Financial Engineering - Five Days of Lectures October 2010 -Rimini

8/8/2019 Financial Engineering - Five Days of Lectures October 2010 -Rimini

1/129

Financial engineering issues

of the Present Valueand term structure of interest rates:

a deterministic introduction

Lectures in Rimini,

UNIVERSIT DI BOLOGNA

October 2010Prof. Dr. Sergey Smirnov

Head of the Department of Risk Management and Insurance

Director of the Financial Engineering and Risk Management Lab

Higher School of Economics, Moscow


2/129

About these lectures

Main objective: develop critical thinking

concerning of present value interpretation, use

and calculation.

Discussion is welcome: lecturer poses questions

to students, students pose questions to the

lecturer, we try together to answer the questionsin the most adequate way.

2


3/129

Topics to discuss

1. Economic assumptions for a concept of Present Value of cash flows

leading to a discounting formula.

2. Discounting function. Time value of money derived from risk free

(coupon) bonds quotes. Stripping of bonds. Par yield invariance

3. No arbitrage principle. Approximate evaluation of Present Valueand accuracy. Forward discounting.

4. Interest rates conventions. Continuous versus periodic

compounding.

5. Term structure of interest rates. Zero coupon yield curve. Typicalshapes. Forward curves.

6. Sensitivities: of the value to the passage of time, of the value to

the perturbation of zero coupon yield curve.

3


4/129

Topics to discuss

7. Duration and convexity, invariance propriety for continuouscompounding. Modifying slope of the yield curve.

8. Dependence of duration on coupon size and on time to maturity.

9. Immunization: is it a reasonable approach?

10. Internal rate of return. Application to bonds: yield to maturity.Yield curve, time to maturity vs. duration. Coupon effect.

11. Working with real data: Filtering, missing data reconstitution.Choosing initial data: bonds, swaps etc.

12. Parametric vs. non parametric approach for zero yield curveconstruction. Regularization: smoothness vs. accuracy.

13. Standard of the European Bond Commission for determining riskfree zero coupon yield curve for the Euro zone.

Note : topics 11, 12 and 13 are not eligible for exam questions

4


5/129

1. Economic assumptions

for a concept of

Present Value of cash flowsleading to a discounting formula.

5


6/129

Relevant economic concepts

Present Value is

the current worth of a future sum of money or

stream of cash flows

Time Value of Money and Time Preference

(money today is better then tomorrow)

6


7/129

Importance of PV

The present value method is widely used fordifferent practical needs of financial analysis,applicable when it the case ofgenerating future

cash flows e.g. to evaluate a capital investment project

as one of the business valuation approaches

to evaluate a financial contract or a financial

instrument, to evaluate a portfolio of financial instruments

in derivatives pricing

7


8/129

The idea for PV interpretation

What measures PV?

Related to the notion ofopportunity cost

(the cost of an alternative that must beforgone in order to pursue a certain action.

Put another way, the benefits you could have

received by taking an alternative action).

Benchmarking to risk-free financial

instruments

8


9/129

Risk free instruments

Risk-free interest rate is the theoretical rate ofreturn of an investment with zero risk, includingdefault risk.

The risk-free rate represents the interest that aninvestor would expect from an absolutely risk-free investment over a given period of time

Therefore, a rational investor will reject all theinvestments yielding sub-risk-free returns.

though a truly risk-free asset exists only in theory,in practice most professionals and academics usegovernment bonds of the currency in question

9


10/129

Central concept to develop

The key concept in the interest rate behavior isreally the term structure of interest rates. Theinterest rate on a loan will normally depend on

the maturity of the loan, and on the bondmarkets there will usually be differences betweenthe yields on short-term bonds and long-termbonds.

Loosely, the term structure of interest rates isdefined as the dependence between interestrates (or yields) and maturities

10


11/129

Economic assumptions

1. Adapting a static (snapshot) approach.

2. Simplifying the world: deterministic (risk-free)

3. Benchmarking to an ideal bond marketof zero-coupon bonds (the present value ofsuch zero-coupon bonds that is traded on this

bond market, with a spot price for eachmaturity date that is determined by the offerand demand from the investors):

11


12/129

Ideal bond market

Ideal market of zero-coupon bonds means:a) all maturities are present

b) all face (nominal) values present (i.e. this bonds

can be regarded as infinitely divisible securities)c) Market is ofperfect liquidity (ability of an asset

to be transformed, almost immediately, into cashor any other asset without any loss in value).

d) No trading restrictions applied (no marginrequirements). Any short positions are freelypossible.

