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Equity securities and markets SMM940 Financial Markets & Financial Intermediation Joe Gong Autumn 2024 1 / 32 Module outline What to expect from this module: Part I: Financial markets Risk & returns; Asset Pricing; Pricing model:...

Equity securities and markets SMM940 Financial Markets & Financial Intermediation Joe Gong Autumn 2024 1 / 32 Module outline What to expect from this module: Part I: Financial markets Risk & returns; Asset Pricing; Pricing model: CAPM; Other security classes. Part II: Financial intermediaries Why banks? Types of banking; Fintech; Bank regulation. 2 / 32 Assessment Group coursework (25%): Week 1 - 5; Portfolio optimisation; Deadline: 4 PM Monday 2nd December. Final exam (75%): 40% from Week 1 - 5 and the rest from Week 6 - 10; contains both qualitative and quantitative questions; Options of questions for you to choose; Date: TBD (probably in January). 3 / 32 About me Here to contribute to your learning experience! have taught both UG and PG finance and banking modules at WBS for five years; But this is a new module for all of us.... How to contact me: Office hours (in person or virtual): Wednesday afternoon 1500 to 1600; Office: TBD (in construction); Email: [email protected] What do I expect from you: Don’t be hesitated to ask any questions! Try the tutorial questions before the release of answers and videos. 4 / 32 Today Basics Example: diversification Summary 5 / 32 Roadmap Stock and portfolio returns: definitions and data Portfolio analysis for risk-averse investors Application to real world data 6 / 32 Today Basics Example: diversification Summary 7 / 32 Stock returns To understand where portfolio returns come from, we first need to define stock returns. The realised return on stock j over a historical period is; (Pj + Dj ) − P0,j (Pj + Dj ) Rj = = −1 P0,j P0,j where P0,j is the beginning of period price of the stock, Pj is the end of period price of the stock and Dj is the dividend on the stock (assumed to be paid at the end of the period). When looking forward in time, the return on stock j, which is unknown as of today is; e j = (Pj + Dj ) − 1 e e R Pj where Pej is the (unknown) future price of the stock and D e j is the (unknown) future dividend. The expected return is just the expected value of this expression. 8 / 32  Prices 20 10 12 14 0 2 4 6 8 01 20 02 20 03 SP500 EXXON 20 LLOYDS MSCIEM FTSE100 04 20 MICROSOFT 05 MSCIWORLD 20 06 20 07 Recent stock market data 20 08 20 09 20 10 20 11 20 12 20 13 20 14 20 15 20 16 20 17 20 18 20 19 20 20 20 21 20 22 9 / 32 Recent stock market data 100 FTSE100 SP500 80 MSCIWORLD MSCIEM 60 LLOYDS EXXON MICROSOFT 40 Returns 20 0 -20 -40 -60 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 10 / 32 Recent stock market data FTSE-100 S&P-500 MSCI World MSCI EM LLOY XOM MSFT Mean 0.43 0.73 0.67 1.00 0.21 0.52 1.30 σ 4.65 5.03 5.10 6.64 12.93 6.32 7.11 Skew -0.47 -0.29 -0.35 -0.14 1.61 -0.05 0.30 Kurtosis 5.07 6.26 5.68 4.28 18.62 5.86 4.72 Min -19.30 -19.02 -18.71 -19.76 -59.94 -21.75 -19.80 Max 14.63 23.66 22.46 26.68 96.53 28.62 29.50 11 / 32 Portfolio weights Assume that our investor spread her wealth of W0 across N assets. She allocated a fraction wj of her wealth to asset j and, as she invested all of her wealth, we must have; N ∑ wj = 1 j =1 We call wj the portfolio weight on stock j. We allow any stock to have; Positive portfolio weight i.e. she has purchased, or gone long, the stock. Zero portfolio weight i.e. she has ignored a stock Negative portfolio weight i.e. a short sale of a stock. 12 / 32 Short sales A short sale is accomplished as follows; Borrow a unit of stock from a lender Sell the stock in the market and receive cash today At some later point, pay cash to buy the stock in the market....... and return it to the lender In this trade; You make money if the price of the stock falls while you’re in the short position and....... you lose money if the stock price rises when you’re short. 