Bond Mathematics

April 10, 2023 Sankarshan Basu, IIM Bangalore 1

Basics of Bond MathematicsBasics of Bond Mathematics

Sankarshan Basu


Basics of Bond Mathematics


What is Bond?

• A bond is a debt instrument • It is used to raise funds for a defined period

of time at a specified interest rate by corporate entities or government.

• In exchange for the funds, the issuer of the bond issues a certificate, or bond, that states all the details about the interest rate as well as maturity date of the bond.

• Interest on bonds is usually paid every six months (semiannually).


Some Preliminaries

• Price - Yield Relationship

• Measures of Price Sensitivity– PVBP– Duration:

• Macaulay Duration

• Modified Duration

– Convexity


Some Preliminaries (continued)• Yield Curves and Term Structure

– Spot Rate

– Forward Rate

– Yield Curve

• Term Structure of Interest Rates– Fitting Term Structure to Bond Price Data

– Conventional Theories of Term Structure• Expectations Theory

• Market Segmentation Theory

• Liquidity Premium Theory


Price Value of a Basis Point (PVBP)

Definition: The change in the price of a given bond if the required yield changes by one basis point (that is one-hundredth of a percentage point).

It is also called Dollar Value of a Basis Point (DV 01) as it indicates a measure of a bond's price/yield relationship that specifies the dollar value of a basis point change in yield.


Duration of a Bond

• Duration is a weighted measure of when the investment on a bond is recouped.

• Duration considers coupon payments. • For a zero-coupon bond, the duration is the

same as the bond's maturity. • For two bonds that mature at the same time,

the one with a higher coupon has the lower duration.


Types of Duration

There are two types of Duration:• Macaulay’s Duration: This is the traditional,

and more intuitive, measure of “duration".• Modified Duration: This expresses the

measurable change in the value of a security in response to a change in interest rates. This is usually used to see what effect a change in interest rates will have on the price of bonds.


Macaulay’s Duration

where,

k = no. of payments per year

N = no. of periods = no. of years * k

t = timing of cash flow CFt

PVCFt = PV of all cash flows from bond

Thus, % change in Bond Price

N

t t

tM PVCFk

tPVCFD

1

100***1

1ChangeYieldD

kyield M


Modified Duration

Thus,

% change in bond price

= - Md * Yield Change * 100

kYieldD

MD M

1


Properties of Duration

• Duration decreases with increase in coupon rate• Duration decreases with increase in YTM• Duration and Maturity

– ZC Bonds: D = T; Increases with T

– Coupon bonds: D 1 + 1/i as T – Premium and Par bonds: Coupon Yield to maturity;

D increases with maturity and approaches (1+ 1/i)

– Discount bonds: Coupon < yield to maturity; D first increases with maturity. reaches a maximum then decreases ,approaches (1+1/i)


Convexity of a Bond

Duration summarizes the most important information about a bond’s or a portfolio’s sensitivity to interest rate changes (small changes).

Convexity on the other hand measures the second most significant information (for medium to large changes in the interest rate)


Formula for Convexity

Convexity =

Dimension of convexity: (Years)2

Also,

where B is the price of the bond and i is the continuously compounded interest rate.

N

t t

t

PVCFk

PVCFtt

kYield 1

22 *

1

1

1

2

2

2

2

*2

1*

2

11

diConvexitydiMD

didi

Bddi

di

dB

BB

B


The Yield Curve

• The Yield to maturity (YTM) is the true rate of return an investor would receive if the security were held until maturity.

• Yield rates (spot interest rate) expressed as a function of maturity is known as the term structure of interest rates.The graphical plotting of this function is known as THE YIELD CURVE.

• The yield curve is one of the most important indicators of the level and changes in the interest rates in the economy.


Zero Rates

A zero rate (or spot rate), for maturity T, is the rate of interest earned on an investment that provides a payoff only at time T

Thus suppose the 5 year zero rate is 5% and the amount invested is $100. Thus, the payoff after 5 years is:

$(100 * e(0.05*5)) = $128.40


Zero Rates (continued)

Note that most of the interest rate observed in the market is not the pure zero rate. Consider a 5-year government bond that provides 6% coupon. Thus the price of the bond does not exactly determine the 5-year Treasury zero rate as some of the return on the bond is realized in the form of coupons before the bond matures.