12


13/129

More assumptions

4. Additivity of the portfolio value

5. The most vicious assumption:

given ideal market prices at the present moment,the only significant information for the fair value of acontract (instrument, portfolio) concerns thegenerated cash flows.

6. The law of one price ( an economic law stated as: "Inan efficient market all identical goods must have onlyone price.")

13


14/129

Market value vs. market price

International valuation Standards defines market (fair)value as "the estimated amount for which a propertyshould exchange on the date of valuation between awilling buyer and a willing seller in an arms-length

transaction after proper marketing wherein the partieshad each acted knowledgeably, prudently, and withoutcompulsion.

Market value is a concept distinct from market price,

which is the price at which one can transact, whilemarket value is the true underlying value accordingto theoretical standards.

14


15/129

When it is the same thing

The concept is most commonly invoked ininefficient markets or disequilibrium situationswhere prevailing market prices are not

reflective of true underlying market value. Formarket price to equal market value, themarket must be informationally efficient andrational expectations must prevail

But what is the market price?A marketconvention.

15


16/129

Liquidation value

Important distinction: Trading vs. investmentportfolio

Fair value for trading portfolio is liquidation

value. liquidation value depends on the strategy of

liquidation

For highly liquid market liquidation value canbe regarded as a result of the closing positionsat market prices

16


17/129

Financial engineering idea for PV

replication of cash flow by a portfolio of long

and short positions in ideal bond market

Not irreproachable: consider the future

development for such a portfolio, i.e. long and

short positions have different implications for

the future market operations (to meet

obligation for closing positions)

17


18/129

Replicating portfolio

Replicating portfolio

of zero-coupon

bonds in

Ideal market

18


19/129

2. Discounting function.Time value of money derived from

risk free (coupon) bonds quotes.

Stripping of bonds. Par yield

invariance.

19


20/129

Numraire

Numraire is a basic standard by which values

are measured, such as gold in a monetary

system. Acting as the numraire is one of the

functions of money: to measure the worth ofdifferent goods and services relative to one

another. "Numraire goods" are goods with a

fixed price of 1 used to facilitate calculations,when only the relative prices are relevant

20


21/129


22/129

Zero-coupon bond with unit face

value in the ideal market

22


23/129

Standard formula for PV:

sum of discounted cash flows

It is a result of all 6 assumptions

23

( )n n

n

P F d s


24/129

Notes:

We use risk-free discounting factors (in

deterministic world)

Interest rates (or yields) were not used to

derive the PV, only discount factors (or

discount function) needed

No assumptions about discount function

behavior were imposed in order to derive PV

formula, i.e. sum of discounted cash flows

24


25/129

Net present value (NPV)

The net present value (NPV) or cash flows, both incoming and

outgoing, is defined as the sum of the present values (PVs) of

the all individual cash flows, not only future cash flows, but

also including cash flow at the present time.

NPV is a central tool in discounted cash flow (DCF) analysis,

and is a standard method for using the time value of money to

appraise long-term projects. Used for capital budgeting, and

widely throughout economics, finance, and accounting, it

measures the excess or shortfall of cash flows, in presentvalue terms, once financing charges are met.

25


26/129

Decision making based on NPV

A project with negative NPV should be

rejected

Appropriately risked projects with a positive

NPV could be accepted, but other proprieties

of the project (for example the payback

period) should be taken into account.

26


27/129

Example

Consider a market where two bullet bonds are

traded both expiring in two years, and with

periodicity of coupon payment of one year.

Coupons are 10% and 20%, market prices arerespectively 90% and 110% (of nominal value)

Problem: determine implied discount factors,

corresponding to available market data.

