BUSM 411: Derivatives and Fixed Income 3. Uncertainty and Risk Uncertainty and risk lie at the core of everything we do in finance. In order to make intelligent investment and hedging decisions, we need ways to model the uncertainty inherent in financial markets (probability distributions), as well as ways to quantify how we feel about that risk (utility functions). For this we look to the tools of probability and statistics, treating outcomes like future stock and commodities prices or other economic conditions as random events. If we understood the universe well enough, we might be able to see everything as deterministic and predictable. To the extent that we do not fully understand the workings of the universe, or of financial markets in particular, we use the mathematical concept of probability distributions to describe the seemingly random outcomes of variables that concern us, such as securities prices, interest rates, and so forth. We will use these probability models for the prices of underlying assets in order to derive models and formulae for pricing derivatives. 3.1. Probability distributions Probability distributions are simply mathematical functions for describing random events. At its essence, a probability distribution is simply a list of the possible outcomes of some random variable, together with the probability of each outcome. Perhaps the simplest example is the probability distribution for a coin toss: Outcome Probability Heads 0.5 Tails 0.5 A more general representation of this is the binomial distribution: { 1 with probability p X = 0 with probability 1 p 1
Our bread-and-butter probability distribution in this course is the normal distribution. 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0-4 -3-2 -1 0 1 2 3 4 68.26% 95.54% 99.74% Figure 1: Standard normal distribution 3.2. A random walk down Wall Street In his 1900 dissertation titled The Theory of Speculation, Louis Bachelier searched for a formula to express the likelihood of a market price fluctuation. He ended up with a mathematical formula that describes what we know call Brownian motion. Einstein came up with the same formula five years later in a different context. In the finance world, Brownian motion came to be known as a random walk, the path a drunken man might follow at night in the light of a lamp post. Using the geometric Brownian motion to describe the random fluctuations in stock prices, Fisher Black, Myron Scholes, and Bob Merton worked out the Black-Scholes option pricing formula, which we ll cover in depth later in the course. 2
A simple random walk: S t+1 = µs t + ε where S t is the stock price at time t, µ is a drift term (to allow for an upward trend in prices over time), and ε N(0, 1). Problem: stock prices can t be negative! A multiplicative random walk: S t+1 = u t S t where u t is a random variable that is i.i.d. over time. Taking logarithms gives us: ln S t+1 = ln S t + ln u t. Now we we can assume that ln u t (the log-return!) is normally distributed, which makes successive prices lognormally distributed. The lognormal distribution cannot go below zero, as desired. If you take the limit of this process as the time interval goes toward zero, you get the geometric Brownian motion mentioned above. Implication: if stock prices follow a random walk, then they are essentially unpredictable (at least based on the sequence of past prices). The change in price from today to tomorrow is random and independent of past price changes. There are theoretical reasons why this should be the case in a well-functioning market, and it seems to be a pretty good description of actual stock prices. 3.3. Why the normal distribution? Model the random fluctuation of stock prices using geometric Brownian motion. This implies that (continuously compounded) stock returns as normally distributed. We can conveniently characterize stock returns as being normally distributed with a certain mean (µ) and standard deviation (σ). For annualized S&P 500 stock returns, µ is roughly 8%, while σ is roughly 15%. The latter is also called volatility. Pick a time horizon, say t. The stock return over t is normally distributed with mean µ t and standard deviation σ t. Given the annual statisitcs mentioned above, what is the distribution of daily returns (assuming 252 trading days per year)? 3.4. Events that are not normal A negative surprise: on October 19, 1987, the S&P 500 index dropped more than 23% in one day 3
A positive surprise: on January 3, 2001, the Nasdaq composite index gained more than 14% in one day Suppose we use a normal distribution to characterize stock returns. What are the probabilities of such surprises? What is the probability of a Black Monday sized crash? 3.5. What the normal distribution fails to capture... There are large movements (both up and down) in stock prices that cannot be captured at all by the normal distribution In mathematical terms, the tail distribution of a normal random variable is too thin. Historical stock returns exhibit fat tails. If we make financial decisions based on the normal distribution, we underestimate the probability of large movements. The consequences can be catestrophic! This is especially important for leveraged investments over a short time horizon Tail fatness can be an particularly important issue in risk management 3.6. Data analysis Preliminaries for data analysis: When given raw data, first look for trends. If there are any, the first step is always to de-trend the data. Why? This is why we typically work with stock returns rather than stock prices. Unlike prices, returns are reasonably stationary over time (i.e., stable mean and variance) When we estimate average stock returns or return variability, we are implicitly assuming that returns are independent and identically distributed. The longer we observe, the more we know about the probability distribution ldots but do not forget structural changes! (shifts in the return-generating process) Sample statistics: Mean: N ˆµ = 1 N i=1 r i 4
