Lecture 3

LECTURE THREE

a. Fixed income markets
b. Fixed income derivatives
c. Other instruments

1

Part 1

FIXED INCOME
MARKETS

a. Overview
b. Foundations
c. Spot and forward rates
d. Term structure
e. Forward rates as forward contracts

2

1. Overview
Kind of bonds
•Government: sovereign bonds
•Agency bonds: Guaranteed by central government, such as Fannie Mae
•Municipal bonds: local government
•Corporate bonds
Basic Bonds: interest rate and principal

Special bonds
Part 1. Fixed income markets

Convertible bonds: option to exchange the bond for a specific number of
shares of common stock of a company
Callable bonds: can be redeemed by the issuer prior to its maturity.
•Early payment is good for the bond issuer, but not good for the bond buyer
•Puttable bonds: gives the option to the bondholder to demand early
repayment of the principal.
•Floating rate bonds: tied to rate. (T+2%). Sometimes these bonds do not
show the real situation of the company, because is following an external
Lecture 3

variable.

2. Foundations
Pricing
Bond Value = PV coupons + PV par value
𝑇
𝐶𝑜𝑢𝑝𝑜𝑛 𝑃𝑎𝑟
𝐵𝑜𝑛𝑑𝑉𝑎𝑙𝑢𝑒 = +
(1 + 𝑟) 𝑡 (1 + 𝑟) 𝑇
𝑡<1

yield

Yield V Current yield
Coupon 8%
30 Y, semi-annual
P:rice 1276.76
FV: 1000

• Yield is the r in the equation: 6.09%
• Current yield: annual payment / price. (It is in fact today’s rate of return).
Lecture 3

80
= 6.27%
1276
4

2. Foundations
Realized compound return VS YTM
Measures the return when the coupon Shows what was the
is reinvested rate at which the
investment was made
𝑃 𝑓 = 𝑃0 (1 + 𝑟)2

Bond prices in the long term
The price of a bond converges toward its par value as it approaches

maturity.

PREMIUM. Coupon > Interest rate
• Coupon will provide more than the
compensation given by the market
Price goes to
par because
PAR. Coupon = Interest rate less coupons
are remaining
Lecture 3

DISCOUNT. Coupon < Interest rate
• Coupon will not provide the
compensation given by the
market

2. Foundations
The yield curve
Very useful to investment ideas

1. Plot the bonds
2. Add a log-trend

3. If bond>trend
BUY
Lecture 3

Duration or maturity

3. Spot and forward rates
Why the yield curve has an upward trend?
Two strategies:

A. Interest rate 6% B. Interest rate 5%
2Y 1Y
Zero coupon Zero coupon
FV: 1000

But reinvesting returns
PV: 890

Return: 12.36% ( 𝑓;𝑖 𝑖)

890 890 * 1.062 890 890*1.05 (890*1.05)* r2
890 * 1.062 = 890 * 1.05* (1+r2)
Lecture 3

r2 = 7%
Starting with 2 portfolios that are similar, Next year rate > this year rate

When this year’s rate is too high, the curve’s slope is inverted

890 * 1.062 = 890 * 1.05* (1+r2) Forward rate
concept
Only rates
(1 + R2)2= (1+R1) (1+F1,2)

Forward rates
Spot rates t=2 Spot rates 1 t=1,2

1 + R2 = {(1+R1) (1+F1,2)}1/2
R1<R2: F1,2>R1 UP
Geometric Mean
of today and tomorrow R1>R2: F1,2<R1 DO
Three periods

1 + R3 = {(1+F1) (1+F2) (1+F3)}1/3
Lecture 3

The spot rate of a long term bond reflects the path of short rates
anticipated by the market

Forward rates
The generalization implies that

(1 + Rn)n= (1+Rn-1)n-1 (1+Fn-1,n)

Solving for the forward rate

(1 + Rn)n/ (1+Rn-1)n-1 = (1+Fn-1,n)