Bond Pricing

• Most bonds provide coupons periodically

• The owner also receives the principal or face value of the bond on maturity.

• The theoretical price can be calculated as the present value of all the cash flows that shall be received using appropriate zero rates as discount rates.

Let us now consider an example and price a 2-year Treasury Bond using appropriate zero rates with continuous compounding.


Example

Maturity(years)

Zero Rate(% cont comp)

0.5 5.0

1.0 5.8

1.5 6.4

2.0 6.8


Bond Pricing (continued)

• In our example, the theoretical price of a two-year bond providing a 6% coupon semiannually is

39.98$)103

333$(0.2068.0

5.1064.00.1058.05.005.0

e

eee


Yield of a Bond• The bond yield is the discount rate that makes the

present value of the cash flows on the bond equal to the market price of the bond

• Suppose that the market price of the bond in our example equals its theoretical price of 98.39

• The bond yield is given by solving

to get y=0.0676 or 6.76%.

3 3 3 103 98 390 5 1 0 1 5 2 0e e e ey y y y . . . . .


Par Yield of a Bond• The par yield for a certain maturity is the coupon rate that

causes the bond price to equal its face value.• Suppose, in our example the coupon on the 2 year bond is

c% per annum and the face value is $100. Then

This implies c = 0.0687 or 6.87% per annum with semi annual compounding

1002

100

222

0.2068.0

5.1064.00.1058.05.005.0

ec

ec

ec

ec


Determining Treasury Zero Rates

• As discussed earlier, the market interest rates are not pure zero rates.

• We now discuss how to calculate Treasury Zero Rates from prices of traded instruments.

• One (and the most common) approach is the Bootstrap Method.

Consider 5 bonds – the first 3 pay no coupons. The zero rates corresponding to these bonds are easy to calculate. Using this information, we then calculate the zero rates corresponding to the other bonds. Data is in the next slide.


Sample Data

Bond Time to Annual Bond

Principal Maturity Coupon Price

(dollars) (years) (dollars) (dollars)

100 0.25 0 97.5

100 0.50 0 94.9

100 1.00 0 90.0

100 1.50 8 96.0

100 2.00 12 101.6


The Bootstrap Method

• An amount 2.5 can be earned on 97.5 during 3 months.

• The 3-month rate is 4 times 2.5/97.5 or 10.256% with quarterly compounding

• This is 10.13% with continuous compounding• Similarly the 6 month and 1 year rates are 10.47%

and 10.54% with continuous compounding


The Bootstrap Method continued

• To calculate the 1.5 year rate we solve

to get R = 0.1068 or 10.68%

• Similarly the two-year rate is 10.81%

4 4 104 960 1047 0 5 0 1054 1 0 1 5e e e R . . . . .


Zero Curve Calculated from the Data

9

10

11

12

0 0.5 1 1.5 2 2.5

Zero Rate (%)

Maturity (yrs)

10.127

10.469 10.536

10.681

10.808


Forward Rates

The forward rate is the future zero rate implied by today’s term structure of interest rates. Alternately, Forward interest rates are the rates of interest implied by current zero rates for periods of time in the future.

Term structure of interest rates shows the relationship between the interest rates on various fixed income products (namely bonds) and their maturities


Formula for Forward Rates

• Suppose that the zero rates for time periods T1 and T2 are R1 and R2 with both rates continuously compounded.

• The forward rate for the period between times T1 and T2 is

R T R T

T T2 2 1 1

2 1


Calculation of Forward Rates

Zero Rate for Forward Rate

an n -year Investment for n th Year

Year (n ) (% per annum) (% per annum)

1 10.0

2 10.5 11.0

3 10.8 11.4

4 11.0 11.6

5 11.1 11.5


Instantaneous Forward Rate

The instantaneous forward rate for a maturity T is the forward rate that applies for a very short time period starting at T. It is given by

where R is the T-year rate

R TR

T


What is Yield of a Bond?

