8 BONDSWITHEMBEDDEDOPTIONS

(1)

ANALYSIS OF BONDS

WITH

(2)

(3)

A BOND WITH AN EMBEDDED OPTION IS ONE IN WHICH EITHER THE ISSUER OR THE BONDHOLDER

HAS THE OPTION TO CHANGE A BOND’S CASH FLOWS

CALL OPTION

(4)

Issue

Price

YTM

(%)

Treasury

C=8.8%

96.61

9.15

Corporate

C=8.8%

87.07

10.24

Yield Spread = 109 BP

This simple analysis does not take into consideration • _{The term structure of interest rate}

(5)

STATIC SPREAD

Will the cash flow analysis be the same for : •a zero coupoun 25-year corporate bond •a 8.8% coupon, 25-year corporate bond

?

NO

(6)

STATIC SPREAD Zero Volatility spread Z-spread

Spread that will make the PV of the cash flows from the corporate bond, when discounted at the Swap zero-rates + spread , equal to the corporate’s bond price

Static spread in our example would therefore be 120BP and not 109BP

(7)

On Bloomberg, when hitting YAS on a bond,

(8)

(9)

CALLABLE BONDS

•The holder of a callable bond has given the issuer

the right…to call (buy back) the issue prior to expiration .

Disadvantage for the bondholder :

Reinvestment

(10)

Lack of price

appreciation potential

(11)

YIELD PRICE

a’

a

b

Y’

Callable Bond (a-b)

PRICE/YIELD RELATIONSHIP FOR A CALLABLE BOND

Bullet Bond

Price Compression

P’

(12)

YIELD PRICE

a

b

Y’

Callable Bond (a-b)

Price Compression

(13)

A bond with an embedded option (call) can be considered as a portfolio of :

A bullet bond _{A call option}

(14)

(15)

NONCALLABLE BOND - CALL OPTION PRICE

CALLABLE BOND

(16)

YIELD PRICE a’ a b Y’ Callable Bond (a-b)

Non callable Bond

Y’’

P_NC

P_C

(17)

VALUATION MODEL

NONCALLABLE BOND - CALL OPTION PRICE

(18)

VALUATION MODEL

(19)

The price of an option free bond is the present value of the cash flows discounted at the spot rates. What is the bond price ?

YEAR ZERO

RATES

COUPON RATE (yearly)

Mkt VALUE

1 3.5% 5.25% 100

2 4.01% 5.25% 100

3 4.541% 5.25% 100

5.25/1.035 + 5.25/(1.0401)2 + 105.25/(1.451)3 = 102.047

(20)

When analysing embedded options, consideration must be given to :

INTEREST RATE VOLATILITY

We are trying to determine how the 1-period forward rate can

vary over time based on some assumption about interest rate

volatility

(21)

OBJECTIVE

Determine whether the forward rates are

correctly reflected in the price of a bond

An interest rate model makes assumptions

about the relationship between the level of

short term interest rates and interest rate

(22)

r

₀

N

r

_1H

N

_H

r

_1L

N

_L

r

_2HL

N

_HL

r

_2HH

N

_HH

r

_2LL

N

_LL

r

_3HHH

N

_HHH

r

_3LLL

N

_LLL

r

_3HHL

N

_HHL

r

_3HLL

N

_HLL

TODAY 1 year 2 years 3 years

(23)

r

₀

N

r

_1H

N

_H

r

_1L

N

_L

r

_2HL

N

_HL

r

_2HH

N

_HH

r

_2LL

N

_LL

r

_3HHH

N

_HHH

r

_3LLL

N

_LLL

r

_3HHL

N

_HHL

r

_3HLL

N

_HLL

H : higher of the

two forward rates

L : lower of the

(24)

r

₀

N

r

_1H

N

_H

r

_1L

N

_L

r

_2HL

N

_HL

r

_2HH

N

_HH

r

_2LL

N

_LL

r

_3HHH

N

_HHH

r

_3LLL

N

_LLL

r

_3HHL

N

_HHL

r

_3HLL

N

_HLL

H : the higher 1-year rate one year from now

(25)

r

₀

N

r

_1H

N

_H

r

_1L

N

_L

r

_2HL

N

_HL

r

_2HH

N

_HH

r

_2LL

N

_LL

r

_3HHH

N

_HHH

r

_3LLL

N

_LLL

r

_3HHL

N

_HHL

r

_3HLL

N

_HLL

H : the higher 1-year rate two year from now

(26)

N is the root of the tree and is nothing more than the current

1-year forward rate which is denoted by r₀

The next year 1-year forward rate can take 2 possible values

of equal probability of occuring. One rate will be higher than the other.

It is assumed that the 1-year rate can evolve over time based on a random process called Lognormal Random Walk with a certain volatility.

 _{= assumed volatility of the 1-year forward rate}

r_1,H= the higher 1-year rate one year from now

r_1,L = the lower 1-year rate one year from now

(27)

r

1,H

= r

1,L

(e

2



)

If

r

1,L

= 4.074% with a 10% volatility…

(28)

•3 different outcomes in the second year for the 1-year rate.

