Zvi WienerContTimeFin - 6 slide 1 Financial Engineering Interest Rates and Fixed Income Securities Zvi Wiener [email protected] tel: 02-588-3049

Zvi Wiener ContTimeFin - 6 slide 1

Financial Engineering

Interest Rates and Fixed Income Securities

Zvi [email protected]

tel: 02-588-3049


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.


Term Structure of IR

time to maturity

r

short term IR

long term IR

spot rate


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


Known IR

If r is not constant, but not risky r(t)

dV = r(t)Vdt

If there is a continuous dividend stream

dV+c(t)Vdt = r(t)Vdt

T

t

dr

etV )(

)(


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

t

dr

eTtV )(

),(


Known IR

T

t

dr

eTtV )(

),(

T

t

drTtV )(),(log

T

V

TtVTr

),(

1)(


Yield

tT

TtVTtY

),(log),(

1),( )( tTYeTtV


Typical yield curves

time to maturity

yield increasing

decreasing

humped


Typical yield curves

increasing - the most typical.

decreasing - short rates are high but expected to fall.

humped - short rates are expected to fall soon.


Term Structure Explanations

Expectation hypothesis states F0=E(PT)

this hypothesis is be true if all market participants were risk neutral.



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



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


8% Coupon BondYield to Maturity T=1 yr. T=10 yr. T=20 yr.

8% 1,000.00 1,000.00 1,000.00

9% 990.64 934.96 907.99

Price Change 0.94% 6.50% 9.20%

Yield to Maturity T=1 yr. T=10 yr. T=20 yr.8% 924.56 456.39 208.29

9% 915.73 414.64 171.93

Price Change 0.96% 9.15% 17.46%

Zero Coupon Bond


DurationF. 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?


Macaulay Duration

A weighted sum of times to maturities of each coupon.

What is the duration of a zero coupon bond?

T

tt

tT

tt y

CFt

iceBondwtD

11 )1(Pr

1


Macaulay Duration(1)

Time untilpayment

(in Years)

(2)

Payment

(3)Payment

Discountedat 5%

(4)

Weight

(5)column (1)multiplied

by (4)Bond A 0.5 $40 $38.095 0.0395 0.01988% 1.0 $40 $36.281 0.0376 0.0376

1.5 $40 $34.553 0.0358 0.05372.0 $1,040 $855.611 0.8871 1.7742

Sum: $964.540 1.000 1.8853


Macaulay Duration(1)

Time untilpayment

(in Years)

(2)

Payment

(3)Payment

Discountedat 5%

(4)

Weight

(5)column (1)multiplied

by (4)Bond A 0.5 $40 $38.095 0.0395 0.01988% 1.0 $40 $36.281 0.0376 0.0376

1.5 $40 $34.553 0.0358 0.05372.0 $1,040 $855.611 0.8871 1.7742

Sum: $964.540 1.000 1.8853

Bond B 0.5-1.5 0 $0 0 0zero 2.0 $1,000 $822.70 1 2Sum $822.70 1 2


Duration

Sensitivity to IR changes:

Long term bonds are more sensitive. Lower coupon bonds are more sensitive. The sensitivity depends on levels of IR.


Duration

The bond price volatility is proportional to the bond’s duration.

Thus duration is a natural measure of interest rate risk exposure.

y

yD

P

PMC 1

)1(


Modified Duration

The percentage change in bond price is the product of modified duration and the change in the bond’s yield to maturity.

yDP

P

y

DD

*

1*


Comparison of two bonds

Coupon bond with duration 1.8853

Price (at 5% for 6m.) is $964.5405

If IR increase by 1bp

(to 5.01%), its price will fall to $964.1942, or

0.359% decline.

Zero-coupon bond with equal duration must have 1.8853 years to maturity.

At 5% semiannual its price is

($1,000/1.053.7706)=$831.9623

If IR increase to 5.01%, the price becomes:

($1,000/1.05013.7706)=$831.66

0.359% decline.


Duration

Maturity

D

0 3m 6m 1yr 3yr 5yr 10yr 30yr

15% coupon, YTM = 15%

Zero coupon bond


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?


ExampleA bond with 30-yr to maturityCoupon 8%; paid semiannuallyYTM = 9%

P0 = $897.26D = 11.37 Yrsif 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


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.


What Determines Duration? Holding the coupon rate constant, duration not always increases with ttm.


Duration

T

tt

t

y

CFP

1 )1(

T

tt

tMC y

CFtD

1 )1(


T

tt

t

T

tt

t

y

CFtD

y

CFP

1

1

)1(

)1(


y

PD

y

CFt

dy

dP

y

CFtD

y

CFP

T

ttt

T

tt

t

T

tt

t

1)1(

)1(

)1(

11

1

1


y

PD

y

CFt

dy

dP T

ttt

1)1(11

T

tt

t

T

tt

t

y

CFtD

y

CFP

1

1

)1(

)1(

Dyyd

PdP

)1()1(


DfactordiscountinchangePercent

pricebondinchangePercent

Dyyd

PdP

)1()1(

Duration can be regarded as the discount-rate elasticity of the bond price

Modern Approach


dyDP

dP

y

dyD

P

dP

Dyyd

PdP

*

1

)1()1(

Duration can be used to measure the price volatility of a bond:

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?

Modern Approach


Duration: Modern Approach

*1

1

Ddy

dP

P

Ddy

dP

P

y


Duration of a Portfolio

dy

dP

P

yD

definitiondy

dP

P

yD

BA

BABA

1

)(1


Duration of a Portfolio

BBAABA

B

BB

A

AA

BA

BA

BA

BA

BABA

DPDPP

dy

dP

P

yP

dy

dP

P

yP

P

dy

dP

dy

dP

P

y

dy

dP

P

yD

1

111

1

1


Simon Benninga, Financial Modelling, the MIT press, Cambridge, MA, ISBN 0-262-02437-3, $45

MIT Press tel: 800-356-0343http://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.