27


28/129

Cash flows generated by

coupon bonds in this example

1 year

1 year

2 years

2 years

28


29/129

Linear algebra problem

0.1d(1) + 1.1d(2) = 0.9

0.2d(1) + 1.2d(2) = 1.1

d(1) = 1.3 , d(2) = 0.7

The market data (market prices of coupon bonds)

was specially chosen is in this example to beinconsistent with the desired properties of the

discount function, see next slide.

29


30/129

Law of Money Preference formalized

by discount function dproprieties

(axioms)1) Time value law:

dis monotonically decreasing

2) Normalization:

d(0)=1

3)Positivity:

d(s) > 0

4) Vanishing

when( ) 0d s s

30


31/129

Stripping

Investment banks or dealers may separate couponsfrom the principal of coupon bonds, which is known asthe residue, so that different investors may receive theprincipal and each of the coupon payments. This

creates a supply of new zero coupon bonds. The coupons and residue are sold separately toinvestors. Each of these investments then pays a singlelump sum. This method of creating zero coupon bondsis known as stripping and the contracts are known as

strip bonds. "STRIPS" stands for Separate Trading ofRegistered Interest and Principal Securities. USTreasury Department introduced STRIPS program in1985 .

31


32/129

Par yield

Par yield (or par rate) denotes in finance, the coupon

rate Cfor which the price of a plain vanilla coupon

bond is equal to its nominal value (or par value). It is

used in the design of fixed interest securities and inconstructing interest-rate swaps. Get C from

equation

Note the invariance with respect to definition of

interest rates (yields), as well as all theory before:

only discount function is needed

1

1

( ) ( 1) ( ) 1n

k nk

Cd t C d t

32


33/129

3. No arbitrage principle.

Approximate evaluation of

Present Value and accuracy.Forward discounting.

33


34/129

Arbitrage

Arbitrage is a polysemantic word, of different use. Practitioners understand it as an advantage of a

price difference between two or more markets:striking a combination of matching deals that

capitalize upon the imbalance, the profit beingthe difference between the market prices.

In financial engineering arbitrage is a transactionthat involves no negative cash flow at any

temporal state and a positive cash flow in at leastone state; in simple terms, it is the possibility of arisk-free profit at zero cost.

34


35/129

Definition of arbitrage

If there is no trading restrictions,

arbitrage is an opportunity, given marketquotes, to take profit by immediate (the

fastest) transactions

without risk without initial capital

35


36/129

No arbitrage principle

If the market prices do not allow for profitablearbitrage, the prices are said to constitute

an arbitrage equilibrium or arbitrage-free market.

An arbitrage equilibrium is a precondition for

a general economic equilibrium.

The assumption that there is no arbitrage is

fundamental for financial engineering purposes. In

particular, it is used to calculate a risk neutral pricefor derivatives.

36


37/129

Offset portfolio

Offest portfolio of zero-

coupon bonds from

ideal market

37


38/129

Application of no arbitrage principle

to find bounds

The aggregate financial result (determined by

the earning or the loss which results from

financial affairs) of

initial cash flow

and offset portfolio(formed at no cost)

cannot be positive to avoid arbitrage.

38


39/129

Example 2

Consider a non-ideal market where two bonds

are traded:

First bond is expiring in two years, with

coupon of 20% paid annually; bid quote is

80%, ask quote is 90%.

Second bond is zero-coupon with maturity 1

year; bid quote is 110%, ask quote is 120%.

39


40/129

Deriving bounds: step 1

Consider the cash flows resulting from the

sale (according to bid quote) of the first bond

110%1 year 2 years

-20%

-120%

40


41/129


Now form the offset portfolio of two zero-coupon bonds in ideal market:

1) Long position in zero-coupon bond of maturity 1

with nominal 0.2 units; to open this position wepay at the present moment 0.2 d(1) units

2) Long position in zero-coupon bond of maturity 2with nominal 1.2 units; to open this position we

pay at the present moment 1.2 d(2) units So to form the offset portfolio we pay at the

present moment 0.2 d(1)+1.2 d(2) units

41


42/129


The aggregate financial resulting from selling

coupon bond and forming offset portfolio

should be not positive to avoid arbitrage:

