FINC3017: Investment and Portfolio Management
FINC3017: Investment and Portfolio Management
Topic 1: Introduction Investment Funds •
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Unit Trusts: investor’s funds are pooled, usually into specific types of assets. o Investors are assigned tradeable units in the fund (e.g. listed property trusts) o Closed end funds sell a fixed number of units to investors o An unlisted trust can issue new units at any time; the value of each unit depends on the value of the underlying investments – open ended funds – known as mutual funds in the U.S. Pension funds: accept and manage contributions from employers/employees to provide retirement income benefits o Structure can be defined benefit (retirement payout is determined on a formula), defined contribution (payout depends on investments of contributions, also known as accumulation funds) Exchange traded funds: listed on the stock market and trade daily unlike other managed funds o Hybrid between a listed security and an open-ended fund o Ease of access and low cost of entry and exit o Often have an explicit objective and benchmark (e.g. index tracking) Hedge Funds – seek to hedge against risky price movements via short-selling, arbitrage trading, derivatives, distressed securities, low-grade bonds and high leverage portfolios so as to maximise the expected return-risk of the portfolio o Access is limited Infrastructure funds – focus on public infrastructure assets – often originate from government privatisations
Asset Allocation • • • • •
Strategic asset allocation is a benchmark allocation between asset classes: cash, fixed interest, property, alternative investments and equity Managers will generally have a range of portfolio weights for each asset class which depends on the objective of the fund – balanced, conservative, imputation or inflation Australian funds have highest weight in equities, then foreign assets Tactical asset allocation is active between asset classes – it takes portfolio holdings away from the strategic asset allocation Managers attempt to exploit temporary mispricing by adjusting exposure to different asset classes – typically move between a maximum and minimum bound on the amounts invested in each class
Topic 2: Investment Decisions under Uncertainty Utility Utility functions provide a means to rank alternatives. Investors choose among alternatives to maximise expected utility. Axioms of expected utility theory: -
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Comparability: if an investor can state whether they prefer A to B, B to A or indifference Transitivity: if an investor prefers A to B, B to C, then they prefer A to C Independence: investor is indifferent between two certain outcomes, G and H. If J is uncertain, they will be indifferent between: o G with prob. P and J with prob. (1-p) o H with prob. P and J with prob. (1-p) Certainty equivalent: for every gamble there is a value that the investor will be indifferent between the gamble and that value (certainty equivalent) Properties of utility functions o More is preferred to less (non-satiation) therefore the fist derivative of the utility function is possible (increasing function) o Adding a constant to a utility function or multiply utility functions does not change rankings A fair gamble: expected value of the gamble is equal to its cost. This could be a risky investment whose expected return equals the risk free rate of return, or a risky investment with zero risk premium Risk averse investor: will reject a fair gamble; they require a premium to accept risk o The expected utility from the risky investment must be less than the expected utility of wealth from the risk free investment o Concave utility function – negative second derivative o Diminishing utility of wealth Risk neutral investor: will be indifferent to a fair gamble o Linear utility function – second derivative is zero Risk seeking investor: will prefer a fair gamble to risk free investment; prepared to pay a premium to take risk o Convex utility function – second derivative is positive
We assume that investors are risk averse, maximise expected utility of wealth, prefer more wealth to less and have diminishing marginal utility of wealth. Absolute risk aversion
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• Measures shifts in investor preferences in response to wealth If the amount invested in risky assets increases as wealth increases then the investor has decreasing absolute risk aversion.