• Definition 1: It is the annual rate of return on an investment, expressed as a percentage.

Definition 2: For bonds and notes, the coupon rate divided by the market price. This is not an accurate measure of total return, since it does not factor in capital gains.


Yield Curve





Upward vs. Downward SlopingYield Curve

• For an upward sloping yield curve:

Fwd Rate > Zero Rate > Par Yield

• For a downward sloping yield curve

Par Yield > Zero Rate > Fwd Rate


Types of Bonds

Bonds can be generally classified into two categories:– Zero Coupon Bonds: These are generally bonds issued

for very short periods of time and have a very small chance of default.

– Coupon Bearing Bonds: Most bonds are of this category. These bonds are issued for relatively longer periods of time and have a higher probability of default as compared to Zero coupon bonds.


Pricing of Bonds

The simplest way to price a bond (value = B) with coupon payments C per year, principal P on maturity and an interest rate of r is given by:

nn r

CP

r

C

r

C

r

C

r

CB

11......

111 132


Price of a Zero Coupon Bond

The price of a zero coupon bond is given by

where

rt = the short term interest rate at time t; 0 t T

T = the time to maturity

b = discount factor (constant)

Note: rt can be a constant or stochastic in nature.

T

sdsrb

eE 0


Coupon bearing bonds

Coupon bearing bonds could have the following features:– Constant coupons – Here the coupon payments are

constant over the entire life of the bond

– Variable coupons – Here the coupon payments vary depending on the remaining tenor of the bond. These are usually offered on bonds with options; i.e. the holder / issuer has the right to return (put option) / recall (call option) at some predetermined points before the maturity of the bond.


Non-defaultable (Sovereign) Bonds

• Here the probability of default by the issuer of the bond is 0 all coupons and principal on maturity is paid on time

• Typically all government issuances in the domestic market are of this type – assumption being that governments cannot default in the domestic market as they have the right of seniorage

• Generally liquid instruments and can easily be traded in the secondary market


Defaultable (Corporate) Bonds

• These bonds have a non-zero probability of default

• At default all payments (coupons and principal) ceases. Less liquid than non-defaultable bonds

• Lot of work on the default probability model has been done by Duffie, Singleton, Kan and Lando – they build in a default probability model in the bond price equation


One Defaultable Bond Price Equation

where• D is the payment at default and C is the coupon rate• the first term is payment at default• the second term is final payment at maturity, no default• the third term represents coupon payments in case of default• the fourth term represents coupon payments, no default u is the default rate – see Duffie, Singleton, Kan, etc.

Note:Price of a non-defaultable bond = second term + fourth term

T duru

s

duT sru

dsbr

T

s

dursT

u

s

u

T

s

s

u

edueCdsedueCeedseDE00 00

0000 )())(()()(


Sensitivity & Impact of Change

• As is obvious, Price and Yield of a bond is inversely related.

• However, this relationship is highly non-linear, especially towards the short end of the yield curve.

• Duration affects the price and yield of a bond (as discussed earlier)


Interest Rate Features

• Typically the benchmark rate is determined by the central bank – this is called the Bank Rate

• Changes in the bank rate reflect the changes in the economic scenario – market determined

• Not changed too frequently – say on a daily basis• Based on the bank rate number of other rates exist

– PLR – Prime Lending Rate– Overnight / Call Rate – the rate at which banks lend to each other

on a daily basis– Repo Rate – the rate at which the central bank refinances banks

and financial institutions


Interest Rate Dynamics

• Typically interest rate is assumed to be positive• Difficult to get a term structure of interest rate in

an emerging market scenario due to lack of historical data

• Interest rate is assumed to be mean reverting – that is in the long run it should revert back to a stable equilibrium value– However, this does not ensure positivity of interest

rates

• One simple way to ensure positive interest rate is to take the exponential process (ex)


