Financial Mathematics III Theory summary

Financial Mathematics III Theory summary Table of Contents Lecture 1... 7 1. State the objective of modern portfolio theory... 7 2. Define the return of an asset... 7 3. How is expected return defined?... 7 4. How do we maximise return if there are no risk constraints?... 7 5. Define positive definiteness and use it to identify covariance matrices... 7 6. What is a covariance matrix?... 7 7. What special properties does a covariance matrix have?... 7 8. Define semi-variance and shortfall semi-variance... 7 Lecture 2... 8 1. What are the assumptions of mean-variance portfolio theory?... 8 2. Define mean-variance efficiency... 8 3. Define the opportunity set and efficient frontier. How do they relate to each other?... 8 4. If a portfolio is efficient and we add in/discard a asset, will the original portfolio remain efficient?... 8 5. State what sort of curve the opportunity set takes... 8 6. Describe the shape of the graph of expected return of two asset portfolios as a function of the investment fraction... 10 7. Describe the shape of the graph of standard deviation of two asset portfolios as a function of the expected return... 10 8. Will the minimum variance portfolio always be efficient?... 10 9. Discuss convexity in the context of efficiency... 10 Lecture 3... 11 1. Define a riskless asset... 11 2. Identify the opportunity set with a riskless asset and a portfolio of risky assets... 11 3. Define and compute the market price of risk... 11 4. Prove that an efficient portfolio containing a riskless asset remains efficient after the riskless asset has been discarded... 11 5. State and prove the Tobin separation theorem... Error! 6. Discuss why tangency is required for optimality... Error! 7. Show how the investment line with three assets meets the opportunity set with two assets... Error! 8. Show that if a portfolio consisting of X units of A and (1-X) units of the riskless assets is efficient then A is efficient if we cannot invest in the riskless asset. Error! 9. Show that if a portfolio consisting of X units of A and (1-X) units of the riskless assets is efficient then A by itself is efficient if we can invest in the riskless asset... Error! 10. Sketch the efficient frontier with and without the riskless asset on the same graph... Error! 1

11. Describe the shape of the efficient frontier in the space of weights when we have a riskless asset... Error! Lecture 4... Error! 1. How would borrowing and lending rates vary for most investors?... Error! 2. What is the shape of the efficient frontier with two risky assets and one riskless asset, if no borrowing is possible?... Error! 3. What is the shape of the efficient frontier with two risky assets and one riskless asset, with different borrowing and lending rates?... Error! Bookmark not Lecture 5... Error! 1. Sketch the efficient frontier in the multi-asset case in return/sd space... Error! 2. What shape does the efficient frontier take if there are n>2 risky assets and no risk-free asset in weight space and in expected return/sd space?.. Error! Bookmark not 3. What shape does the efficient frontier take if there are n>2 risky assets and a risk-free asset in weight space and in expected return/sd space?.. Error! Bookmark not 4. Two fund theorem:... Error! Lecture 6... Error! 1. What are the data problems with mean-variance analysis... Error! Bookmark not 2. Discuss the problem with obtaining the data... Error! 3. Define a single factor model mathematically... Error! 4. Compare the amount of data required for a single factor model with a general one... Error! 5. Define specific, systematic and diversifiable risk... Error! 6. Discuss and compute variances of large portfolios in single factor models Error! Lecture 7... Error! 1. Derive the expression for linear regression... Error! 2. Find alpha and beta given time series of returns... Error! 3. Discuss the issues with fitting a single factor model to time series data... Error! 4. Derive the relationship between idiosyncratic risk and beta estimation... Error! 5. Discuss the accuracy of beta for well-diversified portfolios.. Error! Bookmark not 6. Show how to find the β of a stock in a single factor model given variances and covariances of returns... Error! 7. How do we find the α of a stock from market data in a single factor model Error! 8. How can we improve the stability of α and β estimates when using a single factor model?... Error! 9. If we linearly regress the betas in one period against a previous period, what properties would we expect of the coefficients found? Error! 10. Describe Blume s technique, discuss briefly why it is plausible and use Blume s technique for improving beta estimation... Error! Lecture 8... Error! 1. Motivate the use of multi-factor models... Error! 2