13 / 32 Short sales Note: the lender of the stock will charge you a (usually) small fee for the duration of the loan of stock. Why does a short sale create a position with a negative portfolio weight? Assume you start with £1,000 You sell £250 of MKS short today. You now have £1,250 and you invest all of that in BARC Your portfolio weights are equal to the size of the position you have taken divided by your initial wealth so; −250 1, 250 wMKS = = −0.25 , wBARC = = 1.25 1, 000 1, 000 Note that the weights, as always, sum to 1.0 14 / 32 Portfolio returns Realised portfolio returns: given her investment strategy (i.e. her choice of weights), the return on her portfolio RP is; N RP = ∑ wj Rj = w1 R1 + w2 R2 +... + wN RN j =1 where Rj is the return on stock j. Again this is a realised return. Future portfolio returns: if we denote by R e the future one period return on a stock or portfolio, then the return on the portfolio over the coming period is; N eP = R ∑ wj Re j = w1 Re 1 + w2 Re 2 +... + wN Re N j =1 This is random variable as all of the individual stock returns are random variables. 15 / 32 Risk and return: 2 stock portfolios Consider an investor who has built a portfolio with weight w on stock A and weight (1 − w ) on stock B. If we denote by E(X ) the expectation of X , then the investor’s expected portfolio return and portfolio risk are; E(RP ) = wE(RA ) + (1 − w )E(RB ) Var(RP ) = w 2 σA2 + (1 − w )2 σB2 + 2 w (1 − w )σA,B = w 2 σA2 + (1 − w )2 σB2 + 2 w (1 − w )ρA,B σA σB We see that; Expected portfolio returns are linear combinations of expected stock returns Portfolio return variance depends on stock return variances and the stock return correlation 16 / 32 Portfolio return and risk: N stock portfolios When the investor builds a portfolio of N stocks we have; N E(RP ) = ∑ wj E(Rj ) = w1 E(R1 ) + w2 E(R2 ) +... + wN E(RN ) (1) j =1 The (expected) variance of the investor’s portfolio return is; ! N N Var(RP ) = Var ∑ wj Re j = ∑ wj2 σj2 + ∑ wi wj σi,j (2) j =1 j =1 i ̸ =j N = ∑ wj2 σj2 + 2 ∑ wi wj σi,j (3) j =1 i >j where σj2 is the variance of returns on stock j and σi,j is the covariance between the returns on stocks i and j 17 / 32 Diversification Take the two-stock portfolio return variance and assume that the correlation is exactly 1.0; Var(RP ) = w 2 σA2 + (1 − w )2 σB2 + 2 w (1 − w )σA σB = [w σA + (1 − w )σB ]2 So portfolio return standard deviation is just a linear combination of the two stock return standard deviations; σP = [w σA + (1 − w )σB ] What would be true if the correlation was less than 1.0? For any value of w, σP would be below the value implied by the linear combination In words, portfolio risk would be below weighted average stock risk The size of the risk reduction would be greater when the correlation was smaller This is diversification: when correlations are below 1, building portfolios removes risk. 18 / 32 Limits to diversification In large portfolios, the benefit of diversification is limited by the fact that, on average, stocks are positively correlated with one another. This means that portfolio variance is unlikely ever to reach zero. Total Risk = Non-diversifiable Risk + Diversifiable Risk = Systematic Risk + Specific Risk = Market Risk + Idiosyncratic Risk Bottom line: if you’re invested in a portfolio of stocks; You won’t be able to diversify the market or systematic risk that causes all stocks to move in the same direction at the same time. But you can get rid of individual stocks’ idiosyncratic or specific risks by spreading your money across stocks. 19 / 32 Limits to diversification: maths Consider a well diversified portfolio of N stocks where you have taken equal weight in each stock. Thus each stock has weight 1/N and the variance of the portfolio return is; N 1 1 1 Var(RP ) = ∑ N 2 σi2 + ∑ N × N × σi,j i =1 i ̸ =j Denote the average covariance between pairs of stock returns C̄ and the average variance across stocks V̄. Then we can write; 1 1 1 Var(RP ) = 2 × N × V̄ + × × (N 2 − N )C̄ N N N where we use the fact that the definition of the sample mean implies that for any variable X , we have ∑N j =1 Xj = N × X̄. 