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FINC3017: Investments and Portfolio Management Guanglian Hu University of Sydney [S2 2024] Administrative Details Lecturer/Course coordinator: Guanglian Hu, weeks 1 to 6 Pramod Kumar Yadav, weeks 7 to 12 Lectures, 4 – 6pm on Thursdays, H70.B2010 Tutorials start in week 2 Consultation: 12-1pm on Wednesdays or by appointment Contact: guanglian.hu@sydney.edu.au, H69 534 Assessment: Assignments Two individual assignments, each accounting for 30% of the final grade. They are due by week 8 (20 Sep) and 12 (25 Oct). You will be assessed on your technical application to quantitative questions as well as your critical discussion of key issues. Assessment: Final Exam The final exam is scheduled in the final exam period, 40% of your final grade. Closed book exam. It covers entire course, a mix of quantitative and conceptual questions Learning Materials Course slides Textbook: Investments, by Bodie, Z., Kane, A. and Marcus, A.J.. You have the free access to the ebook via library. Additional readings such as journal articles and other online materials Overview: Weeks 1 to 6 We will cover the following topics in weeks 1 to 6: Overview of Asset Classes and Financial Instruments Portfolio Theory CAPM Asset Pricing Theory The unit emphasizes quantitative methods An Overview of Asset Classes and Financial Instruments Debt securities, equities, and derivatives marked to market, buying on margin, and short selling Mean-Variance Portfolio Theory The theory seeks to find an optimal multi asset allocation Derive and understand portfolio theory This theory has huge impact on practice and forms the cornerstone of a large industry that focuses on diversified investments. Professor Harry Markowitz won the 1990 Nobel Prize in Economics for developing the mean-variance portfolio theory. CAPM The Capital Asset Pricing Model (CAPM) is an extension of modern portfolio theory. It is an equilibrium outcome of everybody applying the portfolio theory. The CAPM has many deep implications. William Sharpe won the 1990 Nobel Prize in Economics for developing the CAPM. Empirical tests and performance of the model Anomalies Asset Pricing Theory Consumption-based asset pricing model Stochastic discount factor (SDF) State prices Math Preliminaries Measuring Returns Matrix Algebra Probability and Statistics Regressions Risk Preferences Measuring Returns Denote the price of an asset at date t by Pt . Ignoring the dividend, the simple net return Rt on the asset between dates t 1 and t is defined as: Rt = Pt Pt 1 1 The simple gross return on the asset is given by 1+ Rt The asset’s gross return over the most recent k periods from date t k to date t, written 1+ Rt(K ), is 1+ Rt(K ) = (1+ Rt)× (1+ Rt 1)× …× (1+ Rt k+1) Example Suppose you invest $100 into stock XYZ. In the first year, you lose 10% and in the second year you make 10%. What is the value of your investment at the end of the second year A = 100 B = 99 C = 101 Measuring Returns The continuously compounded return or log return rt is defined as the natural log of its gross return 1+ Rt : rt = log(1+ Rt) = log( Pt Pt 1 ) = pt pt 1 where pt = log(Pt). Continuously compounded multiperiod return is the sum of continuously compounded single period returns rt(K ) = log(1+ Rt(K )) = log((1+ Rt)× (1+ Rt 1)× …× (1+ Rt k+1)) = log(1+ Rt) + log(1+ Rt 1) + …..