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The derivative of ARA indicates how absolute risk aversion changes as wealth changes It is generally assumed that investors have decreasing absolute risk aversion
Relative Risk Aversion •
The percentage of wealth invested in risky assets changes as wealth changes
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The derivative indicates how relative risk aversion changes as wealth changes – no consensus on how relative risk aversion changes as wealth changes
Types of Utility Functions Log utility functions: U(W) = ln(W) -
Decreasing ARA, constant RRA
2 Quadratic utility functions: U(W) = W - cW
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Increasing ARA, increasing RRA
Exponential utility: U(W) = 1- e-!W where y is the risk aversion coefficient -
Constant ARA, decreasing RRA
Power utility: U (W ) =
W 1! A 1! A
where A>0
Mean-Variance Expected return and standard deviation are appropriate to approximate investor preferences if: -
The distribution of expected returns is normal – investments can then be ranked according to risk and return Utility functions are quadratic – expected utility is determined by expected wealth and standard deviation of expected wealth
Defining expected wealth by means and variances for quadratic functions:
U(W) = W - cW 2 -
However, quadratic utility has some problems
E[U(1+ r)] = E(1+ r) - cE(1+ r)2 ! c" 12+r
o It implies investors become satiated (at a certain point an increase in wealth reduces utility) o It implies increasing absolute risk aversion ! Therefore risky assets are inferior goods, instead of normal goods
-
The mean variance criterion states that portfolio A is preferable to B if its expected return is higher and standard deviation is lower
Topic 3: Efficient Portfolio Selection Mean-Variance Opportunity Set The combinations of risky assets that minimise portfolio variance for a given level of portfolio expected return.
Once an investor identifies these portfolios they will pick the one that maximises E(U). -
Investors that desire to maximise expected returns, without regard to variance, places all their funds in the single security with the highest expected return. Investors that wish to exclusively minimise variance will build a portfolio of the two stocks –requires two expected returns, two variances and one covariance/correlation
Gains from Diversification • •
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The degree to which a two-security portfolio reduces variance depends on the correlation between the returns of the two assets The smaller the correlation, the higher the gains from diversification o Investors can stabilise the rate of return, without reducing portfolio expected return o Gains from diversification occur as long as the securities are not perfectly correlated Removes idiosyncratic (firm-specific) risk Portfolio variance always decline as more stocks are added, however there are diminishing marginal returns
Portfolio with n risky assets -
For 3 securities, 3 expected returns, 3 variances and 3^2 covariance terms are required to estimate For 200 securities, over 20,000 parameters need to be estimated If you are investing in a global portfolio or trying to track an index this is incredibly difficult
Efficient portfolio: no other portfolio will have the same expected return and a lower variance of returns The result of minimising portfolio variance at each expected level of portfolio return is called the minimum-variance set. This includes efficient and inefficient sets.
Single Index Model The SIM reduces the problem of the Markowitz framework that requires a significant number of inputs. Individual returns are generally correlated with each other as a common set of factors or a common response to market changes drive returns. Therefore this correlation can be captured by relating the stock return to a market index. -
SIM describes an asset return as made up of a constant and a sensitivity to a factor (often the market index)
Sharpe assumes that returns to the factor = At + ct (A is constant and c is a random residual) Markowitz - Involves much more data, and computationally difficult - Makes no assumptions about return generating process
SIM - Much easier to compute - Makes assumptions about return generating process
Both approaches are only as good as the inputs into the model – are your estimates accurate? Capital Allocation between Risk-Free and Risky Assets The optimal risky portfolio is selected by investors based on their own preferences. It will depend on their utility function in expected return-standard deviation space. All selected portfolios will be on the efficient part of the mean-variance frontier – it will provide the maximum return for a given level of risk. Capital allocation line: a combination of a risky portfolio and a risk free asset (E(rP ) ! rf ) ! E(ri ) = rf + i !P
Sharpe Ratio Reward to variability ratio: S =
E(rP ) ! rf !P
This is the gradient of the CAL. It gives an indication of the additional reward that an investor will receive for each additional unit of risk they take on.