Variance (standard deviation is the square root of variance) ˆσ 2 = 1 N N (r i µ) 2 i=1 Skewness (the degree of asymmetry): ˆ skew = 1 N N i=1 (r i µ) 3 σ 3 Kurtosis (the degree to which the distribution has a skinny middle and fat tails : ˆ kurt = 1 N N i=1 (r i µ) 4 σ 2 Covariance (the degree to which two variables, say the returns of stock A and the returns of stock B, move together): ˆ Cov(r A, r B = 1 N N (r A µ A )(r B µ B ) i=1 Standard errors: Take the sample mean as an example: N ˆµ = 1 N i=1 r i We assume that the r i s are random draws from a stationary distribution This implies that the sample mean µ is itself a random variable: if we observed a different sample of returns drawn from the same distribution we would get a different estimate of the mean What is the mean of ˆµ? What is the standard deviation of ˆµ? We call this the standard error. In this case: s.e. = ˆσ N Standard error is a measure of the precision of the estimates 5
4. Risk Aversion and Pricing We ve just talked about how we model risk (variability and uncertainty in the future values of assets that are of interest to firms or investors), using probability distributions described by mean and variance. Now we need a way to represent the decision makers preferences. That is, how do the you feel about this risk? 4.1. Utility and Risk Aversion Utility is a concept developed by economists to measure the relative satisfaction from or desirability of the consumption of goods and services Given this measure, one may speak meaningfully of increasing or decreasing utility, and thereby explain economic behavior in terms of attempts to increase one s utility. We will be particularly concerned with utility of money or wealth. Based on some reasonable assumptions about human preferences, derived from observation and introspection: We prefer more wealth to less wealth (non-satiability) We get less satisfaction or utility out of an additional dollar if we are already relatively wealthy, vs. if we are relatively poor (diminishing marginal utility) We prefer a fixed sum of money to a gamble with the same expected payoff (risk aversion) It turns out that we can capture all of these features by thinking of utility as a concave, increasing function of wealth. 4.2. Risk aversion and asset pricing The prices (or equivalently, the expected returns) of the basic financial securities like stocks and bonds are determined by investors risk aversion. This is easily seen in the Capital Asset Pricing Model (CAPM). 4.2.1 A quick review of the CAPM For a given set of risky assets, there is a unique portfolio on the efficient frontier that has the highest possible Sharpe ratio. If everyone observes the same set of risky assets and has the same expectations about the returns on those assets (that is, everyone agrees on the expected returns and variances of the risky assets), then everyone chooses the same tangency portfolio. 6
If everyone chooses the same risky portfolio, then that risky portfolio must be the market portfolio! In equilibrium, prices adjust until supply equals demand. In this case, the supply is the number of shares of each stock or security existing in the market. The demand is the amount of each risky security that investors want to hold in their portfolio. Furthermore, because the degree of risk aversion varies across investors, some will want to lend (hold some of their wealth in the risk-free asset), while some will want to borrow at the risk-free rate to invest more in the tangency portfolio. The risk-free rate will adjust until the amount of lending and the amount of borrowing cancel out. Thus, the value of the aggregate risky portfolio will equal the entire wealth of the economy! 4.2.2 The market price of risk Suppose there are N investors in the economy, and each investor i has utility U i = E[r] 1 2 A iσ 2 where U i is utility (of the ith investor), E[r] is the expected return (of the investor s portfolio), σ 2 is the return variance, and A i is a parameter that captures the investor s degree of risk aversion. For concreteness, suppose that each has $1 to invest How much will each investor put in the market portfolio? y i = E[r M] r f A i σ 2 M If we add up the amount invested in the market portfolio by all investors, we get: $1 E[r M] r f σ 2 M ( 1 + 1 + + 1 ) A 1 A 2 A N In equilibrium, the total wealth invested in the stock market must be $1 N 7
This implies where E[r M ] r f = Āσ2 M Ā is the average risk aversion of investors in the market (to be precise, it s actually the inverse of investors average risk tolerance): Ā 1 N ( 1 + 1 + + 1 ) A 1 A 2 A N 4.2.3 The main idea In market equilibrium, investors are only rewarded for bearing systematic risk the type of risk that cannot be diversified away. They should not be rewarded for bearing idiosyncratic risk, since this uncertainty can be mitigated through appropriate diversification. William Sharpe (one of the originators of the CAPM and namesake of the Sharpe ratio), in an interview with the Dow Jones Asset Manager: But the fundamental idea remains that there s no reason to expect reward just for bearing risk. Otherwise, you d make a lot of money in Las Vegas. If there s reward for risk, it s got to be special. There s got to be some economics behind it or else the world is a very crazy place. - Sharpe (1998) 4.2.4 Risk aversion and derivatives pricing Like the games in Las Vegas, derivatives contracts are essentially bets bets that the value of some underlying asset will be high or low on a given date in the future. As such, they are not special, to use Bill Sharpe s words. Their riskiness is easily eliminated by taking an offsetting position, and is not systematic. Therefore, derivatives prices are NOT driven by risk aversion. That is, derivatives prices do not incorporate any premium for bearing risk. Instead, derivatives prices are determined by the principle of no arbitrage. Because the payoff of a derivative is contingent on the value of some underlying asset, we can replicate the payoff of the derivative using a portfolio of the underlying asset itself together with the risk-free security. By no arbitrage arguments, the price of the derivative should be equal to the cost of the replicating portfolio. 8
We will spend the remainder of the course focusing on developing models or formulae for derivatives prices based on this idea of arbitrage-free pricing. 9