Slide Analisa Pasar Modal 210910

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An Introduction to Capital Marketsand Investment

Dr. Ir. Budhi Arta Surya

September 21, 2010

The materials of this course are based on the references listed at the end of this slide.


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An Introduction to Capital Markets and Investment

Definitions and Conventions

Before we proceed with the content materials of this course, let us first make some

definitions and conventions that will be used throughout this course.

Denote by Pt the price of an asset at date t. Assume that this asset pays no dividends.The simple net return, Rt, on the asset between dates t 1 and t is defined as

Rt =Pt

Pt1 1.

The simple gross return on the asset is 1 + Rt.

The assets multiyear gross return 1 + Rt(n) over recent n periods is defined by

1 + Rt(n) :=Pt

Ptn=

Pt

Pt1 Pt1

Pt2 Pt2

Pt3 Ptn+1

Ptn= (1 + Rt) (1 + Rt1) (1 + Rtn+1)

=n1

j=0(1 + Rtj).

An Introduction to Capital Markets and Investment, SBM-ITB, August 2010 2


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Definitions and Conventions: Continued

Multiyear returns are normally annualized to make investments with different horizons

comparable. The annualized multiyear return is defined by

Ra

t(n) =

n1

j=0(1 + Rtj)1/n

1.

If single-period returns Rts are relatively small in magnitude, first-order Taylor

approximation simplifies the annualized multiyear return as

Rat (n) 1nn1j=0

Rtj.

Continuous Compounding The continuously compounded return or log return rt of

an asset is defined to be the natural algorithm of its gross return (1 + Rt):

rt := log(1 + Rt) = logPt

Pt1= pt pt1, where pt := log Pt.

To distinguish Rt from rt, we shall refer to Rt as a simple return.



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Continuous Compounding: Continued

The continuously compounded returns become advantageous when considering

multiperiod returns, since

rt(n) = log(1 + Rt(n)) = logn1

j=0(1 + Rtj) =n1

j=0 log(1 + Rtj) =n1

j=0 rtj.Drawbacks of Continuously Compounded Returns

Consider a portfolio of N assets whose value is given by

Vt =N

j=1

#j(t)Pj

t Rpt =N

j=1

j(t)Rjt ,

where #j(t) is the number of assets j held at time t and j(t) =#j (t)P

jt

Vt. Hence,

the simple return Rpt of the portfolio is a weighted average of the returns of individualassets. However, continuously compounded returns do not have this property, i.e.,

rpt =N

j=1j(t)r

jt .



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Distributional Properties of Asset Returns

Normal Distributional PropertiesOne of the most common models for asset returns is the independent and identically

distributed (IID) normal model, in which returns are assumed to be independent over

time, identically distributed over time, and normally distributed. The original

formulation of the CAPM employed this assumption of normality.

Remarks The normality assumption suffers from at least two important drawbacks:

The smallest net return achievable is 1 whereas the normal distributions supportis the entire real line. Hence, the lower bound of1 is clearly violated.

If single-period returns are assumed to be normal, then multiperiod returns cannotalso be normal since they are the products of the single-period returns.

Log-Normal Distributional PropertiesAs alternative, log-returns rit = log

1 + Rit

are assumed to be normal, i.e.,

rit N

(i, 2

i).



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Log-Normal Distributional Properties: Continued

Under the lognormal assumption, if the mean and variance of rit are i and 2i ,

respectively, then the mean and variance of simple returns are given by

ERit = ei+

2i /2

1

Var

Rit

= e2i+2i

e2i 1. (1)

The lognormal model has a long history dating back to the dissertation of Bachelier

(1900) and the work of Einstein (1905), containing the math of Brownian motion and

heat conduction. The model has underpinned the financial asset pricing theory.

Non-Normal Distributional Properties of Log-Returns

The stylized fact of log-returns exhibit a deviation from the normality assumption. At

short horizons, historical returns show weak evidence of skewness and strong evidenceof excess kurtosis. The normal distribution has skewness 1 equal to zero, as do all

other symmetric distributions, and kurtosisequal to 3, but fat-tailed distributions with

extra probability mass in the tail areas have higher or even kurtosis.1Normalized third and fourth moment of a r.v. X is defined by E

(X)3

3 and E(X)4

4 , respectively.



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Distributional Properties of Asset Returns: Continued

Histogram of data

data

Density

0.15 0.05 0.05

0

5

10

15

20

25

30

Asymm NIGGaussian

0.15 0.05 0.05

3

2

1

0

1

2

3

ghyp.data

log(Density)

0.15 0.05 0.05 0.15

0.1

5

0.1

0

0.0

5

0.0

0

0.0

5

0.1

0

Generalized Hyperbolic QQ Plot

Theoretical Quantiles

SampleQuantiles

Asymm NIGGaussian

Figure 1: Histogram and density fits of Normal and NIG distributions to log-return of

CS data. Observe that the assumption on normality of the log-return is violated.



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Efficient Capital Markets



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Efficient Market Hypothesis

The origins of the Efficient Market Hypothesis2 can be traced back to

the pioneering theoretical contribution of Bachelier (1900) and the empiricalresearch of Cowles (1933).

In modern theory of economics, the work of Samuelson (1965) whose contributioncan be neatly summarized by the title of his article:

Proof that Properly Anticipated Prices Fluctuate Randomly.

Referring to Samuelson, in an informationally efficient market price changes must

be unforecastable if they are properly anticipated, i.e., if they fully incorporate the

expectations and information of all market participants.

Fama (1970) who summarizes this idea in his survey:A market in which prices always fully reflect available information is called

efficient.

2We refer to Bernstein (1992) and Lo (1996) for further literature study on the contributions of Bachelier, Cowles,

Samuelson, and many other early authors.



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Efficient Market Hypothesis: Continued

More recently, Malkiel (1992) has offered the following more explicit definition:

Definition 1. A capital market is said to be efficient if it fully and correctly reflectsall relevant information in determining security prices. Formally, the market is said to

be efficient with respect to some information set...if security prices would beunaffected by revealing that information to all market participants. Moreover,

efficiency with respect to an information set...implies that it is impossible to make

economic profits by trading on the basis of that information set.

Remarks on Malkiel definition of EMH

The first sentence of Malkiel definition repeats Famas definition. The second sentence suggests that market efficiency can be tested by revealing

information to market participants and measuring the reaction of security prices.If prices do not move when information is revealed, then the market is efficient

with respect to that information.

Malkiels third sentence suggests an alternative way to judge the efficiency of amarket, by measuring the profits that can be made by trading on information.



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Malkiels idea on measuring the market efficiency by measuring the profits that can be

made by trading on information lays the foundation of almost all the empirical work on

market efficiency. It has been used in two main ways.

First, many researchers have tried to measure the profits earned by marketprofessionals such as mutual fund managers. If these managers achieve superior

returns (after adjustment for risk) then the market is not efficient with respect to

the information possessed by the fund managers.

This approach has the advantage that it concentrates on real trading by realmarket participants.

But is has the disadvantage that one cannot directly observe the informationused by the managers in their trading strategies3.

Alternatively, one can ask whether hypothetical trading based on an explicitlyspecified information set would earn superior returns. To implement this approach,

one must first choose an information set.

3see Fama (1970,1991) for a thorough review of this literature.



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According to Roberts (1967), the classic taxonomy of information sets can be divided

into three forms:

Weak-form Efficiency The information set includes only the history of prices orreturns themselves.

Semistrong-form Efficiency The information set includes all information knownto all market participants (publicly available information).

Strong-form Efficiency The information set includes all information known to anymarket participant.



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The Law of Iterated Expectations

As discounted present-value model of security prices is found to be entirely consistent

with randomness4 in the security returns, the key to understanding the randomness

feature/behavior of security returns is the so-called Law of Iterated Expectations.