The point at which utility is maximised is by differentiating and maximising the following:
max U = E[rC ] " 12 A! C2 x
= rf + x(E[rP " rf ]) " 12 Ax 2! P2 As A (risk aversion) increases, x decreases
We use the mean-variance theory to choose which risky portfolio of N-assets we wish to invest in and combine this with a risk free asset. The equations for this combination are given by: E[rC ] = rf + x(E[rP ] ! rf )
! C = x! P This is a straight line connecting the risk free asset to a portfolio on the efficient frontier One Fund Theorem: When a risk free asset is available there is only one risky fund that people will want to invest in. -
The efficient frontier now becomes the straight line which is tangential to the efficient frontier
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This may be found by maximising the Sharpe ratio
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Topic 1: Introduction Investment Funds •
•
•
•
•
Unit Trusts: investor’s funds are pooled, usually into specific types of assets. o Investors are assigned tradeable units in the fund (e.g. listed property trusts) o Closed end funds sell a fixed number of units to investors o An unlisted trust can issue new units at any time; the value of each unit depends on the value of the underlying investments – open ended funds – known as mutual funds in the U.S. Pension funds: accept and manage contributions from employers/employees to provide retirement income benefits o Structure can be defined benefit (retirement payout is determined on a formula), defined contribution (payout depends on investments of contributions, also known as accumulation funds) Exchange traded funds: listed on the stock market and trade daily unlike other managed funds o Hybrid between a listed security and an open-ended fund o Ease of access and low cost of entry and exit o Often have an explicit objective and benchmark (e.g. index tracking) Hedge Funds – seek to hedge against risky price movements via short-selling, arbitrage trading, derivatives, distressed securities, low-grade bonds and high leverage portfolios so as to maximise the expected return-risk of the portfolio o Access is limited Infrastructure funds – focus on public infrastructure assets – often originate from government privatisations
Asset Allocation • • • • •
Strategic asset allocation is a benchmark allocation between asset classes: cash, fixed interest, property, alternative investments and equity Managers will generally have a range of portfolio weights for each asset class which depends on the objective of the fund – balanced, conservative, imputation or inflation Australian funds have highest weight in equities, then foreign assets Tactical asset allocation is active between asset classes – it takes portfolio holdings away from the strategic asset allocation Managers attempt to exploit temporary mispricing by adjusting exposure to different asset classes – typically move between a maximum and minimum bound on the amounts invested in each class
Topic 2: Investment Decisions under Uncertainty Utility Utility functions provide a means to rank alternatives. Investors choose among alternatives to maximise expected utility. Axioms of expected utility theory: -
-
•
•
•
• •
Comparability: if an investor can state whether they prefer A to B, B to A or indifference Transitivity: if an investor prefers A to B, B to C, then they prefer A to C Independence: investor is indifferent between two certain outcomes, G and H. If J is uncertain, they will be indifferent between: o G with prob. P and J with prob. (1-p) o H with prob. P and J with prob. (1-p) Certainty equivalent: for every gamble there is a value that the investor will be indifferent between the gamble and that value (certainty equivalent) Properties of utility functions o More is preferred to less (non-satiation) therefore the fist derivative of the utility function is possible (increasing function) o Adding a constant to a utility function or multiply utility functions does not change rankings A fair gamble: expected value of the gamble is equal to its cost. This could be a risky investment whose expected return equals the risk free rate of return, or a risky investment with zero risk premium Risk averse investor: will reject a fair gamble; they require a premium to accept risk o The expected utility from the risky investment must be less than the expected utility of wealth from the risk free investment o Concave utility function – negative second derivative o Diminishing utility of wealth Risk neutral investor: will be indifferent to a fair gamble o Linear utility function – second derivative is zero Risk seeking investor: will prefer a fair gamble to risk free investment; prepared to pay a premium to take risk o Convex utility function – second derivative is positive
We assume that investors are risk averse, maximise expected utility of wealth, prefer more wealth to less and have diminishing marginal utility of wealth. Absolute risk aversion
•
• Measures shifts in investor preferences in response to wealth If the amount invested in risky assets increases as wealth increases then the investor has decreasing absolute risk aversion.