YEAR 2

R_2,LL = 1-year rate in the second year assuming the lower rate in

the first year and the lower rate in the second year

R_2,HH = 1-year rate in the second year assuming the higher rate in

the first year and the higher rate in the second year

R_2,HL = 1-year rate in the second year assuming the higher rate in

the first year and the lower rate in the second year(or vice versa)

r

2,HH

= r

2,LL

(e

4



)

r

(29)

r

₀

N

r

₁

e

2

N

_H

r

₁

N

_L

r

₂

e

2

N

_HL

r

₂

e

4

N

_HH

r

₂

N

_LL

r

₃

e

6

N

_HHH

r

₃

N

_LLL

r

₃

e

4

N

_HHL

r

₃

e

2

N

_HLL

(30)

DETERMINING THE

VALUE AT A NODE

Components to price a bond ?

•Coupon (C)

•_{Forward rate ( r )}

•Maturity ( t )

(31)

r

₀

N

r

₁

e

2

N

_H

r

₁

N

_L

r

₂

e

2

N

_HL

r

₂

e

4

N

_HH

r

₂

N

_LL

r

₃

e

6

N

_HHH

r

₃

N

_LLL

r

₃

e

4

N

_HHL

r

₃

e

2

N

_HLL

•The appropriate rate to use is

(32)

r

₁

e

2

N

_H

r

₂

e

4

N

_HH

r

₂

e

2

N

_HL

•The appropriate rate to use is

the 1-year forward rate at the node

V_H = Bond’s value for the higher rate

V_L = Bond’s value for the lower rate

(33)

The cash flow at each node is either :

• V

_H

+ C for the higher rate

• V

_L

+ C for the lower rate

What is the present value of V

_H

+ C ?

V

H

+ C

1 + r

V

_L

+ C

1 + r

(34)

V

_H

+ C

1 + r

+

V

_L

+ C

1 + r

---2

(35)

EXAMPLE

• 2 YEAR BOND

•TRADING AT 100 TODAY

•VOLATILITY =



=

10%

•ANNUAL COUPON = 4%

(36)

Step by step process….

Step 1 : Select a value for r₁ , lowest 1-year rate one year from now

Let’s select r₁ arbitrarily = 4.5%

Step 2 : Determine the corresponding value for the higher 1-year

forward rate.

r

_1,H

= 0.045e

(2 *0.10)

=

5.496%

Step 3 : Compute the bond’s value one year from now

(at maturity for us, therefore 100 + 4 = 104)

Step 4 : Calculate the bond’s value in step3 using the higher rate

V _H= 104/1+0.05496 = 98.585

Step 5 = Calculate the bond’s value in step3 using the lower rate

(37)

Step 6 = Add the coupon to V _H and V _L to get the cash flow at N _H and N _L

Step 7 =

V _H+ C = 102.582 V _L+ C = 103.522

Calculate the PV of those 2 values using the root rate of 3.5% 102.582 / 1.035 = 99.13

103.522 / 1.035 = 100.021

Step 8 = Calculate the average of the two PV

(38)

WHAT WAS THE PRICE OF OUR BOND TODAY ?

100

Remember step 1 : lowest 1-year rate one year from now let’s select r₁ = 4.5%

What is needed is to find the exact 1-year forward rate,

one year from now, so that our bond price becomes 100 instead of 99.567

(39)

Next step is to determine the low 1-year rate two years from now. It needs to be done by trail and error on Excel.

For this, we analyse a 3-year 4 ½ coupon bond that trades at par.

We know from previous calculations that the 1-year, one year from now, is at 4,074% and that the 1-year rate today is 3,50%.

R_1,0 = 3,5%

R_1,1 = 4,074%

(40)

Vol 10% Year 0 1 2 3 Face Value 100

Coupon 4,50% V 100

C 4,50% Data based on the market

V 97,88497 C 4,50% YTM 6,758%

V 98,07298 V 100

C 4,50% C 4,50%

Check 100,00 R 4,976%

V 102,075 V 99,0212

C 4,50% C 4,50%

R 3,50% YTM 5,533%

V 99,92529 V 100

C 4,50% C 4,50%

(41)

Now that we have all three low rates, R_1,0 = 3,5%

R_1,1 = 4,074%

R_1,2 = 4,53%

…..it is easy to determine the other rates on the binomial tree with the formula :

R_1,H = R_1,Le2∞

(42)

Valuing a Callable Corporate Bond

Same process as an option free bond except :

•When the call option may be exercised by the issuer

the bond value at the node must be changed to reflect the lower of its value if it is not called and call price.

(43)

The price of an option free bond is the present value of the cash flows discounted at the spot rates. What is the bond price ?

YEAR ZERO

RATES

COUPON RATE (yearly)

Mkt VALUE

1 3.5% 5.25% 100

2 4.01% 5.25% 100

3 4.54% 5.25% 100

(44)

Suppose this same bond is callable at 100 in year 2…..

Any bond valuation above 100 (node N_L an N_LL) must be called at 100.

Call option = non callable bond – callable bond

(45)

On Bloomberg, when hitting YAS on a bond,

(46)