1.1 0.2d 1 1.2 d 2 0 Or, equivalently

1.1 0.2d 1 1.2 d 2 42


43/129

Arbitrage bounds

Similar reasoning give us 3 more inequalities,

so that we have get 4 inequalities, obtained

by application of no arbitrage principle:

1.1 0.2d 1 1.2 d 2 1.2

0.8 1d 1 0d 2 0.9

43


44/129

No arbitrage set of possible values

for both d(1) and d(2)

The 4 inequalities define a parallelogram :

d(1)

0

d(2)

A

B

C

D

44


45/129

Time preference bounds

Additionally we get 3 strict inequalities:

1 (0) (1) (2) 0d d d

45


46/129

Non-active bounds

In general 7 inequalities define a polyhedron

with maximum 7 vertex

In our case two inequalities are fulfilled for

the set constructed using no arbitrage

bounds:

(2) 0d

1 (1)d

46


47/129

Exercise

Problem: get a polyhedron set of possible

values for both d(1) and d(2) resulting from

both no arbitrage principle and time

preference law

It is sufficient to find 4 vertices of the

parallelogram and check position of each,

whether it belongs to the half-plane of pointssatisfying (1) (2)d d

47


48/129

Vertices of parallelogram

Four points of values (d(1),d(2)):

A= (0.8, 0.783)

B= (0.8, 0.867)

C =(0.9, 0.850)

D= (0.9, 0.767)

All vertices except B satisfy d(1) d(2)

48

Set of possible values for d(1) and d(2)


49/129

Set of possible values for d(1) and d(2)

derived from No Arbitrage Principle

and Time Preference Law

In our case we get a pentagon

d(1)

0

d(2)

A

B

C

D

E

D

49


50/129

Example 3

Let us consider a contract generating the

following (deterministic) cash flows:

First year pay 2 million EUR

Second year receive 5 million EUR

Suppose that we would like to evaluate the

range of the possible Present Values of the

contract given market information, presentedin example 2

50


51/129

Linear programming

Our problem can be stated as follows:

Find minimum an maximum value of

-2d(1)+5d(2) (millions of euro)

for (d(1),d(2))

It is a linear programming problem.

Linear programming is a simplest type of the

conditional optimization problem: of a linear objective function,

subject to linear inequality constraints.

51


52/129

Some history

The founder of the subject is Leonid Kantorovich, a

Russian (Soviet) mathematician, the winner of the

Nobel Prize in Economics in 1975, who developed

linear programming in 1939 , in his book TheMathematical Method of Production Planning andOrganization.

It was reinvented and much advanced by George

Dantzig, who published the simplex method in 1947,and John von Neumann, who developed the theory

of the duality in the same year.

52


53/129

Where is attained extremum

It can be shown that the minimum and

maximum of linear function on the bounded

closed convex set S is attained in extreme

points. That is a point in S which does not lieinside in any open line segment joining two

points ofS. Intuitively, an extreme point is a

"corner" ofS. In our case (of convex polygon) extreme

points are vertices.

53


54/129

Vertices of the pentagon

Points

(d(1),d(2))

Objective function

-2d(1)+5d(2)

Extremum

A= (0.8, 0.783) 2.315

C =(0.9, 0.850) 2.45

D= (0.9, 0.767) 2.035 min

E=(0.8, 0.8) 2.4

F=(0.857, 0.857) 2.571 max

54


55/129

Range for PV

2.035 PV< 2.571

PV= 2.303 11.6%

We get therefore not only theapproximate PV but also its accuracy

55


56/129

Forward discounting

Consider a forward contract with, a market maker(an intermdiaire), regarded as an absolutely safe

counterpart in deterministic world, for buying in thefuture, at time t, a zero coupon bond of time to

maturity sand unit nominal.

