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Computational Finance
Dr Nalan Gulpinar
Asset Pricing Models
381 Computational Finance
20-21 February 2006Imperial College
London
Computational Finance
Problem Types in Investment Science
Determining
correct, arbitrage free price of an asset:
price of a bond, a stock
the best action in an investment situation:
how to find the best portfolio –
how to devise the optimal strategy for managing an investment
Single period Markowitz model
Computational Finance
Topics Covered
The Capital Asset Pricing Model (CAPM)
Single and Multi Factor Models
CAPM as a Factor Model
The Arbitrage Pricing Theory (APT)
Computational Finance
M-V model
investor chooses portfolios on the efficient frontier –
deciding if given portfolio is on efficient frontier or not
no guarantee that a portfolio that was efficient ex ante
will be efficient ex post
statistical considerations regarding time period over
which to estimate & which assets to include are non-trivial
not mention implications of m-v optimisation on asset
pricing
CAPM describes MV portfolios and provides asset pricingCAPM describes MV portfolios and provides asset pricing
Computational Finance
CAPM: Capital Asset Pricing Model
developed by Sharpe, Lintner and Mossin
single period asset pricing model
determines correct price of a risky asset within
the mean-variance framework
highlights the difference between systematic &
specific risk
Computational Finance
Assumptions
All investors
– are mean variance optimisers –portfolios on efficient frontierare mean variance optimisers –portfolios on efficient frontier
– plan their investments over a single period of timeplan their investments over a single period of time
– use the same probability distribution of asset returns: the same use the same probability distribution of asset returns: the same
mean, variance, & covariance of asset returnsmean, variance, & covariance of asset returns
– borrow and lend at the risk free rate borrow and lend at the risk free rate
– are price-takers: investors’ purchases & sales do NOT influence are price-takers: investors’ purchases & sales do NOT influence
price of an asset price of an asset
– There is no transaction costs and taxesThere is no transaction costs and taxes
Computational Finance
Market Portfolio
Everyone purchases single fund of risky asset, borrows (lends) at risk-free rate.
Form a portfolio that is a mix of risk free asset and single risky fund
Mix of the risky asset with risk free asset will vary across individuals according to their individual tastes for risk
Seek to avoid risk – Seek to avoid risk – have high percentage of the risk free asset in their portfoliohave high percentage of the risk free asset in their portfolio More aggressive to risk – More aggressive to risk – have a high percentage of the risky assethave a high percentage of the risky asset
What is the fund that everyone purchases?This fund is Market Portfolio and defined as summation of
all assets – total invested wealth on risky assets
An asset weight in market portfolio is the proportion of that asset’s total An asset weight in market portfolio is the proportion of that asset’s total capital value to total market capital value capital value to total market capital value – capitalization weights– capitalization weights
Computational Finance
The Capital Market Line (CML)
Consider single efficient fund of risky assets (market portfolio) and a risk free asset (a
bond matures at the end of investment horizon):
If a risk free asset does not exist, investor would take positions at various points on
the efficient frontier. Otherwise, efficient set consists of straight line called CML.
Pricing Line: prices are adjusted so that efficient assets fall on this line
CML describes all possible mean-variance efficient portfolios that are a combination
of the risk free asset and market portfolio
Investors take positions on CML by
– buying risk free asset (between buying risk free asset (between MM and and rrff)) or or
– selling risk free assetselling risk free asset (beyond point (beyond point MM) ) and and
– holding the same portfolio of risky assetsholding the same portfolio of risky assets
Computational Finance
The Capital Market Line
Equation describes all portfolios on CML
CML relates the expected rate of return of an efficient portfolio to its
standard deviation
The slope the CML is called the price of RISK!
– How much expected rate of return of a portfolio must increase if the risk of the portfolio How much expected rate of return of a portfolio must increase if the risk of the portfolio
increases by one unit?increases by one unit?
M
fMf
rrrrrE
)(
Expected Value of market rate of return
Standard Deviation of market rate of return
Computational Finance
The Pricing Model
How does the expected rate of return of an individual asset relate to
its individual risk?
If the market portfolio M is efficient, then the expected return of an
asset i satisfies
The beta of an asset (risk premium):
fMifi
fMifi
rrrr
rrErrE
][][
2M
iMi
Computational Finance
expected excess rate of return of an asset is proportional to the expected excess rate of return of the market portfolio: proportional factor is the beta of asset..