Samuelson (1965) was the first to show the relevance of the Law of Iterated

Expectations for security market analysis5.

To state the law, let us define two information sets (sometime known as filtrations)

Ft and Gt that are available to market participants at time t, where Ft Gt so thatall the information contained in Ft is also contained in Gt, i.e., Gt contains extrainformation compared to Ft. (In this case Gt is also known as enlarged filtration.)

Let us now consider expectation of a random variable X conditional on these

information sets. The Law of Iterated Expectations says that

E

XFt = EEXGtFt (2)

4See among others, Samuelson (1965) and Black (1971).5See LeRoy (1989) for a literature review of the argument.



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The Law of Iterated Expectations: Continued

In other words, if one has limited information Ft, the best forecast one can make of arandom variable X is the forecast of the forecast one would make of X if one had

superior information

Gt. The expression (2) can rewritten as

E

X EXGtFt = 0, (3)which has an intuitive interpretation that one cannot use limited information Ft topredict the forecast error one would make if one had superior information

Gt.

Suppose that a security price at time t, Pt, can be written as the rational expectationof some fundamental value V, conditional on information Ft available at time t,

Pt = EV

Ft.The same reasoning applies to the security price Ps at time s > t, that is

Ps = E

VFs

.



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The Law of Iterated Expectations: Continued

As Ft Fs, for all t < s, the expectation of the price change Ps Pt over theinterval [t, s) is therefore given following the Law of Iterated Expectation (2) by

E

Ps PtFt

= E

Ps

Ft

Pt= EEVFsFt Pt= E

V

Ft Pt= Pt Pt= 0.

Thus, realized changes in prices are unforecastable given information set Ft. From thisrelatively elementary exercise we deduce martingale property of security prices:

E

Ps

Ft

= Pt.

The essential passage behind the martingale concept is the notion of a fair game, a

game which is neither in the favor of an investor nor his/her opponents.

Alternatively, a game is fair if the expected incremental investment wealth at any

stage is zero when conditioned on the history of the game.



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Asset Dynamics: The Random Walk Hypothesis

The notion of martingale has become a powerful tool and important applications

modern theories of asset prices - it has revolutionized the pricing of complex financial

instruments such as fixed income, options, swaps, and other derivative securities.

Needless to say, the martingale has become an integral part of every scientific

discipline concerning with asset model of asset prices: the random walk hypothesis.

1. The Random Walk I: IID Increments

The simplest version of the RW hypothesis is the IID increments case

Pt = + Pt1 + t, where t IID(0, 2), (4)

where is the expected price change or drift, and IID(0, 2) denotes that t is

independently and identically distributed with mean 0 and variance

2

.



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Exercise Conditional on P0, show that

E

PtP0 = P0 + t

Var

PtP0 = 2t. (5)

It is apparent from (5) that the RW (4) is nonstationary and conditional mean andvariance are both linear in time.

Remarks The independence of the increments of {t} implies that the randomwalk is also a fair game, but in a much stronger sense than the martingale:

Independence implies not only that increments are uncorrelated, but that anynonlinear functions of the increments are also uncorrelated.

Perhaps the most common distributional assumption for the innovations or increments

t is normality. If ts are IID

N(0, 2), then (4) is equivalent to an arithmetric

Brownian motion, sampled at regularly spaced unit intervals.

This distributional assumption simplifies many calculations, but violates the limited

liability condition: the price must be nonnegative. (If the conditional distribution of Ptis normal, then there will always be a positive probability that P

t< 0.)



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However, when natural logarithm of prices pt := log Pt follows a random walk with

normally distributed increments t, i.e.,

pt = + pt1 + t, t are IID N(0, 2),

the returns pts follow the lognormal model of Bachelier (1900) and Einstein (1905).

2. The Random Walk II: Independent Increments

This model relaxes the assumptions of RW1 to include processes with independent but

identically distributed increments. An example of this type of process, consider

Pt = + Pt1 + t1t, where t IID(0, 1), (6)

where t accounts for the time-variation in volatility of the asset price Pt.



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Capital Asset Pricing Model (CAPM)


A I d i C i l M k d I


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The Economic Models of Asset Returns

The Economic model is a statistical model which gives the opportunity to calculate

more precise measures of the return of any given security using proper economic

indicators.

Assumption LetR be an (N 1) vector of asset returns for calendar time periodt. Rt is independently multivariate normally distributed with mean and

covariance matrix for all t.

1. Constant-Mean-Return Model

Let i, the ith element of , be the mean return for asset i. The

constant-mean-return model is defined by

Rit = i + it

where E

it

= 0 and Var

it

= 2i

.

where Rit, the ith element ofRt, is the period-t return on security i, it is the

zero-mean disturbance term, and 2i

is the (i, i) element of.


A I t d ti t C it l M k t d I t t


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2. Market Model

The model is a statistical model which relates the return of any given security to the

return of the market portfolio. For any security i, the model is specified by

Rit = i+iRmt + it

where E

it

= 0 and Var

it

= 2i

.

where

Rit and Rmt are the period-t returns on security i and the market portfolio,respectively,

it is the zero-mean disturbance term,

and i, i and

2

i

are the parameters of the market model.

In applications, a broad based stock index is used for the market portfolio, with the

S&P500 index, etc. (incl. CRSP value-weighted index, and the CRSP equal-weighted

index being popular choice.)




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Statistical Estimation of the Economic Models

Recall the market model for security i and observation time

Ri

= i

+ iR

m+

i(7)

The estimation-window observations can be expressed as a regression system,

Ri = Xii + i (8)

where

Ri = (Rit1,...,Ritl) is an (l 1) vector of estimation-window returns,

Xi = (1,Rm) is an (l 2) matrix with a vector of ones in the first columnreturns,

Rm = (Rmt1,...,Rmtl) is the vector market return observations,

and i = (i, i)

is the (2

1) parameter vector.




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Under general conditions ordinary least squares (OLS) is a consistent estimation

procedure for the market-model parameters.

Statistical Estimation: Continued

The OLS estimators of the market-model parameters using an estimation window of lobservations are given by

i = (X

iXi)

1X

iRi

2i = 1(l 2)iii = Ri Xii

Var

i

= (X

iXi)

1

2i




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Capital Asset Pricing Model

Markowitz (1959) laid the groundwork for the CAPM. Sharpe (1964) and Lintner(1965) improved the Markowitz model to develop economy-wide implications.

The Sharpe and Lintner derivations of the CAPM assume the existence of lendingand borrowing at a risk-free rate of interest. The Sharpe and Lintners CAPM model is

E

Ri

= Rf + imE

Rm Rf (9)

im =Ri, RmRm, Rm

:=Cov

Ri, Rm

Var

Rm

(10)

Define Zi := Ri Rf. Then, for the Sharpe and Lintners CAPM model we haveE

Zi

= imE

Zm

im =Zi, ZmZm, Zm

(11)

where Zm := Rm Rf is the excess return on the market portfolio of assets.In the case where the risk-free interest rate Rf is nonstochastic

6 (10) = (11).6implying that

Ri, Rf

= 0.




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Capital Asset Pricing Model: Continued

In the absence of riskfree asset, Black (1972) derived a more general version of CAPM,

known as the Black version. In this version, the expected return of asset i in excess of

the zero-beta return is linearly related to its beta. Specifically, the model is given by

ERi = ERom + imERm ERom,where Rom is the return on the zero-beta portfolio

7 associated with market portfolio

m. The analysis of the Black version of CAPM treats the zero-beta portfolio return asunobserved quantity, making the analysis more complicated than the Sharpe-Lintner.

