Uncertainty, Risk, and Risk Management

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  • 1. Outline Goals Risk Risk measurement: Single Bond Risk Measurement: A Portfolio Sources of Risk Capital Asset Pricing Model (CAPM) CAPM EconometricsUncertainty, Risk, and Risk Management SFC - jhuato.sfc@gmail.com Fall 2012 SFC - jhuato.sfc@gmail.com Uncertainty, Risk, and Risk Management
  • 2. Outline Goals Risk Risk measurement: Single Bond Risk Measurement: A Portfolio Sources of Risk Capital Asset Pricing Model (CAPM) CAPM EconometricsGoalsRiskRisk measurement: Single BondRisk Measurement: A PortfolioSources of RiskCapital Asset Pricing Model (CAPM)CAPM Econometrics SFC - jhuato.sfc@gmail.com Uncertainty, Risk, and Risk Management
  • 3. Outline Goals Risk Risk measurement: Single Bond Risk Measurement: A Portfolio Sources of Risk Capital Asset Pricing Model (CAPM) CAPM EconometricsGoals My goals are to help you address the following questions: What is risk? What’s risk aversion? How do we measure the risk of a bond? How can we use statistics to quantify the risk of a bond? What is the effect on risk of diversifying a bond portfolio? What are the different sources of risk? What’s the beta of a bond? What is the capital asset pricing model (CAPM) and what is it for? SFC - jhuato.sfc@gmail.com Uncertainty, Risk, and Risk Management
  • 4. Outline Goals Risk Risk measurement: Single Bond Risk Measurement: A Portfolio Sources of Risk Capital Asset Pricing Model (CAPM) CAPM EconometricsUncertainty and Risk The future is fundamentally unknown. Therefore, there is uncertainty about the future consequences of choices we make today. The past is a guide to the future only if the future looks like the past in some way. But if the future doesn’t look like the past, then the past is not necessarily a good guide. One way to measure risk in bonds is to look at the variability of returns. Think of bonds as lotteries. For example, a lottery gives you return X if the state of the world is A and a return Y if the state of the world is non − A. SFC - jhuato.sfc@gmail.com Uncertainty, Risk, and Risk Management
  • 5. Outline Goals Risk Risk measurement: Single Bond Risk Measurement: A Portfolio Sources of Risk Capital Asset Pricing Model (CAPM) CAPM EconometricsUncertainty and Risk We know that return is the total gain/loss experienced on investing in a bond over a period of time expressed as a percentage of the value of the investment at the beginning of the period. Ct + Pt − Pt−1 kt = Pt−1 where kt is the actual, expected, or required return rate (or just return) over period t, Ct is the cash flow (coupon, income, dividend, rent, or interest) received from the investment in the time period from t − 1 to t, Pt is the price (value) of the bond at time t, and Pt−1 is the price (value) of the bond at time t − 1. SFC - jhuato.sfc@gmail.com Uncertainty, Risk, and Risk Management
  • 6. Outline Goals Risk Risk measurement: Single Bond Risk Measurement: A Portfolio Sources of Risk Capital Asset Pricing Model (CAPM) CAPM EconometricsRisk Preferences People have “preferences” for risk. Some like it or tolerate it better than others (“risk lovers”). Some are “risk neutral” (indifferent to bonds of same return but different return variation). The observed behavior of crowds (e.g. markets) indicates that most people are risk averse as they are willing to pay a return premium on risky bonds. SFC - jhuato.sfc@gmail.com Uncertainty, Risk, and Risk Management
  • 7. Outline Goals Risk Risk measurement: Single Bond Risk Measurement: A Portfolio Sources of Risk Capital Asset Pricing Model (CAPM) CAPM EconometricsMeasuring risk of a single bond To measure the risk of a single bond, we need to learn about statistics, a branch of applied math that uses a mathematical theory of uncertainty (probability theory) plus some additional assumptions to extract information from a data set. Let’s introduce the concept of a probability distribution. SFC - jhuato.sfc@gmail.com Uncertainty, Risk, and Risk Management