Amount that rate of return is expected to exceed risk free rate is proportional the amount that market portfolio return is expected to exceed risk free rate
fMifi rrrr
describes relationship between risk and expected return of asset
The Pricing Model
Computational Finance
Beta of an Asset
beta of an asset measures the risk of the asset with
respect to the market portfolio M.
high beta assets earn higher average return in
equilibrium because of
beta of market portfolio: average risk of all assets
fMi rr
1)(
),(2
2
M
M
M
MMM rVar
rrCov
Computational Finance
The Beta of Portfolio
If the betas of the individual assets are known,
then the beta of the portfolio is
This can be shown by using
rate of return of the portfolio
covariance
n
iiip w
1
n
iiip rwr
1
n
iMiiMp rrwrr
1
),cov(),cov(
Computational Finance
Systematic and Specific Risk
CAPM divides total risk of holding risky assets into two parts:
systematic (risk of holding the market portfolio) and specific risk
Consider the random rate of return of an asset i:
Take expected value and the correlation of the rate of return with rM
The total risk of holding risky asset i is
ifMifi errrr )(
0),cov( and 0)( Mii reeE
risk specific
2
risk ystematic
22
risk total
2 ie
s
Mii
Computational Finance
Summary: CAPM
The capital market line: expected rate of return of an efficient
portfolio to its standard deviation
The pricing model: expected rate of return of an individual asset
to its risk
The risk of holding an asset i is
risk specific
2
risk ystematic
22
risk total
2 ie
s
Mii
M
fMf
rrrr
2
whereM
iMifMifi rrrr
Computational Finance
Beta of the Market
Average risk of all assets is 1 (beta of the market portfolio)
Beta of market portfolio is used as a reference point to measure risk of other assets.
– Assets or portfolios with betas greater than 1 are above average risk: tend to move more Assets or portfolios with betas greater than 1 are above average risk: tend to move more
than market. than market. Example:Example:
If risk free rate is 5% per year and market rises by 10 %, then assets with a beta of 2 will If risk free rate is 5% per year and market rises by 10 %, then assets with a beta of 2 will
tend to increase by 15%. tend to increase by 15%.
If market falls by 10%, then assets with a beta of 2 will tend to fall by 25% on average. If market falls by 10%, then assets with a beta of 2 will tend to fall by 25% on average.
– Assets or portfolios with betas less than 1 are of below average risk: tend to move less Assets or portfolios with betas less than 1 are of below average risk: tend to move less
than market. than market.
rr
fr fr
),cov( Mrr2M 1
Capital Market Line Security Market LineM M
M
fMf
rrrr
2M
iMi
fMifi rrrr
Computational Finance
CAPM as a Pricing Formula
CAPM is a pricing model.
standard CAPM formula only holds expected rates of return
suppose an asset is purchased at price P and later sold at price S.
rate of return is substituted in CAPM formula
CAPMin asset of Price 1
formula CAPM
return of Rate
fMf
fMf
rrr
SP
rrrP
PSP
PSr
Computational Finance
Discounting Formula in CAPM
CAPMin asset an of Price 1 fMf rrr
SP
rate interest adjusted-risk
factor discount case, randomthe In
factor discount case,tic deterministhe In
)(1
1
1
1
fMf
f
rrr
r
Computational Finance
Single-Factor Model
Consider n assets with rates of return ri for i=1,2,…,n and one factor f
which is a random quantity such as inflation, interest rate
Assume that the rates of return and single factor are linearly related.
Errors 1.1. have zero meanhave zero mean
2.2. are uncorrelated with the factorare uncorrelated with the factor
3.3. are uncorrelated with each otherare uncorrelated with each other
niefbar iiii ,,2,1 RandomConstantConstant
Intercept Factor Loadings
Error
0ieE
0ifeE
jieeE ij ,0
Computational Finance
Multi-Factor Model
Single factor model is extended to have more than one factor. For two factors f1 and f2 the model can be written as
For k number factors
niefbfbar iiiii ,,2,12211 RandomConstantConstantConstant
niefbar i
k
jjjiii ,,2,1
1
Computational Finance
How to Select Factors?
Factors are external to securities: consumer price index, unemployment rateconsumer price index, unemployment rate
Factors are extracted from known information about security returns:
the rate of return on the market portfoliothe rate of return on the market portfolio
Firm characteristics: price earning ratio, dividend payout ratioprice earning ratio, dividend payout ratio
How to select factors: It is part science and part art!Statistical approach – Statistical approach – principal component analysis Economical approach – Economical approach – its beta, inflation rate, interest rate, industrial production etc.
