bond basics

January 2012

Bond Basics

Dr. Gunther Hahn, CFA

Frankfurt, January 2012

January 2012

Overview

• Discounting and the time travelling machine

(compounding vs. discounting)

• Value of a Bond

(pricing formula)

• Special Bonds

(Zero Coupon, Consol, Floater)

• Price Quotation in the market

(Clean vs. Dirty Price, Day Count Conventions)

• Price Behaviour of bonds

(Discount vs. Premium Bond, Price vs. time, Price vs. yield)

January 2012

Overview II

• Yield Changes and Performance of Bonds

(Duration)

• A closer look at Duration

(Performance Approximation)

• McCauley Duration

(Average time, Price elasticity, Immunization)

• Application to an immunizing Strategy

(Insurance company)

January 2012

Literature

Bond Basic:Fabozzi, F. (1993): „Fixed Income Mathematics“, McGraw-Hill

Bonds and Yield Curves :Luenberger, D. (1998): „Investment Science“, Oxford, pp. 40 – 101

Bonds and xls examples:Benninga, S. (2008): „Financial Modelling“, 3rd edition, MIT press, pp. 669-717

January 2012

Discounting and the time travelling machine

• Assume you invest today 100€ at 10% interest.

Which amount can you expect after one year?

Amount + Interest

100 + 100 * 10% = 100 * (1 + 10%) = 110

• And after 2 years ?

Amount + Interest

100 * (1 + 10%) + 100 * (1 + 10%) * 10% = 100 * (1 + 10%)2 = 121

• And after n years ?

100 * (1 + 10%)n = Amount * (1 + interest)n

January 2012

• Now assume you receive 110€ in 1 year from today.

How much is this worth today, if the interest level is at 10% ?

Amount + Interest = 110

? + ? * 10% = ? * (1 + 10%) = 110

? = 110 / (1 + 10% ) = 100

• Assume you receive X € in n years. How much is this worth at y % interest?

Todays Value = X / (1 + y)n

January 2012

Value of a Bond

A Bond represents the right to receive future Cash Flows.

The Cash Flows consists out of Coupon and principal payment.

Today 1 st Coupon date

2 nd Coupon date

… Maturity

Pay for bond CouponPrincipal +

CouponCoupon

January 2012

Example: Assume you buy a 5% Bond for 80 € with a maturity of 4.3 years.

0 0.3 1.3

-80 1055

2.3 3.3 4.3

5 5 5

January 2012

Idea of Valuation: Each individual Cash Flow can be valued and aggregated to the total value !

0 0.3 1.3

-80 1055

2.3 3.3 4.3

5 5 5

5 / (1+ 10%)0.3 = 4,86

5 / (1+ 10%)1.3 = 4,42

5 / (1+ 10%)2.3 = 4,02

5 / (1+ 10%)3.3 = 3,65

105 / (1+ 10%)4.3 = 69,69

86,64

January 2012

Pricing Formula

The Value of the bond consists out of the sum of the individual values.

3.43.33.23.13.0 %101

105

%101

5

%101

5

%101

5

%101

5

Or in a more formal way.

T

tt

t

y

CFP

1

Notation

P Price (dirty) of Bond

T Time to maturity

t index

CFt Cash Flow at time t

y interest (yield) of bond

January 2012

Special Bonds

TyP

1

100

• British Consol

Bond that never matures. The Bond pays its coupon forever and needs to be bought back by the issuer in order to mature.

0 0.3 1.3

5

2.3 3.3 …

5 5 5 …

• Zero Coupon Bond

Bond that pays no Coupon. Only at maturity the principal is repaid.

0 0.3 1.3

0

2.3 Maturity

0 0 100

y

CouponP

January 2012

Special Bonds II• Floater

Bond that pays a floating rate (on a quartely basis) depending on the level of the interest rate. At the beginning of the period the rate is observed and at the end the rate is paid and the new rate is

observed.