The price agreed upon, called the delivery price, isequal to the forward price fb(t,s) at the time tthecontract is entered into.

fb(t,s)t+s

56


57/129

No arbitrage reasoning

The value of the offset portfolio is:

-fb(t,s)d(t)+d(t+s)

The value of the forward contract is zero,

we get so

-fb(t,s)d(t)+d(t+s)0

or( , ) ( )

( )bf t s d t s

d t

57


58/129

No arbitrage reasoning

The similar reasoning is applicable for selling

bond in the future so that we get

fs(t,s)d(t)-d(t+s)O

or

( , ) ( )( )s

f t s d t sd t

58


59/129

Forward discounting factor

Therefore

If we can neglect the difference between buyand sell prices, e.g. for deal without we getforward discounting factor as a fair forward

pricef(t,s):

( )( , ) ( , )

( )

s b

d t s f t s f t s

d t

( )( , )

( )

d t s f t s

d t

59


60/129

4. Interest rates conventions.

Continuous versus periodic

compounding.

60


61/129

Rate of return

Rate of return R, is the ratio of moneyV

gained on an investment relative to the

amount of money invested Vfor a certain

(fixed) period of time.

The amount of money gained may be referred

to as interest, also may be referred to as the

yield(1 )V V V R

61


62/129

Annualization

Usually one year is considered as universal

time period unit.

An annualized rate of return is the hypothetic

one-year return on an investment over aperiod other than one year (such as a month,

or two years).

The problem is that there is a lot of ways ofdoing it.

62


63/129

Compounding

Simple interest is calculated on the principle

amount by the linear growth of return with

respect to the time.

For compound interest unpaid interest isadded to the balance due, assuming that it is

possible to reinvest under the same

conditions; it leads to exponential growth. Both assumptions are artificial, but compound

interest has more clear interpretation

63


64/129

How many days in the year?

Day count convention determines how interestaccrues over time for a variety of investments,

including bonds, notes, loans, mortgages, medium-

term notes, swaps, and forward rate agreements

(FRAs).

This determines the amount transferred on interest

payment dates, and also the calculation of accrued

interest for dates between payments. The day count is also used to quantify periods of

time when discounting a cash-flow to its present

value.64


65/129

Some conventions

30/360

Actual/Actual

There are some nuances according the source

of convention, for example ICMA or ISDA

There is no central authority defining day

count conventions, so there is no standard

terminology. Certain terms, such as "30/360",

"Actual/Actual must be understood in the

context of the particular market.65


66/129

Periodic compounding

Given a time period of cash flows

where k is a number of periods

66


67/129

Divide interval (0,s) into n small periods,i.e. subintervals of length

Limit of periodic compounding

( )( ) (1 ( ) )

n s ssd s s en

n

assume reinvestment rate per period is

always the same:

We get than

67


68/129

Continuous compounding

The most convenient convention

Note that can be also called spot rateLet us use this convention in what follows

68

( )s

( )

( )s s

d s e


69/129

5.Term structure of interest rates.

Zero-coupon yield curve.Typical shapes.

Forward curves.

69


70/129

Term structure of interest rates

relationship between (spot) rates of zero-

coupon securities and their term to maturity.

Can be described by: Discount function (given convention relating

to the interest rates);

Zero-coupon yield curve

Instantaneous forward curve

70


71/129

Shapes of yield curves

Shape is invariant with respect to the conventionrelating discount function and interest rates

71


72/129

Shapes

Curve 1 corresponds to the flat term structure

of interest rates (flat TSIR)

Curve 2 corresponds to the increasing TSIR

Curve 3 corresponds to the decreasing TSIR

Curve 4 corresponds to the humped TSIR

72


73/129

Forward rate

Definition of forward rate at time t in the

future and time to maturity s is the following:

Reflects market opinion about interest rates in

the future

( , )( , ) sR t s f t s e

73

Relation between forward and


74/129

Relation between forward and

spot rates

But as

One can get

( ) ( ) ( )( )

( , ) ( )

t s t s t t d t s f t s ed t

( , ) ( ) ( ) ( )sR t s t s t s t t

74


75/129

Instantaneous forward rates

Finally

Ifs tend to zero, we obtain instantaneous

forward rates

( ) ( ) ( )( , )

t s t s t t R t s

s

( ) ( , 0) ( ( )) ( ) ( )R t R t t t t t t

( )R t

75


76/129

Exercise 1 (not obligatory)

Verify that

In particular

and

1( , ) ( )

t s

t R t s R u du

s

( )( , )

t s R u dutf t s e

0

1( ) (0, ) ( )

s

s R s R u dus

76


77/129

6. Sensitivities:

to the passage of time,

to the perturbation of

zero-coupon yield curve.

