Financial Engineering
Interest Rates and Fixed Income
Securities
Zvi Wiener
mswiener@mscc.huji.ac.il
tel: 02-588-3049
Zvi Wiener
ContTimeFin - 6
slide 1
Bonds
A bond is a contract, paid up-front that yields
a known amount at a known date (maturity).
The bond may pay a dividend (coupon) at
fixed times during the life.
Additional options: callable, puttable,
indexed, prepayment options, etc.
Credit risk, recovery ratio, rating.
Zvi Wiener
ContTimeFin - 6
slide 2
Term Structure of IR
r
long term IR
short term IR
spot rate
time to maturity
Zvi Wiener
ContTimeFin - 6
slide 3
Known IR
V - value of a contract.
r(t) - short term interest rate.
If there is no risk and no coupons then
dV = rVdt
V(t) = V(T)e-rt
if there is a continuous dividend stream
dV+cVdt = rVdt
Zvi Wiener
ContTimeFin - 6
slide 4
Known IR
If r is not constant, but not risky r(t)
T
dV = r(t)Vdt
V (t )  e

 r ( ) d
t
If there is a continuous dividend stream
dV+c(t)Vdt = r(t)Vdt
Zvi Wiener
ContTimeFin - 6
slide 5
Known IR
Assume that there are zero coupon bonds for
all possible ttm (time to maturity).
Denote the price of these bonds by V(t,T).
T
V (t , T )  e
Zvi Wiener

 r ( ) d
t
ContTimeFin - 6
slide 6
Known IR
T
V (t , T )  e

 r ( ) d
t
T
log V (t , T )    r ( )d
t
 1 V
r (T ) 
V (t , T ) T
Zvi Wiener
ContTimeFin - 6
slide 7
Yield
Y (T t )
V (t , T )e
1
log V (t , T )
Y (t , T )  
T t
Zvi Wiener
ContTimeFin - 6
slide 8
Typical yield curves
increasing
yield
humped
decreasing
time to maturity
Zvi Wiener
ContTimeFin - 6
slide 9
Typical yield curves

increasing - the most typical.
decreasing - short rates are high but
expected to fall.

humped - short rates are expected to fall
soon.

Zvi Wiener
ContTimeFin - 6
slide 10
Term Structure Explanations
Expectation hypothesis states F0=E(PT)
this hypothesis is be true if all market
participants were risk neutral.
Zvi Wiener
ContTimeFin - 6
slide 11
Term Structure Explanations
Normal Backwardation (Keynes),
commodities are used by hedgers to reduce
risk. In order to induce speculators to take the
opposite positions, the producers must offer a
higher return. Thus speculators enter the long
side and have the expected profit of
E(PT) – F0 > 0
Zvi Wiener
ContTimeFin - 6
slide 12
Term Structure Explanations
Contango is similar to the normal
backwardation, but the natural hedgers are the
purchasers of a commodity, rather than
suppliers. Since speculators must be paid for
taking risk, the opposite relation holds:
E(PT) – F0 < 0
Zvi Wiener
ContTimeFin - 6
slide 13
8% Coupon Bond
Yield to Maturity
8%
9%
Price Change
T=1 yr. T=10 yr. T=20 yr.
1,000.00 1,000.00 1,000.00
990.64 934.96
907.99
0.94% 6.50%
9.20%
Zero Coupon Bond
Yield to Maturity
8%
9%
Price Change
Zvi Wiener
T=1 yr. T=10 yr. T=20 yr.
924.56 456.39
208.29
915.73 414.64
171.93
0.96% 9.15% 17.46%
ContTimeFin - 6
slide 14
Duration
F. Macaulay (1938)
Better measurement than time to maturity.
Weighted average of all coupons with the
corresponding time to payment.
Bond Price = Sum[ CFt/(1+y)t ]
suggested weight of each coupon:
wt = CFt/(1+y)t /Bond Price
What is the sum of all wt?
Zvi Wiener
ContTimeFin - 6
slide 15
Macaulay Duration
T
T
CFt
1
D   t wt 
t

