted for a given credit and a given sector (e.g. BBB US Industrials).

The spread contains all information about the price, given the benchmark.

Therefore, all the effects determining the bond price enter in the spread. These include the perception of risk due to potential default and credit downgrades, technical supply/demand factors, market psychology, and generally all the information used by bond tradersi6.

I 5 Numerical Instabilities and DVOl: Numerical code involves discretization and if the yield change is too small, the code gets two nearly equal prices. The price error may be relatively small, but the difference between the nearly equal prices is small enough to be very sensitive to these price errors, sometimes unfortunately resulting in instabilities in DVOI . If this happens, increasing the shift 6y can help. All this is hard to explain to some people.

l 6 Spreads and Implied Probabilities of Default: Some analysts assume that bond

spreads are entirely due to the possibility of default. Actually, the logic is turned around to get “implied probabilities of default” pimp-defaul, from the spreads. This is done using bond spreads in the discount factors along with logic that eliminates bonds that happen to default in the future based on the probability pimp-default

.

Then pimp-default is varied until the bond price is obtained. However, historical statistics give actual default probabilities that are quite different from the theoretical probabilities pimp-default.

Bonds: An Overview 121

Option-Adjusted Spreads (OAS)

The option-adjusted spread (OAS) for bonds with embedded options is defined as a calculated spread added to the benchmark curve, such that the bond model price is the same as the market price. The benchmark curve is the curve off which spreads are defined, e.g. US Treasury. The model needs to use whatever logic is needed to include the embedded options. The model numerical algorithm can be a formula, a numerical calculation using a discretized lattice, a Monte-Carlo simulation, etc. Given the benchmark curve, the OAS is therefore a translation of the bond price, including the effect of the options.

Callable bonds, mortgage products, etc. are commonly quoted using OAS.

Duration and convexity are often defined such that the OAS is held constant.

General, Specific, and Idiosyncratic Risks

Risk can be classified in successive degrees of refinement. If an average risk is taken over many bonds, such as a large bond index17, the risk is called “general”.

As refinements are made, narrowing the focus to an average over bonds in a sector, the risk becomes more “specific”. If the details of a specific issuer are specified, the risk is highly specific. The difference between specific and general risks is also a form of “idiosyncratic risk”. The problem in getting idiosyncratic risk essentially is that the market prices are not known for all bonds. Hence, various approximations have to be made.

Matrix Pricing and Factor Models

There are various approximate pricing methods. One is called “matrix pricing“

where known prices of some bonds in a given sector are marked on a matrix of coupon vs. maturity, and interpolation is used to get other prices.

“Factor models“ assign components of bond spreads to various issuer credit and sector characteristics etc., and thereby arrive at an approximate theoretical price for a bond.

References

i Bonds (General)

Sharpe, W. and Alexander, G., Investments, 4‘h Ed. Prentice Hall, 1990. See Ch.12-14.

Fabozzi, F. and Zarb, F., Handbook of Financial Markets, 2”d Ed.. Dow Jones-Irwin Fabozzi, F. , Fabozzi, T. D., Bond Markets, Analysis and Strategies. Prentice Hall 1989.

Fabozzi, F., Bond and Mortgage Markets., Probus Publishing, 1989.

1986.

” Bond Indices: A standard general bond index used as a benchmark is the Lehmann Brothers Aggregate Bond Index. Sector bond indices for corporates, munis, etc. exist.

Allen, S. L. and Kleinstein, A. D., Valuing Fixed-Income Investments and Derivative Series 7, General Securities NYSE/NASD Registered Representative Study Manual,

Securities. New York Institute of Finance, 199 1.

Securities Training Corp., 1999.

i J Money Market

Stigum, M., The Money Market. Business One Irwin, 1990.

''I Step-up Bonds

Derivatives Week Editors, Leaning Curves Vol II. Institutional Investors, 1995. See p. 78.

Convertibles

Zubulake, L., Convertible Securities Worldwide. John Wiley & Sons, Inc. 1991

" Mortgage-Backed Securities

Bartlett, W., Mortgage-Backed Securities. NY Institute of Finance, 1989.

Davidson, A., Herskovitz, M., Mortgage-Backed Securities. Probus Publishing Co., 1994.

Davidson, A., Ho, T. and Lim, Y., Collateralized Mortgage Obligations. Probus Fabozzi, F. (Ed), The Handbook of Mortgage-Backed Securities. Probus Publishing Co.

Hayre, L. (Ed), Guide to mortgage-backed and asset-backed securities. Salomon Smith The Mortgage-Backed Securities Workbook. Mc-Graw Hill, 1996.

Publishing Co., 1994.

Barney, John Wiley & Sons, 2001.

vi Asset Backed Securities

Pavel, C. Securitization: The Analysis and Development of the Loan-BasedAsset-Backed Securities Markets, Probus Publishing, 1989.

vii Non USD Bonds

Bowe, M., Eurobonds. Dow Jones-Irwin 1988.

Viner, A., Inside Japanese Financial Markets. Dow Jones-Irwin 1988.

v''' Brady Bonds

Govett, H., Brady Bonds - Past, Present and Future. www.bradvnet.co1n/e24.htni, 1996.

Molano, W. T., From Bad Debts to Healthy Securities? The Theory and Financial Graicap Fl Research, Introduction to Brady Bonds. www.bradvnet.com/e52.htm, 1997.

Techniques of the Brady Plan. www.bradynet.conl/n025.htni, 1996.

ix Flight to Quality and LTCM

Lowenstein, R., When Genius Failed. Random House, 2000.

Bloomberg, L. P., Meriwether Apologizes for LTCM Collapse, Cites Flaws, 812 1/00;

Long-Term Capital CEO Meriwether's Letter to Investors, 9/2/98; Long-Term Capital's Problems Loom over Markets, 10/5/98; Long-Term Capital Management's Investments, 10/9/98.

New York Times, At Long-Term Capital, a Victory of Markets over Minds, 1011 1/98.

10. Interest-Rate Caps (Tech. Index 4/10)