For the Black model, returns are generally stated on an inflation-adjusted basis. The

model can be tested as a restriction on the real-return market model, i.e.,

E

Ri

= im + imE

Rm

im = Ri, RmRm, Rmim = E

Rom

1 im

7This portfolio is defined to be the portfolio that has the minimum variance of all portfolios uncorrelated with m. (Any

other uncorrelated portfolio would have the same expected return, but a higher variance.)




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p


Assumptions underlying the CAPM model

investors behave so as to maximize the expected utility of their wealth at the end

of a single period. investors choose among alternative portfolios according to expected return andvariance (or standard deviation) in return

borrowing and lending are unlimited and take place at an exogeneously determinedrisk-free rate.

all investors share identical subjective estimates of the means, variances, andcovariances of return of all assets.

assets are completely divisible and perfectly liquid, with no transactions costsincurred in their purchase or sale.

all investors are priced takers8, there are no taxes, and the quantities of all assetsare fixed.

8in a perfectly competitive market, no individuals decisions to buy or sell will affect the market price; hence, individuals

must simply accept the market price when they trade.




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The CAPM is a single-period model; there is no time dimension involves in thedynamics of asset returns. Consider a portfolio of N risky assets whose value is given

by

Rp

=N

j=1 xjpR

j:= xp

R, with xp

1 = 1.

Technical assumptions

Assume that returns are IID through time and jointly multivariate normal.

The expected returns of at least two assets differ

The covariance matrix := ERR is of full rank - invertible.Denote := E

R

and := ERR

. Portfolio p has mean return and variance

p := xp and variance 2

p = xpxp

The covariance Rp, Rq between any two portfolios p and q is given by

Rp, Rq = xpxq.




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Definition 2. Portfolio p is the minimum-variance portfolio of all portfolios withmean return p if weights vectorxp is the solution to the CO problem:

minx

xx, subject to

x = px1 = 1

(12)

To solve the problem, let us consider the Lagrangian function:

L(x, 1, 2) := xx + 1

p x

+ 2

1 x1The first order Euler condition gives us the following system of equations

2x 1 21 = 0 (13)x

= p (14)

x1 = 1 (15)

By multiplying the equation (13) both side by 1 we get

x =1

2

11

+1

2

211. (16)




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CAPM with Risk-Free Asset

By imposing the conditions (14) and (15), we have

1

2

1

1

1 +1

2

1

11

2 =1

1211 + 1

2112 =p.

(17)

Now let us define the constants: A = 11, B = 1, C = 111, andD = BC

A2. Solving the equations (17) for 1 and 2, we obtain from (16):

xp = g + hp,

where g and h are (N 1)vectors defined by

g = 1D

B11 A1h =

1

D

C

1 A11,




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CAPM with No Risk-Free Asset

Figure 2: Minimum-Variance Portfolios Without Risk-free Asset.

g is the minimum variance portfolio. op is the zero-beta portfolio w.r.t the portfolio

p, as this portfolio has a zero covariance with the portfolio p, i.e., Rop, Rp = 0.




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Let Rf be the return of risk-free asset. The portfolio optimization amounts to solving

minx

xx, subject to x

+

1 x1

Rf = p (18)

The corresponding Lagrangian function is defined by

L(x, ) := xx + p x 1 x1Rf.Differentiating L w.r.t x and , we obtain

2x Rf1 = 0 (19)x

+

1 x1

Rf = p (20)

Notice that the equation (20) can be rewritten as

p Rf = x

Rf1

. (21)




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CAPM with Risk-Free AssetMultiplying both sides of (19) by 1 to get

x =1

2

1

Rf1

. (22)

By inserting x (22) in (21), we get

=(p Rf)( Rf1)

1

Rf1

,

from which we finally have

xp =(p Rf)

( Rf1)1

Rf1

1 Rf1.Note that we can express xp as a scalar which depends on the mean of p times aportfolio weight vector which does not depend on p, i.e.,

xp = Cp

x.




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CAPM with No Risk-Free Asset

Figure 3: Minimum-Variance Portfolios Wit Risk-free Asset.

With a risk-free asset, all efficient portfolios lie along the line from the risk-free asset

to the tangency portfolio, whose slope measures the market price of risk. The

tangency portfolio can be characterized as the portfolio with the maximum Sharpe

ratio of all portfolios of risky assets.




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Thus, with a risk-free asset all minimum-variance portfolios are a combination of a

given risky asset portfolio with weights proportional to x and the risk-free asset.This portfolio of risky assets is called the tangency portfolio and has weight vector

xq =1

Rf1

11 Rf1 . (23)

With a risk-free asset, all efficient portfolios lie along the line from the risk-free asset

to the tangency portfolio, whose slope measures the market price of risk.

The tangency portfolio can be characterized as the portfolio with the maximum

Sharpe ratio of all portfolios of risky assets.

In the next section below we will derive the Sharpe-Lintner CAPM model (9).




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The Sharpe-Lintner CAPM model

The model can be derived as follows. Replace x by the tangency portfolio xq (23) in

the equation (20) so that 1xq = 1 and multiply xm to both sides to get

2xmxm = xm

Rf1

=

m Rf

from which we obtain1

=

m Rf1

2xmxm

=

m Rf1

22m

.

Substituting back to the equation (20), we have

Rf1 =

m Rf1

2mxm.

More specifically in terms of individual asset i, for i = 1, 2,...,N, we have

i Rf =ip

2m

Rm Rf

= i

Rm Rf

.




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Fixed Income Analysis :Discrete-Time Framework




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The Basic Theory of Interest Rate

Simple Interest rate: The general rule is that If an amount A (referred to as principal) is left in account at simple interest y, the value after n

years is V = (1 + yn)A.

If the proportional rule holds for fractional years, then after any time t (measured in years), theaccount value is V = (1 + yt)A.

Compound Interest rate Compounding at various intervals The general method is that a year is divided into a fixed

number of equally spaced periods - say m periods. The interest rate for each of the m periods isthus ym. Then after any time t (measured in years), the account value is (1 +

ym)

mt.

Continuous compounding To determine the yearly effect of continuous compounding, we use ofthe fact that limm(1 + ym)

mt = eyt.

Debt: money borrowed from the bank will grow according to the same formulas.

Money market In reality there are many different rates9

each day applied todifferent circumstances, different user classes, and different periods.

Examples: US Treasury bills and notes, LIBOR rate, Mortgage rate, inflation rate, etc.9Most rates are established by the forces of supply and demand in broad market to which they apply




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Net Present and Future Values of Streams

Definition 3. [Future value of a stream] Given a cash flow stream (x0, x1,...,xn)and interest rate r each period, the future value of the stream is defined by

FV = x0(1 + y)n + x1(1 + y)

n1 + ... + xn

.Example Consider the cash flow stream (2, 1, 1, 1) when the periods are years andthe interest rate is 10%. The future value is given by

FV = 2 (1.1)3 + 1 (1.1)2 + 1 (1.1) + 1 = +0.648

.Definition 4. [Net present value of a stream] Given a cash flow stream

(x0, x1,...,xn) and interest rate y each period, the present value of this stream is

PV = x0 +x1

1 + y

+x2

(1 + y)2

+ ... +xn

(1 + y)n

.

Example Consider the cash flow stream (2, 1, 1, 1). Using an interest rate of 10%,

PV = 2 + 11.1

+1

(1.1)2+

1

(1.1)3= +0.487.



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Applications and Extensions

Cycle Problems. When using interest rate theory to evaluate ongoing (repeatable)activities, it is essential that alternatives be compared over the same time horizon.

Net Flows. In conducting a cash flow analysis using either NPV or IRR, it isessential that the net of income minus expenses be used as the cash flow each period.

The net profit usually can be found in a straightforward manner, but the process can be subtle incomplex situations, e.g., certain tax-accounting costs and profits are not always equal to actual cashoutflows or inflows.