t
Bond Pr ice t 1 (1  y)
t 1
A weighted sum of times to maturities of each
coupon.
What is the duration of a zero coupon bond?
Zvi Wiener
ContTimeFin - 6
slide 16
Macaulay Duration
(1)
(2)
(3)
(4)
(5)
Time until
Payment
column (1)
payment Payment Discounted Weight multiplied
(in Years)
at 5%
by (4)
Bond A
8%
Sum:
Zvi Wiener
0.5
1.0
1.5
2.0
$40
$40
$40
$1,040
$38.095
$36.281
$34.553
$855.611
$964.540
ContTimeFin - 6
0.0395
0.0376
0.0358
0.8871
1.000
0.0198
0.0376
0.0537
1.7742
1.8853
slide 17
Macaulay Duration
(1)
(2)
(3)
(4)
(5)
Time until
Payment
column (1)
payment Payment Discounted Weight multiplied
(in Years)
at 5%
by (4)
Bond A
8%
0.5
1.0
1.5
2.0
$40
$40
$40
$1,040
$38.095
$36.281
$34.553
$855.611
$964.540
0.0395
0.0376
0.0358
0.8871
1.000
0.0198
0.0376
0.0537
1.7742
1.8853
0.5-1.5
2.0
0
$1,000
$0
$822.70
$822.70
0
1
1
0
2
2
Sum:
Bond B
zero
Sum
Zvi Wiener
ContTimeFin - 6
slide 18
Duration
Sensitivity to IR changes:
Long term bonds are more sensitive.
 Lower coupon bonds are more sensitive.
 The sensitivity depends on levels of IR.

Zvi Wiener
ContTimeFin - 6
slide 19
Duration
 (1  y ) 
P
  DMC 

P
 1 y 
The bond price volatility is proportional to the
bond’s duration.
Thus duration is a natural measure of interest
rate risk exposure.
Zvi Wiener
ContTimeFin - 6
slide 20
Modified Duration
D
D* 
1 y
P
  D * y
P
The percentage change in bond price is the
product of modified duration and the change
in the bond’s yield to maturity.
Zvi Wiener
ContTimeFin - 6
slide 21
Comparison of two bonds
Coupon bond with duration Zero-coupon bond with equal
1.8853
duration must have 1.8853
years to maturity.
At 5% semiannual its price is
Price (at 5% for 6m.) is
$964.5405
($1,000/1.053.7706)=$831.9623
If IR increase by 1bp
If IR increase to 5.01%, the
price becomes:
(to 5.01%), its price will
fall to $964.1942, or
($1,000/1.05013.7706)=$831.66
0.359% decline.
0.359% decline.
Zvi Wiener
ContTimeFin - 6
slide 22
Duration
D
Zero coupon bond
15% coupon, YTM = 15%
Maturity
0
Zvi Wiener
3m
6m
1yr
3yr
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5yr
10yr
30yr
slide 23
Example
A bond with 30-yr to maturity
Coupon 8%; paid semiannually
YTM = 9%
P0 = $897.26
D = 11.37 Yrs
if YTM = 9.1%, what will be the price?
Zvi Wiener
ContTimeFin - 6
slide 24
Example
A bond with 30-yr to maturity
Coupon 8%; paid semiannually
YTM = 9%
P0 = $897.26
D = 11.37 Yrs
if YTM = 9.1%, what will be the price?
P/P = - y D*
P = -(y D*)P = -$9.36
P = $897.26 - $9.36 = $887.90
Zvi Wiener
ContTimeFin - 6
slide 25
What Determines Duration?
Duration of a zero-coupon bond equals maturity.
 Holding ttm constant, duration is higher when
coupons are lower.
Holding other factors constant, duration is higher
when ytm is lower.
 Duration of a perpetuity is (1+y)/y.

Zvi Wiener
ContTimeFin - 6
slide 26
What Determines Duration?
Holding the coupon rate constant, duration not
always increases with ttm.

