Principles of Finance Risk and Return. Instructor: Xiaomeng Lu

Principles of Finance Risk and Return Instructor: Xiaomeng Lu 1

Course Outline Course Introduction Time Value of Money DCF Valuation Security Analysis: Bond, Stock Capital Budgeting (Fundamentals) Portfolio Choice, Asset Pricing Models, Behavioral Finance Capital Budgeting (Advanced) Derivatives: Options 2

Lecture Outline Return: Facts and Basics Risk: Systematic vs Idiosyncratic The Effect of Diversification on Risk Risk vs Return: Efficient Portfolio Efficient portfolio with stocks Efficient portfolio with stocks and risk-free borrowing and savings The Efficient Portfolio and the Cost of Capital 3

Value of $100 Invested at the End of 1925 4 Source: Chicago Center for Research in Security Prices (CRSP), Standard and Poor s, MSCI, and Global Financial Data. Returns were calculated at year-end assuming all dividends and interest are reinvested and excluding transactions costs.

Empirical Distribution of Annual Returns for Different Securities 1926-2011 5

Risk and Return Single period (Simple) Return defined: r t = D t + P t P t 1 P t 1 = D t + P t P t 1 1 Holding period return: The holding period return over 2 periods is r t 2 = P t P t 2 = P t 1 = P t P t 1 1 P t 2 P t 2 P t 1 P t 2 = 1 + r t 1 + r t 1 1 Note: Assume no div. Try the more generalized case with div at home. 6

Return Defined Sometimes for modeling or computing purposes, it is easier to work with continuous returns: r t c = ln 1 + r t = ln r t c 2 = ln 1 + r t (2) = ln = r c c t + r t 1 P t P t 1 = ln P t ln(p t 1 ) P t P t 2 = ln P t P t 1 P t 1 P t 2 7

Historical Return Statistics The history of capital market returns can be summarized by describing the: Average return r = t=1 T Standard deviation (volatility) of those returns T r t SD = σ 2 = T t=1 r t T 1 r 2 8

Average Return: Arithmetic vs. Geometric People talk about two different average returns: Arithmetic average return: T r a = 1 T t=1 r t Geometric average return: r g = T 1 + r t 1 For the S&P500 Index, from 1926 to 2004 Annual arithmetic average return = 7.29% Annual geometric average return = 5.30% Why are these two different? And which average return should we use? (Depends on the purpose) 9

Arithmetic vs. Geometric Return Geometric return is better in describing the past Suppose we have an asset whose return is either 50% or - 50%, with equal probability. Suppose we observe a return of 50% followed by a return of -50% Arithmetic average = Geometric average = 10

Arithmetic vs. Geometric Return Geometric return is better in describing the past If we started with $100, it turned into $100 x 1.5 = $150 after the first year $150 x 0.5 = $75 after the second year 50% then -50% left us worse off than two returns of 0% $75 = $100 1 + 13.4% 2. Growing at the geometric average return each period results in the same terminal wealth as actually observed. (Arithmetic return is incorrect for this purpose.) 12

Arithmetic vs. Geometric Return Arithmetic average return is an unbiased estimate of expected return over a future horizon based on its past performance. (What is the assumption for this statement?) Suppose I invest $100 today: Year 1 Year 2 sdafsd 13

Arithmetic vs. Geometric Return Arithmetic average return is an unbiased estimate of expected return over a future horizon based on its past performance. Suppose I invest $100 today: Year 1 Year 2 correct incorrect Note: The underlying assumption is that past returns can be viewed as independent draws from the same distribution 14

Empirical Distribution of Annual Returns for Different Securities 1926-2011 Remember : r = r f + r p expected return = risk free rate +risk premium Asset Average Annual Return Volatility (Standard Deviation) Average Excess Return Small Stocks 18.7% 39.2% 15.1% Large Stocks (S&P500) 11.7% 20.3% 8.1% Corporate Bonds 6.6% 7.0% 3.0% Treasury Bills 3.6% 3.1% 0% 15

Risk-Return Tradeoff (Portfolios): 1926-2011 16 Source: CRSP

The Returns of Individual Stocks Is there a positive relationship between volatility and average returns? More on this (important!) topic later Data Source: CRSP 17