0 0.25 0.5

X1=3 Month-Rate

Maturity

100 + Xn / 4X1 / 4 is paid

X2=3 Month-Rate

X2 / 4 is paid

X3=3 Month-Rate

...

...

resetnexttotimei

y

XP

1

4/100

On each of the reset days the value of the floater is 100.

The idea behind this logic is that the cash flow from a floater can be duplicated easily. On each of the reset days a fixed term deposit for 3 Month earning the 3 Month-Rate is opened. At the end of the period the 3 Month-Rate is earned and the 100 are recieved back.

resetnexttotimei

y

XP

1

4/100

resetnexttotimei

y

XP

1

4/100

resetnexttotimei

y

XP

1

4/100

On each of the reset days the value of the floater is 100.

The idea behind this logic is that the cash flow from a floater can be duplicated easily. On each of the reset days a fixed term deposit for 3 Month earning the 3 Month-Rate is opened. At the end of the period the 3 Month-Rate is earned and the 100 are recieved back.

January 2012

Price Quotation in the market

• So far the valuation was equal to the amount which needs to be paid. This amount is called the dirty price.

• The price which is quoted on Bloomberg or in the newspaper is the clean price of the bond, which accounts for the accrued interest.

Dirty Price = Clean Price + Accrued Interest

Next coupon

payment

CF

Last coupon

payment

CF

today

Accrued Interest = (1-t) * CF

t1 - t

January 2012

Example: Assume you buy a 8% Coupon Bond with 4.25 years to maturity. The clean Price is 90€. How much do you pay to receive the Bond?

Clean Price 90 €

Accrued Interest 6 € (1 – 0.25) * 8

Dirty Price 96 €

January 2012

Example Accrued Interest

January 2012

Example Accrued Interest continued

Clean Price 1.000 € * 103.19% = 1031.90

Accrued Interest (230 Days) 1.000 € * 7.125% *(230+1)/365 = 45.09

Dirty Price (107.699%) 1076.99 €

Next coupon

payment

20.04.2012

Last coupon

payment

20.04.2011

today

6.12.2011

230 Days230 Days230 Days

January 2012

Day Count Conventions

The difference between two dates can be calculated according to different market standards.

• Actual / Actual : real Number of days are counted.

• Actual / 365 : real Number of days are counted; the number of days in a year is counted as 365 (even if it is a leap year).

• Actual / 360 : real Number of days are counted;the number of days in a year is counted as 360.

• 30 / 360 : every month is counted as 30 days and every year as 360 days;- If the period starts on the 31st then the start is moved on the 30th- If the period ends on the 31st then the end is moved on the 1st- If the period ends on the 31st and starts on the 31st then the end is moved on the 30th.

• 30 E / 360 : every month is counted as 30 days and every year as 360 days;- If the period starts on the 31st then the start is moved on the 30th- If the period ends on the 31st then the end is moved on the 30th.

January 2012

Example Day Count Conventions

January 2012

Example: Pricing of a Bond

7.12.2011

7,125 7,125 107,1257,1257,125

20.04.2012 20.04.2013 20.04.2014 20.04.2015 7.12.2016

-107,699

tt

t

y

CFAccruedP

1

January 2012

Discount vs. Premium Bonds

• Discount Bond

Bond which a coupon rate below the market interest rate. Consequently the Price of the bond is cheaper than 100.

January 2012

• Premium Bond

Bond which a coupon rate above the market interest rate. Consequently the Price of the bond is greater than 100.

January 2012

Yield Changes and Performance of Bonds

The following picture shows how the dirty price changes if we vary the market interest rate.

Dirty Price

Interest Rate

January 2012

In order to compute the price change approximately, we calculate the first derivative of the dirty price function. Using the derivative we can approximate the change in price.

T

t

tt

T

tt

t yCFy

CFP 1

1

• We start with the pricing function …

• And calculate the first derivative with respect to the interest rate y.

T

tttt

T

tt yy

CFtytCF

y

P

1

1

11 1

T

ttt

y

CFt

yPyP

P

11

11

• Changing to percentage change in Price gives:

January 2012

Using the modified Duration we can approximate percentage price change.