Taxes. If a uniform tax rate were applied to all revenues and expenses as taxes andcredits, respectively, then recommendations from before-tax and after-taxes analyses

would be identical.

Inflation is another economic factor that causes confusions, arising from the choicebetween using actual dollar values to describe cash flows and using values expressed in

purchasing power, determined by reducing inflated future dollar values back to a

nominal level.




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Inflation: Continued

Inflation is characterized by an increase in general prices in time. Inflation can bedescribed quantitatively in terms of an inflation rate f. Inflation compoundsmuch like interest does. So, after n years of inflation at rate f, prices will be

(1 + f)n times their original values.

Another way to look at inflation is that it erodes the purchasing power of money.A dollar today does not purchase as much bread or milk as a dollar did ten years

ago. If the inflation rate is f, then the value of a dollar next year in terms of

purchasing power of todays dollar is 1/(1 + f).

In reality, inflation rates do not remain constant, but may fluctuate randomly. we define a new interest rate, termed the real interest rate r0, which is the rate at

which real dollars increase if left in a bank that pays the nominal interest rate r.

If one dollar is left in a bank for a year, it will increase nominally by (1 + r) atthe end of the year, but its purchasing power is deflated by 1/(1 + f). So the

real value of the dollar is given by 1 + r0 =1+r1+f

.

Solving the above equation for r0, we arrive at r0 = rf1+f.




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Bonds and Forward Rates

This part of the course is aimed at providing answer to the question like what is the

sixth-month interest rate that the market expects to prevail two years in the future?

Definition 5. [Zero-coupon bond]A zero-coupon bond with maturity date T, also

called T

bond, is a contract which guarantees the holder $1 to be paid on date T.The price at time t of a Tbond is denoted by P(t, T). We denote by P(t) the value at time zero of tbond, i.e., P(t) := P(0, t).

We impose the following assumption:

There exists a (frictionless) market for Tbonds for every T > 0. The relation P(t, t) = 1 holds for all t. For each fixed t, the bond price P(t, T) is differentiable w.r.t T.

[Remarks]

For a fixed value of t, P(t, T) (as a function of T) is a very smooth graph11. For a fixed value of T, P(t, T) (as a function of t) is a random process.11also called the term structure at time t.




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What is forward rate?

Suppose that at time t I decide to write a financial contract which allows me to makea $1 investment at time S > t at a deterministic rate of return L over a period oftime [S, T] in the future. The rate of return is determined at current time t.

How can I replicate this contract?

Borrow one unit of Sbond at time t and sell it to get $P(t, S). Use the $P(t, S) cash to buy at time t P(t, S)/P(t, T) units of Tbonds. At time S, the Sbond matures, so we have to pay back $1 at time S.

At time T, the Tbonds mature and worth $P(t, S)/P(t, T). Based on the financial contract written at time t, $1 investment at time S willgive me $P(t, S)/P(t, T) at time T.

Thus, at time t we have made a contract which guarantees a riskless rate ofinterest L over the future interval [S, T]. Such an interest is called a forward rate.

So, the simple forward rate L satisfies the relation:

1 + (T S)L = P(t, S)P(t, T)

or eR(TS) =P(t, S)

P(t, T).




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Varieties of Interest Rates in Fixed Income

Definition 6.

Simple forward rate (LIBOR rate) for [S, T] contracted at time t is defined as

L(t; S, T) = P(t, T) P(t, S)(T S)P(t, T) .

Simple spot rate (LIBOR spot rate) for the period [S, T] is defined as

L(S, T) = P(S, T) 1(T S)P(S, T). (25)

Continuously compounded forward rate for [S, T] is defined as

R(t; S, T) = log P(t, T) log P(t, S)(T S) .

Continuously compounded spot rate for the period [S, T] is defined as

R(S, T) = log P(S, T)(T

S)

.

Instantaneous forward rate with maturity T contracted at time t is defined as

f(t, T) = log P(t, T)T

. (26)

Instantaneous short rate at time t is defined as r(t) = f(t, t).



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Duration and Convexity

Duration measures the sensitivity of cash flow to parallel shift of the yield curve12

.

For continuously compounded yield, the PV of streams X is given by

P V =n

i=1P(ti)X(ti) =

n

i=1ey(ti)tiX(ti).

As the yield curve y(t) makes parallel shift of amount y for each t, i.e.,ynew(t) = y(t) + y, the PV changes accordingly to

P V

y = n

i=1

tiey(t

i)t

iX(ti) = n

i=1

tiP(ti)X(ti).

Macaulay duration13 DX (of flows X)defined as the percentage change of PV, i.e.,

DX = 1

PV

P V

y = ni=1 tiP(ti)X(ti)n

i=1 P(ti)X(ti). (29)

Hence, Duration can also be seen as the PV-weighted average of time to maturity.12referring to the 1930 work of Macaulay.13Also known as the Fisher-Weil duration.




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Examples of Duration

Consider a three-year 10% coupon bond with a face value of $100. Suppose that theyield on the bond is 12% per annum with continuous compounding. This means that

y = 0.12. Coupon payments of $5 are made every six months.

Time (yrs) Payment ($) Present Value Weight Time Weight0.5 5 4.709 0.05 0.025

1.0 5 4.435 0.047 0.047

1.5 5 4.176 0.044 0.066

2.0 5 3.933 0.042 0.084

2.5 5 3.704 0.039 0.098

3.0 105 73.256 0.778 2.334

Total 130 94.213 1.00 2.654

Following equation (29), we have that P = DPPy = 2.654 94.213y,that is, P = 250.04y. If y = +0.001 so that the yield increases to12.1%, we expect that the bond price to go down to 94.213 0.250 = 93.963.




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Examples of Duration: Continued

Case Study A firm X is subject to paying an obligation of 1$ million in 10 years. Thefirm is interested in finding an alternative investment that would meet its obligation.

The purchase of a zero-coupon bond would provide one of the solution. However,zero-coupon bonds are not always available for the desired maturity. Alternatively, the

firm may invest in corporate bonds having the same yield of 9%

Security Maturity Price Yield Duration

Bond 1 30 69.04 9% 11.44Bond 2 10 113.01 9% 6.54

Bond 3 20 100.00 9% 9.61

Investment on Bonds 2 and 3 would create a serious problem as any positiveweighted average of D2 and D3 will be less than the required duration 10 years.

A bond with a longer duration is required. So, the firm decides to use bonds 1 and 2.



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Duration-Based Hedging Strategies

Consider a portfolio W consisting of an underlying security with PV X1 and a units

of hedging instrument with PV X2. The total value of this portfolio is given by

W = X1 + X2.

The changes of the hedging portfolio to the parallel shift in the term structure of

interest rate can be written in terms of duration of each security as

W

y = X1

X1

X1y + X2

X2

X2y = X1DX1 X2DX2,from which the amount of the hedging instruments to hold is given by

=

X1DX1

X2DX2, (31)

which makes the duration of the entire position zero.

This is called the duration-based hedge ratio, or price sensitivity hedge ratio.




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Discrete Compounded Macaulay Duration

Suppose that the discount factors (P(ti), i = 1, 2,...,n are replaced by the one

that has the same interest rate (yield to maturity) at each maturity, i.e.,

P(ti) =1

[1 + (y/m)]i1+x,

where the time ti is directly calculated as ti =i1+x

m . Using this formulation, theconventional duration for bond with a dollar coupon C and discrete compoundingyield to maturity y is defined as

Conventional Duration = C

ni=1 tiP(ti) + tnP(tn)F

Present Value

=1

PV

C n

i=1

(i 1 + x)m

1[1 + (y/m)]i1+x

+

(n

1 + x)

m F

[1 + (y/m)]n1+xHowever, the formula doesnt measure the percentage change in PV for a small

change in the discrete yield to maturity, as explained further on the next page.