2. State the definition of multi-factor models for stock returns Error! Bookmark not 3. What are the advantages of multi-factor models over single-factor models for stock returns?... Error! 4. How many parameters are there in a multi-factor model?... Error! Bookmark not 5. Give three different ways to choose factors for a multifactor model... Error! 6. How can we assess which of two models for return covariances is better? Rank the methods of historical covariance, single factor model, multi-factor model, and single-factor with Blume s technique, with best first.... Error! 7. Describe fundamental anaylsis... Error! 8. What is the advantage of using fundamental analysis to estimate betas?... Error! Lecture 9... Error! 1. What is an inner product?... Error! 2. Does covariance define an inner product on the space of random variables?... Error! 3. What is the purpose of Gram-Schmidt algorithm?... Error! 4. Does the output of the Gram-Schmidt algorithm depend on the order of the inputs?... Error! Lecture 10... Error! 1. Why mean-variance analysis not enough to decide between investments? Error! 2. What is the St Petersburg paradox?... Error! 3. Define a utility function... Error! 4. Explain how utility functions are used to choose between investments... Error! 5. What properties would you expect a utility fu nction to have and why?... Error! 6. What does it mean for two utility functions to be equivalent?... Error! Bookmark not 7. Give examples for three typical utility functions... Error! Lecture 11... Error! 1. What sort of utility function does a mean-variance investor use?... Error! 2. Define a quadratic utility function and discuss the issues with it... Error! 3. Define the indifference price... Error! 4. Define risk premium of an investment... Error! 5. What is utility indifference curve?... Error! 6. Use indifference curves to find the portfolio of maximal utility.. Error! Bookmark not 7. Explain why quadratic utility functions are approximations to general utility functions... Error! Lecture 12... Error! 1. Define and derive the absolute risk aversion function associated to a utility function... Error! 2. Define and derive the relative risk aversion function associated to a utility function... Error! 3. How do we compute the indifference price given the absolute risk aversion?... Error! 3

4. How do we compute the indifference price given the relative risk aversion?... Error! 5. Suppose an investor has constant absolute risk aversion, what does this tell us about his behaviour?... Error! Lecture 13... Error! 1. What does utility theory imply about risk aversion for small sums of money?... Error! 2. What four axioms does a rational investor s behaviour satisfy?. Error! Bookmark not 3. What does the rational expectations theorem say?.. Error! 4. What does the axiom of comparability say? Show that an investor deciding according to expected utility satisfies this axiom.... Error! 5. What does the axiom of transitivity say? Show that an investor deciding according to expected utility satisfies this axiom.... Error! 6. What does the axiom of independence say? Show that an investor deciding according to expected utility satisfies this axiom.... Error! 7. What does the axiom of certainty equivalence say? Show that an investor deciding according to expected utility satisfies this axiom.... Error! Bookmark not Lecture 14... Error! 1. Formulate the problem of maximizing the long term growth of a portfolio Error! 2. Define the geometric mean return of an asset... Error! 3. What does Kelly s theorem say?... Error! 4. Samuelson s objection... Error! 5. Which utility function is equivalent to maximizing long-term utility?... Error! 6. If we have iid returns every year and we want to maximise expected return for precisely 1000 years away, what quantity should be maximised?.. Error! Bookmark not 7. If we have iid returns every year and we want to maximise returns in the very long term, what quantity should we maximise?... Error! Lecture 15... Error! 1. What does it mean for an investment to be dominant, first order stochastically dominant and second order stochastically dominant to another investment?... Error! 2. If X is dominant to Y, must it be more efficient in a mean-variance sense?. Error! 3. Under reasonable assumptions which should be clearly stated, prove that a portfolio that is first order stochastically dominant to another investment will be preferred.... Error! 4. Does first order stochastic dominance imply second order stochastic dominance? Justify your answer. What about other way round?... Error! Bookmark not 5. If X is first order stochastically dominant to Y, what can we say about their expected returns? What about second order? Justify your answers.... Error! 6. Procdusure to find stochastic dominance... Error! Lecture 16... Error! 1. What is the CAPM equation?... Error! 2. If we know the covariance of an asset with the tangent portfolio, how do we find its return? Give the derivation.... Error! 4