Computational Finance
The CAPM as a Factor Model
Special case of a single-factor model f = rM
iiiM
Mii
MiMi
fMiifi
ifMii
ifMifiifi
iMiii
iiii
br
rrb
rbrr
rrEbrrE
errb
errbrbarr
erbar
efbar
0 and ]var[
],cov[
]var[],cov[
)][(][
)(
)()1(
Computational Finance
The CAPM as a Factor Model: Example
Single Index Model applied to Lloyds
-0.1
-0.05
0
0.05
0.1
0.15
-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05
Market Returns
Re
turn
s o
n L
lloyd
s
slope = Beta = 1.77
intercept = alpha = 0 (approx)
•Single factor model equation defines a linear fit to data
• Imagine several independent observations of the rate of return and factor• Straight line defined by single factor model equation is fitted through these
points such that average value of errors is zero. • Error is measured by the vertical distance from a point to the line
Computational Finance
Arbitrage: “The law of one price”
Arbitrage relies on a fundamental principle of finance : the law of one price
sayssays – – security must have the same price regardless of security must have the same price regardless of
the means of creating that securitythe means of creating that security..
implies – implies – if the payoff of a security can be synthetically if the payoff of a security can be synthetically
created by a package of other securities, the price of the created by a package of other securities, the price of the
package and the price of the security whose payoff package and the price of the security whose payoff
replicates must be equal.replicates must be equal.
Computational Finance
Arbitrage – Example
How can you produce an arbitrage opportunity involving securities A,
B,C?
Replicating Portfolio: – combine securities combine securities AA and and BB in such a way that in such a way that – replicate the payoffs of security replicate the payoffs of security CC in either state in either state
Let wA and wB be proportions of security A and B in portfolio
Security PricePayoff in State 1
Payoff in State 2
A £70 £50 £100
B £60 £30 £120
C £80 £38 £112
Computational Finance
Example Continued
Payoff of the portfolio
Create a portfolio consisting of A and B that will reproduce the payoff of C regardless of the state that occurs one year from now.
Solving equation system, weights are found wB = 0.6 and wA = 0.4
An arbitrage opportunity will exist if the cost of this portfolio is different than the cost of security C.
Cost of the portfolio is 0.4 x £70 + 0.6 x £60 = £64 - price of security C is £80. The “synthetic” security is cheap relative to security C.
BA
BA
ww
w w
120100 :2 statein
3050 :1 statein
112120100
383050
BA
BA
ww
w w
Computational Finance
Example – Continued
Security Investment State 1 State 2
A -400000 5714 x 50 = 285700 5714 x 100 = 571400
B -600000 10000 x 30 = 300000 10000 x 120 =1200000
C 1000000 12500 x 38 = -475000 12500 x 112 = -1400000
Total £0 £110,700 £371,400
The outcome of forming an arbitrage portfolio of £1m
Riskless arbitrage profit is obtained by “buying A and B” in these proportions and “shorting” security C. Suppose you have £1m capital to construct this arbitrage portfolio.
Investing £400k in A £400k £70 = 5714 shares
Investing £600k in B £600k £60 = 10,000 shares
Shorting £1m in C £1m £80 = 12,500 shares
Computational Finance
The Arbitrage Pricing Theory
CAPM is criticised for two assumptions: The investors are mean-variance optimizersThe investors are mean-variance optimizers
The model is single-periodThe model is single-period
Stephen Ross developed an alternative model based purely on
arbitrage arguments
Published Paper:
““The Arbitrage Pricing Theory of Capital Asset Pricing”, The Arbitrage Pricing Theory of Capital Asset Pricing”,
Journal of Economic Theory, Dec 1976.Journal of Economic Theory, Dec 1976.
Computational Finance
APT versus CAPM
APT is a more general approach to asset pricing than CAPM. CAPM considers variances and covariance's as possible measures of risk while APT allows for a number of risk factors. APT postulates that a security’s expected return is influenced by a variety of factors, as opposed to just the single market index of CAPM APT in contrast states that return on a security is linearly related to “factors”. APT does not specify what factors are, but assumes that the relationship between security returns and factors is linear.
Computational Finance
Simple Version of APT
Consider a single factor model.
Assume that the model holds exactly; no error
The uncertainty comes from the factor f
APT says that ai and bi are related if there
is no arbitrage
nifbar iii ,,2,1for
Computational Finance
Derivation of APT
Choose another asset j such thatForm a portfolio from asset i and j with weights of w and (1-w)
Choose w so that the coefficient of factor is zero; so
ji bb
fbwwbawwar jijip ])1([)1(
ji
ij
ij
jip
jiij
j
bb
ba
bb
bar
bwwbbb
bw
0)1( and
Computational Finance
Derivation of APT
cba
cb
a
b
a
b
a
bb
ba
bb
ba
ii
i
i
i
i
j
j
ji
ij
ij
ji
0
0
00
0
ai and bi are not independent
Computational Finance
Arbitrage Pricing Formula
Once constants are known, the expected rate of return of an asset i is determined by the factor loading.
The expected rate of return of asset i
CAPM?
10
0
0
][
][
][][
i
i
ii
iii
iii
b
cfEb
fEbcb
fEbarE
fbar
Computational Finance
CAPM as a consequence of APT
The factor is the rate of return on the market
APT is identical to the CAPM with
fMifi
ii
fM
f
M
rrEbrrE
brE
rrE
r
rf
][][
][
][
10
1
0
iib