Week 1: Risk and Return Securities: why do we buy them? To take advantage of future cash flows (in the form of dividends or selling a security for a higher price). How much should we pay for this, considering that the cash flows will be ours in the future (the future is unknown). We thus need to find the present value Companies issue or sell securities o Company value is the value of the securities it has issued Individuals acquire/buy securities o Individual wealth is the value of the securities that they have bought/invested in On average, what all investors expect will be realised (this is an assumption) E[future cash flows] Price = PV[Expected Future cash flows] = discount factor Different models make different assumptions about: o The future cash flows e.g. magnitude and timing o The probability associated with each future cash flow o The PV calculation the discount rate Prices affect the value of companies and individuals o Say you own 100 shares worth $10 each, then wealth = $1000 o From prices, we move easily to returns: r1 = P1 - Returns can be backward looking o t = 1 today and t = 0 last year, therefore r1 is the return realised over the past year Or forward looking o t = 1 next year and t = 0 today, therefore r1 is the expected return over the next year Now, assuming that P 0 is known and thus its expected value is just P 0 : E(r)%=%E P1# = E(P1)& When we model expected returns, we implicitly model expected prices Let X be a random variable, say Intel s possible returns Let x be one realisation, say Intel s return over January 2009 (historical/backward looking) or Intel s return over January 2014 (forward looking) E(X) is called the expectation operator to find E(X), multiply all the possibly values of x by their respective probabilities and then add them up to find E(X) If a quantity is certain then its probability is 1 (guaranteed to happen) Obviously an event with a probability of 0 means that the event will never happen Arithmetic Average: For historical returns, add and divide by n For future returns: multiply by the probability and then add (1/n) is a proxy for the probability of a return 1
Geometric Average: Multiply and then raise to the power of 1/n GA = A1#x#A2#x#.#x#An# For example, GA of 9 and 4 = 9x4=36 A company worth $100 on Monday and after some thought (analysis), you decide that on Wednesday: $90 1/3 $110 1/3 $130 1/3 What is the arithmetic expected return? What is the geometric expected return? Which are we meant to use? The arithmetic average is frequently used as another term for the expected value. Thus: -10% (1/3) +10% (1/3) +30% (1/3) The arithmetic mean/expected return is thus: E(r) = (1/3)(-10) + (1/3)(10) + (1/3)(30) = 10% The geometric version of the expected return calculation: 10% 30% = (-111.111%) 1/3 [note the negative term nonsense] 10% The compound rate of return: geometric mean/average ((90%)(110%)(130%)) 1/3 = 108.77% [increase in investment of 8.77%] WHICH DO WE USE? Whichever describes the changes in wealth Use arithmetic average when estimating costs of capital from historical returns Use arithmetic mean when estimating the opportunity cost of capital Use geometric mean when estimating compounding changes in wealth There are many risk measures each has strengths and weaknesses A natural concept of risk is a deviation from that is expected o This suggests: E(x E(X)) # If x > E(X) = good (+) # If x < E(X) = bad (-) A problem with this suggestion is that there is a 50% chance of +10% and a 50% chance of -10% o E(X) = 0 but is this really riskless? No this quantity provides a poor measure of risk If we square the difference and call this the variance or σ 2 σ 2 = E[(x E(X)) 2 ] o Now, our measure of risk is: o 0.5(10 0) 2 + 0.5(-10 0) 2 2
Note that returns and expected returns are in terms of % and variances are in terms of % 2 Hence, we often standardise and use standard deviation= Advantage risk and return measures are in comparable units Covariance is a measure of co-movement : the expression for the covariance between x and y is covar(x,y) or σ x,y : E[(x E(X))(y E(Y))] If x achieves below average returns at the same time when y achieves above average returns (covariance is negative) and is x achieves below average returns at the same time when y achieves below average returns (covariance is positive) Another way of writing covariance: σ x σ yρ xy o ρ xy is the correlation (coefficient) between x and y o σ x is the standard deviation of security x o σ y is the standard deviation of security y Portfolio variance (σ 2 p) can be calculated in two ways: o Take the portfolio as ONE security and then you can calculate its average return and variance following what we did just now, or: o Calculate as the weighted sum of individual security variances and covariances: Where N is the number of securities wi,wj are the weights (%) invested in each security σ i, σ j is the covariance between security i and security j Suppose we invest 60% of your portfolio in Exxon Mobil and 40% in Coke. The expected dollar return on Exxon is 10% and on Coke it is 15%. The expected return on your portfolio is: E(R) = (.6x10) + (.4x15) = 12% The standard deviation of their annualised daily returns at 18.2% and 27.3% respectively. Assume a correlation coefficient of 1.0 and calculate portfolio variance: Exxon Mobil Coke Exxon Mobil Coke i.e.: [(.6) 2 x (18.2) 2 ] + [(.4) 2 x (27.3) 2 ] + 2(0.4 x 0.6 x 18.2 x 27.3) = 477.0 Standard deviation of portfolio = square root of 477.0 = 21.8% Stocks Standard deviation % of portfolio Av. Return ABC Corp 28 60% 15% Big Corp 42 20% 21% With a correlation coefficient of 0.4 3
Standard deviation = weighted average = (0.6 x 28) + (0.4 x 42) = 33.6 Real standard deviation (i.e. portfolio standard deviation) = (28 )(0.6 )+(42 )(0.4 )+2(0.4)(0.6)(28)(42)(0.4) = 28.1 Expected return (weighted average) of portfolio: (0.15)(0.60) + (0.21)(0.40) = 17.4% We can achieve a higher return with lower risk through diversification Note that when i = j, the covariance expression the variance expression. For example, the covariance of company A with itself is just its variance: o E[(r A E(r A ))(r A E(r A ))] = E[(r A E(r A )) 2 ] = variance THE RISK OF A WELL-DIVERSIFIED PORTFOLIO DEPENDS ON THE MARKET RISK of the securities included in the portfolio β i is the beta of security i and this measures the sensitivity of the security to market movements i = i,m σ m The beta of a security is a measure of the relative risk of that security to the market Beta is a measure of non-diversifiable risk risk that cannot be eliminated by diversification i.e.: market risk = systematic risk = beta risk = diversifiable risk unique risk = unsystematic risk = residual risk = idiosyncratic risk = diversifiable risk How should we build portfolios that are optimal for investors minimise risk/maximise returns If returns are normally distributed then mean and standard deviation are the only quantities needed to describe the probabilities of all returns o Investors thus only need to worry about these two qualities 4
Week 2: Risk and Return (The CML and CAPM) Markowitz Portfolio Theory and Normality If returns are normally distributed they are completely described by their mean and variance So investors can choose portfolios based solely on their mean and variance Investors will prefer portfolios with high means and low variances Expected returns and standard deviations vary given different weights for shares in the portfolio But what about combinations for more than two securities? Now, combining portfolio of AB* with N The Efficient Frontier The jellyfish shape contains all possible combinations of risk and return: the feasible set The red line constitutes the efficient frontier the highest return for given risk Market Equilibrium Assumptions Everyone is a mean-variance optimiser There are no transaction costs i.e. frictionless capital markets There exists a unique risk-free rate for borrowing and lending (the same rate) All investors have homogenous (identical) expectations/beliefs about the distribution of returns offered by all assets all investors will perceive the same efficient set 5