History of Indian Bank Rate

0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.007

/5/3

5

7/5

/38

7/5

/41

7/5

/44

7/5

/47

7/5

/50

7/5

/53

7/5

/56

7/5

/59

7/5

/62

7/5

/65

7/5

/68

7/5

/71

7/5

/74

7/5

/77

7/5

/80

7/5

/83

7/5

/86

7/5

/89

7/5

/92

7/5

/95

7/5

/98

7/5

/01

Date

Ba

nk

Ra

te


Interest rate models

• Number of models have been used to describe the interest rate process assuming stochastic term in the interest rate process – some are:– Vasicek (1977)

– Ho and Lee (1986)

– Hull and White (1990)

– Heath, Jarrow and Morton (1992)

– Brace, Gatarek and Musiela (1997)

– Cox, Ingersoll and Ross (1985)


Yield Curve Modeling


The Yield Curve





Why do we need a ZCYC

• Strictly, speaking, the most suitable bonds for yield curve generation and analysis are the pure discount bonds such as the treasury bills and zero coupon bond as these securities do not involve periodic coupon payments.

• The reason behind this is in coupon bonds there is intermediate cash flow which is supposed to be re-invested at the same discount rate.


Importance of yields on Government securities ?

• Government securities are usually free from credit risk, hence, it serves as a benchmark for setting yields in all other segments of the debt market. Example: Corporate debt, Bank loans,

• Usually the government bond market happens to be the largest and the most liquid segment of the debt market. Accordingly the data are more current compared to data on, say, yield on corporate debt.


Mechanics of the construction of The Yield Curve

• STRICTLY speaking, most suitable bonds for yield curve analysis are the pure discount bonds such as the treasury bills and zero coupon bond as these securities do not involve periodic coupon payments.

• The reason behind this is in coupon bonds there is intermediate cash flow which is supposed to be re-invested at the same discount rate.

• So for coupon bonds we construct the yield curve that comes close to the ideal curve by following an approach that treats each coupon bond as a sum of individual zero coupon bonds.


Mechanics of the construction of The Yield Curve (continued)

• The yield on a coupon bond then can be viewed as a mixture of spot rates of various maturities.

• The choice of the function usually depends upon a compromise between :

i) smoothness of the curve

ii) the goodness of fit

• In general it is better to have a yield curve that fits the data reasonably well and makes sense economically.


Different Models

• Term Structure Models:– Vasicek (1979)– Hull and White (1990)– Heath, Jarrow and Morton (1992)– Cox, Ingersoll and Ross (1985)– Brace, Gatarek and Musiela (1997)

• Alternative Models:– Nelson and Siegel (1987)– Nelson, Siegel and Svensson (1994)

• Apart from these are are also spline fitting models advocated by McCullough and Nelder (1971) and others.


Which class of models to use?

In the Indian context, the Term Structure of Interest Rate models are difficult to use because of lack of data. Even though we have interest rate data for about 70 years, most of this data is administered and hence not applicable to market scenarios.

The most commonly used models in the Indian context are the Nelson and Siegel (1987) model or its extension suggested by Svensson in 1994. These models fit the Indian context very well – as shall be shown later.


Nelson Siegel Model

• This model proposes a class of models, motivated by but not dependent on the Expectations theory of the term structure.

• It offers representation of the shapes traditionally associated with yield curves.

• This model introduces enough flexibility to represent the range of shapes associated with yield curves: monotonic, humped or S-shaped.


Nelson Siegel Model (continued)

• Instantaneous forward rate at maturity m is given by:

r(m) = ß0 + ß1 e-m/τ + ß2[(m/τ) e-m/τ]

• YTM on a bill is the average of forward rates:

R(m) = ß0 + (ß1+ß2).[1-e-m/τ]/(m/τ) - ß2.e-m/τ

• Discount function: d(m) = e-(R(m).m/100)


Nelson Siegel Model - Parameters

• The four parameters of the Nelson Siegel Model (described in the earlier slide) are:

ß0 is long-term component

ß1 is short-term component

ß2 is medium-term component

τ is the time constant of the curve


Nelson Siegel Model – Estimation Procedure

1. Select a vector of starting parameters (ß0 ß1 ß2 τ).2. Using this starting vector of parameters, calculate the discount

factor function, present value of the bond cash flows and thereby determine a vector of starting ‘model’ bond prices.