The Volatility of Individual Stocks 19 Data Source: CRSP

The first rule is not to lose. The second rule is not to forget the first rule. ~ Warren Buffett 20

Systematic vs. Idiosyncratic Risk Roulette wheels are typically marked with numbers 1~36 plus 0 and 00. Each outcome is equally likely every time the wheel is spun. If you place a bet on any one number and are correct, the payoff is 35:1; that is, if you bet $1, you will receive $36 if you win ($35 plus your original $1) and nothing if you lose. Suppose you place a $1 bet on your favorite number. What is the casino s expected profit? What is the SD of this profit for a single bet? Suppose 9 million similar bets are placed throughout the casino in a typical month. What is the SD of the casino s average revenues per dollar bet each month? 22

Systematic vs. Idiosyncratic Risk E[payoff] = SD(payoff)= SD(average payoff)= What is the 95% confidence interval for the average payoff? What is the key assumption? 24

Systematic vs. Idiosyncratic Risk The previous example illustrates the power (and the limit) of diversification. You can eliminate volatility by holding a large portfolio (e.g. insurance company). Idiosyncratic vs. Systematic risk Idiosyncratic risk; Firm specific news Good or bad news about an individual company Also known as unique risk, unsystematic risk, diversifiable risk Systematic risk; Market-Wide News News that affects all stocks, such as news about the economy Also known as undiversifiable risk, market risk 26

Systematic vs. Idiosyncratic Risk Which of the following risks are likely to be firm-specific, diversifiable risks, and which are likely to be systematic risks? (a) The risk that the founder and CEO retires (b) The risk that interest rate rises (c) The risk that a product design is faulty and the product must be recalled (d) The risk that the economy slows, reducing demand for the firm s products 27

Diversification in Stock Portfolios When many stocks are combined in a large portfolio, the firm-specific risks for each stock will average out and be diversified. Intuition: Will firm-specific risk earn a risk premium (above r f )? Why or why not? The systematic risk, however, will affect all firms and will not be diversified. 29

Expected Return and Variance: Portfolio of Two Assets Let s begin by looking at a portfolio of two assets with returns r A and r B, with weights w A and w B. First some statistics: σ AB = cov AB = E[(r A E[r A ])(r B E[r B ])] σ AB = 1 (r T 1 A,t r A )(r B,t r B ) t=1 ρ AB = corr AB = cov AB σ A σ B Portfolio return (w A + w B = 1) r p = w A r A + w B r B Portfolio expected return E r p = w A E r A + w B E r B T 30

Expected Return and Variance: Portfolio of Two Assets Portfolio variance σ p 2 = var r p = w A 2 σ A 2 + w B 2 σ B 2 + 2w A w B σ AB = w A 2 σ A 2 + w B 2 σ B 2 + 2w A w B σ A σ B ρ AB When would there be diversification effects? 31

Expected Return and Variance: Portfolio of Multiple Stocks Return on portfolio with N stocks r p = N i=1 Variance of portfolio σ p 2 = var N i=1 w i r i = cov N i=1 w i r i, r p = N i=1 w i r i w i cov r i, r p Portfolio variance is the weighted average covariance of each stock with the portfolio. σ p 2 = N i=1 w i cov r i, r p = N i=1 w i cov r i, N j=1 w j r j = N N i=1 j=1 w i w j cov r i, r j Portfolio variance is the sum of the covariances of the returns of all pairs of stocks in the portfolio multiplied by each of their portfolio weights. 32

The Effect of Diversification on Portfolio Risk Suppose we invest our money equally in the N stocks w = 1 N We can show that: var 1 N r 1 + 1 N r 2 + + 1 N r N = 1 N Avg. Var + 1 1 N Avg. Cov (see next slide for details on the derivation) Let s look at this in more detail Assume each stock has variance σ 2. Assume correlation between each pair of stocks is ρ. So covariance of each pair of stocks is ρσ 2. Then the portfolio variance is σ 2 N + ρσ2 1 1 N = σ2 ρ + 1 N 1 ρ = σ2 1 N + ρ 1 1 N 33