Zvi Wiener
ContTimeFin - 6
slide 27
Duration
T
CFt
P 
t
t 1 (1  y )
T
t CFt
DMC  
t
t 1 (1  y )
Zvi Wiener
ContTimeFin - 6
slide 28
T
CFt
P 
t
t 1 (1  y )
T
t CFt
D 
t
t 1 (1  y )
Zvi Wiener
ContTimeFin - 6
slide 29
T
CFt
P 
t
t 1 (1  y )
T
t CFt
D 
t
t 1 (1  y )
dP T  t CFt
DP



t 1
dy t 1 (1  y )
1 y
Zvi Wiener
ContTimeFin - 6
slide 30
T
CFt
P 
t
t 1 (1  y )
T
t CFt
D 
t
t 1 (1  y )
dP T  t CFt
DP


t 1
dy t 1 (1  y )
1 y
dP P
 D
d (1  y ) (1  y )
Zvi Wiener
ContTimeFin - 6
slide 31
Modern Approach
dP P
 D
d (1  y ) (1  y )
Percent change in bond price
 D
Percent change in discount factor
Duration can be regarded as the discount-rate
elasticity of the bond price
Zvi Wiener
ContTimeFin - 6
slide 32
Modern Approach
Duration can be used to measure the price
volatility of a bond:
dP P
 D
d (1  y ) (1  y )
dP
dy
 D
P
1 y
dP
  D * dy
P
Zvi Wiener
ContTimeFin - 6
slide 33
Modern Approach
What are the natural bounds on duration?
Can duration be bigger than maturity?
Can duration be negative?
How to measure duration of a portfolio?
Zvi Wiener
ContTimeFin - 6
slide 34
Duration: Modern Approach
1  y dP

D
P dy
1 dP

 D*
P dy
Zvi Wiener
ContTimeFin - 6
slide 35
Duration of a Portfolio
1  y dP
D
(definition )
P dy
1  y dPA B
D A B  
PA B dy
Zvi Wiener
ContTimeFin - 6
slide 36
Duration of a Portfolio
D A B
1  y dPA B


PA B dy
1  y  dPA dPB 

 


PA B  dy
dy 
1
PA B

1  y dPA
1  y dPB 
  PA
 
 PB
PA dy
PB dy 

1
PA DA  PB DB 
PA B
Zvi Wiener
ContTimeFin - 6
slide 37
Modern Approach to Duration
Simon Benninga, Financial Modelling, the MIT
press, Cambridge, MA, ISBN 0-262-02437-3, $45
MIT Press tel: 800-356-0343
http://mitpress.mit.edu/book-home.tcl?isbn=0262024373
see also my advanced lecture notes on duration
Convexity is a similar measurement but with
second derivative.
Zvi Wiener
ContTimeFin - 6
slide 38
Financial Modelling
by Simon Benninga
Implementation in Excel
 Duration Patterns
 Duration of a bond with uneven payments
 Calculating YTM for uneven periods
 Nonflat term structure and duration
 Immunization strategies
 Cheapest to deliver option and Duration

Zvi Wiener
ContTimeFin - 6
slide 39
Passive Bond Management
Passive management takes bond prices as
fairly set and seeks to control only the risk of
the fixed-income portfolio.