Modern Approach to Duration


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

Financial Modellingby Simon Benninga


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)


Bond-Index Funds

Similar to stock indexing.

Major indices: Lehman Brothers, Merill Lynch, Salomon Brothers.

Include: government, corporate, mortgage-backed, Yankee bonds (dollar denominated, SEC registered bonds of foreign issuers, sold in the US).


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.


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%


Immunization

Immunization techniques refer to strategies used by investors to shield their overall financial status from exposure to interest rate fluctuations.


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?


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.


Target Date Immunization

Important for pension funds and insurances.

Price risk and reinvestment risk.

What is the correlation between them?



Accumulatedvalue

0 t* t

Original plan



Accumulatedvalue

0 t* t

IR increased at t*



Accumulatedvalue

0 t* D t



Accumulatedvalue

0 t* D t

Continuous rebalancingcan keep the terminal value

unchanged


Good Versus Bad Immunization

value

0 8% r

Single payment obligation

$10,000



value

0 8% r


Good immunizing strategy$10,000



value

0 8% r





value

0 8% r



Bad immunizing strategy


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).


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 difficult.


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.


Active Bond Management

Mainly speculative approach based on ability to predict IR or credit enhancement or market imperfections (identifying mispriced loans).


Contingent Immunization

0 5 yr t

value

$10,000

$12,000



0 5 yr t

value

$10,000

$12,000

Stop boundary



0 5 yr t

value

$10,000

$12,000

Stop boundary



0 5 yr t

value

$10,000

$12,000

Stop boundary


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?


Interest Rate Swap

Company A Company BSwap dealer

6.95% 7.05%

LIBOR LIBOR

No need in an actual loan.Can be used as a speculative tool or for hedging.


Interest Rate Swap

Can not be priced as an exchange of two loans (old method).

Why?


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.


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?


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.


Modeling a Swap

0 1 2 3 60

6%

Floating IR


Modeling a Swap

0 1 2 3 60

6%

Floating IR

1 1 0 1 0


Modeling a Swap

0 1 2 3 60

6%

Floating IR

1 1 0 1 0


Modeling a Swap

0 1 2 3 60

6%

Floating IR

1 1 0 1 0


Modeling a Swap

0 1 2 3 60

6%

Floating IR

1 1 0 1 0


Modeling a Swap

0 1 2 3 60

6%

Floating IR

1 1 0 1 0


Modeling a Swap

0 1 2 3 60

6%

Floating IR

Arithmetical BM – all jumps of the same size,direction is defined by the sequence of randomvariables that you have prepared.

1 1 0 1 0


Modeling a Swap

0 1 2 3 60

6%

Floating IR

Geometrical BM – for an up jump you multiplythe current level by a constant u > 1, for a downward jump you multiply by d < 1.

1 1 0 1 0


Modeling a Swap

0 1 2 3 60

6%

Floating IR

Geometrical BM – jumps have different sizesbut up*down = down*up – an important property!

1 1 0 1 0


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.


Financial Engineering

New securities created:

IO (negative duration)

PO

CMO

Swaptions

Caps and Caplets

Floors

Ratchets


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.


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


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.


Why TS is not flat?

r

price

rnow


Why TS is not flat?

So lve[ { -e-r - 2be-2r - 3ce-3r == 0,

-e-r - 2be-2r - 3ce-3r == 0}, {b,c}]

r

price

rnow


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

)(),()(),( TTTVEtTtV Qt


Term Structure Models

)(),()(),( TTTVEtTtV Qt

)(

)(),(),(

t

TTTVETtV Q

t

T

t

Qt dssrTTVETtV )(exp),(),(


One Factor Models

Assume that the short rate is the only factor.

dZtrdttrdr ttt ),(),(


One Factor Models

),(),( 21 TtVTtV

Consider a riskless portfolio consisting of two bonds: V1, and V2 (with ttm T1 and T2).

The riskless portfolio can be formed as

How to choose and so that the portfolio is riskless?


One Factor Models

x

V

x

V

12 ,

x

V

x

V

x

P

21

This portfolio is riskless, so it earns the risk free interest.


One Factor Models

21

12 V

x

VV

x

VP

x

V

x

V

x

V

x

V

x

P

2112 0


One Factor Models

dtt

Pdx

x

Pdx

x

PdP

22

2

)(2

1

rPdtdPE


One Factor Models

dtt

V

x

Vdt

t

V

x

V

dxx

V

x

Vdx

x

V

x

V

dxx

V

x

Vdx

x

V

x

VdP

2112

2

22

212

21

22

2112

2

1

2

1


One Factor Models

dtVx

VV

x

Vrdt

t

V

x

V

t

V

x

V

dtx

V

x

V

x

V

x

V

21

122112

222

21

21

22

2

1

2

1


One Factor Models

22

22

221

11

21

222

2

2

rVt

V

x

V

x

V

rVt

V

x

V

x

V


One Factor Models

xV

rVt

VxV

xV

rVt

VxV

22

222

22

11

121

22

2

2


One Factor Models

qxV

rVt

VxV

1

11

21

22

2

x

VqrV

t

V

x

V

1

11

21

22

2

1),(1 ttV

Documents

Zvi WienerContTimeFin - 6 slide 1 Financial Engineering Interest Rates and Fixed Income Securities Zvi Wiener [email protected] tel: 02-588-3049