AFIN3052
APPLIED PORTFOLIO MANAGEMENT - WEEK 1
What is Portfolio Management?
• Portfolio management in Australia is regulated by ASIC.
• They refer portfolio management to Managed Investment Schemes.
Managed Investment Scheme
• Manages investment schemes are also known as 'managed funds', 'pooled investments' or
'collective investments'.
• People are brought together to contribute money to get an interest in the scheme ('interests' in
a scheme are a type of 'financial product' and are regulated by the Corporations Act 2001)
• Money is pooled together with other investors or used in a common enterprise
• A 'responsible entity' operates the scheme. Investors do not have day to day control over the
operation.
Investment Companies
• Record keeping and administration
• Investment companies issue periodic status reports, keeping track of capital gains
distributions, dividends, investments, and redemptions, and they may reinvest dividend and
interest incomes for shareholders
• Diversification and divisibility
• By pooling their money, investment companies enable investors to hold fractional shares of
many different securities. They can act as large investors even if any individual shareholder
cannot.
• Professional management
• Investment companies can support full-time staff of security analysts and portfolio managers
who attempt to achieve superior investment results for their investors.
• Lower transaction costs
• Because they trade large blocks of securities, investment companies can achieve substantial
savings on brokerage fees and commissions.
What is a Portfolio Manager?
• Responsible for delivering investment performance;
• Full authority to make at least some investment decisions;
• Accountable for investment results.
Portfolio Management
• Specifying Investor Objectives - traditionally done with reference to an economic model of
preferences (utility)
• Specifying the Dynamics of Investable Asset Values - assuming a model of randomness and
estimating it’s parameters
• Optimisation - construction of a portfolio to meet the objectives specified in step 1 given the
distribution of future outcomes estimated in step 2
• Empirical Evaluation an Revision - involves evaluating the model and outcomes with reference to
expectations and revising accordingly
, AFIN3052
What Decision does a Portfolio Manager Make?
• Asset Allocation
• Determination of weights in asset classes in light of objectives
• Key determinants of cash flow pattern and performance particularly long-term performance
• Client-dependent
• Manage Money within a Style or Asset Class
• Performance is usually measured with respect to clearly defined benchmarks
• Client-independent
Modelling Returns
• QQ-Plots (Quantile-Quantile plots) is a useful first step tool for us to check whether our data fit
the model (or vice versa)
• We can see below that the distribution derivates at both tails (returns are not normally distributed)
•
• Although we know that log-return does not follow normal distribution very well, there are any
good properties that normal distribution has.
• Sum of normally distributed random variables is still normal.
• If log-return follows normal distribution, the price follows log-normal distribution. This applies to
multi period returns.
• Suppose we have independent normally distributed random (i.i.d) returns:
• Sum of returns are still normal:
• If they are i.i.d [independent and identically distributed random variables],
• Then,
• A log-normal random variable follows normal distribution after taking natural log.
• The price:
•
Portfolio Returns
• A portfolio consists of different assets with different return distributions.
• We can call the underlying assets Xi, with specific weightings Wi.
• Assume returns (ri) on the assets follow normal distributions, we have the return of the portfolio
follows normal distribution.
• Where is the covariance of asset i and asset j, and is the variance of asset i.
, AFIN3052
Portfolio of Two Assets
• The return on a two-asset portfolio is:
•
• The variance of this portfolio is:
•
•
Matrix Notation
• A vector/matrix is a collection of scaler numbers.
• A vector is a one-row or one-column matrix.
• Transpose is an operation to turn a row vector into a column vector, or vice versa.