Notes on the first equation in the previous slide var 1 N r 1 + 1 N r 2 + + 1 N r N = 1 N Avg. Var + 1 1 N Avg. Cov cov 1 N r i, 1 N r j = 1 N 2 cov(r i, r j ) So the second term on the right of the original equation is: i,j cov 1 N r i, 1 N r j = 1 N 2 i,j cov(r i, r j ) (i j) We have N (N 1) pairs of covariances between different assets. So Avg. Cov = 1 N N 1 i,j cov(r i, r j ) So the second term of the equation can be written as 1 cov(r i, r j ) = 1 N 2 N N 1 Avg. Cov = 1 1 Avg. Cov N N 2 i,j 34

Actual correlation of stock returns GS MS APPL FB MCD GS 1.00 0.90 0.47 0.39 0.35 MS 1.00 0.44 0.35 0.36 APPL 1.00 0.33 0.28 FB 1.00 0.40 MCD 1.00 Correlation are computed using daily stock returns from Oct. 6 th 2015 to Oct. 3 rd 2016. Data source: Yahoo finance. 35

Portfolio Variance vs. Number of Stocks 36

Example Stocks within a single industry tend to have a higher correlation than stocks in different industries. Likewise, stocks in different countries have lower correlation on average than stocks within the US. What is the volatility of a very large portfolio of stocks within an industry in which the stocks have a volatility of 40% and a correlation of 60%? What is the volatility of a very large portfolio of international stocks in which the stocks have a volatility of 40% and a correlation of 10%? 37

Diversification As we add more stocks, the variance (or SD) of the portfolio declines Diversification No effect when stocks are perfectly correlated. Why? Effect stronger for lower correlation between stocks. Why? Diversification eliminates all risk except the average covariance of stocks 38

Diversification Since people can do this for themselves The market only rewards people for holding market risk (covariance between all stocks) No reward for firm specific risk (non-market based variance of a single stock) 39

No Arbitrage and the Risk Premium If the diversifiable risk of stocks were compensated with an additional risk premium, then investors could buy the stocks, earn the additional premium, and simultaneously diversify and eliminate the risk. By doing so, investors could earn an additional premium without taking on additional risk. This opportunity to earn something for nothing would quickly be exploited and eliminated. Because investors can eliminate firm-specific risk for free by diversifying their portfolios, they will not require or earn a reward or risk premium for holding it. This implies that a stock s volatility, which is a measure of total risk (that is, systematic risk plus diversifiable risk), is not especially useful in determining the risk premium that investors will earn. 40

The Effect of Diversification: No Arbitrage Company A sells umbrellas. With 1 probability, tomorrow will rain, so the payoff for A 2 will be 2. With 1 probability, tomorrow will be sunny, and 2 in that case, the payoff for A is 0. If you are risk averse, how much are you willing to pay for the stock of A? A. 1 a B. 1 C. 1 + a Assume that risk free rate is 0 41

The Effect of Diversification: No Arbitrage Company B sells sunglasses. With 1 probability, tomorrow will rain, so the payoff for B 2 will be 0. With 1 probability, tomorrow will be sunny, and 2 in that case, the payoff for B is 2. If you are risk averse, how much are you willing to pay for the stock of B? A. 1 a B. 1 C. 1 + a 42

The Effect of Diversification: No Arbitrage If you buy one share of A and one share of B, what is the payoff? 2 for sure You paid 2-2a to obtain 2. So, an arbitrage opportunity arise in the market. Arbitrage will not occur, because investors will be continue to take this opportunity until it disappears. So, in equilibrium, there will be no compensation for taking idiosyncratic risk, even though investors are risk averse. 43

Reconsider the Risk and Return relationship for Individual Stocks 44

Diversification in N-stock Portfolios Rewrite the portfolio variance formula: Unless all of the stocks in a portfolio have a perfect positive correlation of +1 with one another, the volatility of the portfolio will be lower than the weighted average volatility of the individual stocks. 45

Risk vs Return: Choosing an Efficient Portfolio An efficient portfolio is a portfolio with no way to further reduce the volatility of the portfolio without lowering its expected return. For an inefficient portfolio, it is possible to find another portfolio that is better in terms of both expected return and volatility. Let s consider a portfolio with two stocks: Intel and Coca-cola 47