+ log(1+ Rt k+1) = rt + rt 1 + …+ rt k+1 It is much easier to derive the statistical properties of additive process than of multiplicative process Measuring Average Returns Assume returns for stock XYZ over the past 4 years are 10%, 25%, -20%, 20% respectively. What is the average return of the stock Arithmetic Average: sum of returns in each period divide by the total number of periods 0.1+ 0.25 0.2+ 0.2 4 = 8.75% Geometric Average: single per-period return that gives the same cumulative performance as the sequence of actual returns (1+ rG)4 = (1+ 0.1)× (1+ 0.25)× (1 0.2)× (1+ 0.2) rG = 7.19% Compounding Suppose James graduated from college at 25 and he invested $10,000 into the S&P 500. Assuming that the S&P 500 would return 10% per year going forward, what would this investment worth when James retired at 65 Compounding The answer is: $10,000*(1+10%)40 =$452,593 If the investment horizon is 50 years, he will get $10,000*(1+10%)50 =$1,173,909 Each dollar you spend now is expensive in terms of the opportunity cost. Matrix Algebra A matrix is a set of elements (e.g., real numbers), organized into rows and columns. Information is described by data. Matrix is a nice tool to organize the data. Matrices are like plain numbers in many ways: they can be added, subtracted, and, in some cases, multiplied and inverted (divided). Matrix Algebra Examples A = [ a11 a12 a21 a22 ] b = b1b2 b3 C = [ c11 c12 c13c21 c22 c23 ] Dimensions of a matrix: the number of rows by the number of columns. A is a 2×2 matrix, b is a 3×1 matrix, and C is a 2×3 matrix. A matrix with only 1 column or only 1 row is called a vector. b is a column vector. If a matrix has an equal number of rows and columns, it is called a square matrix. Matrix A, above, is a square matrix. In matrix A, a11 and a22 are diagonal elements and a12 and a21 are off-diagonal elements. Matrix Addition and Subtraction Matrix addition and subtraction is only defined for the matrices that are of the same order, or, in other words, share the same dimensionality. Matrix addition[ a b c d ] + [ e f g h ] = [ a+ e b + f c + g d + h ] Matrix subtraction[ a b c d ] [ e f g h ] = [ a e b f c g d h ] Matrix Addition and Subtraction: Examples matrix addition[ 2 1 7 9 ] + [ 3 1 0 2 ] = [ 5 2 7 11 ] matrix subtraction[ 2 1 7 9 ] [ 1 0 2 3 ] = [ 1 1 5 6 ] Matrix Multiplication Multiplication of matrices requires a conformability condition: the column dimension of the lead matrix A (NxT) must be equal to the row dimension of the lag matrix B (TxK). The product of AB is a NxK matrix When the matrices do conform, we multiply rows of the first matrix (pre multiplier) with columns of the second matrix (post multiplier)[ a b c d ] × [ e f g h ] = [ ae + bg af + bh ce + dg cf + dh ] For matrices, AB = BA. For example, suppose A is 2×3 and B is 3×2, then AB is a 2×2 matrix and BA is a 3×3 matrix. Scalar multiplication[ a b c d ] ×m = [ am bm cm dm ] Matrix Multiplication: Examples Matrix multiplication 2 13 6 7 9 3×2 × [ 1 0 2 2 3 1 ] 2×3 = 4 3 515 18 12 25 27 23 3×3 Scalar multiplication[ 2 4 6 1 ] × 2 = [ 4 8 12 2 ] Transpose Matrix The transpose of a matrix A is another matrix AT (also written as A′) created by swapping rows and columns Formally, the (i,j) element of AT is the (j,i) element of A. In other words, if A is a m x n matrix, then AT is a n x m matrix A = a b cd e f g h i A′ = a d gb e h c f i Transpose Matrix: Examples Example 1 A = [ 3 8 9 1 0 4 ] A′ = 3 18 0 9 4 Example 2 A = [ 2 1 1 2 ] A′ = [ 2 1 1 2 ] If A′=A, then A is called a symmetric matrix. Note that only square matrices can be symmetric. Inverse of a Matrix The inverse of a matrix A is also a matrix, written as A 1, where AA 1 = A 1A = I. I is the identity matrix: a square matrix with all diagonal elements equal to one and off-diagonal elements equal to zero. For example, I(3) A = 1 0 00 1 0 0 0 1 In scalar algebra, a number times its inverse equals one The inverse of a matrix is usually very difficult to compute by hand, but can be calculated easily with computer For a 2×2 matrix, it works as follows[ a b c d ] 1 = 1 ad bc [ d b c a ] Probability and Statistics: Random Variables A random variable can take on values randomly. Two types of random variables: A discrete random variable has a countable number of possible values A continuous random variable takes an infinite number of values We model stock returns as random variables. For example, the gross return on a stock might be one of the following four values: R = Value Probability 1.10 1/5 1.05 1/5 1.00 2/5 