Indexing strategy
– attempts to replicate a bond index

Immunization
– used to tailor the risk to specific needs
(insurance companies, pension funds)
Zvi Wiener
ContTimeFin - 6
slide 40
Bond-Index Funds
Similar to stock indexing.
Major indices: Lehman Brothers, Merill
Lynch, Salomon Brothers.
Include: government, corporate, mortgagebacked, Yankee bonds (dollar denominated,
SEC registered bonds of foreign issuers, sold
in the US).
Zvi Wiener
ContTimeFin - 6
slide 41
Bond-Index Funds
Properties:
many issues
not all are liquid
replacement of maturing issues
Tracking error is a good measurement of
performance. According to Salomon Bros.
With $100M one can track the index within
4bp. tracking error per month.
Zvi Wiener
ContTimeFin - 6
slide 42
Cellular approach
ttm\Sector Treasury Agency MBS
< 1yr
12.1%
1-3 yrs
5.4%
4.1% 3.2%
3-5 yrs
9.2% 6.1%
Zvi Wiener
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slide 43
Immunization
Immunization techniques refer to strategies
used by investors to shield their overall
financial status from exposure to interest rate
fluctuations.
Zvi Wiener
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slide 44
Net Worth Immunization
Banks and thrifts have a natural mismatch
between assets and liabilities. Liabilities are
primarily short-term deposits (low duration),
assets are typically loans or mortgages
(higher duration).
When will banks lose money, when IR
increase or decline?
Zvi Wiener
ContTimeFin - 6
slide 45
Gap Management
ARM are used to reduce duration of bank
portfolios.
Other derivative securities can be used.
Capital requirement on duration (exposure).
Basic idea:
to match duration of assets and liabilities.
Zvi Wiener
ContTimeFin - 6
slide 46
Target Date Immunization
Important for pension funds and insurances.
Price risk and reinvestment risk.
What is the correlation between them?
Zvi Wiener
ContTimeFin - 6
slide 47
Target Date Immunization
Accumulated
value
Original plan
0
Zvi Wiener
t*
t
ContTimeFin - 6
slide 48
Target Date Immunization
Accumulated
value
IR increased at t*
0
Zvi Wiener
t*
t
ContTimeFin - 6
slide 49
Target Date Immunization
Accumulated
value
0
Zvi Wiener
t*
D
ContTimeFin - 6
t
slide 50
Target Date Immunization
Accumulated
value
0
Zvi Wiener
Continuous rebalancing
can keep the terminal value
unchanged
t*
D
ContTimeFin - 6
t
slide 51
Good Versus Bad Immunization
value
$10,000
Single payment obligation
0
Zvi Wiener
8%
ContTimeFin - 6
r
slide 52
Good Versus Bad Immunization
value
Good immunizing strategy
$10,000
Single payment obligation
0
Zvi Wiener
8%
ContTimeFin - 6
r
slide 53
Good Versus Bad Immunization
value
Good immunizing strategy
$10,000
Single payment obligation
0
Zvi Wiener
8%
ContTimeFin - 6
r
slide 54
Good Versus Bad Immunization
value
Bad immunizing strategy
Good immunizing strategy
$10,000
Single payment obligation
0
Zvi Wiener
8%
ContTimeFin - 6
r
slide 55
Standard Immunization
Is very useful but is based on the assumption of
the flat term structure. Often a higher order
immunization is used (convexity, etc.).
Another reason for goal oriented mutual funds
(retirement, education, housing, medical
expenses).
Zvi Wiener
ContTimeFin - 6
slide 56
Duration Immunization
Duration protects against small IR changes.
 Duration assumes a parallel change in the
TS.
 Immunization is based on nominal IR.
 Immunization is very conservative and is
inappropriate for many portfolio managers.
 The passage of time changes both duration
and horizon date, one need to rebalance.
 Duration changes if yields change.
 Obtaining bonds for immunization can be