Dirty Price (P)

Interest Rate (y)

T

ttt

y

CFt

yPD

yP

P

11

11mod

Current Interest

yDPP

mod

January 2012

Example: Assume you have a bond with a modified duration of 6. The dirty price is 120€. Suddenly the yield decreases from 4% to 3.5%. Will you gain or loose? How much is the percentage change in price and absolute change?

• Since the yield decreases the price of the bond will increase. This way investors are compensated for a lower yield level.

• Percentage change in dirty price = - modified Duration * change in yield

Percentage change in dirty price = - 6 * -0,5% = 3%

• Absolute change in dirty price = 3% * 120€ = 3,6€

The Price will increase approximately from 120€ to 123,6€.

January 2012

A closer look at Duration

Using the modified Duration and yield curve we can approximate the Performance of a bond over a period of time.

yDtyePerformanc mod

Interest Rate / yield curve

Time to MaturityTodayToday - Δt

Δt

Δy

January 2012

Example: Assume you hold a bond for half a year. When you buy the bond, the Duration was 6 and the yield 3%. At the end of the period the yield increased to 3.5%. Which approximate Performance did you earn?

• The formula gives:

Performance = 3% * 0,5 - 6 * 0,5% = 1,5% - 3% = -1,5%

January 2012

McCauley Duration

Besides the modified Duration, the McCauley Duration is often used as well. For its computation we start with the modified Duration:

T

ttt

y

CFt

yPyP

PD

11

11mod

Now we multiply both sides with (1+y) to obtain the McCauley Duration:

yD

y

CFt

Py

yP

PD

T

tt

tMcCauley

1

1

1

1mod

The McCauley Duration represents the percentage price change over the percentage yield change. So the McCauley Duration is an elasticity

(% Change / % Change).

January 2012

McCauley Duration – Calculation Example

The following table is helpful to calculate the McCauley Duration:

yD

y

CFt

Py

yP

PD

T

tt

tMcCauley

1

1

1

1mod

January 2012

McCauley Duration – Interpretation

The McCauley Duration has 3 interpretations (average Time; price elasticity; Immunization).

• Average Time to maturity (balances discounted Cash Flows)

t

T

tt

T

t

tt

McCauley wtPy

CF

tD1

Disc. CF

• Price elasticity (can be used to calculate percental price changes)

yyP

PDMcCauley

1

If yields rise from 5% to 6% the denominator is not 1%,

but 1%/1.05 = 0.95%. For this reason of complexity

modified duration is more often used.yyP

PDMcCauley

1

January 2012

Immunization

The McCauley Duration represents the “immunization” time period.

The “immunization” time period, is reached if the effect of an interest rate change is offset.

T

tt

tMcCauley

y

CFt

PD

1

1

Time

Final Value

McCauley Duration

January 2012

Immunization - ProofWe start with the final value of the amount invested.

t

tTtT iCFFV 1

0111

1

t

tt

t

tt

T

iCFtiCFTi

i

t

tt

t

tt

t

tt

iCFtPiCF

iCFtT 1

1

1

1

= McCauley Duration!

0

i

FVT

The point where the final value is immune to interest rate changes is characterized as:

0

i

FVT

t

tTt

T iCFtTi

FV01 1

Calculating the first derivative gives:

t

tTt

T iCFtTi

FV01 1

January 2012

Application to an immunizing Strategy

The “SURE” insurance company has to pay 1 Mio € in 10 years to its policyholders. The current yield curve is flat at 6%. The treasurer calculates that today an amount of 558.395 € is needed to have a terminal value of 1 Mio € (Present Value of 1 Mio € at 6% over 10 years).

At the capital market there are only 3 Bonds to invest in:

Which bond should the company invest in?

Bond A Bond B Bond C

Coupon 6,7% 6,99% 5,9%

Time to Maturity (years) 10 15 30

Price 105,15% 109,6% 98,62%

McCauley Duration 7,67 10 14,64

bond basics

Documents