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Limitations of Duration : Convexity measure

The duration only measures the linear relationship between the percentage changeP/P in present value and change y in yield of a bond portfolio.

Figure 4: Bonds X and Y with different convexity.

For large yield changes, the portfolio may behave differently as the relationship maynot be linear. A factor known as convexity measures this curvature and can therefore

be used to improve the relationship in (29).



F i l d d f i i d fi d b


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For a continuously compounded, a measure of convexity is defined by

CX =1

PV

2P V

y2=

ni=1 t

2i P(ti)X(ti) n

i=1 P(ti)X(ti). (32)

Using Taylor series expansions, we find

P =dP

dyy +

1

2

d2P

dy2(y)

2

=

DPy +

1

2

CP(y)2

.

By matching convexity as well as duration, a portfolio can be made immune to

relatively large parallel shifts in the curve. However, it is exposed to nonparallel shifts.



Yi ld t M t it d F d R t


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Yield-to-Maturity and Forward Rates

What is Yield-to-Maturity?

The yield to maturity of a bond is a discount rate which equates the PV of the bonds

payments to its price. Thus if Pnt is the time t price of a discount bond that makes a

single payment of $1 at time t + n, and Ynt is the bonds YTM, we then have

Pn,t =1

(1 + Yn,t)n, (33)

or, equivalently, we can express YTM in terms of the bond price as

(1 + Yn,t) = P1/n

n,t . (34)

Taking the natural algorithm on both sides of the above equation, we have

yn,t = 1

npn,t. (35)

The term structure of interest rates is the set of YTMs, at a given time, on bonds of

different maturities. The yield curve is a plot of the term structure, that is the plot of

Yn,t or yn,t against n on some particular date t.



Yi ld t M t it


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Yield-to-Maturity

1950 1955 1960 1965 1970 1975 1980 1985 1990 19950

2

4

6

8

10

12

14

16

18

Annualizedpercentagepoints

10year yield

1month yield

Figure 5: Short-and Long-Term Interest Rates 1952 to 1991.




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Holding-Period Returns

The holding-period return on a bond is the return over some holding period less than

the bonds maturity. Define Rn,t+1 as the one-period holding-period return on an

nperiod bond purchased at time t and sold at time t + 1, i.e.,

(1 + Rn,t+1) =Pn1,t+1

Pn,t=

(1 + Yn,t)n

(1 + Yn1,t+1)n1. (36)

Remarks The holding period-return (36) is high if the bond has a high yield when it is

purchased at time t, and if it has a low yield when it is sold at time t + 1.

Taking the logs, the log holding-period return, rn,t+1 := log(1 + Rn,t+1), is given by

rn,t+1 = pn1,t+1 pn,t (37)= nyn,t (n 1)yn1,t+1= yn,t (n 1)

yn1,t+1 yn,t

. (38)

Following (37), we obtain a relation between log bond prices and log holding-period



tn1

i t f th i ld


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return as pn,t =

n 1i=0 rni,t+1+i., or in terms of the yield:

yn,t =1

n

n1i=0

rni,t+1+i,

showing that the log YTM on a zero-coupon bond equals the average log return per

period if the bond is held to maturity.

Forward Rates

The forward rate is defined to be the return on the time t + n investment of

Pn+1,t/Pn,t, i.e.,

(1 + Fn,t) =1

(Pn+1,t/Pn,t)=

(1 + Yn+1,t)n+1

(1 + Yn,t)n.

Moving to the logs scale, the n

period ahead log forward rate is

fn,t = pn,t pn+1,t= yn+1,t + n(yn+1,t yn,t)= yn,t + (n + 1)(yn+1,t

yn,t). (39)



Term Structure Models


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Term Structure Models

All the models start from the general asset pricing martingale condition (??)14

:

Pn,t = EQ

t

Pn1,t+1

= Et

Lt+1Pn1,t+1

. (40)

where the stochastic process Lt, known as the stochastic discount factor for

transforming the pricing kernel from P to Q, is defined by

Lt :=dQ

dP

Ft

and has the property that E

Lt

= 1.

The equation (40) lends itself to the recursive equation for the n

period ZC bond:

Pn,t = Et

Lt+1 . . . Lt+n

.

Assumption: We assume that the distribution of the stochastic discount factor L is

conditionally lognormal, and that bond prices are jointly lognormal with L, i.e.,

Et

LtPn,t

= expEt

lt + pn,t

+

1

2Vart

lt + pn,t

. (41)

14we have used the notation EQt

Pn1,t+1Pn,t

:= EQ

Pn1,t+1Pn,t

Ft

.



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Single-Factor Term Structure Model


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As the stochastic discount factor Lt is defined in Ft, we assume that the variation ofthe process L could be explained by some macro factors xt modeled by

lt+1 = xt + t+1, (44)

with t+1 being an innovation process normally distributed with constant variance.

The model is a discrete-time version of the Vasicek single-factor term structure model.

The model assumes that the macro factors xt evolves according to a univariate AR(1)

process with mean and persistence 15,

xt+1 = (1 ) + xt + t+1 . (45)

The innovations t+1 may be correlated with t+1. To capture this, we can write

t+1 = t+1 + t+1,

where t+1 and t+1 are normally distributed with constant variances and are

uncorrelated with each other.15If = 1 the process is a unit root process.




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Single-Factor Term Structure Model: Continued

The presence of the uncorrelated shock t+1 only affects the average level of the term

structure and not its average slope or its time series behavior.

To simplify the notation, we accordingly drop it and assume that

t+1 = t+1.

The equation (44) can therefore be rewritten as

lt+1 = xt + t+1. (46)

The innovation t+1 is now the only shock in the system.

Equations (46) and (45) imply that lt+1 can be written as an ARMA(1,1) process asit is a sum of an AR(1) process and white noise.




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We can determine the price of a one-period bond by noting that

when n = 1, pn1,t+1 = p0,t+1 = 0

so that the term pn1,t+1 in (42) drop out. Substituting (46) and (45) into (42),

p1,t = Et

lt+1

+1

2Var

lt+1

= xt + 22/2. (47)

The one-period bond yield y1,t =

p1,t can be written as

y1,t = xt 22/2.

Remarks:

The short rate equals the state variable less a constant term, so it inherits the AR(1)

dynamics of the state variable. So, we can think of the short rate as measuring the

state of the economy in this model. Notice that there is nothing in (47) that rules out

a negative short rate.





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Our guess for the price function (43) implies that the two terms in (42) are given by

Et

lt+1 + pn1,t+1

= xt An1 Bn1(1 ) Bn1xt,Vart

lt+1 + pn1,t+1

= ( + Bn1)

2

2. (48)

Substituting (43) and (48) into (42), we get

An + Bnxt xt An1 Bn1(1 ) Bn1xt+

1

2( + Bn1)

2

2= 0.

As this must hold for any xt, so the coefficients on xt must sum to zero and the

remaining coefficients must also sum to zero. This implies that

Bn = 1 + Bn1 =(1 n)

(1 )An = An1 + (1 )Bn1

1

2

2( + Bn1)

2, (49)

with A0 = B0 = 0, A1 = 22/2, and B1 = 1.



Heteroskedastic Single-Factor Term Structure Model


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The Vasicek model discussed previously is a homoskedastic model. Apart from itsappealing simplicity, the model has several unattractive features.

First, it assumes that interest changes have constant variance.

Secondly, the model allows interest rates to be negative. Given these, the model isapplicable to real interest rates, but less appropriate for nominal interest rates.

Thirdly, the model implies that the the ratio of the expected excess return on a bondto the variance is constant over time.