  • 8. Outline Goals Risk Risk measurement: Single Bond Risk Measurement: A Portfolio Sources of Risk Capital Asset Pricing Model (CAPM) CAPM EconometricsProbability distributions The return on an investment is a random variable because, in advance, we don’t know for sure what its value will be. Its value is contingent upon the particular state of the world realized. Say we can list all the possible states of the world. Say there are only three equally-possible states of the world: (a) Good, (b) Regular, and (c) Bad. And, under each of th.ese states of the world, we know (or can guess) the return rate. Then we can form expectations on the return. Also, we can compute different measures of dispersion or variability that could give us a measure of risk. Discrete and continuous distributions. Show examples. SFC - jhuato.sfc@gmail.com Uncertainty, Risk, and Risk Management
  • 9. Outline Goals Risk Risk measurement: Single Bond Risk Measurement: A Portfolio Sources of Risk Capital Asset Pricing Model (CAPM) CAPM EconometricsProbability distributions Expected return: Mean of returns. n ¯ k= ki pi i=1 ¯ where k is the expected return, kj for j = 1, . . . , n is the list of returns observed under different states of the world, and Prj is the probability that kj occurs – being the probability a number between 0 and 1, where 0 means absolute impossibility of occurrence and 1 means absolute certainty that it will occur. The measures of dispersion or variability used as proxies for risk are formulas (3) the variance, (4) the standard deviation, and (5) the coefficient of variation – of the returns. The higher each of these is, the greater the variability of returns and the higher the risk. n 2 σ = SFC - jhuato.sfc@gmail.com ¯ − k)2 p (k Uncertainty,, Risk, and Risk Management
  • 10. Outline Goals Risk Risk measurement: Single Bond Risk Measurement: A Portfolio Sources of Risk Capital Asset Pricing Model (CAPM) CAPM EconometricsPortfolio risk A portfolio is a collection of bonds. Correlation is the statistical measure of the association between any two series of numbers, e.g. returns of two bonds under different states of the world. The degree of correlation is measured by the Pearson coefficient: −1 ≤ ρ ≤ 1, where ρ = −1 means perfect negative correlation, ρ = 0 means no correlation at all, and rho = 1 means perfect positive correlation. Show in Excel. SFC - jhuato.sfc@gmail.com Uncertainty, Risk, and Risk Management
  • 11. Outline Goals Risk Risk measurement: Single Bond Risk Measurement: A Portfolio Sources of Risk Capital Asset Pricing Model (CAPM) CAPM EconometricsSources of risk Say, we measure the risk of a portfolio by its standard deviation, σk . Start with a portfolio with one single bond and add bonds randomly to the portfolio, one at a time. What happens to risk as you keep adding bonds? It tends to decline towards a lower limit or baseline risk. On average, portfolio risk approaches the lower limit when you’ve added 15-20 randomly selected securities to your portfolio. So, total risk can be viewed as the sum of two types of risk: Total security risk = Nondiversifiable risk + Diversifiable risk. Diversifiable risk is also called ‘unsystematic,’ because it comes from random factors that can be eliminated by diversifying the portfolio. Nondiversifiable risk is also known as ‘systematic’ and it comes from market-wide factors that affect all securities. It may also be called ‘market’ or ‘macroeconomic’ risk. SFC - jhuato.sfc@gmail.com Uncertainty, Risk, and Risk Management
  • 12. Outline Goals Risk Risk measurement: Single Bond Risk Measurement: A Portfolio Sources of Risk Capital Asset Pricing Model (CAPM) CAPM EconometricsDiversification However correlated the returns of two bonds may be, the expected return of a portfolio with the two bonds will fall in some midpoint between the returns of the two bonds held in isolation. If the bonds are highly positively correlated, the risk is some midpoint between the risk of each bond in isolation. If the bonds are largely uncorrelated, the risk is some midpoint between the risk of the most risky bond and less than the risk of the least risky bond, but greater than zero. If the bonds are highly negatively correlated, the risk is some midpoint between the risk of most risky bond and zero. In no case will a portfolio be riskier than the riskiest bond included in it. SFC - jhuato.sfc@gmail.com Uncertainty, Risk, and Risk Management