So, the forward rate will be a function of the nearly periods

(1 + 𝑅4 )4
1 + 𝐹3,4 =
(1 + 𝑅3 )3
Lecture 3

4.Term structure
Expectations Hypothesis
• Buyers of bonds do not prefer bonds of one maturity over another:
bonds with different maturities are perfect substitutes
• Liquidity premiums are 0

𝐹1,2 = 𝐸[𝑅2 ] 𝐹2,3 = 𝐸[𝑅3 ]
(1 + R2)2= (1+R1) (1+F1,2) (1 + R3)3= (1+R1) (1+F1,2) (1+F2,3)

(1 + R2)2= (1+R1) (1+E[R,2]) (1 + R3)3= (1+R1) (1+E[R,2])(1+E[R,3])
(1 + R2)= (1+R1) (1+E[R,2])1/2 (1 + R3)= {(1+R1)(1+E[R,2])(1+E[R,2])}1/3
• According to Expectations theory, long-term rates are all averages of
expected future short-term rate: If the short term rate changes so will
long term rates
FACT: interest rates of different maturities will move together
• The movement Rn will be less than proportional:
Lecture 3

FACT: short term rates are more volatile

• But, Expectations theory cannot explain why long-term yields are
normally higher than short-term yield

4.Term structure
Segmented market theory
• Markets for different-maturity bonds are completely segmented
• Longer bonds that have associated with them inflation and interest rate
risks are completely different assets than the shorter bonds.
• Bonds of shorter periods have lower inflation and interest rate risks that are
different from longer bonds (these factors will be higher)

FACT: yield curve is usually upward sloping

• But, this theory cannot explain fact 1 and fact 2
Liquidity premium theory
• Bonds of different maturities are substitutes, but not perfect substitutes
Short term bonds
free of inflation and ≫≫ long term bonds
interest rate risks
Lecture 3

Pay a liquidity premium

4.Term structure
• Short term bond buyers will prefer long term bonds if
𝐹1,2 > 𝐸[𝑅2 ] Expected short term interest
• Long term bond buyers will prefer short term bonds if
𝐹1,2 < 𝐸[𝑅2 ] Expected short term interest
Expectations H. Liquidity premium H.

R1= 5% E(R2)=5% E(R3)=5% R1= 5% E(R2)=5% E(R3)=5%
𝐹1,2 > 𝐸[𝑅2 ]
(1 + R2)2= (1+R1) (1+E[R,2])
(1 + R2)2= (1+5%)(1+5%) (1 + R2)2= (1+5%)(1+6%)
Yield to maturity R2= 5% R2= 5.5%
3Y YTM will be 5.6%
(1 + R3)3= (1+5%)(1+6%)(1+6%)
R3= 5.67%
Lecture 3

Yield curve will be flat Yield curve will have an upward slope

4.Term structure
Expectation theory will predict a flat yield curve, while the liquidity premium
theory will predict an upward sloping yield curve

If short rates are expected to fall in the
future.
• ET: Yield curve predicted will be
Lecture 3

downward sloping
• LPT: Yield curve predicted can still be
upward sloping.

5. Forward rates as forward contracts
Purpose: make a loan in the future (and receive it in the future)

Bond One year Bond: Two years Forward rate: 7%
FV: 1000 FV: 1000 Using the formula:
Yield: 5% Yield: 6% (1 + 𝑅4 )2
PV: 952 PV: 890 1 + 𝐹1,2 =
(1 + 𝑅3 )1

1000
𝐵0 1 =
(1 + 𝑦1 )

1000
𝐵0 2 =
(1 + 𝑦1 )(1 + 𝐹2 )
Lecture 3

Part 2

FIXED INCOME
DERIVATIVES

a. Forward rate agreements (FRA’s)
b. Swaps
c. Interest rate options

15

2. FRA

Forward rate agreements:

General definition
Two parties swapping a future payment
The underlying is an interest rate
Lecture 3

1. Forward rate agreements (FRA’s)
Foundations
VS Traditional forwards: payoff
Definition:
based on price
• Underlying: interest rate
• two parties agree to make interest payments at future dates

• lends a notional sum • borrows a notional sum
of money of money
• locks a lending rate • locks a borrowing rate.