•
Matrix Operations
• Summation or Subtractions - only defined to equal size operands
• Multiplication - Only defined to agreed dimension operands
Matrix Notation for Portfolio Construction
• Suppose we have a portfolio consisting of two assets, we can represent our portfolio return as:
•
• The portfolio variance is:
• =
• = =
, AFIN3052
APPLIED PORTFOLIO MANAGEMENT - WEEK 2
Mean and Variance of Portfolio Return
• We treat the investment return R as discrete random variable, characterised by its probability
distribution. For example, in the discrete case, we have:
•
• Two important statistics (discrete random variable)
• Mean =
• Variance =
Statement of the Problem
• A portfolio is defined by allocating fractions of initial wealth to individual assets. The fractions
(weights) must sum to one (some weights may be negative, corresponding to short selling)
• Return is quantified by portfolio’s expected rate of return;
• Risk is quantified by variance of portfolio’s rate of return.
• Goal: maximise return for a given level of risk, minimise risk for a given level of return
Limitations in the Mean Variance Portfolio
• Only the mean and variance (or sd) are taking into consideration. The higher order moments
(skewness) of the probability distribution are irrelevant in the formulation.
• Only the Gaussian (normal) distribution is fully specified by its mean and variance.
Unfortunately, the rates of return of risky assets are not Gaussian in general.
• Calibration of parameters in the ,model is always challenging. Using historical data:
• Sample mean:
• Sample variance:
• Where rt is the historical rate of return observed at time t.
Short Sales
• It is possible to sell an asset that you do not own through the process of short selling, or shorting
the asset. You then sell the borrowed asset to someone else, receiving the current price of the asset
in cash. At a later date, you repay your loan of the asset by purchasing the asset and return the
asset to your lender. Short selling is profitable if the asset price declines.
• Positive number = long (buy)
• Negative number = short (sell)
Portfolio Weights
• Suppose now that n different assets are available. We form a portfolio of these n assets. Suppose
that this is done by apportioning an amount of X0 among the n assets. We then select amounts;
Xi, I = 1, 2, etc, to invest into each asset, such that the sum of Xi = X0.
Markowitz Mean-Variance Formulation
• We consider a single-period investment model. Suppose there are N risky assets, whose rates of
return are given by the random variables, r1, …, rN.
• Let be the weighting vector with the proportion of wealth invested in each
asset summing to 1.
• The rate of return of the portfolio rp is:
APPLIED PORTFOLIO MANAGEMENT - WEEK 1
What is Portfolio Management?
• Portfolio management in Australia is regulated by ASIC.
• They refer portfolio management to Managed Investment Schemes.
Managed Investment Scheme
• Manages investment schemes are also known as 'managed funds', 'pooled investments' or
'collective investments'.
• People are brought together to contribute money to get an interest in the scheme ('interests' in
a scheme are a type of 'financial product' and are regulated by the Corporations Act 2001)
• Money is pooled together with other investors or used in a common enterprise
• A 'responsible entity' operates the scheme. Investors do not have day to day control over the
operation.
Investment Companies
• Record keeping and administration
• Investment companies issue periodic status reports, keeping track of capital gains
distributions, dividends, investments, and redemptions, and they may reinvest dividend and
interest incomes for shareholders
• Diversification and divisibility
• By pooling their money, investment companies enable investors to hold fractional shares of
many different securities. They can act as large investors even if any individual shareholder
cannot.
• Professional management
• Investment companies can support full-time staff of security analysts and portfolio managers
who attempt to achieve superior investment results for their investors.
• Lower transaction costs
• Because they trade large blocks of securities, investment companies can achieve substantial
savings on brokerage fees and commissions.
What is a Portfolio Manager?
• Responsible for delivering investment performance;
• Full authority to make at least some investment decisions;
• Accountable for investment results.
Portfolio Management
• Specifying Investor Objectives - traditionally done with reference to an economic model of
preferences (utility)
• Specifying the Dynamics of Investable Asset Values - assuming a model of randomness and
estimating it’s parameters
• Optimisation - construction of a portfolio to meet the objectives specified in step 1 given the
distribution of future outcomes estimated in step 2
• Empirical Evaluation an Revision - involves evaluating the model and outcomes with reference to
expectations and revising accordingly
, AFIN3052
What Decision does a Portfolio Manager Make?