20 / 32 Limits to diversification: maths Cancelling terms in N we get; 1 (N − 1) Var(RP ) = V̄ + C̄ N N Finally, as N gets large, the first term disappears and the second converges to C̄ so; lim Var(RP ) = C̄ N →∞ Interpretation: the risk of well diversified portfolios is controlled by the average covariance between stock returns, so as stocks covary positively on average well diversified portfolios will have positive risk. 21 / 32 Limits to diversification: maths 0.5 Assume that the average return standard deviation is 0.4 40%, so average variance is 0.16 0.3 Assume that average return correlation is σP 0.2 0.25 Then you can 0.1 eliminate half of the average standard deviation by 0 10 20 30 40 50 diversifying. N 22 / 32 Today Basics Example: diversification Summary 23 / 32 Portfolio analysis: data Data: 10 years of monthly (total) returns on 3 stocks UU. : United Utilities, a UK water company GSK: Glaxosmithkline, a global pharmaceuticals firm RIO: Rio Tinto, a global mining firm All listed on the London Stock Exchange and traded in GBP 24 / 32 Stock-level descriptive statistics Table: Return descriptive stats Statistic N Mean St. Dev. Pctl(25) Median Pctl(75) UU. 120 0.8 4.4 −2.1 1.1 4.0 GSK 120 1.1 5.3 −2.9 1.3 5.2 RIO 120 1.5 6.7 −1.9 1.8 5.3 All of these statistics are on a monthly basis. To get annual means multiply the monthly means by 12 and to get√(approximate) annual standard deviations multiply the monthly standard deviations by 12. 25 / 32 Stock-level correlations Here is the correlation matrix for the three return series; UU. GSK RIO UU. 1.00 0.43 0.18 GSK 0.43 1.00 0.21 RIO 0.18 0.21 1.00 All correlations are positive, but there is plenty of scope for diversification here. 26 / 32 Some simple 3 stock portfolios Equally weighted: if we set the weights on each stock to be 1/3 and use the empirical correlation between the returns we get; E(RP ) = 1.14 , Var(RP ) = 15.31 , σP = 3.91 Non-equally weighted: I have manually chosen another set of weights for the three stocks, those being 0.25, 0.30, and 0.45. Applying to our data gives; E(RP ) = 1.22 , Var(RP ) = 17.54 , σP = 4.19 Diversification: comparing the portfolio data to the stock-level data; Both portfolios have smaller standard deviations than any of the stocks This is diversification in action Any rational risk averse investor will exploit this effect. 27 / 32 The feasible set of portfolios Finally, for now, let’s get a rough idea of the entire set of (long-only) portfolios available from these three stocks. I do the following; Loop over the weight on UU., starting at 0.0 and ending at 1.0 with increments of 0.01 At each value in the loop over UU.’s weight, also loop over the weight on GSK. If the value of the weight on UU. is w1 then loop over the weight on GSK from 0.0 to 1 − w1 (with increments of 0.01). Call the weight on GSK w2. At each step in the loop over GSK’s weight, set the weight on RIO to be 1 − w1 − w2. Finally, compute the mean portfolio return and the variance of the portfolio return for these weights Note that this imposes no short sales (no stock ever has a weight less than zero) and also imposes that the weights always sum to exactly 1.00. 28 / 32 Graph of the feasible set of long-only portfolios RIO 1.4 Portfolio µ 1.2 GSK 1.0 0.8 UU. 4 5 6 7 Portfolio σ 29 / 32 Feasible set: interpretation Comments; The points representing the individual stocks are visible on the plot Again, benefits of diversification are clearly visible A set of frontier portfolios has become clear These portfolios have the lowest risk for their particular level of expected return There are lots of inefficient portfolios These are portfolios that lie to the right of the frontier and are not risk minimising There is a set of efficient portfolios These portfolios are on the upward sloping part of the frontier They have the lowest risk for their particular level of expected return and the highest level of expected return for their level of risk. 30 / 32 Today Basics Example: diversification Summary 31 / 32 Summary We have; Shown how to compute properties of stock portfolios Demonstrated the value of, and limits to, diversification Applied concepts to a simple data set 32 / 32