In order to handle these problems, while retaining the simplicity of the basic structure

and tractability of the model, one can alter the model by allowing the state variable xtto follow a conditionally lognormal but heteroskedastic square-root process.

The heteroskedastic square-root model is a discrete-time version of the Cox, Ingersoll,

and Ross (1985) continuous-time model, replaces (46) and (45) with

lt+1 = xt + xtt+1 = xt + xtt+1 (50)xt+1 = (1 ) + xt + xtt+1. (51)



Heteroskedastic Single Factor Term Structure Model: Continued


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Heteroskedastic Single-Factor Term Structure Model: Continued

The new term appearing in the model is that the shock t+1 is multiplied byxt.

Proceeding with the recursive analysis as before, we can determine the price of a

one-period bond by substituting (50) into (42) to get

p1,t = Et

lt+1

+1

2Vart

lt+1

= xt

1 22/2. (52)

As we can see, the one-period bond yield y1,t = p1,t is proportional to the statevariable xt; the short rate measures the state of the economy in the model.

Since the short rate is proportional to the state variable, it inherits the property that

its conditional variance is proportional to its level16. As a result, interest rate volatility

tends to be higher when interest rates are high.

This property makes it difficult for the interest rate to go negative, since the upwarddrift in the state variable tends to dominate the shocks as xt declines towards zero.

16implying the risk premia to be time-varying.



Heteroskedastic Single-Factor Term Structure Model: Continued


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Heteroskedastic Single Factor Term Structure Model: Continued

The price function for nperiod bond has the same linear form as before:

pn,t = An + Bnxt

with A0

= B0

= 0, A1

= 0, and B1

= 1

22/2.

It is straightforward to show that

Bn = 1 + Bn1 ( + Bn1)22/2,

An = An1 + (1 )Bn1. (53)

Remarks:

Comparing (53) to (49), we see that the term in

2

has been moved from theequation describing An to the equation describing Bn. This is due to the fact that

the variance is now proportional to the state variable, so that it affects the slope

coefficient rather than the intercept coefficient for the bond price. The slope

coefficient Bn is positive and increasing in n.



Two-Factor Term Structure Model


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Two Factor Term Structure Model

We have considered a single-factor model for term structure of interest rate. Such

models imply that the all bond returns are perfectly correlated. The model is a

discrete-time version of the model of Longstaff and Schwartz (1992). The model takes

the following form:

lt+1 = x1,t + x2,t + x1,tt+1. (54)We notice that the model nests the single-factor model by setting x2,t = 0, but does

not nests the single-factor homoskedastic model. In this model, minus the log SDF is

forecast by two state variables x1,t and x2,t, whose dynamics are

x1,t+1 = (1 1)1 + 1x1,t + x1,t1,t+1 (55)x2,t+1 = (1 2)2 + 2x2,t + x2,t2,t+1. (56)

The relation between shocks is specified according to

t+1 = 1,t+1

meaning that the variance of the innovation to the log SDF is proportional to the level

of x1,t. The log SDF is conditionally correlated with x1,t but not with x2,t.




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Two-Factor Term Structure Model: Continued

The shocks 1,t+1 and 2,t+1 are uncorrelated with each other. We denote by 21 for

the variance of 1,t+1 and 22 for the variance of 2,t+1.

Following the usual way, we find that the price of a one-period bond is given by

p1,t = Et

lt+1

+1

2Vart

lt+1

(57)

=

x1,t

x2,t + x1,t

2

21/2. (58)

We observe that the one-period bond yield y1,t = p1,t is no longer proportional tothe state variable x1,t as it also depends on 2,t. Said differently, the short rate is no

longer sufficient to measure the state of the economy in this model.

As pointed out by Longstaff and Schwartz, the variance of the short rate is a differentlinear combination of the two state variables:

Vart

y1,t+1

= (1 221/2)221x1,t + 22x2,t.




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Two-Factor Term Structure Model: Continued

The bond price is assumed to be an affine function of the two state variables, i.e.,

pn,t = An + B1,nx1,t + B2,nx2,t, (59)with A0 = B1,0 = B2,0 = 0, A1 = 0, B1,1 = 1 2 /2, and B2,1 = 1.

It is straightforward to show that An, B1,n, and B2,n satisfy the recursive equations:

B1,n = 1 + 1B1,n1 ( + B1,n1)221/2, (60)B2,n = 1 + 2B2,n1 B22,n122/2, (61)

An = An1 + (1 1)1B1,n1 + (1 2)2B2,n1 (62)



Fitting Term Structure Models to Real Data


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10 20 30 40 50 60 70 80 90 100 110 120

5.4

5.6

5.8

6

6.2

6.4

6.6

Maturity in Months

MeanYield

Comparing one and twofactor models

twofactor model

onefactor model

bond yields data

Figure 6: Fitting single-factor and two-factor term structure models to real data.




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Fixed Income Analysis :Continuous-Time Framework



Term Structure Model of Interest Rate


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In the previous lectures we have assumed through the concept of duration that interestrate at all maturities move at the same direction - parallel shifts.

In contrast to this assumption, the interest rate in fact moves randomly in time.

[Term Structure Equation] We assume that there is a arbitrage-free market forTbonds for every choice of T and that the price of a Tbond

P(t, T) := P(t, r(t); T),

is a smooth function of the three variables with the maturity condition

P(T, r; T) = 1.

[Short Rate Dynamics] We assume that the dynamics of short rate r(t) is driven by

the following stochastic differential equations (SDE):

dr(t) = (t, r(t))dt + (t, r(t))dB(t), (63)

where B is a standard Brownian motion having the following properties.



Standard Brownian motion (SBM)

Definition 7. [Standard Brownian motion] is a stochastic process with properties:


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Definition 7. [Standard Brownian motion] is a stochastic process with properties:

For any t1 and t2 s.t. 0 t1 t2 T,

B(t2) B(t1) N(0, t2 t1).

For any t1, t2, t3, and t4 s.t. 0 t1 t2 t3 t4 T,B(t2) B(t1) is statistically independent of B(t4) B(t3).

The sample paths of B(t) are continuous.

Multiplication rules for stochastic differentials dB(t)

dB(t) dtdB(t) dt 0

dt 0 0

(64)

If we have a stochastic differential equation (SDE):

dX(t) = (t, X(t))dt + (t, X(t))dB(t),



then due to the multiplication rule (64), we have


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(dX(t))2 = 2(t, X(t))dt2 + 2(t, X(t))dB(t)2+ 2(t, X(t))(t, X(t))

dtdB(t)

= 2(t, X(t))(dB(t))2

= 2(t, X(t))dt.

Simulating Stochastic Differential Equations

Assume that we are given a stochastic differential equation (SDE) of the form

dX(t) = (t, X(t))dt + (t, X(t))dB(t).

We are interested in simulating the solution in discrete time. The most common

scheme and easiest way to implement is the so-called Euler scheme:Xkh = X(k1)h + (k 1)h, X(k1)hh + (k 1)h, X(k1)hhNk,where Nk are IID N(0, 1) standard Gaussian random variable.



Itos formula


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Theorem 1. [Itos Formula] Assume that the process X solves the following SDE

dX(t) = (t, X(t))dt + (t, X(t))dB(t).

Let F be a C1,2

smooth function. Define the process Z by

Z(t) = P(t, X(t)).

Then the process Z has the following SDE:

dP = Pt + (t, x)Px + 12

2

(t, x)Pxxdt + (t, x)PxdB(t). (65)[Heuristic proof] Applying the Taylor expansion including second order terms, we have

dP = Ptdt + PxdX +1

2Pxx(dX)

2+

1

2Ptt(dt)

2+ Ptx(dtdX).

The result in (65) is obtained after applying the multiplication rules (64).