Efficient Portfolio with Two Stocks The lower the correlation, the lower the volatility we can obtain. As the correlation decreases, the volatility of the portfolio falls. The curve showing the portfolios will bend to the left to a greater degree. 48

Efficient Portfolio with Two Stocks 49

Two Stock Portfolio with Short Sales Long Position A positive investment in a security Short Position A negative investment in a security In a short sale, you sell a stock that you do not own and then buy that stock back in the future. Short selling is an advantageous strategy if you expect a stock price to decline in the future. 50

Short Sales: An Example Suppose you have $20,000 in cash to invest. You decide to short sell $10,000 worth of Coca-cola stock and invest the proceeds from your short sale, plus your $20,000, in Intel. At the end of the year, you decide to liquidate your portfolio. If the two stocks have the following realized returns, what is the return on your portfolio? P 0 Div 1 + P 1 Return Intel 25 31.50 26% Coca-Cola 40 42.40 6% 51

Short Sales: An Example Suppose Intel has a volatility of 50%, Coca-Cola has a volatility of 25%, and the stocks are uncorrelated. What is the volatility of a portfolio that is short $10,000 of Coca- Cola and long $30,000 of Intel? 52

Efficient Portfolio with Two Stocks Allowing for Short Sales 53

Efficient Portfolio with Multiple Stocks Consider adding Bore Industries to the two stock portfolio: 54

Efficient Portfolio with Multiple Stocks 55

Efficient Portfolio with Multiple Stocks 56

Efficient Portfolio with Multiple Stocks The efficient portfolios, those offering the highest possible expected return for a given level of volatility, are those on the northwest edge of the shaded region, which is called the efficient frontier for these three stocks. In this case none of the stocks, on its own, is on the efficient frontier, so it would not be efficient to put all our money in a single stock. 57

Efficient Frontier with Ten Stocks vs. Three Stocks 58

Efficient Frontier with Risk-Free Saving and Borrowing Consider an arbitrary risky portfolio and the effect on risk and return of putting a fraction of the money in the portfolio, while leaving the remaining fraction in risk-free Treasury bills. The expected return would be: E r xp = 1 x r f + xe r P = r f + x E r p r f Notation: Here r p denotes the return of the risky portfolio, and r xp denotes the return of putting x fraction of the money in the risky portfolio, while leaving the remaining fraction in risk-free Treasury bills. 59

Efficient Frontier with Risk-Free Saving and Borrowing The standard deviation of the portfolio would be calculated as: SD r xp = 1 x 2 Var r f + x 2 Var r P + 2 1 x xcov(r f, r P ) = x 2 Var r P = xsd r P Note: The standard deviation is only a fraction of the volatility of the risky portfolio, based on the amount invested in the risky portfolio. 60

Efficient Frontier with Risk-Free Saving and Borrowing: The Risk-Return Combinations 61

Efficient Frontier with Risk-Free Saving and Borrowing: Borrowing and Buying Stocks on Margin Buying Stocks on Margin Borrowing money to invest in a stock. A portfolio that consists of a short position in the risk-free investment is known as a levered portfolio. Margin investing is a risky investment strategy. Example: Suppose you have $10,000 in cash, and you decide to borrow another $10,000 at a 5% interest rate to invest in the stock market. You invest the entire $20,000 in portfolio Q with a 10% expected return and a 20% volatility. What is the expected return and volatility of your investment? What is your realized return if Q goes up 30% over the course of the year? What return do you realize if Q falls by 10% over the course of the year? 62

Identifying the Tangent Portfolio On the previous graph, to earn the highest possible expected return for any level of volatility we must find the portfolio that generates the steepest possible line when combined with the risk-free investment. Sharpe Ratio Measures the ratio of reward-to-volatility provided by a portfolio Portfolio Excess Return Sharpe Ratio = = E r P r f Portfolio Volatility SD r P The portfolio with the highest Sharpe ratio is the portfolio where the line with the risk-free investment is tangent to the efficient frontier of risky investments. The portfolio that generates this tangent line is known as the tangent portfolio. 64