0.00 1/5 probabilities must sum up to one often times we don’t know the true probabilities. We have a prior (guess). Probability and Statistics: Probability Distribution A listing of the values a random variable can take on and their associated probabilities is a probability distribution. For example, the distribution of the returns in the above example, Probability and Statistics: Normal Distribution Of course stock returns can take on a much wider range of values. It is common in finance to assume that stock returns are normally distributed. Probability and Statistics: Normal Distribution However, this assumption is inappropriate for financial data Strong evidence of excess kurtosis (fat tails) for stock returns: a higher probability of extreme observations than a normal distribution would suggest Negative skewness at the index level and positive skewness at individual stock level Probability and Statistics: Moments The behavior of a random variable can be characterized by its moments Mean: measures the central tendency Median: middle observation, also measures the central tendency Variance: dispersion around mean Standard deviation: the square-root of variance Skewness: symmetry of the distribution Covariance and correlation: comovements between two random variables Probability and Statistics: Population and Sample Population v.s. sample: We don’t know the true probability distribution of stock return (population); we only observe the realizations (sample) It is common to use sample statistics (sample mean, sample standard deviation, etc.) to proxy for population values. But this approach can be problematic Understanding the Difference Between Population Mean and Sample Mean Consider a coin toss game ($1 for heads, $-1 for tails). The expected value of your payoff in the population is 0. The sample mean can be different and varies across samples. For example, a realized sequence of coin tosses might be H,T,T,H,H. In that sample, the sample mean is $0.2. How much does the sample mean vary from sample to sample Understanding the Difference Between Population Mean and Sample Mean Suppose you observe a sample of returns for a stock (r1, r2…rt …rT ). By definition, the sample mean is rˉ = 1T T∑ 1 rt The variance of the sample mean is Var ( 1T T∑ 1 rt) = 1 T 2Var ( T∑ 1 rt) = 1 T 2 T∑ 1 Var (rt)+ covariance terms; Assuming i.i.d, Var (rˉ ) = Var (r )T Regression The commonly used linear regression model is Y = Xβ + As an example, y1 y2 . . yT = x11 x12 x21 x22 . . . . xT1 xT2 [ β1 β2 ] + 1 2 . . T Regression It is quite common in finance to use OLS estimates, but consider more sophisticated estimates of standard error: β = (X ′X ) 1X ′Y σ2(β ) = (X ′X ) 1X ′ X (X ′X ) 1 (1) where takes into account various forms of autocorrelation and heteroskedasticity in residuals. OLS standard error is a special case of (1) with = σ2 I, σ2(β ) = (X ′X ) 1σ2 Risk Preferences We capture risk preferences with the expected utility framework The expected utility framework assumes that U(W ) (W denotes wealth) is increasing and twice differentiable and that an investor maximizes E [U(W )] when considering risky investments In a two-state example, E [U(W )] = π1U(W1) + π2U(W2) where π1 and π2 are probabilities of respective states. Consider an investor with initial wealth of W0, offered a gamble that pays +h or h with probability 1/2. It is a fair gamble because the expected payoff is zero. An investor is said to be risk averse if she rejects a fair gamble W0 or W0 + h W0 W0 or W0 h 1/2 1/2 Risk Preferences Risk aversion puts some discipline on the utility function Investor rejects the fair gamble on the previous slide if U(W0) > 1 2U(W0 + h) + 1 2U(W0 h) which implies U(W0) U(W0 h) > U(W0 + h) U(W0) For investors to be risk averse, the utility function must satisfy the above inequality Equivalently, this means that U”(W ) < 0, or U(W ) is a concave function. The marginal utility falls as wealth increases.