Dynamics of Tbond


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Consider a hedging portfolio V consisting of Tbond and Sbond, i.e.,V(t) = PT(t) + (t)PS(t). (66)

Assume that the hedging strategy (t) is predictable17, and the short rate r(t) follows

(63) so that by (65) the dynamics of the Tbond price PT(t, r) can be written asdPT

PT= T(t)dt + T(t)dB(t), (67)

T(t) = (PT

)1PTt + (t, r)PTr + 122(t, r)PTrr, (68)

T(t) = (t, r)(PT

)1

PT

r . (69)

Likewise for Sbond. By inserting (67) and the corresponding equation for Sbond,

dV(t) = T(t)PT(t) + (t)S(t)PS(t)dt+

T(t)P

T(t) + (t)S(t)PS(t)

dB(t).

(70)

17that is to say (t) = (t + dt) - the hedging strategy for the period of time [t, t + dt) has been specified at time t.



In order to make the portfolio risk-neutral, the hedging strategy should be

T T


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(t) = T(t)PT

(t)S(t)PS(t)

= PT

r

PSrfor all t 0 (71)

By substituting this portfolio strategy back to the equation (70) gives

dV(t) = V(t) S(t)T(t) T(t)S(t)(T(t) S(t)) dt.By applying the no arbitrage argument, we must have the condition

S(t)T(t)

T(t)S(t)

(T(t) S(t)) = r(t) for all t 0,with probability 1. Equivalently, we can rewrite the above equation as

S(t)

r(t)

S(t) =

T(t)

r(t)

T(t) for all t 0 . (72)

Remark Notice that the two quotients are equal regardless of the choice of the bond18.18It is the fixed income equivalent of the Sharpe ratio.



The Market Price of Risk and Term Structure Equation


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Proposition 1. If the bond market is free of arbitrage, then there exist a process such that it holds true for all t 0 and every choice of maturity time T that

T(t) r(t)T(t)

= (t). (73)

Furthermore, by inserting the expressions (68) and (69) for T and T, respectively,

we obtain a so-called the term structure equation (TSE) given in the eqn. (74) below.

Proposition 2. Assume that the short rate evolves according to (63). If the bondmarket is free of arbitrage, then the price of the Tbond PT solves the TSE:

PTt +

PTr + 122PTrr rPT = 0P

T

(T, r) = 1.

(74)

In order to solve the term structure equation, we must specify the market price of risk

as well as the drift and and the volatility of the short rate (63).



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Affine Term Structure Models


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We are interested in finding a closed form solution to the TSE (74). We assume thatunder the risk-neutral pricing measure Q the dynamics of the short rate r is given by

dr(t) = (t, r(t))dt + (t, r(t))dB(t).

Question Under which forms of and such that the T

bond price is of the form

P(t, r; T) = eF(t,T)G(t,T)r(t). (76)

Proposition 4. Assume that the drift and the volatility of r are of the form

(t, r) = (t)r + (t)

(t, r) =

(t)r + (t)(77)

Then, the Tbond price has the form (76) where the functions F and G solve

Gt(t, T) + (t)G(t, T) 12

(t)G2

(t, T) = 1, G(T, T) = 0,

Ft(t, T) (t)G(t, T) +1

2(t)G

2(t, T), = 0, F(T, T) = 0.

(78)



Merton Term Structure Model


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In his famous 1970 paper, Merton proposed a simple model of short rate:

dr(t) = dB(t). (79)

Following the TSE (74), the T

bond price solves the following equation:

Pt Pr +1

2

2Prr rP = 0, P(T, r; T) = 1

After some tedious calculations, it turns out that the solution to this PDE is

P(t, r; T) = P(, r) = er+2/2+1/62a3, = T t.

r Duration of coupon bearing bonds

Consider PV of streams (Xi, i = 1, 2,...,n). Duration under Merton is

Merton r Duration =n

i=1 iP(i, r)Xi

ni=1 P(i, r)Xi

,



Hedge ratio


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The hedge ratio of the hedging portfolio is given by

= (T t)P(t, r; T)(S t)P(t, r; S) .

Yield curve

For continuous compounding, the ZC bond yield implied by the model is

y() = 1

ln P(, r) = r 12

16

2

2

.

Remarks

The yield achieves its maximum value r + 382 at = 32 . The yield equals zero at = 32 + 324 + 2r3 . The yields become negative as time to maturity increases. The yield tends to be largely negative for long-maturity bond.



Extended Merton Model of Term Structure

Ho and Lee Model


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The model introduces a deterministic drift to the Merton model (79):

dr(t) = (t)dt + dB(t).

Solving the TSE (74) we arrive at

P(t, r; T) = eF(t,T)G(t,T)r(t)

, (80)

where the functions F and G are defined by

G(t, T) = T t and F(t, T) = T

t (s)(s T)ds +2

6 (T t)3

+

2 (T t)2

(81)

following which the yield to maturity implied is

y(t, T) = r

2

(T

t)

2

6

(T

t)

2

1

(T t) T

t

(s)(s

T)ds. (82)

The function (t) is chosen such that the theoretical zero-coupon bond prices att = 0 (P(0, T); T 0) fits the observed initial term structure (P(0, T); T 0).We thus want to find (t) such that P(0, T) = P(0, T) for all T 0.



The Vasicek Term Structure Model


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0 2 4 6 8 100.02

0.03

0.04

0.05

0.06

0.07

0.08

Time

r(t)

Short Rate

In his 1977 work, Vasicek proposed a model that avoids the certainty of having

negative yields in the Merton model of term structure, which is of the form

dr(t) = ( r(t))dt + dB(t). (83)



There are compelling economic arguments in favor of mean reversion:

Wh h h h d l d h l d d f


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When rates are high, the economy tends to slow down as there is low demand forfund from borrowers, which results in declining rates.

When rates are low, there is a high demand for funds from borrowers, resultingincreasing in rates.

When rates rise above the expected long-term rate , the rates is pulled back below at the speed , vice versa.

Exercise Show using Itos formula that solution to the Vasicek model (83) is

r(t) = + et

r0

+ et

t0

eu

dB(u) . (84)

Following this solution, it is readily to see that

r(t) N

+ et

(r0 ), 2

2

1 e2t. (85)



The governing equation for the T bond price is given by

P ( ) P1

2P P 0 P (T T ) 1 (86)


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Pt + ( r) Pr + 22Prr rP = 0, P(T, r; T) = 1. (86)Exercise Assume that solution to the equation (86) is of the form

P(, r) = eF()

G()r

,

where = T t. By inserting this in (86), the functions f and g are found to be

G() =1

1 e

F() =

+

2

22

G() 2G2()

4.

r Duration of coupon bearing bondsConsider PV of streams (Xi, i = 1, 2,...,n). Duration under Vasicek is

Vasicek r Duration =n

i=1 G(i)P(i, r)Xi

ni=1 P(i, r)Xi

.



Hedge ratio

Th h d ti f th h d i tf li i i b


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The hedge ratio of the hedging portfolio is given by

= G(T t)P(t, r; T)G(S t)P(t, r; S) .

Yield curve

For continuous compounding, the ZC bond yield implied by the model is

y() =1

f() + g()r

as y() = +

2

22.

Remark

The duration and hedge ratio under the Merton model can be seen as the limit ofthe Vasicek models as the speed of mean reversion tends to zero.

The yield is positive at perpetual time, correcting one of major concern of theMerton term structure model (79). Since the short rate r(t) is normally distributed, see equation (85), there is a

positive probability that the short rate r(t) can become negative.



Extended Vasicek Model of Term Structure

H ll d Whit M d l


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Hull and White Model

Hull and White (1990) proposed a model which improves the Vasicek model Under the

risk-neutral measure, the Hull and White model of short rate is defined as follows

dr(t) = (t) ar(t)dt + dB(t),where a and are constants while (t) is a deterministic function of time.