77


78/129

Sensitivity

Sensitivity analysis is the study of how a smallvariation (uncertainty) in the output of a

mathematical model can be apportioned,

qualitatively or quantitatively, to different sources of

small variation in the input of the model.

Put another way, it is a technique for slightly

changing parameters in a model to determine the

effects of such changes Differential calculus is applicable.

78

l


79/129

Example: NPV sensitivity

Given the uncertainty inherent in project forecasting andvaluation, analysts will wish to assess the sensitivityof project

NPV to the various inputs (i.e. assumptions) to the DCF model.

In a typical sensitivity analysis the analyst will vary one key

factor while holding all other inputs constant, ceteris paribus. The sensitivity of first order of NPV to a change in that factor

is then observed, and is calculated as a "slope":

NPV / x, where x is a factor (partial derivative)

The sensitivity of the second order is the sensitivity ofsensitivity (second partial derivative)

79

l f l


80/129

Taylor formula

First order (linear) approximation

Second order (quadratic) approximation

80

Changes in pricing after a small


81/129

Changes in pricing after a small

period of timet

Difference between perturbed PV and initial PV:

( ) ( ) ( ) ( )

( )

( )

( ( ) ( ) ( ) ( )

i i i i

i

i i

s t s t s s s s

s s

s s

i i i i

de e e t ds

e s s s t d s R s t

Using first order Taylor approximation we get:

81


82/129

W i h


83/129

Weights qi

If all future cash flows are of the same sign,say they are all positive,

than , so that can be

regarded as weights

The weight reflect the contribution of the

discounted cash flow number i to the total

sum of discounted cash flows

iq iq

iq

83

Ti i i i


84/129

Time sensitivity (relative change in PV perlength of (small) time period) can be

interpreted in case when

as a weighted sum of instantaneous forwardrates at maturities of cash flows:

Time sensitivity

iq

( )i ii

PP q R s

t

84

E l f fl t TSIR


85/129

Example for flat TSIR

For the special case offlat term structure ofinterest rates

implies , so that( )s ( )R s

PP

t

85

Perturbation of


86/129

zero-coupon yield curve

86

Changes in pricing due


87/129

a small perturbation of

zero-coupon yield curveDifference between perturbed PV and initial PV:

Using first order Taylor approximation we get:

87

Sensitivity to a perturbation


88/129

y p

of the curve

( )i i i

i

PP q s s

h

88


89/129

7.Duration and convexity,

Modifying slope of the yield curve.

Invariance propriety for continuous

compounding.

89

Parallel shifts of zero-coupon yield


90/129

p y

curve and duration

( ) 1,s h

P P D

i i

i

D q s , where is duration

90

Ch i l f th


91/129

Changing slope of the curve

( )s s s

s

91

Sensitivity to changing slope


92/129

Sensitivity to changing slope

where is convexity (coefficient)

( )P

P C s Dh

2

i i

i

C q s

92

Second order sensitivity


93/129

Second order sensitivity

Using second order Taylor approximation weget:

( ) 2 212

s s se e se s

2 2 212

( ) ( )i i i i i i

i i

Ph q s s h q s s

P

93

Second order sensitivity for


94/129

y

parallel shifts (convexity)

( ) 1,s h

For parallel shifts

We get

21

2 ( )P D CP

94

Invariance


95/129

with respect to convention

Note the invariance of weights, duration and

convexity with respect to convention for

definition of interest rates (yields): onlydiscount function is needed, not interest

rates.

95

Exercise 2: conditions


96/129

Exercise 2: conditions

Consider continuous compoundingconvention and TSIR given by zero-coupon

yield curve

for years

and a serial bond of maturity 2 years paying

interest rate of 10% on the outstanding debtannually.