Zvi Wiener
ContTimeFin - 6
slide 57
Cash Flow Matching and
Dedication
Is a very reasonable strategy, but not always
realizable.
Uncertainty of payments.
Lack of perfect match
Saving on transaction fees.
Zvi Wiener
ContTimeFin - 6
slide 58
Active Bond Management
Mainly speculative approach based on ability
to predict IR or credit enhancement or market
imperfections (identifying mispriced loans).
Zvi Wiener
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slide 59
Contingent Immunization
value
$12,000
$10,000
0
Zvi Wiener
5 yr t
ContTimeFin - 6
slide 60
Contingent Immunization
value
$12,000
$10,000
Stop boundary
0
Zvi Wiener
5 yr t
ContTimeFin - 6
slide 61
Contingent Immunization
value
$12,000
$10,000
Stop boundary
0
Zvi Wiener
5 yr t
ContTimeFin - 6
slide 62
Contingent Immunization
value
$12,000
$10,000
Stop boundary
0
Zvi Wiener
5 yr t
ContTimeFin - 6
slide 63
Interest Rate Swap
One of the major fixed-income tools.
Example: 6m LIBOR versus 7% fixed.
Exchange of net cash flows.
Risk involved: IR risk, default risk (small).
Why the default risk on IR swaps is small?
Zvi Wiener
ContTimeFin - 6
slide 64
Interest Rate Swap
7.05%
6.95%
Company B
Swap dealer
Company A
LIBOR
LIBOR
No need in an actual loan.
Can be used as a speculative tool or for hedging.
Zvi Wiener
ContTimeFin - 6
slide 65
Interest Rate Swap
Can not be priced as an exchange of two loans
(old method).
Why?
Zvi Wiener
ContTimeFin - 6
slide 66
Currency Swap
A similar exchange of two loans in different
currencies.
Subject to a higher default risk, because of the
principal.
Is useful for international companies to hedge
currency risk.
Zvi Wiener
ContTimeFin - 6
slide 67
Modeling a Swap
A simple fixed versus floating swap.
Current fixed rate on a 30 years loan is 7%
with semi annual payments for simplicity.
Current floating rate is 6%. Notional amount
is 1,000.
How can we model our future payments?
Zvi Wiener
ContTimeFin - 6
slide 68
Modeling a Swap
There are two flows of cash. At maturity they
cancel each other. The fixed part has
payments known in advance. The only
uncertainty is with the floating part.
We need a simple model of interest rates.
Zvi Wiener
ContTimeFin - 6
slide 69
Modeling a Swap
Floating IR
6%
0
1
Zvi Wiener
2
3
60
ContTimeFin - 6
slide 70
Modeling a Swap
Floating IR
6%
0
1
1
0
1
2
3
Zvi Wiener
1
0
60
ContTimeFin - 6
slide 71
Modeling a Swap
Floating IR
6%
0
1
1
0
1
2
3
Zvi Wiener
1
0
60
ContTimeFin - 6
slide 72
Modeling a Swap
Floating IR
6%
0
1
1
0
1
2
3
Zvi Wiener
1
0
60
ContTimeFin - 6
slide 73
Modeling a Swap
Floating IR
6%
0
1
1
0
1
2
3
Zvi Wiener
1
0
60
ContTimeFin - 6
slide 74
Modeling a Swap
Floating IR
6%
0
1
1
0
1
2
3
Zvi Wiener
1
0
60
ContTimeFin - 6
slide 75
Modeling a Swap
Floating IR
6%
1
1
0
1
0
Arithmetical BM – all jumps of the same size,
direction is defined by the sequence of random
variables that you have prepared.
0
1
Zvi Wiener
2
3
60
ContTimeFin - 6
slide 76
Modeling a Swap
Floating IR
6%
1
1
0
1
0
Geometrical BM – for an up jump you multiply
the current level by a constant u > 1, for a
downward jump you multiply by d < 1.
0
1
Zvi Wiener
2
3
60
ContTimeFin - 6
slide 77
Modeling a Swap
Floating IR
6%
1
1
0
1
0
Geometrical BM – jumps have different sizes
but up*down = down*up – an important property!
0
1
Zvi Wiener
2
3
60
ContTimeFin - 6
slide 78
Home Assignment
Evaluate the swap with your sequence of
random or pseudo-random numbers using
both approaches arithmetical and geometrical.
Up jumps are 10 bp., and 1.1
Down -10bp., ans 0.9
Your side is fixed, discount at 7% annually.
You do not have to submit, but bring it to the
class, we will discuss it.
Zvi Wiener
ContTimeFin - 6
slide 79
Financial Engineering
New securities created:
IO (negative duration)
PO
CMO
Swaptions
Caps and Caplets
Floors
Ratchets
Zvi Wiener
ContTimeFin - 6
slide 80
Why TS is not flat?
Assume that TS is flat, but varies with time.
Then the price of a zero coupon bond
maturing in time  is e-r.
How one can form an arbitrage portfolio?
Requirements:
zero investment,
never losses,
sometimes gains.
Zvi Wiener
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slide 81
Why TS is not flat?
Take 3 bonds, maturing in 1,2, and 3 years.
The current prices are:
P1 = e-r,
P2 = e-2r,
P3 = e-3r.
We want to form a portfolio with a one-year
bonds, b two-years, c three-years.
So the first requirement is
ae-r + be-2r + ce-3r=0
Zvi Wiener
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slide 82
Why TS is not flat?
So the second requirement is that there are no
possible losses
Equate duration of long and short sides.
-ae-r - 2be-2r - 3ce-3r=0
The two equations can be solved
simultaneously.
Solution is a zero-investment, zero-loss
portfolio - arbitrage.
Zvi Wiener
ContTimeFin - 6
slide 83
Why TS is not flat?
price
rnow
Zvi Wiener
ContTimeFin - 6
r
slide 84
Why TS is not flat?
So lve[ { -e-r - 2be-2r - 3ce-3r == 0,
-e-r - 2be-2r - 3ce-3r == 0}, {b,c}]
price
rnow
Zvi Wiener
ContTimeFin - 6
r
slide 85
Term Structure Models
Let V(t,T) be the price at time t of an asset
paying $1 at time T.
Obviously V(T,T) =1.
Under the equivalent martingale measure
the discounted price is a martingale, so
V (t , T )  (t )  E V (T , T )  (T )
Q
t
Zvi Wiener
ContTimeFin - 6
slide 86
Term Structure Models
V (t , T )  (t )  E V (T , T )  (T )
Q
t