The parameters a and are typically chosen to obtain a nice volatility structure,

whereas the function (t) is chosen such that the theoretical zero-coupon bond pricesat t = 0 (P(0, T); T 0) fits the observed initial term structure (P(0, T);T 0). We thus want to find (t) such that P(0, T) = P(0, T) for all T 0.

P(t, T) = eG(t,T)r(t)+T

t

12

2G2(s,T)(s)G(s,T)

ds, (87)

where the function G(t, T) is defined as

G(t, T) =1

a

1 ea(Tt).



Alternative Term Structure Models

The aim is to review other alternative term structure models that exclude the


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The aim is to review other alternative term structure models that exclude thepossibility of having negative short rate found in the Merton and Vasicek models.

One-factor interest rate models

Cox-Ingersoll-Ross model The model is very influential in market practisedr(t) =

r(t)dt + r(t)dB(t).

This model also exhibits mean reverting behavior as Vasicek model does, causing

interset rate cycles. The distinct feature is that the model induces positive short rate.

CIR shows that the price of ZC bond is given by

P(, r) = eF()G()r, (88)

where = T t and the functions F and G are defined by

F() =2

2ln

2e(++) /2H()

, G() = 2

1 e

H(),



where the function H and parameter are defined by

H( ) = 2 + + + e

1


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H() = 2 + + + e 1 =

( + )2 + 22.

The Longstaff modeldr(t) =

r(t)

dt +

r(t)dB(t).

The model results in a complex function for the ZC bond price of the form

P(, r) = eF()+G()r+H()r

.

Remarks on Longstaff model

Longstaff argues that the non-linearity of the yield curve implied by the model doesbring extra explanatory power to the model.

However, the pricing of even intermediate term discount bond can be morecomplex than can be accommodated within the context of one factor model.



Black, Derman and Toy model

This model also avoids the problem of negative interest rates and allows for timef


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This model also avoids the problem of negative interest rates and allows for timedependent parameters. The model is of mean reverting type in the natural algorithm

of interest rate:

d ln r(t) =

(t) (t) ln r(t)

dt + (t)dB(t).

The model has desirable features of positive interest rates, ability to model the

volatility curve observable in the market, and thus has the ability to fit the observable

yield curve. However, the models liability is the lack of tractable analytical solutions.

Black and Karasinski modelIn the natural algorithm of interest rate, this model is a time-varying version of the

Vasicek model in which the speed of mean reversion, the expected long term interest

and the volatility are all time varying. The model is of the form:

d ln r(t) = (t) ln (t) ln r(t)dt + (t)dB(t).As the model is quite complicated to get a closed form expression for the price of ZC

bond, Black and Karasinski used lattice approach.



Two-factor interest rate models

Brennan and Schwartz model


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Brennan and Schwartz modelBrennan and Schwartz (1979) introduces a two-factor model where both a long-term

rate and a short-term rate follow a diffusion process. The long-term rate is defined as

the yield on a consol (perpetual) bond. The log of the short-rate evolves as follows

d ln r(t) = ln (t) lnp ln r(t)dt + 1dB1(t),where p the target value of ln r relative to the level of ln , and is the consol rate,

d(t) = (t)(t) r(t) + 22 + 22dt + 2(t)dB2(t).

Two-factor Vasicek modelOther extensions of mean-reverting type short-rate model have been proposed by many

authors, see among others, Chen and Scott (1992), Hull and White (1993), Longstaff

and Schwartz (1992). Hull and Whites model assumes the nominal short rate r as the

sum of two factors x1 and x2 each of which is modeled after the Vasicek model:

dxi(t) = i

i xi(t)

dt + idBi(t),



i.e., r = x1 + x2, where B1 and B2 are two independent Brownian Motions.

Due to the independence structure of the factors the price of ZC bond is of the form:


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Due to the independence structure of the factors, the price of ZC bond is of the form:

V(t, x1, x2; T) = P1(t, x1; T)P2(t, x2; T),

where Pi(t, xi; T) has the same functional form with that of Vasicek one-factor

model.

Three-factor Term Structure model

Chen (1994) extends the CIR model where the speed of mean reversion , long-run

interest rate level and volatility are all stochastic,

dr(t) = ((t) r(t))dt +

(t)

r(t)dB1(t)

d(t) = (

(t))dt +

(t)dB2(t)

d(t) = ( (t))dt + (t)dB3(t).



Calibration of Ho and Lee Model to Data

As there is one-to-one correspondence between forward rates and bond prices, i.e.,


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f(t, T) = T

log P(t, T), (89)

we may bridge the gap between the theoretical forward rate curve (f(0, T); T > 0)

and to the observed forward rate curve (f

(0, T); T > 0), where f

is definedsimilarly as in (89), i.e., f(t, T) = T

log P(t, T). Following (87) and (89),

f(0, T) = r0 +

T0

(s)ds 2

2T

2 T.

The function (t) is obtained by solving the above equation to get

(t) =

Tf(0, t) + 2t + .

By inserting (t) in equations (80)-(81), the bond prices is given by

P(t, T) =P(0, T)

P(0, t)e(Tt)r(t)+(Tt)f

(0,t)22 t(Tt)2

(90)



Forward Rate Models

Heat-Jarrow-Morton Framework


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We have previously discussed interest rate models in which the short-rate r is the only

state variable. The main advantages of such models:

allowing us to use partial differential equations (PDE) to get the price of ZC bond. allowing us to get analytical formulas for the bond prices, etc.

The main drawbacks of the short rate models:

from the view point of economics, it seems unreasonable to assume that the entiremoney market is governed by only one state variable.

it is hard to get a realistic volatility structure for the forward rates withoutintroducing a very complicated short rate model.

the inversion of yield curve (through the concept of parallel shift of the yield) asdiscussed before becomes increasingly more difficult to get a realistic one.



In the section where we discuss the calibration of Ho and Lee model to the terms

structure data, we find it more convenient to work with forward rate.

[Assumption] Assume that for every fixed T > 0 the forward rate f (t T ) follows


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[Assumption] Assume that for every fixed T > 0, the forward rate f(t, T) follows

f(t, T) = f

(0, T) +

t0

(s, T)ds +

t0

(s, T)dB(s) (91)

The Proposition 5 below provides condition on the drift of the forward rate model(91) so that the price of ZC bond under the forward rate model

P(t, T) = eT

t f(t,u)du, (92)

coincides with the one under the risk-neutral pricing measure Q

P(t, T) = EQ

e Tt r(u)duFt.

Proposition 5. HJM drift conditionUnder risk-neutral measureQ, the drift and

the volatility of the HJM model (91) must satisfy the following relation:

(t, T) = (t, T)

Tt

(t, s)ds (93)



In order to apply the HJM model, we need to

specify the structure of the volatility (t, T).


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p y y ( , )

the drift parameter of the forward rate is specified by (93). collect the forward rate structure (f(0, T); T 0) from the market.

take the forward rate model as in (93).

compute the ZC bond price using the formula (92).

Exercise Consider HJM forward rate model with constant volatility, i.e.,(t, T) = , with > 0. On account of the fact that r(t) = f(t, t), show that

the corresponding short rate dynamics is given by the Ho and Lee model

dr(t) = (t)dt + dB(t),

where the drift (t) is specified by

(t) =

Tf(0, t) +

2t.

As discussed earlier, this model perfectly fits the initial term structure. See eqn (90).



Parameters Estimation of Term Structure Models

We are interested in estimating the parameters of the Vasicek model


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dr(t) = ( r(t))dt + dB(t).

Due to the expression (85), we know that

r(t) N

+ et

(r0 ), 2

2

1 e2t. (94)

There are many approaches in doing s

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