3. The best-fitting values of ß0,ß1,ß2, τ are then compared using linear least squares.

4. Determine the spot rate function from the estimated set of parameters.

5. Now create a series of maturity values and use the estimated parameters for the required day to derive corresponding spot rates.

6. With maturity on the X-axis, plot the spot rates on Y-axis. 7. The curve thus obtained is the Yield Curve for the day.


Nelson-Siegel-Svensson Model

• In 1994 Svensson suggested a flexible form for the forward rate function, an extension of Nelson and Siegel model, using 6 parameters.

• Instantaneous forward rate at maturity m ;

r(m) = ß0+ß1e-m/τ1+ß2[(m/τ1)e-m/τ1]+ß3[(m/τ2)e-m/τ2]

• YTM on a bill is the average of forward rates ;

R(m)=ß0 +ß1[1-e-m/τ1]/(m/τ1) +ß2[{1-e-m/τ1}/(m/τ1)- e-m/τ1]

+ß3[{1-e-m/τ2}/(m/τ2)- e-m/τ2]


Parameters of NSS Model• ß0 is a constant and must be positive to ensure that long forward

rates are positive.

• ß1e-m/τ1is an exponential term.When the time to settlement approaches zero,forward rate approaches (ß0+ß1), which must be non-negative.

• ß2[(m/τ1)e-m/τ1] generates a hump-shape (or U-shape) as a function of the time to settlement.

• ß3[(m/τ2)e-m/τ2 ] generates another hump or U-shape : thus this curve allows up to two hump shapes,whereas,the original NS function has only one hump shape.

• τ1 & τ2 are parametric time constants of the curve.


Estimation Procedure for NSS Model

The estimation procedure is the same as the Nelson and Siegel model (as discussed earlier).

The only issue that has to be considered while estimating the parameters of Nelson, Siegel and Svensson model is the fact that there are 2 extra parameters here and hence estimation is more complicated and time consuming.


Estimated Curves for 21st May, 2003

4.7

4.9

5.1

5.3

5.5

5.7

5.9

6.1

6.3

6.5

6.7

6.9

0 2 4 6 8 10 12 14 16 18 20 22

Terms to Maturity(in yrs)

YT

M(a

nnua

lised

in %

)

Yield using NS Yield using Svensson NSE curve


Comparing the two models

• In most cases the Nelson & Siegel model gives a satisfactory fit, but sometimes with complex shape of forward rate curve the extended variant results is a better fit.

• Computation of linear least square can hang if τ1 becomes equal to τ2 in Svensson model, in this case model is over-parameterized, this however does not happen with original NS model.

• Due to an additional term the Svensson model is more capable of handling more complex and volatile environment.


• Nelson Siegel approach is widely used by most financial institutions – at least in India

• The curve obtained by the Nelson Siegel method is very close to the curve published by NSE everyday (as shown in the example with the date on 21st May).

• Nelson-Siegel model is regarded to produce more regular or smoother functions than a splines interpolation.

Remarks…


Some sources of the ZCYC

• National Stock Exchange of India

• Reuters

• Fixed Income Money Market Dealers Association (FIMMDA)

• Internally constructed based on market data (most commonly used by major players)


Some Issues to Think About!!!


Remarks

• Should there be one benchmark yield curve that is followed by all participants in the market for uniformity

• Should issues like Duration and Convexity be given more importance than they are right now – extend this information top the public domain

• Should stochastic interest rate models be used for interest rate modeling and bond pricing

• Should detailed economic analysis be carried out either by market players or an independent agency funded by the market as to the correct mean reversion level of interest rates in India


Remarks (continued)

• Appropriate regulation and control by the exchange of interest rate products – what is the level of acceptable level

• Uses of Interest Rate Futures – hedging and/or speculation


Thank You!!!

Bond Mathematics

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