3. If everyone holds the same tangent portfolio, what can we say about its composition?... Error! 4. What are the assumptions of CAPM?... Error! 5. Using CAPM... Error! 6. What do we typically use as the market portfolio when using CAPM?... Error! Lecture 17... Error! 1. Derive the two-factor CAPM equation... Error! 2. Show that a zero beta portfolio is not efficient in the two-factor CAMP... Error! 3. What are the problems with testing CAPM?... Error! 4. Give three different methods of testing CAPM and discuss result from testing CAPM... Error! 1) What was Sharpe and Cooper s test of CAPM and what did they find?... Error! 2) What was Lintner s test of CAPM and what did they find?... Error! Bookmark not 3) Explain Roll s objection to tests of the two-factor CAPM model... Error! Bookmark not 4) How did Shanken deal with the problems of testing CAPM and what did he find?... Error! 5. What is the principle of falsification?... Error! 6. Contrast the Tobin separation theorem and CAPM... Error! Lecture 18... Error! 1. Define and classify arbitrage... Error! 2. What is the principle of no arbitrage?... Error! 3. What is the law of one price?... Error! 4. State the assumptions for the APT and prove that under these the factor loadings determine returns... Error! 5. To what extent are CAPM and APT compatible?... Error! 6. What indices did Chen, Roll and Ross use for their work and why? What did they find? Are their results compatible with CAPM?... Error! Lecture 19... Error! 1. What are the issues with testing the APT?... Error! 2. What form does the expected return of a general portfolio take in the APT?... Error! 3. Loadings versus weights... Error! 4. Using APT... Error! Lecture 20... Error! 1. What are the three forms of stock market efficiency?... Error! Bookmark not 2. A study shows that insider trading is an effective way to beat the market. What forms of market efficiency does this contradict?... Error! 3. What is meant by market rationality?... Error! 4. What are the size, rebound and January effects?... Error! 5. What is the equity risk premium puzzle?... Error! 6. Why is excess volatility an argument against market rationality?... Error! 7. Why are crashes an argument against market rationality?... Error! Bookmark not 8. Name three famous market bubbles... Error! 9. What is survivorship bias?... Error! 5

Lecture 21... Error! 1. Discuss the shortcomings of variance as a risk measure... Error! Bookmark not 2. Define value at risk... Error! 3. What does it mean for a risk-measure to be sub-additive? Prove or disapprove that each of VAR and variance is sub-additive.... Error! 4. What does it mean for a risk-measure to be monotone? Prove or disapprove that each of VAR and variance is monotone.... Error! 5. Discuss the shortcomings of VAR... Error! 6. What does it mean for a distribution to be fat-tailed? How will the VAR of such a distribution compare to that of a normal distribution?... Error! Bookmark not 7. What is a VAR excess? What form does the distribution of the number of excess over a fixed period of time take?... Error! 8. If we change the size of a loss below the VAR level, what effective will it have on the VAR?... Error! 9. For each of the following risk-measures, discuss how they relate to utility functions: short fall, (short fall) semi-variance, VAR, CES.... Error! Bookmark not Lecture 22... Error! 1. What is the log-normal model for stock price returns?... Error! Bookmark not 2. What are the mean and variance of a log-normally distributed asset price?... Error! 3. What is the definition of the market price of risk?... Error! 4. Compute VAR for log-normal models... Error! 5. What are the disadvantages of using a log-normal model for stock prices? Error! 6. What is a mean-reverting process?... Error! 7. Use log-normal models to compute probability that liabilities are covered... Error! 8. Define ARCH and GARCH processes... Error! Lecture 23... Error! 1. Explain the need for long-term asset price models... Error! 2. What are the variables in the Wilkie model?... Error! 3. How do the variables in the Wilkie model depend on each other?... Error! 4. What is the Wilkie model use for?... Error! 5. Describe how the Wilkie model might be used to assess a long-term liability... Error! 6. Order the methodologies of mean-variance analysis, utility theory, CAPM, geometric means and stochastic dominance, in terms of how strong the assumptions needed on the investor are, with weakest first... Error! Bookmark not 6

