Computational Finance
Lecture 1
Introduction to Financial Engineering
Computational Finance
Computational Finance (quantitative finance,
financial engineering, or mathematical finance):
A cross-disciplinary field which uses quantitative
methods developed

Risk Aversion and Risk Premium
The degree to which investors are willing to
commit funds to stocks depends on their risk
aversion (), the reluctance to accept risk.
In the presence of risk aversion, risky assets must
offer higher return in order to induce

Portfolio and Diversification
One important weapon to reduce the investment
risk is through diversification, or investing in
portfolios.
Portfolio and Diversification
Example:
Suppose that we form a portfolio with 40%
invested in the stock fund and 60% in

The Risk-Return Tradeoff
We can assess the benefit from diversification by
comparing the risk and expected return of a betterdiversified portfolio to a less-diversified
benchmark.
Example:
Suppose that an investor estimates the following input
data:
E(rB

Is Normal Approximation Good?
(Continued)
Kurtosis: a sample Kurtosis of n values can be
defined as
n
( r - r )
i
4
/n
i=1
1
2
ri - r
n i=1
n
(
)
2
- 3.
The kurtosis of a normal sample should be
close to 0. Positive kurtosis indicate higher
frequency o

Normal Distribution (Continued)
Normal distributions have many nice
mathematical properties. For instance,
If X ~ N(m, s ) , it can be decomposed into
X = m +sY
where Y ~ N(0,1).
For two independent normal
X1 ~ N(m1, s 1 ) and X2 ~ N(m2 , s 2 ),
the sum

Normal Distribution (Continued)
We may use the probability density function to
calculate probabilities of a normal random number.
For instance,
a
1
P(X a) =
e
- s 2p
( x-m )2
2s 2
dx
and
a
1
P(b X a) =
e
b s 2p
( x-m )2
2s 2
dx
Normal Distribution (Cont

Scenario Analysis and Probability
Distributions
From the sample data, we also can estimate the
probabilities associated with each possible return
rate. These probabilities is called the probability
distribution function of returns.
Examine the IBM stock s

Measuring Risk: Scenario Analysis and
Probability Distributions
When we attempt to quantify risk, we often use an
approach known as scenario analysis: we devise a
list of possible economic outcomes, or scenarios,
and specify both the likelihood (i.e., the

Population and Samples
An ideal scenario analysis should be based on all
possible outcomes of a random variable
(population). But we usually only have access to
a subset (sample) of the population. Therefore,
statistical inference are made from a sample t

Computational Finance
Lecture 3
Risk and Return
Part III
The Capital Asset Pricing Model (CAPM)
Main Contents
The Capital Asset Pricing model (CAPM)
Derivation of CAPM
Beta and alpha
Applications
Performance evaluation
Pricing
Arbitrage Pricing Theory (AP

Computational Finance
Lecture 3
Risk and Returns
Part I: Risk Quantification
Main Contents
Portfolio returns
Risk characterization:
Variance and standard deviation
Probability Distribution
Normal approximation
Diversification
Covariance and correlation
Ef

Computational Finance
Lecture 3
Risk and Returns
Part II: Mean-Variance Portfolio Theory
Main Contents
Diversification and Portfolio theory
Mean-standard deviation diagram of portfolios
Feasible set and efficient frontier
Optimal risky portfolio with the

Portfolio Mean and Variance
Suppose that there are n assets with (random)
rates of return r1, r2 , , rn.
These have expected values
E(r1 ) = r1, E(r2 ) = r2, , E(rn ) = rn
The variance of the return of asset i is s i , the
covariance of the return of asse

Example: CUHK Fund Analysis
(Continue)
Step 4. Finally we need to conduct an efficiency check
on the fund by calculating its Sharpe ratio S
r - rf = Ss
Compare it with the Sharpe ratio of the Hang Seng
Index.
Pricing Form of the CAPM
The CAPM is a pricing

Example: CUHK Fund Analysis
The Chinese University organizes a mutual fund
for its employees. The fund has the 10-year record
of return shown in the table in the next slide.
We would like to evaluate this funds performance
in terms of mean-variance portfo

CAPM and Its Investment Implications
The CAPM theory is widely used in the
investment management industry to measure the
performance of various investment portfolios.
The security market line.
Given the beta of an individual asset, we call it is
underpric

Market Equilibrium
Suppose that
Investors cannot affect prices by their individual trades.
All investors plan for one identical holding period.
Investors form portfolios from a common universe of
publicly traded financial assets, and have access to
unlimi

The Pricing Model
The capital market line relates the expected rate of
return of an efficient portfolio to its standard
deviation, but it does not show how the expected
rate of an individual asset return relates to its
individual risk.
The capital asset p

Example: The Oil Venture
(Ex. 7.6 in Investment Sciences)
Consider an oil drilling company. The stock price
of that company is $875. It is expected to yield the
equivalent of $1,000 after 1 year, but due to high
uncertainty about how much oil is at the dr

The Two-Fund Theorem
In the previous example, changing r will sweep
out all the possibility of efficient portfolios. It is a
parabolic curve in the mean-standard deviation
diagram.
Alternatively, we can obtain the efficient frontier
through the following

The Markowitz Model
We can extend the two-risky-assets portfolio
construction methodology to cover the case of
many risky assets and a risk-free asset.
Input data:
Stock expected return:
Stock covariance: s i, j
E(ri )
Efficient Diversification with Many

Mathematical Solution of the Markowitz
Problem
We can find the conditions for a solution to the
above optimization problem using Lagrange
multipliers.
Form a Lagrangian
n
n
1 n
L =
wi w js i, j - l1 wi E(ri ) - r - l2 wi -1
2 i, j=1
i=1
i=1
Differentiat

The Optimal Risky Portfolio with a RiskFree Asset
Suppose that investors pursue highest Sharpe ratio
in their investments.
Notice that the slope of the CAL is the Sharpe
ratio of the risky portfolio. The optimal risky
portfolio should be the one the CAL r

Inclusion of a Risk-Free Asset
Risk-free asset offers a deterministic rate of
return rf. Its variance and standard deviation
should be 0, and its covariance and correlation
with other assets also should be 0.
Suppose that two assets, one risk-free and the

Minimum-Variance Portfolio
We can determine explicitly the weights of the
minimum-variance (least risky) portfolio:
Minimizing
s = a 2s B2 + (1- a )2 s S2 + 2a (1- a )rBSs Bs S
The optimal weights are
s S2 - s Bs S r BS
a= 2
s S + s B2 - 2s Bs S rBS
The M

The Expectations Theory of Term
Structure
For investments in 2-year bonds to be competitive
with the strategy of rolling over 1-year bonds, the
two strategies must offer comparable returns.
(1+ s2 )2 = (1+ s1 )(1+ f1,2 )
Forward Rates ()
Forward rates are

Bond Sensitivity to Interest Rate:
Duration
The duration of a bond is a weighted average of
the times when each coupon or principal payment
are made.
The weight applied to each time is equal to the
proportion of the bonds total value accounted for
by that