VS Traditional market: to buy (a
Notional: the amount on which interest bond or equity is to LEND
payment is calculated

i changes between t0 (FRA is traded) and t1:(FRA comes into effect)
Lecture 3

One party has to pay the other party the difference as percentage of the notional sum

Rise in interest rates, the buyer will be protected
Fall in interest rates, the buyer must pay the difference between t0,i and t1,I

Foundations
Definition:
• Netting: only the payment that arises as a result of the difference
in interest rates changes hands. There is no exchange of cash at the
time of the trade
• Quotation: FRA (A x B)
A: the borrowing time period. B: the time at which the FRA
matures.

• The terminology quoting FRAs refers to the borrowing time period
and the time at which
a 3-month loan starting in 3 months’ time
3x6
a 3-month loan in 1month’s time
1x4
a 6-month loan in 3 months
3x9
Lecture 3

Important dates

Notional loan or
The notional loan deposit expires.
becomes effective, or
FRA is dealt BEGINS
The reference rate
is determined. The
rate to which the
FRA dealing rate is
compared 2 days before settlement
Lecture 3

Settlement payment

Extra interest payable in the cash market, and then discounts the amount

because it is payable at the start of the period

90 day libor expires in 30 days
M=20M
rFRA= 10%

LIBOR 8% In 30 days LIBOR 10%
Lecture 3

Upfront= -98.039 Upfront= 97.08
Long position hast to pay Short position has to pay

Pricing. How rFRA is defined?
Main idea: Both loans must have the same price to avoid arbitrage
    m  
   1 F  
 1   360  0
  h   hm
 1  L0 (h)    1  L0 (h  m) 
  360     360  
PV Loan we PV Loan we receive.

made for $1 Maturing in h+m
And solve for F
 hm 
 1  L0 (h  m)  
F   360   1  360  
 
  h    m  
 
 1  L0 (h)  
  360  
We want to find the price for a 30 day FRA
Underlying 90 day LIBOR   120  
 1  L(120)  
Lecture 3

h=30
 360    360  
m=90 F  1      10%
  30    90  
 
Find 30 day Libor  1  L(30)  
Find 120 day Libor   360  

2. Swaps

Swap: General definition

Two parties swapping a series of
payments
Lecture 3

2. Swaps
Definition
• Two parties swapping payments.
• Derivative in which two parties make a series of payments to each other
at a specific dates, at a some future dates.

Varieties
• One party makes fixed payments and the other variable payments
• Both parties making variable payments

• Both parties make fixed payments but in different currencies (at the end
payments are variables).

Types according to the underlying
• Interest rate swaps: fixed or variable in same currency
• Currency swaps: fixed or variable payments in different currencies
• Equity swaps: some stock price or index involved
• Commodity swaps: one set of payments involves prices of commodities
Lecture 3

2. Swaps
Structure
• Do not involve up-front payment
• Profit and loses are netted (no principal is changed) EXCEPT currency
• Their price is zero at the beginning of the transaction (pricing foundation).
How is the market?
• Dealers determine fees at which they will enter in a swap (either side) and
dealers hedge themselves.  They provide market liquidity
A). Interest rate swaps

• Payments based on a specific notional (N) that is not changed in the
transaction
• Most common. Plain vanilla swap: fixed V floating
Payoff
Has three parts:
1. amount of money in which the calculation is based on
2. Rates comparison
3. Accrual period: fraction of the year
Lecture 3

𝒅𝒂𝒚𝒔
(𝑵𝒐𝒕𝒊𝒐𝒏𝒂𝒍)(𝑳𝒊𝒃𝒐𝒓 − 𝒓𝒂𝒕𝒆 𝑭𝑰𝑿 )( 𝟑𝟔𝟎𝒐𝒓𝟑𝟔𝟓)

Determined by the rate in the previous settlement date

2. Swaps
Interest rate swaps-payoff (cont)
Example:
• Two companies:
• XYZ, and the dealer Aexchange that has to make payments for 1 year
based on 90 days LIBOR based on a N of 50M.
• XYZ has to pay a rate of 7.5%
• Libor: 7.68% So, 4 payments
per year