• Asset Allocation
• Determination of weights in asset classes in light of objectives
• Key determinants of cash flow pattern and performance particularly long-term performance
• Client-dependent
• Manage Money within a Style or Asset Class
• Performance is usually measured with respect to clearly defined benchmarks
• Client-independent
Modelling Returns
• QQ-Plots (Quantile-Quantile plots) is a useful first step tool for us to check whether our data fit
the model (or vice versa)
• We can see below that the distribution derivates at both tails (returns are not normally distributed)
•
• Although we know that log-return does not follow normal distribution very well, there are any
good properties that normal distribution has.
• Sum of normally distributed random variables is still normal.
• If log-return follows normal distribution, the price follows log-normal distribution. This applies to
multi period returns.
• Suppose we have independent normally distributed random (i.i.d) returns:
• Sum of returns are still normal:
• If they are i.i.d [independent and identically distributed random variables],
• Then,
• A log-normal random variable follows normal distribution after taking natural log.
• The price:
•
Portfolio Returns
• A portfolio consists of different assets with different return distributions.
• We can call the underlying assets Xi, with specific weightings Wi.
• Assume returns (ri) on the assets follow normal distributions, we have the return of the portfolio
follows normal distribution.
• Where is the covariance of asset i and asset j, and is the variance of asset i.
, AFIN3052
Portfolio of Two Assets
• The return on a two-asset portfolio is:
•
• The variance of this portfolio is:
•
•
Matrix Notation
• A vector/matrix is a collection of scaler numbers.
• A vector is a one-row or one-column matrix.
• Transpose is an operation to turn a row vector into a column vector, or vice versa.
•
Matrix Operations
• Summation or Subtractions - only defined to equal size operands
• Multiplication - Only defined to agreed dimension operands
Matrix Notation for Portfolio Construction
• Suppose we have a portfolio consisting of two assets, we can represent our portfolio return as:
•
• The portfolio variance is:
• =
• = =
, AFIN3052
APPLIED PORTFOLIO MANAGEMENT - WEEK 2
Mean and Variance of Portfolio Return
• We treat the investment return R as discrete random variable, characterised by its probability
distribution. For example, in the discrete case, we have:
•
• Two important statistics (discrete random variable)
• Mean =
• Variance =
Statement of the Problem
• A portfolio is defined by allocating fractions of initial wealth to individual assets. The fractions
(weights) must sum to one (some weights may be negative, corresponding to short selling)
• Return is quantified by portfolio’s expected rate of return;
• Risk is quantified by variance of portfolio’s rate of return.
• Goal: maximise return for a given level of risk, minimise risk for a given level of return
Limitations in the Mean Variance Portfolio
• Only the mean and variance (or sd) are taking into consideration. The higher order moments
(skewness) of the probability distribution are irrelevant in the formulation.
• Only the Gaussian (normal) distribution is fully specified by its mean and variance.
Unfortunately, the rates of return of risky assets are not Gaussian in general.
• Calibration of parameters in the ,model is always challenging. Using historical data:
• Sample mean:
• Sample variance:
• Where rt is the historical rate of return observed at time t.
Short Sales
• It is possible to sell an asset that you do not own through the process of short selling, or shorting
the asset. You then sell the borrowed asset to someone else, receiving the current price of the asset
in cash. At a later date, you repay your loan of the asset by purchasing the asset and return the
asset to your lender. Short selling is profitable if the asset price declines.
• Positive number = long (buy)
• Negative number = short (sell)
Portfolio Weights
• Suppose now that n different assets are available. We form a portfolio of these n assets. Suppose
that this is done by apportioning an amount of X0 among the n assets. We then select amounts;
Xi, I = 1, 2, etc, to invest into each asset, such that the sum of Xi = X0.
Markowitz Mean-Variance Formulation
• We consider a single-period investment model. Suppose there are N risky assets, whose rates of
return are given by the random variables, r1, …, rN.
• Let be the weighting vector with the proportion of wealth invested in each
asset summing to 1.
• The rate of return of the portfolio rp is:
