Fixed income introduction, Monografías, Ensayos de Finanzas

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Introduction

Required reading:

IntroNotes.pdf

Suggested readings: Veronesi book pp.

3-7, 9-21, 32-38; Tuckman pp. 4-6, 23-25,

53-59, chapter 15

Importance

of Fixed Income Markets



Much larger than equity markets



US treasury is the world’s most liquid

market



Large movements in stock markets

often driven by fixed income markets

(cf. monetary policy, subprimes!)



Virtually all households and

businesses participate in fixed income

markets



Better technical tools than for stocks

US Market Size

What we are going to do



Survey of the main types of bonds

and fixed income derivatives



Quantitative tools for:

Valuing bonds and fixed income

derivatives

Managing the risk of bonds and fixed

income portfolios

Determining the optimal exercise policy

for the options that are embedded in

many fixed income securities

Compounding frequency



Take three investments that all offer

12% interest / year:

One makes one interest payment of 12%

of principal after one year (e.g. Eurobond)

One makes two coupon payments of 6%

in 6 and 12 months (e.g. corporate bond)

One makes an interest payment of 1%

every month (e.g. mortgage loan)



Are these equivalent?

Compounding frequency (r(m)=12% in all

cases)

m Compounding

periodicity

Annually comp.

equivalent (r)

1 annual 12%

2 semi-annual 12.36%

4 quarterly 12.55%

12 monthly 12.68%

52 weekly 12.73%

8760 hourly 12.7496%

∞ continuous 12.7497%

Continuous compounding



In the limit, as m→∞, lim where

e≈2.



So if we define the continuously compounded

rate, r(∞), such that 1+r = e

r(∞)

, or r(∞) = ln (1+r),

where r is the annually compounded rate, we

have equivalent ways to discount a cash-flow of

K occurring at t:

(same result for all m)

r

m

r e

m

r

  











1  exp



















  



 

 mt

m

r m

r t

t r t

K

Ke

e

K

r

K

( )

( )

( )

1

( 1 )

11

Continuous compounding



Why use continuously compounded

rates?



With annually compounded rates, if

return is r

1

in yr. 1 and r

2

in yr. 2, total

return after 2 years is (1+r

1

)(1+r

2

and the annual rate of return is [(1+r

1

(1+r

2

)]

1/



With continuously compounded rates,

total return is e

r

1

(∞)+r

2

(∞)

-1 and the

annual rate of return is (r

(∞)+r

• A 20-year bond issued in

Quotes and prices

Example: US Treasury bonds



Coupon rate c, maturity M, par F (makes

payment of $cF/2 twice a yr., plus $F at

maturity)



Quantity of bonds bought or sold measured

by total par value (not price paid or number

of bonds).



Price quoted for $100 par.



If c=7.25% and M=May 15, 2016, bond

called “seven and a quarter’s of May 15,



Trade date: parties agree. Settlement date:

next business day

Exercise:



What is the value of the following bond:



coupon=10%, yield=5%, Time to maturity=

years, payments semi-annual, notional amount

Answer: Bond price = 152.

T T

(1 y/ 2 )

(1 y/ 2 )

y/

c/

Bond Value 100 *

US Treasury Bonds

Price between settlements



Price at t=today, where today is between

t=0 and t=1:

t=0 ( L ast) t=1 ( N ext)

c/

……………………………

t=today ( S ettlement)

nLN

nLS nSN

SN LN

LN

n LS

n

n n

t

t today t

y

P

P P y

/

1

0

( 1 / 2 )

• ( 1 / 2 )

/



  



 

Exercise: Bond Pricing



Compute the price of the following bond:



Assume semiannual payment and par=



Inputs of the function are

◦ Coupon, Yield, T

,

n

SN,

n

LN



Example: coupon=10%, yield=11%, Time

to maturity=20 years, Nsn=35, Nln=

Answer: Bond price = 960.

20

US Treasury bonds



Quoted price = “Flat price” or “clean price”

(“full price” or “dirty price”) minus accrued

interest a :

where n

LS

=days from last cp to settlement



Price quoted in 32nds (p.8): e.g., quote =

92-15 means p = 92+15/32=92.



If maturity < one coupon period, yield

computed using “simple interest”. Full price

LN

LS

n

n

a  100 c / 2

LN

SN

n

n

y

c

P

1 / 2

1 / 2

100