𝟗𝟎
(50,000,000)(0.768 − 0.075)( 𝟑𝟔𝟎)

• The same than
90 90
50𝑀 0.768 − 50𝑀 0.75
360 360
=22,500

• And so on…according to the LIBOR. Partial balances are netted
Lecture 3

• When both parties have floating rates, they have to add some spread
according to the rate that they are swapping. Libor V 2Y T’s

2. Swaps q
Interest rate swaps. Pricing
𝒅𝒂𝒚𝒔
How to find rFIX? (𝑵𝒐𝒕𝒊𝒐𝒏𝒂𝒍)(𝑳𝒊𝒃𝒐𝒓 − 𝒓𝒂𝒕𝒆 𝑭𝑰𝑿 )( )
𝟑𝟔𝟎𝒐𝒓𝟑𝟔𝟓
• Avoid arbitrage. Why?
• Obligations of one party = Obligations of other party AT INCEPTION

1 + L270(90)q
Fixed stream = Floating stream

L0(90)q L90(90)q L180(90)q

Day 0 Day 90 Day 180 Day 270 Day 360
R: fixed rate

Rq Rq Rq 1 + Rq Final payment discounted 270
day value
Day 0 Day 90 Day 180 Day 270 Day 360 PV of future payments
The payment x Discount factor
n
ti  ti 1
VFX   R ( ) B0 (ti )  B0 (t n )
90
i 1 360 1:𝐿270 (90)(
360
)
90 =1
Discount factor   1:𝐿270 (90)( )
360
 
Lecture 3

1
B0 (ti )   
 1  L (t )( ti  So, the price of the floating leg
 )
 360 
0 i
@ time 0 or payment date = 1
PV of interest and principal payments on a fixed rate bond

2. Swaps

n
t t Each coupon is multiplied by the
VFX   R ( i i 1 ) B0 (ti )  B0 (t n ) discount factor. Also the payment at
i 1 360
the end of the period (that has a value
  of 1)
 1 
B0 (ti )   

Discount factors
 1  L (t )( ti ) 
 
 360 
0 i

At the end we have the PV of interest and principal


VFL=1
Lecture 3

2. Swaps

  
  1  B (t )  And… solve for R
 n 0 n 
1
R 
 ( ti  ti 1 )   Price of the fixed rate following
   B0 (ti )  no arbitrage assumptions
 360  i 1 

1
720
1 + 0.105( )
360
Lecture 3

360 1;0.8264
𝑅=( ) =9.75%
180 0.9569:0.9112:0.8673:0.8264

2. Swaps
B). Currency swaps
• Two notional principals based on the exchange rate. (Notional change)
• Paid at the beginning and at the end of the period according to the contract
• Not netted

• The idea (It is like): one party issues a bond (including paying coupons),
takes that money and purchases a bond in a foreign currency (receiving a
different coupon)

Make payments in one currency and receive funds in a different one
• Rates can be fixed or floating: It is not only about the currency. It is
about currencies + rates in each market

Currency swaps. Pricing
• How to find the rates?
• Both legs must have the same PV to avoid arbitrage (including exchange
Lecture 3

rate and rates of return)

2. Swaps
Currency swaps (cont)
Example
US market EU market
Discount Discount
Term Dollar rate Term Euro rate
bond price bond price
180 5,50% 0,9732 180 3,80% 0,9814
360 5,50% 0,9479 360 4,20% 0,9597 1
540 6,20% 0,9149 540 4,40% 0,9381 540
1 + 0.044( )
720 6,40% 0,8865 720 4,50% 0,9174 360

And apply the pricing formula

360 1 − 0.8865 360 1 − 0.9174
𝑅$ = ( ) 𝑅€ = ( )
180 0.9732 + 0.9479 + 0.9149 + 0.8865 180 0.9814 + 0.9597 + 0.9381 + 0.9174

The PV of a stream of dollar (euro) payments with a hypothetical notional
principal of $1 (€1) at a rate R$ (R€) is $1 (€1)

+ Rates equalize
Lecture 3

principal 1in both
markets. And notional
The notional follows market currency exchange should be equivalent in