 (T ) 
V (t , T )  E V (T , T )

 (t ) 

Q
t






Q
V (t , T )  Et V (T , T ) exp    r ( s)ds 

 t

T
Zvi Wiener
ContTimeFin - 6
slide 87
One Factor Models
Assume that the short rate is the only factor.
drt   (rt , t )dt   (rt , t )dZ
Zvi Wiener
ContTimeFin - 6
slide 88
One Factor Models
Consider a riskless portfolio consisting of two
bonds: V1, and V2 (with ttm T1 and T2).
The riskless portfolio can be formed as
V (t, T1 )  V (t, T2 )
How to choose  and  so that the portfolio is
riskless?
Zvi Wiener
ContTimeFin - 6
slide 89
One Factor Models
P
V1
V2


x
x
x
V2
V1

,  
x
x
This portfolio is riskless, so it earns the risk
free interest.
Zvi Wiener
ContTimeFin - 6
slide 90
One Factor Models
V2
V1
P
V1 
V2
x
x
P V2 V1 V1 V2


0
x
x x x x
Zvi Wiener
ContTimeFin - 6
slide 91
One Factor Models
P
1 P
P
2
dP 
dx 
(
dx
)

dt
2
x
2 x
t
2
EdP  rPdt
Zvi Wiener
ContTimeFin - 6
slide 92
One Factor Models
V2 V1
V1 V2
dP 
dx 
dx 
x x
x x
2
2
1 V2  V1
1 V1  V2
2
2




dx

dx

2
2
2 x x
2 x x
V2 V1
V1 V2
dt 
dt
x t
x t
Zvi Wiener
ContTimeFin - 6
slide 93
One Factor Models
 1 V2  V1 1 V1  V2  2

 dt 

2
2 
2 x x 
 2 x x
V1 
 V2 V1 V1 V2 
 V2

V1 
V2 dt

dt  r 
x t 
x
 x t
 x

2
Zvi Wiener
2
ContTimeFin - 6
slide 94
One Factor Models


 2
2
V1  

x  2
V2
x
Zvi Wiener
2

 V1 V1



rV

1
2

x
t

2

 V2 V2



rV
2
2

x
t

2
ContTimeFin - 6
slide 95
One Factor Models


 2
2


 2
2
Zvi Wiener

 V1 V1



rV
1
2
x
t

V1

x

 V2 V2



rV
2
2

x
t

V2
x
2
2
ContTimeFin - 6
slide 96
One Factor Models
   V1 V1





rV
1
2

t
 2 x

2
2
V1
q
x
  V1
V1
V1

 rV1  q
2
2 x
t
x
2
2
V1 (t , t )  1
Zvi Wiener
ContTimeFin - 6
slide 97