  • 13. Outline Goals Risk Risk measurement: Single Bond Risk Measurement: A Portfolio Sources of Risk Capital Asset Pricing Model (CAPM) CAPM EconometricsCAPM model The model links nondiversifiable risk to returns for all bonds. First, we will discuss the beta, an element of the model. Second, we will introduce the CAPM equation. Third, we will show how to use it in concrete applications. The beta is a measure of the nondiversifiable risk. It indicates to what extent the return on a bond responds to a change in the market or average return of all bonds. Who knows about ‘all bonds,’ but there are broad indices of securities, e.g. S&P 500. How do we estimate the beta of a given bond? We need data on the returns on that bond and data on the returns of a well-diversified portfolio representative of the market as a whole. We use regression analysis. SFC - jhuato.sfc@gmail.com Uncertainty, Risk, and Risk Management
  • 14. Outline Goals Risk Risk measurement: Single Bond Risk Measurement: A Portfolio Sources of Risk Capital Asset Pricing Model (CAPM) CAPM EconometricsRegression analysis Consider the plot of the returns on two bonds. The horizontal axis shows the return on an bond representative of the whole market, e.g. the S&P 500. The vertical axis shows the return on a particular bond, e.g. the return on Google. Show plot in Excel. SFC - jhuato.sfc@gmail.com Uncertainty, Risk, and Risk Management
  • 15. Outline Goals Risk Risk measurement: Single Bond Risk Measurement: A Portfolio Sources of Risk Capital Asset Pricing Model (CAPM) CAPM EconometricsRegression analysis Consider the simple bi-variate linear equation: y = a + bx The equation says that the value of the variable y (the dependent variable) depends on the value of the variable x (the independent variable). The literals a and b are called, respectively, the intercept and the slope. The intercept indicates the value of y when x = 0. The slope indicates the change in y associated to a unit change in x or b = ∆Y .∆X SFC - jhuato.sfc@gmail.com Uncertainty, Risk, and Risk Management
  • 16. Outline Goals Risk Risk measurement: Single Bond Risk Measurement: A Portfolio Sources of Risk Capital Asset Pricing Model (CAPM) CAPM EconometricsRegression analysis Regression is a statistical procedure to fit a curve in a scatterplot. For example, a straight line. yi = α + βxi + i . Show regression in Excel. The beta coefficient (slope) indicates the change in the return on a particular bond (e.g. Google or GE) when the return on a market portfolio (e.g. S&P 500) changes in one unit. The beta shows how sensitive the return on the bond is to performance of the market as a whole. What do the betas of Google and GE say? The beta of a portfolio is the weighted average of its individual bonds’ betas: n βp = w i βi , SFC - jhuato.sfc@gmail.com Uncertainty, Risk, and Risk Management
  • 17. Outline Goals Risk Risk measurement: Single Bond Risk Measurement: A Portfolio Sources of Risk Capital Asset Pricing Model (CAPM) CAPM EconometricsCAPM Model The capital asset pricing model (CAPM) gives us the return that we would require in order to compensate for the risk involved in holding a given bond i. The model helps us determine the return over and above a risk-free return that would offset the risk inherent to that bond. And by risk inherent to that bond, we mean the risk of holding that bond that is not diversifiable, i.e. the risk that it shares with a well-diversified market portfolio. So, to determine that extra return or risk premium, we need to measure the extent to which an bond is exposed to nondiversifiable risk. But instead of pulling that measure out of our own heads, we consider the way crowds (e.g. markets) determine that extra return. SFC - jhuato.sfc@gmail.com Uncertainty, Risk, and Risk Management