0 2s ( ) 0.005( 1)(7 )s s s

96

Exercise 2: to find


97/129

Exercise 2: to find

1. Discount function

2. Shape of the yield curve

3. Price of the bond and weights

4. Duration and convexity of the bond

5. First order and second order approximation

of the relative price change of the bond for

the upside parallel shift of zero-coupon yield

curve of 50 bp

97

Exercise 2: to find


98/129

Exercise 2: to find

6. First order approximation of the relative pricechange for the increasing slope of zero-

coupon yield curve so that (0) decrease is of

50 bp and (2) increase is of 50 bp7. Instantaneous forward curve

8. Relative price change relative price change

after one day

98

Serial bonds


99/129

Serial bonds

99

Exercise 3


100/129

Exercise 3

The conditions are the same as in Exercise 2

What is expected (implied) discount function

in one year in the future and the

corresponding zero-coupon yield, with term tomaturity up to one year?

Hint : consider forward discounting and rates

100

Parallel shifts of zero-coupon yield


101/129

curve for periodic compounding

(1 ( ) ) (1 ( ))i is s

i i iiP F r s h r s

101

General and flat TSIR.


102/129

Modified duration

P

( )r s r

In general

For fat TSIR

is called modified duration102

Invariance

f h i i i f l


103/129

of the sensitivity formulas

For continuous compounding sensitivity

formulas for a perturbation of the zero-

coupon yield curve are invariant to the shapeof the yield curve and the relative PV change

can be calculated using only discount function

It is not the case for periodic discounting

103


104/129

8. Dependence of duration

on coupon size andon time to maturity.

104

Plain vanilla


105/129

Plain vanilla

We consider in this section exclusively fixedcoupon bonds, also referred to as straight or

plain vanilla coupon bonds, are bonds in

which the rate of interest remains fixed fromthe time of issuance till the date of maturity

105

Dependence of duration


106/129

on time to maturity

In normal (typical) bond market conditionsthe longer is maturity the greater is duration.

In general it is not the case. Let us consider, to

fix ideas, periodic compounding conventionand the flat TSIR. The duration can be shorten

with an increase in maturity in case of very

high yield (such a situation can be aconsequence of hyperinflation).

The reason is a very heavy discounting106

Example


107/129

Example

Consider two coupon bonds paying annuallythe same coupon of 50%, with maturities 2

and 3 years.

Suppose that rate of return for (each) oneyear period is as high as 400%.

107

Exercise 4


108/129

Exercise 4

Calculate durations of the bonds and showthat :

the bond with maturity 2 has duration

the bond with maturity 3 has duration

108

381

1

31

Dependence of duration on


109/129

coupon size

Discounting function is a homographicfunction, i.e. it is of the form

all four coefficients are determined by the

discount function

109

1 1

2 2( )

a c b

D c a c b

Exercise 5


110/129

Exercise 5

Show that for the coupon bond with period ofpayments and n coupon periods

110

1

1

( ) ( )

( )

( ) ( )

n

k

n

k

c kd k nd n

D c

c d k d n

Graphic is a part of hyperbola


111/129

(with asymptote)

111

0 c

D(c)


112/129

9. Immunization: is it a reasonable

approach?

112

Immunization


113/129

Immunization

(Interest rate) immunization is a strategy thatensures that a small change in the level interest rates

will not affect the value of a portfolio.

Similarly, immunization can be used to ensure that

the value of a pension fund's or a firm's assets will

increase or decrease in exactly the opposite amount

of their liabilities, thus leaving the value of the

pension fund's surplus or firm's equity unchanged,regardless of small changes in the level of interest

rates.

113

Immunizing equity


114/129

Immunizing equity

Adopting continuous compoundingconvention, consider parallel shifts impact

on a firm assets and liabilities, generating

future cash flows. Denote Assets byA Liabilities by L, Equity by E; E=A-L

For immunization of equity we need

114

( ) ( )A L

E A L D A D L o

A LD A D L

Immunizing a portfolio


115/129

of a single asset type

Immunization can be done in a portfolio of a singleasset type, such as government bonds, by creating

long and short positions along the yield curve. It is

reasonable to immunize a portfolio against the most

prevalent risk factors.

A principal component analysis of changes along the

U.S. Government Treasury yield curve reveals that

more than 90% of the yield curve shifts are parallelshifts, followed by a smaller percentage of slope

shifts and a very small percentage of curvature shifts.

115

Constructing


116/129

Constructing

Using that knowledge, an immunized portfoliocan be created by creating long positions with

durations at the long and short end of the

curve, and a matching short position with aduration in the middle of the curve.

These positions protect against parallel shifts

and even it is possible adjust positions toprotect against slope changes, in exchange for

exposure to curvature changes.

116

Difficulties


117/129

Difficulties

Users of this technique include banks,insurance companies, pension funds and bond

brokers; individual investors infrequently have

the resources to properly immunize theirportfolios.

The disadvantage associated with duration

matching is that it assumes the durations ofassets and liabilities remain unchanged, which

is rarely the case.

117


118/129

10. Internal rate of return.Application to bonds: yield to

maturity. Yield curve, time to

maturity vs. duration. Coupon

effect.

118

Internal rate of return(IRR)

f i h fl


119/129

for given cash flows

In hypothetical situation, if the TSIR was flat,NPV would be a function of the level of

interest rates, namely of for periodic

compounding and for continuouscompounding, .

IRR is defined as a the root of the equation

for periodic compounding, andfor continuous compounding

119

r

( ) ( ) NPV F r G

( ) 0F r

( ) 0G

Periodic compounding IRR


120/129

Periodic compounding IRR

If is rate of return for (any) one period,the equation is the following:

Introducing the discounting factor z for each

single period that is we get a

standard problem of roots of a polynomial

120

r

0

10

(1 )

n

k kk

Fr

0

0n

k

k

k

F z

1

1

z

r

satisfying 0 1z

But


121/129

But

There could be cases when:

1. Roots satisfying condition do not

exist

2. There are multiple roots

In these two cases the use of IRR consistent

with an economic interpretation is hardly

possible

121

0 1z

Exercise 6


122/129

Exercise 6

Consider cash flows respectively for maturities0,1 and 2:

Case1: -2, -1, 1

Case2: 1, -5, 6

What about IRR in this cases?

122

Sufficient condition


123/129

If future cash flows are positive and currentflow is negative, such that it absolute value is

less then the sum future cash flows values, i.e.

then there is a single root satisfying

because of monotone dependence

123

0

1

n

k

k

F F

0 1z

Yield to maturity (YTM)


124/129

y ( )

Yield to maturity (YTM) for a bond is IRR for itspaid price and promised cash flows.

There is one to one correspondence between

YTM and bond price. The sufficient condition from previous slide is

satisfied for equal to PV of future cash

flows, calculated for arbitrary TSIR, notnecessary for flat TSIR

124

0F

Exercice 7


125/129

Calculate YTM based on periodiccompounding for the bond from exercise 2.

Calculate using periodic compounding (spot)

zero-coupon yields for maturities 1 and 2years. Is the TSIR flat?

125

Exercise 8


126/129

Calculate par yield of for the bond fromexercise 2

Compare it with YTM, try to explain the result

126

Use of IRR


127/129

The internal rate of return (IRR) is used in tomeasure and compare the profitability of

investments.

In general, an investment whose IRR exceedsits cost of capital is regarded as adding value

for the company (i.e., it is economically

profitable).

127

NPV vs.IRR


128/129

Despite academic preference for NPV, surveysindicate that executives prefer IRR over NPV .

Apparently, managers find it easier to compare

investments of different sizes in terms of percentage

rates of return than by dollars of NPV. However, NPVremains the "more accurate" reflection of value to

the business. IRR, as a measure of investment

efficiency may give better insights in capital

constrained situations. However, when comparing

mutually exclusive projects, NPV is the appropriate

measure.128

Yield curve


129/129

The (approximate) dependence of YTM on maturityM is called yield curve

This curve for coupon bonds will be typically shifted

to the right with respect to zero-coupon yield curve,

referred as coupon effect

More economically meaningful is modified yieldcurve which represents the (approximate)

dependence of YTM on duration , much closer tozero-coupon yield curve, so that it can be used as a

Financial Engineering - Five Days of Lectures October 2010 -Rimini

Documents

Transcript of Financial Engineering - Five Days of Lectures October 2010 -Rimini