Lecture 1 1. State the objective of modern portfolio theory To maximise the risk-return trade-off when investing in the markets 2. Define the return of an asset The return on an asset over a time period is the percentage change in its value taking into account all cash in flows and out flows 3. How is expected return defined? E(R) = ( p * R * 4. How do we maximise return if there are no risk constraints? Simply find the asset that maximises R, + and put all money in that 5. Define positive definiteness and use it to identify covariance matrices If C is a symmetric matrix and x. Cx 0, for all x, C is said to be positive semidefinite. It is said to be positive definite if x. Cx > 0, for x 0. All covariance matrices are positive semi-definite and any positive semi-definite matrix is the covariance of some collection of random variables. 6. What is a covariance matrix? A matrix whose element in the i th row and j th column is the covariance between the i th and j th elements of a random vector 7. What special properties does a covariance matrix have? All covariance matrices are positive semi-definite and any positive semi-definite matrix is the covariance of some collection of random variables. 8. Define semi-variance and shortfall semi-variance Semi-variance: E 67X E(X): ; I =>? @ Note: if X is symmetric, then the semi-variance is just the half of the variance Shortfall semi-variance: E((X L) ; I =>B ) The idea is to penalise returns which fall below the fixed benchmark return without penalising over-performance; Shortfall level: returns which fall below some specified threshold level 7

Lecture 2 1. What are the assumptions of mean-variance portfolio theory? Investors only care about the mean and variance of returns Investors prefer higher means to lower means Investors prefer lower variance to higher variance We know the means, variances and covariances of returns for the assets we can invest in So all investors want more money and less risk 2. Define mean-variance efficiency A portfolio is efficient provided No other portfolio has at least as much expected return and lower standard deviation, and No other portfolio has higher return and standard deviation which is smaller or equal 3. Define the opportunity set and efficient frontier. How do they relate to each other? Opportunity set: the set of all possible pairs of standard deviations and returns attainable from investing in collection of assets Efficient frontier: the subset of the opportunity set which is efficient An efficient asset is therefore in the edge of the opportunity set 4. If a portfolio is efficient and we add in/discard an asset, will the original portfolio remain efficient? No. Efficiency is only defined relative to a set of investment opportunities. If we allow an extra asset, then portfolios that were previously efficient are no longer efficient. Similarly, if we throw away an asset both from the set of investment opportunities and from an efficient portfolio, then the portfolio containing the remaining assets may not be efficient. 5. State what sort of curve the opportunity set takes Hyperbola for two risky assets Two perfectly correlated assets: straight line in return/sd space but will have a turn when we pass through zero 8

Anti-correlated assets: the graph of the opportunity set is piecewise linear: linear for X C above and below point where σ is zero but not cross the point where it is zero Uncorrelated assets: hyperbola Correlated assets: a curve somewhere the perfect correlation case and the uncorrelated case hyperbola 9

Comparison 6. Describe the shape of the graph of expected return of two asset portfolios as a function of the investment fraction The expected return is linear in X C (variance is quadratic)- i.e. E7R E : = X C 7E(R C ) E(R F ): + E (R F ) Hence it is a straight line as a function of the investment fraction. 7. Describe the shape of the graph of standard deviation of two asset portfolios as a function of the expected return The variance of two asset portfolios is generally a parabola as a function of expected return for two asset portfolios variance is quadratic in the expected return 8. Will the minimum variance portfolio always be efficient? Yes 9. Discuss convexity in the context of efficiency Convex: if the chord between any two points lie above the graph Concave: if the chord between any two points lie below the graph 10

Straight line is trivially both convex and concave The efficient frontier of two assets is concave since hyperbolas are concave above the turning point (Note: the area below the turning point is not concave and not efficient) Lecture 3 1. Define a riskless asset An asset whose return is known in advance; an asset is risk-free if and only if the variance or standard deviation of returns is zero 2. Identify the opportunity set with a riskless asset and a portfolio of risky assets A straight line in return/sd space: IIII R E = R J + K L IIIIMK N O L σ P. Between the two points, we are dividing our portfolio into risk-free assets and risky assets. Above the risky point, we are short-selling the risk-free asset and putting more money into the risky assets. 3. Define and compute the market price of risk For each extra unit of risk expressed as standard deviation that we take, we get λ = K IIIIMK L N extra units of expected return. λ is the market price of the risk on A. O L 4. Prove that an efficient portfolio containing a riskless asset remains efficient after the riskless asset has been discarded Theorem: If E is efficient then the portfolio A consisting of the risky assets in E is efficient relative to investing solely in risky assets Proof: Assume C is riskless so E(R R ) = R J. If A is not efficient, then either 1) RIII S < RIIII, U σ U σ S 11