• •
The derivative of ARA indicates how absolute risk aversion changes as wealth changes It is generally assumed that investors have decreasing absolute risk aversion
Relative Risk Aversion •
The percentage of wealth invested in risky assets changes as wealth changes
•
The derivative indicates how relative risk aversion changes as wealth changes – no consensus on how relative risk aversion changes as wealth changes
Types of Utility Functions Log utility functions: U(W) = ln(W) -
Decreasing ARA, constant RRA
2 Quadratic utility functions: U(W) = W - cW
-
Increasing ARA, increasing RRA
Exponential utility: U(W) = 1- e-!W where y is the risk aversion coefficient -
Constant ARA, decreasing RRA
Power utility: U (W ) =
W 1! A 1! A
where A>0
Mean-Variance Expected return and standard deviation are appropriate to approximate investor preferences if: -
The distribution of expected returns is normal – investments can then be ranked according to risk and return Utility functions are quadratic – expected utility is determined by expected wealth and standard deviation of expected wealth
Defining expected wealth by means and variances for quadratic functions:
U(W) = W - cW 2 -
However, quadratic utility has some problems
E[U(1+ r)] = E(1+ r) - cE(1+ r)2 ! c" 12+r
o It implies investors become satiated (at a certain point an increase in wealth reduces utility) o It implies increasing absolute risk aversion ! Therefore risky assets are inferior goods, instead of normal goods
-
The mean variance criterion states that portfolio A is preferable to B if its expected return is higher and standard deviation is lower
Topic 3: Efficient Portfolio Selection Mean-Variance Opportunity Set The combinations of risky assets that minimise portfolio variance for a given level of portfolio expected return.
Once an investor identifies these portfolios they will pick the one that maximises E(U). -
Investors that desire to maximise expected returns, without regard to variance, places all their funds in the single security with the highest expected return. Investors that wish to exclusively minimise variance will build a portfolio of the two stocks –requires two expected returns, two variances and one covariance/correlation
Gains from Diversification • •
• •
The degree to which a two-security portfolio reduces variance depends on the correlation between the returns of the two assets The smaller the correlation, the higher the gains from diversification o Investors can stabilise the rate of return, without reducing portfolio expected return o Gains from diversification occur as long as the securities are not perfectly correlated Removes idiosyncratic (firm-specific) risk Portfolio variance always decline as more stocks are added, however there are diminishing marginal returns
Portfolio with n risky assets -
For 3 securities, 3 expected returns, 3 variances and 3^2 covariance terms are required to estimate For 200 securities, over 20,000 parameters need to be estimated If you are investing in a global portfolio or trying to track an index this is incredibly difficult
Efficient portfolio: no other portfolio will have the same expected return and a lower variance of returns The result of minimising portfolio variance at each expected level of portfolio return is called the minimum-variance set. This includes efficient and inefficient sets.
Single Index Model The SIM reduces the problem of the Markowitz framework that requires a significant number of inputs. Individual returns are generally correlated with each other as a common set of factors or a common response to market changes drive returns. Therefore this correlation can be captured by relating the stock return to a market index. -
SIM describes an asset return as made up of a constant and a sensitivity to a factor (often the market index)
Sharpe assumes that returns to the factor = At + ct (A is constant and c is a random residual) Markowitz - Involves much more data, and computationally difficult - Makes no assumptions about return generating process
SIM - Much easier to compute - Makes assumptions about return generating process
Both approaches are only as good as the inputs into the model – are your estimates accurate? Capital Allocation between Risk-Free and Risky Assets The optimal risky portfolio is selected by investors based on their own preferences. It will depend on their utility function in expected return-standard deviation space. All selected portfolios will be on the efficient part of the mean-variance frontier – it will provide the maximum return for a given level of risk. Capital allocation line: a combination of a risky portfolio and a risk free asset (E(rP ) ! rf ) ! E(ri ) = rf + i !P
Sharpe Ratio Reward to variability ratio: S =
E(rP ) ! rf !P
This is the gradient of the CAL. It gives an indication of the additional reward that an investor will receive for each additional unit of risk they take on.
The point at which utility is maximised is by differentiating and maximising the following:
max U = E[rC ] " 12 A! C2 x
= rf + x(E[rP " rf ]) " 12 Ax 2! P2 As A (risk aversion) increases, x decreases
We use the mean-variance theory to choose which risky portfolio of N-assets we wish to invest in and combine this with a risk free asset. The equations for this combination are given by: E[rC ] = rf + x(E[rP ] ! rf )
! C = x! P This is a straight line connecting the risk free asset to a portfolio on the efficient frontier One Fund Theorem: When a risk free asset is available there is only one risky fund that people will want to invest in. -
The efficient frontier now becomes the straight line which is tangential to the efficient frontier
-
This may be found by maximising the Sharpe ratio
-
