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8/8/2019 Financial Engineering - Five Days of Lectures October 2010 -Rimini
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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
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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.
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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.
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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
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1. Economic assumptions
for a concept of
Present Value of cash flowsleading to a discounting formula.
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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)
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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
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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
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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
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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
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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):
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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.
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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.")
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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.
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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.
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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
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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)
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Replicating portfolio
Replicating portfolio
of zero-coupon
bonds in
Ideal market
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2. Discounting function.Time value of money derived from
risk free (coupon) bonds quotes.
Stripping of bonds. Par yield
invariance.
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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
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Zero-coupon bond with unit face
value in the ideal market
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Standard formula for PV:
sum of discounted cash flows
It is a result of all 6 assumptions
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( )n n
n
P F d s
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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
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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.
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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.
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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.
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Cash flows generated by
coupon bonds in this example
1 year
1 year
2 years
2 years
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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.
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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
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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 .
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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
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3. No arbitrage principle.
Approximate evaluation of
Present Value and accuracy.Forward discounting.
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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.
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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
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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.
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Offset portfolio
Offest portfolio of zero-
coupon bonds from
ideal market
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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.
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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%.
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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%
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Deriving bounds: step 2
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
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Deriving bounds: step 3
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
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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
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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
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Time preference bounds
Additionally we get 3 strict inequalities:
1 (0) (1) (2) 0d d d
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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
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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
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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)
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Set of possible values for d(1) and d(2)
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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
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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
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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.
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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.
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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.
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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
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Range for PV
2.035 PV< 2.571
PV= 2.303 11.6%
We get therefore not only theapproximate PV but also its accuracy
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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
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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
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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
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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
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4. Interest rates conventions.
Continuous versus periodic
compounding.
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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
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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.
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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
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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
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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
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Periodic compounding
Given a time period of cash flows
where k is a number of periods
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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
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Continuous compounding
The most convenient convention
Note that can be also called spot rateLet us use this convention in what follows
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( )s
( )
( )s s
d s e
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5.Term structure of interest rates.
Zero-coupon yield curve.Typical shapes.
Forward curves.
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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
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Shapes of yield curves
Shape is invariant with respect to the conventionrelating discount function and interest rates
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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
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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
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Relation between forward and
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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
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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
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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
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6. Sensitivities:
to the passage of time,
to the perturbation of
zero-coupon yield curve.
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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
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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)
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l f l
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Taylor formula
First order (linear) approximation
Second order (quadratic) approximation
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Changes in pricing after a small
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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:
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W i h
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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
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Ti i i i
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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
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E l f fl t TSIR
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Example for flat TSIR
For the special case offlat term structure ofinterest rates
implies , so that( )s ( )R s
PP
t
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Perturbation of
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zero-coupon yield curve
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Changes in pricing due
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a small perturbation of
zero-coupon yield curveDifference between perturbed PV and initial PV:
Using first order Taylor approximation we get:
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Sensitivity to a perturbation
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y p
of the curve
( )i i i
i
PP q s s
h
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7.Duration and convexity,
Modifying slope of the yield curve.
Invariance propriety for continuous
compounding.
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Parallel shifts of zero-coupon yield
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p y
curve and duration
( ) 1,s h
P P D
i i
i
D q s , where is duration
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Ch i l f th
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Changing slope of the curve
( )s s s
s
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Sensitivity to changing slope
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Sensitivity to changing slope
where is convexity (coefficient)
( )P
P C s Dh
2
i i
i
C q s
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Second order sensitivity
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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
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Second order sensitivity for
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y
parallel shifts (convexity)
( ) 1,s h
For parallel shifts
We get
21
2 ( )P D CP
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Invariance
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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
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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
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Exercise 2: to find
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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
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Exercise 2: to find
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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
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Serial bonds
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Serial bonds
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Exercise 3
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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
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Parallel shifts of zero-coupon yield
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curve for periodic compounding
(1 ( ) ) (1 ( ))i is s
i i iiP F r s h r s
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General and flat TSIR.
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Modified duration
P
( )r s r
In general
For fat TSIR
is called modified duration102
Invariance
f h i i i f l
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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
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8. Dependence of duration
on coupon size andon time to maturity.
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Plain vanilla
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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
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Dependence of duration
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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
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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%.
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Exercise 4
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Exercise 4
Calculate durations of the bonds and showthat :
the bond with maturity 2 has duration
the bond with maturity 3 has duration
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381
1
31
Dependence of duration on
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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
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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
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(with asymptote)
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0 c
D(c)
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9. Immunization: is it a reasonable
approach?
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Immunization
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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.
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Immunizing equity
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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
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( ) ( )A L
E A L D A D L o
A LD A D L
Immunizing a portfolio
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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.
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Constructing
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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
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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.
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10. Internal rate of return.Application to bonds: yield to
maturity. Yield curve, time to
maturity vs. duration. Coupon
effect.
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Internal rate of return(IRR)
f i h fl
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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
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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
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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
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0 1z
Exercise 6
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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?
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Sufficient condition
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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)
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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
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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
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Calculate par yield of for the bond fromexercise 2
Compare it with YTM, try to explain the result
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Use of IRR
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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).
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NPV vs.IRR
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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
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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