= currency market

Two streams are equal

2. Swaps
The initial value is zero, because
• Rates and
• Currency

The profit/loss is given with market movements

During life, the change in rates give new discount
bond prices …

180
𝑅𝑎𝑡𝑒 𝐷𝑖𝑠𝑐𝑜𝑢𝑛𝑡𝐵𝑜𝑛𝑑𝑃𝑟𝑖𝑐𝑒𝑠 ∗ 𝑁𝑜𝑡𝑖𝑜𝑛𝑎𝑙 = 𝑝𝑎𝑦𝑚𝑒𝑛𝑡 𝑖𝑛 𝑜𝑛𝑒 𝑐𝑢𝑟𝑟𝑒𝑛𝑐𝑦
360
Lecture 3

Equilibrium Equilibrium

2. Swaps
Term Dollar rate Discount Bond price
90 5,70% 0,986
270 6,10% 0,9563
450 6,40% 0,9259
630 6,60% 0,8965

180
0.61 ( 0.986 + 0.9563 + 0.9259 + 0.8965) + 1 0 − 8965 = 1.011
360
x
Initial N

Market Value
Lecture 3

Same strategy with the other leg

2. Swaps
C). Equity swaps
• Involves stock price, index price or value of a stock portfolio
• Payment: determined by the return of the stock
• Stock payment can be negative
A has a stock that in the period had negative return
 B has a stock that in the period had positive return
A will make TWO payments

Some
differences

 The upcoming payment is never known until the
•
settlement date (in others swaps it is indeed known)
 There is not time adjustment (accrual period)
•

Structure
• Company A Company B
Lecture 3

Pay SP500’s return Pay fixed rate 3.45%
@ 2710
• Each 90 days and maturity 1 year
• N= 25M

This is the fixed interest This is the
Cash flow
part stock part
2. Swaps 𝑡 𝑖 − 𝑡 𝑖;1
Equity swaps 𝑁 𝑅 𝐹𝐼𝑋 ∗ − 𝑆
360
Fixed
Floating leg
Day interest SPX Net payment
payment
payment
0 2.711
90 215.625 2.765 501.282 -285.657 Rate of return
180 215.625 2.653 -1.011.791 1.227.416 of the index
270 215.625 2.805 1.432.341 -1.216.716

360 215.625 2.705 -891.266 1.106.891

Rate 3,45% 𝑆 𝑡:1
𝑁 −1
Notional 25.000.000 𝑆𝑡

Pricing
• Suppose you borrow $1 to buy $1 in stocks 𝑁(𝑅𝑒𝑡𝑢𝑟𝑛 𝑖𝑛𝑑𝑒𝑥 1
• Same idea: Equity leg = Equity leg, Fixed leg, Index leg − 𝑅𝑒𝑡𝑢𝑟𝑛 𝑖𝑛𝑑𝑒𝑥 2)

1  B0 (t n )  Rq i 1 B0 (ti )  0
n
Lecture 3

Principal of Interest payments
Invest $1 in S loan including their rate And solve for R

2. Swaps
Equity swaps
  
  1  B (t ) 
 n 0 n 
1
R 
 ( ti  ti 1 )  
   B0 (ti ) 
 360  i 1 
And the idea is completely the same than previous swaps
Lecture 3

3. Interest rate options

Interest rate options: General

definition
Two parties swapping a series of
payments, but with some protection
Lecture 3

3. Interest Rate options
• Represent the RIGHT to make a fixed interest payment and receive a floating
interest payment
• They have exercise rate or strike rate
Structure
• Call: make a known fixed rate payment
receive an unknown floating payment
• Put: receive a known fixed rate payment

Pays a premium make an unknown floating payment

Receives a premium
Payoff
𝑚
𝑁 𝑀𝑎𝑥(0, 𝐿𝐼𝐵𝑂𝑅 − 𝑋
360 Payoff
𝑚
𝑁 𝑀𝑎𝑥(0, 𝑋 − 𝐿𝐼𝐵𝑂𝑅
360
Lecture 3

