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Computational Finance 1/23
Panos Parpas
Single Period Markowitz Model
381 Computational Finance
Imperial College
London
Computational Finance 2/23
Topics Covered
Notation and Terminology
Random variables, mean, (co)variance, correlationRandom variables, mean, (co)variance, correlation
Asset return, portfolio return and riskAsset return, portfolio return and risk
Portfolio Optimisation
Optimal asset allocation, risk management Optimal asset allocation, risk management
Short sale, diversificationShort sale, diversification
Mean-Variance model, efficient frontier
Computational Finance 3/23
Terminology
Random Variable y is a random variable, takes finite number of values, yj for j=1,2,…,m.
a probability (associated with each event) represents the relative chance of an occurrence of yj.such that
Expected Value (Mean value or mean) average value obtained by regarding probabilities as frequencies
Variance measure of possible deviation from the mean
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relative frequency
finite number ofpossibilities
Expected value of squared variable
how much y tends to vary from its mean
Computational Finance 4/23
Terminology
Covariance of two random variables
Correlation between two random variables
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Computational Finance 5/23
Asset Returns
Asset: investment instrument that can be bought and sold
uncertain asset prices – the return is random uncertainty described in probabilistic terms
Asset return: you purchase an asset today and sell it next year
Total return on this investment is defined as
Rate of return:
Rate of return acts much like an interest rate
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Computational Finance 6/23
Returns
Returns Returns Time Period r1 r2
1 6 82 4 23 7 114 3 -15 8 126 2 -27 11 138 -1 -3
MEAN 5 5VARIANCE 3.77964 6.36209
Computational Finance 7/23
Portfolio Return
Consider n risky assets to form a portfolio and invest among n assetsSelect an amount invested in the ith asset
The amounts invested can be expressed as fractions of the total investment
is fraction or weight of asset i in portfolio
Let the total return of asset i be . The amount of money gained at the end of the period on asset i is
The total return of the portfolio The rate of return of portfolio
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Computational Finance 8/23
Portfolio Return Both the total return and the rate of return of the portfolio of assets are equal to the weighted sum of the corresponding individual asset returns, with the weight of an asset being its relative weight in the portfolio.
Expected return of the portfolio is the weighted sum of the individual expected rates of return
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Computational Finance 9/23
Example
Consider a portfolio of a risk-free asset and two risky stocks and three equally likely states
Find the expected return of the portfolio formed by 3 assets
Scenario tree with 3 states
States
(s) T-Bill
Return% Stock A Return%
Stock B Return %
(i=1) (i=2) (i=3) s1 “boom” 5 16 3 s2 “normal” 5 10 9 s3 “recession” 5 1 15
root represents today
future uncertainty is discretised
by 3 events (states)
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Computational Finance 10/23
Example: Expected Returns
The expected return of risk free asset is For stock A and B, expected return of the portfolio
The expected return of the portfolio
For equally weighted portfolio
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Computational Finance 11/23
Risk
Risk: a chance that investment’s actual return will be different than
expected –includes losing some or all of original investment
risk averse and risk-seeking (loving)
Systematic risk: influences a large number of assets, such as political
events – impossible to protect yourself against this risk
Unsystematic risk: (specific risk) affects very small number of assets.
For example, news that affects a specific stock such as a sudden strike
Diversification is the only way to protect yourself
Others: credit (default) risk, foreign exchange risk, interest rate risk,
political risk, market risk
Computational Finance 12/23
Diversification
Risk management technique
Mixes a wide variety of investments within a portfolio
Objective is to minimize the impact that any one security will
have on overall performance of the portfolio
For the best diversification, portfolio should be spread among many different assets; cash, stocks, bonds
securities should vary in risk and
securities should vary by industry to minimize unsystematic risk to different
companies
Computational Finance 13/23
Portfolio Risk
For n securities, portfolio risk – variance of the portfolio return
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Expected value of squared variable- how much rate of return of portfolio tends to vary from its mean
Computational Finance 14/23
Example: Variance of Portfolio
The variance and covariance are calculated as
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Computational Finance 15/23
Example: Variance of Portfolio
The portfolio risk is
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Computational Finance 16/23
Short Sale
It is possible to sell an asset that you do not own through the short selling (shorting) the asset. In order to implement short selling
borrow the asset from the owner such as brokerage firm
sell the borrowed asset to someone else, receiving an amount x0
at a later date purchase the asset for x1, and return the asset to lender
The short selling is profitable if the asset price declines
Risky since the potential for loss is unlimited –if asset value increases
Although prohibited within certain financial institutions, not universally
forbidden
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0101 then , If xxlossxx
Computational Finance 17/23
Example:Short-sale
Assume that company X has a poor outlook next month. The stock is now trading at £65, but you see it trading much lower than this price in the future.
You decide to take risk and trade on this stock. Two things can happen; stock price can go up or down
Predicted Stock price £40
Predicted Stock price £85
Action taken Price(£) Cost(£) Price(£) Cost(£) Borrowed 100
shares of X & sell 65 6500 65 6500
Bought back 100 shares of X
40 -4000 85 -8500
Profit/Loss 2500 -2000
Make Money
Lose Money
RISKY: no guarantee that the price of a short stock will drop
Computational Finance 18/23
Optimal Portfolio
Portfolio theory • effects of investor decisions on security prices • relationship that should exist between the returns and risk.
• possible to have different portfolios varying levels of risk &return• decide how much risk you can handle and allocate (or diversify) portfolio according to this decision
Harry Markowitz: 1990 Nobel Prize winner in Economic Sciences• published in 1952 Journal of Finance titled “Portfolio Selection”• formalized an integrated theory of diversification, portfolio risks, efficient & inefficient portfolios
Maximize (expected return of portfolio)
(weighted sum of individual expected rates of return)
Minimize (portfolio risk)
(expected value of squared variable- how much rate of return of portfolio
tends to vary from its mean)
Computational Finance 19/23
The Markowitz Model
Construct portfolio under uncertainty
Suppose that there are n assets with random rates of return
with mean values
The portfolio consists of n assets
are the portfolio weights
such that
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Computational Finance 20/23
Maximum Expected Wealth Problem
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•No risk –maximize expected portfolio return –risk-neutral approach
Computational Finance 21/23
Minimum Variance Portfolio
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•Take risk into account
Computational Finance 22/23
Mean-Variance Optimisation Problem
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• trade-off between expected rate of return &variance of rate of return in a portfolio• naturally leads to diversification by taking risk attitudes in to account
Computational Finance 23/23
Efficient Frontier
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um• Varying the desired level of return and repeatedly solving QP problems identifiesthe minimum variance portfolio for each value of : efficient portfoliosr