  • 18. Outline Goals Risk Risk measurement: Single Bond Risk Measurement: A Portfolio Sources of Risk Capital Asset Pricing Model (CAPM) CAPM EconometricsCAPM Model We now remember that the beta of a bond tells us the change in the return on the bond that is due to a one-percent change in the market return. That beta can be used as an index or measure of nondiversifiable risk, since it reflects to covariation of the return on that bond and the market return. So, we plug the beta in the following equation and get the required return, ki : ki = RF + [βi (km − RF )], where, again, ki is the required return on bond i, RF is the risk-free rate of return (usually the return on a U.S. Treasury bill), βi is the beta coefficient or index of nondiversifiable risk for bond i, and km is the return on a market portfolio. SFC - jhuato.sfc@gmail.com Uncertainty, Risk, and Risk Management
  • 19. Outline Goals Risk Risk measurement: Single Bond Risk Measurement: A Portfolio Sources of Risk Capital Asset Pricing Model (CAPM) CAPM EconometricsCAPM Model ki = RF + [βi (km − RF )]. (1) Under the CAPM model, the required return on bond i has two parts: (1) the risk-free rate of return – something like a baseline rate of return (say, the return or interest on a 3-month T bill) – and (2) the risk premium. In turn, (2) has two parts: (a) the beta or index of nondiversifiable risk and (b) (km − RF ) or market risk premium. The market risk premium represents the premium investors require for taking the average amount of risk associated with holding a well-diversified market portfolio of bonds. SFC - jhuato.sfc@gmail.com Uncertainty, Risk, and Risk Management
  • 20. Outline Goals Risk Risk measurement: Single Bond Risk Measurement: A Portfolio Sources of Risk Capital Asset Pricing Model (CAPM) CAPM EconometricsCAPM Model The graphical representation of the CAPM model is called the security market line (SML). We plot our measure of nondiversifiable risk on the horizontal axis and the required return on the vertical axis. Note that beta is our independent variable. Suppose the risk-free return (RF ) is 7% and the expected market return (km ) is 11%. Then, (km − RF ) = 4%. The intercept of our graph will be RF . The slope will indicate the change in our required return when we vary beta in one unit. The slope is given by (km − RF ) = 4%. Example. If beta goes from 1 to 1.5, i.e. ∆β = 0.5, the required return increases from 11% to 13% or 2% = 0.5 × 4%. SFC - jhuato.sfc@gmail.com Uncertainty, Risk, and Risk Management
  • 21. Outline Goals Risk Risk measurement: Single Bond Risk Measurement: A Portfolio Sources of Risk Capital Asset Pricing Model (CAPM) CAPM EconometricsCAPM Model In brief, the CAPM model provides us with a way to determine a return adequate to the (nondiversifiable) risk involved in holding a given bond. And determining an adequate return is essential to value bonds. It translates (nondiversifiable) risk into a risk premium or additional return that would compensate (according to the market) for our taking the risk of holding that given bond. The CAPM is not foolproof. We can only use historical data to estimate the betas. But past variability may not reflect future variability. So, we have to be careful and make adjustments if we have additional information. The CAPM, if it is to function well, requires that the market that prices bonds and, therefore, determines returns, be competitive and efficient – in the sense of being made up by many buyers and sellers, and capable of absorbing available information quickly. It is also assumed that government or other jhuato.sfc@gmail.com SFC - types of restrictions don’t Risk, and that there are no Uncertainty, exist, Risk Management
  • 22. Outline Goals Risk Risk measurement: Single Bond Risk Measurement: A Portfolio Sources of Risk Capital Asset Pricing Model (CAPM) CAPM EconometricsWhat have we learned? What is risk? What is return? What’s risk aversion? How do we measure the risk of a given bond? How can we use statistical measures to quantify the risk of a given bond? What is the effect on risk of diversifying a portfolio of bonds? Are there different types of risk and what are they? What’s the beta of a bond? What is the capital asset pricing model (CAPM)? What is the security market line? SFC - jhuato.sfc@gmail.com Uncertainty, Risk, and Risk Management