3. Interest Rate options
Structure Libor 90 days

90 90
20𝑀 𝑀𝑎𝑥(6% − 10% 20𝑀 𝑀𝑎𝑥(10% − 6%
360 360
Call payoff Put payoff

Pricing
• As all options, these instruments should be priced using B-S model
Lecture 3

3. Interest Rate options. Additional
instruments
Foundations
• A floating rate bond is a bond which has an interest rate linked up to an
index to reduce the interest rate risk
BUT
• Some cap their floating rate obligations to ensure that interest rates do
not rise above a pre-specified rate

• Some floating rate bonds offer buyers some compensation by providing a
floor, below which interest rates will not decline
• If a floating rate bond has a cap and a floor, a collar is created

Cap Example

N: 25M
Libor today is 10%
Company wishes to fix the rate on each payment at no more than 10%
Lecture 3

instruments
Foundations
Cap Example

N: 25M
Libor today is 10%
Company wishes to fix the rate on each payment at no more than 10%

Has to pay less
Lecture 3

When rate rises, the owner is beneficiated

instruments
Foundations
Cap Floor
Used by a borrower who wants Used by a lender who wants
protection against raising rates protection against falling rates
Lecture 3

Price of floating rate bond with cap = Price of floating rate bond with floor =
Price of floating rate bond without cap Price of floating rate bond without cap
- Value of call on bond + Value of put on bond

instruments
Foundations
Collar

Two options - a call option with a
strike price of Kc for the issuer of
the bond and a put option with a

strike price of Kf for
the buyer of the bond.

Price of floating rate bond with
collar =

Price of floating rate bond without
collar

+ Value of call on bond
Lecture 3

- Value of put on bond

Part 3

OTHER INSTRUMENTS

a. Convertible bonds
b. Callable bonds

43

Other Bonds- Convertible Bonds
Foundations
Fixed income features Equity features

Issuer: XYZ Company Inc. Issuer: XYZ Company Inc.
Nominal value: $1000 Stock price: $80
Issue date: today Volatility 20%
Maturity 5 years Dividend: 0
Coupon: 2%
Price at which the shares
Number of shares obtained are bought upon
Conversion features conversion
if one converts $1000 of
Part 3. Other instruments

FV of bond.
Conversion ratio: 10 𝑀𝑎𝑟𝑘𝑒𝑡 𝑃𝑟𝑖𝑐𝑒
This number usually
remains fixed Conversion price: $100 𝐶𝑜𝑛𝑣𝑒𝑟𝑡𝑖𝑜𝑛 𝑟𝑎𝑡𝑖𝑜

Could have Call If share price > conversion
protection price, the bondholder will
convert into shares
a. Market Valuation
Lecture 3

Convertible price: price of the convertible. In this case it is $100
Parity: Market value of the shares into which the bond can be converted at that time
10 x 80 = 800. Quoted as % of Face Value: 80%

Foundations
• How much an invertor has to pay to control the same number of shares via convertible
• Difference between convertible bond price and parity as % of parity

Convertible bond:
Conversion ratio X Conversion price
10 X 100
I can accede to 10 shares by
Market direct purchase:
$80 x 10

I have a conversion premium of $200
200/800=
Pricing
Assumes that
convertible bond = option + traditional bond
Lecture 3

American, out of
the money

Convertible Bonds
Pricing
4. The bond can be called
back by the issuer

Hybrid
2. No conversion. S is too low

Flat because represent
1. Junk or distressed bond the cash flow of the

bond

If S is too low,
the bond also
became
worthless
Lecture 3

Other Bonds- Callable Bonds
Callable Bonds
The issuer preserves the right to call back the bond and pay a fixed price

WHY?
If interest rates drop, the issuer can refund the bonds at the fixed price.

The bond holder is short the call option, and the issuer is long the call option

• Most callable bonds come with an
initial period of call protection,

during which the bonds cannot be
called back.

Pricing
Value of Value of Value of Call
Callable = Straight - Feature in
Bond Bond Bond
Lecture 3

Value of Value of
Callable < Straight
Bond Bond
Valuation is using the Yield to worst

Lecture 3

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Lecture 3