The Tangent or Efficient Portfolio 65

Identifying the Tangent Portfolio Combinations of the risk-free asset and the tangent portfolio provide the best risk and return tradeoff available to an investor. This means that the tangent portfolio is efficient and that all efficient portfolios are combinations of the risk-free investment and the tangent portfolio. Every investor should invest in the tangent portfolio independent of his or her taste for risk Every investor should invest in the tangent portfolio independent of his or her taste for risk Every investor should invest in the tangent portfolio independent of his or her taste for risk. 66

Identifying the Tangent Portfolio An investor s preferences will determine only how much to invest in the tangent portfolio versus the risk-free investment. Conservative investors will invest a small amount in the tangent portfolio. Aggressive investors will invest more in the tangent portfolio. Both types of investors will choose to hold the same portfolio of risky assets, the tangent portfolio, which is the efficient portfolio. 67

Example Your uncle calls and asks for investment advice. Currently, he has $100,000 invested in portfolio P in the previous graph. The portfolio has an expected return of 10.5% and a volatility of 8%. Suppose the risk-free rate is 5%, and the tangent portfolio has an expected return of 18.5% and a volatility of 13%. To maximize your uncle s expected return without increasing his volatility, which portfolio would you recommend? If your uncle prefers to keep his expected return the same but minimize his risk, which portfolio would you recommend? 68

The Efficient Portfolio and the Cost of Capital Our goal is to get the expected (required) return of an individual security. How to Improve a portfolio: Beta and the Required Return Assume there is an arbitrary portfolio of risky securities, P. To determine whether P has the highest possible Sharpe ratio, consider whether its Sharpe ratio could be raised by adding more of some investment i to the portfolio. The contribution of investment i to the volatility of the portfolio depends on the risk that i has in common with the portfolio, which is measured by i s volatility multiplied by its correlation with P. 70

The Efficient Portfolio and the Cost of Capital How to Improve a Portfolio: Beta and the Required Return If you were to purchase more of investment i by borrowing, you would earn the expected return of i minus the risk-free return. Thus, adding i to the portfolio P will improve our Sharpe ratio if: LHS: Additional return from investment i RHS: Additional return from taking the same risk investing in P E r i r f > SD r i Corr r i, r p E r P r f SD r P Additional return Incremental Volatility Return per unit of from investment i from investment i volatility available from portfolio P 71

The Efficient Portfolio and the Cost of Capital How to Improve a Portfolio: Beta and the Required Return Define: Beta of Portfolio i with Portfolio P β i P = SD r i Corr r i, r P SD r P = Cov r i, r P Var r P Increasing the amount invested in i will increase the Sharpe ratio of portfolio P if its expected return E[r i ] exceeds the required return r i, which is given by: r f + β i P E R P r f 72

The Efficient Portfolio and the Cost of Capital How to Improve a Portfolio: Beta and the Required Return Required Return of i The expected return that is necessary to compensate for the risk investment i will contribute to the portfolio. 73

Example You are currently invested in the Omega Fund, a broadbased fund that invests in stocks and other securities with an expected return of 15% and a volatility of 20%, as well as in risk-free Treasuries paying 3%. Your broker suggests that you add a real estate fund to your current portfolio. The real estate fund has an expected return of 9%, a volatility of 35%, and a correlation of 0.10 with the Omega Fund. Will adding the real estate fund improve your portfolio? 74

Expected Returns and the Efficient Portfolio Expected Return of a Security E r i = r f + β i eff E r eff r f A portfolio is efficient if and only if the expected return of every available security equals its required return. 76

Cost of Capital The appropriate risk premium for an investment can be determined from its beta with the efficient portfolio: Cost of Capital for Investment i Cost of Capital i = r f + β i eff E R eff r f The cost of capital of investment i is equal to the expected return of the best available portfolio in the market with the same sensitivity to systematic risk 77

Example Alphatec is seeking to raise capital from a large group of investors to expand its operations. Suppose the S&P 500 portfolio is the efficient portfolio of risky securities (so that these investors have holdings in this portfolio). The S&P 500 portfolio has a volatility of 15% and an expected return of 10%. The investment is expected to have a volatility of 40% and a 50% correlation with the S&P 500. If the risk-free interest rate is 4%, what is the appropriate cost of capital for Alphatec s expansion? 78