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Chapter 5: Risk and Return: Past and Prologue Outline: I. Rates of Returns II. Risk and Risk Premiums III. The Historical Records Motivation: - Reality check I. Rates of Return Investment: the current commitment of resources in the expectation of reaping future benefits that will compensate the investor for (1) the time the resources are committed, (2) the expected rate of inflation, and (3) the uncertainty of the future payoffs. Essential nature of investment: 1. Reduce current consumption. 2. Planned later consumption. Typically, economic agents prefer to consume today than tomorrow. Therefore, incentives (expected returns) need to be provided to induce an investment. The rate of exchange between future consumption (future dollars) and current consumption (current dollars) is the pure rate of interest. Required rate of return = pure (real) rate of interest (expected) inflation rate risk premium = nominal risk-free return risk premium The symbol means an aggregation operation, not an addition operation. The real risk-free rate, also called pure time value of money, is the basic interest rate, assuming no inflation and no uncertainty about future cash flows. - Economists tend to believe that there is a positive relationship between the real growth rate in the economy and the real riskfree rate. - The real risk-free rate is widely believed that it changes only gradually (so it is quite stable) over the long term. The risk-free rate that we observe in the market, e.g., annualized 1-month T-bill rate, is called the nominal risk-free rate. This nominal rate is an aggregation of the real rate and expected inflation rate. - It is not stable over the long term. - Two other factors influence the nominal rate: (1) the relative ease of tightness in the capital markets, and (2) changes in the expected rate of inflation. Example: Suppose that the 12-month T-bill rate is 9% and the expected rate of inflation is 5%. What is the real rate? Nominal risk-free rate = [(1 + real risk-free rate (1 + expected rate of inflation)] 1 Real risk-free rate = [(1 + nominal risk-free rate)/(1 + expected rate of inflation)] 1 = [(1 + 9%)/(1+5%)] 1 = 3.8% Risk premium: the additional return added to the nominal rate of interest. Risk premium is required to compensate the investment risk, the uncertainty of the payoffs from a risky investment. - The risk-free asset, e.g., 1-month T-bill, has a risk premium of zero. Where does uncertainty come from? 1. Business risk: the uncertainty of income flows caused by the nature of a firm’s business and operation. 2. Financial risk: the uncertainty introduced by the use of debt. 3. Liquidity risk: the possible losses associated with an immediate liquidation of an asset. 4. Exchange rate risk: the uncertainty in exchange rate movements. 5. Country (political) risk: the uncertainty of returns caused by the possibility of a major change in the political or economic environment of a country. 6. Timing of payoffs: a potential loss is viewed to be riskier when the economy is in a poor state. Return and Risk can be estimated and measured using (1) historical return data or (2) subjective probability distributions. [1]. Historical returns and risk Consider an investment with the following observations: Year Beginning Value 1 2 3 100.0 115.0 138.0 Ending Value (including cash dividends) 115.0 138.0 110.4 Holding Period: the period during which the investor owns an investment. In this case, the holding period is three years. The 3-year holding period return is: Holding period return (HPR) = (ending value/beginning value) − 1 = (110.4/100.0) − 1 = 0.104 = 10.4% Frequently, we want to estimate a rate of return that indicates this investment’s typical experience and the return rate that we should expect to receive over the next 12 months. We often use the mean historical rate of return to estimate the expected rate of return. To do so, we first define every year a holding period. We then calculate HPR for each year: Year 1 2 3 Beginning Ending Value Value 100.0 115.0 115.0 138.0 138.0 110.4 HPR 0.15 0.20 -0.20 Two methods exist for estimating mean historical return: (1) the arithmetic mean (AM), and (2) the geometric mean (GM). AM = ( HPY )/n = (0.15 + 0.20 + -0.20)/3 = 5% AM is a good indicator of the expected rate of return for an investment during a future individual year. However, AM is biased upward (AM GM; AM = GM only if rates of return are the same for all years) if we are attempting to measure an asset’s long-term performance. GM = ( 1 HPR )1/n 1 = (1.15 1.20 0.80)1/3 1 = 3.353%. GM is a better measure for long-term performance evaluation. Risk: the uncertainty that an investment will earn its expected rate of return. We typically use variance and/or stand deviation to measure the investment’s (total) risk. If we estimate variance and standard deviation using historical return observations: Variance (i2) n = ( HPR AM ) i 2 i 1 n 1 = [(15% 5%)2 + (20% 5%)2 +(-20% 5%)2]/2 = 0.0476 Standard deviation (i) = (i 2) 1/2 = 21.82% II. Risk and Risk Premiums [2]. Risk and return based on a probability distribution An investor can form subjective probabilities based on the historical performance of the investment or similar investments modified by the investor’s expectations for the future. For example, the investor may believe that there are three possible states (S = 3): States of Economics Good Fair Poor Probability Rate of Return 15% 70% 15% 20% 10% -20% Then the expected rate of return: E(Ri) = [ (probabilitys) (possible returns)] S s 1 = [15% 20% + 70% 10% + 15% -20%] = 7% The variance: Variance (i2) = [ (probabilitys) (possible returns E(Ri))2] S s 1 = 15% (20% 7%)2 + 70% (10% 7%)2 + 15% (-20% 7%)2 = 0.0141 Standard deviation (i) = (i2) 1/2 = = 11.87% 0.0141 Risk premium: an expected return in excess of that on risk-free securities = expected return − nominal risk-free rate Excess return: rate of return in excess of the Tbill rate = rate of return − nominal risk-free rate Risk aversion: reluctance to accept risk Holding other factors constant, the higher the risk aversion, the higher the risk premium III. The Historical Record, 1926-2001 Table 5.3, pp. 140-141 Small stocks (Russell 2000): mean 18.29%, std 39.28% Large stocks (S&P 500): mean 12.49%, std 20.30% Long-term bonds: mean 5.53%, std 8.18% T-bills: mean 3.85%, std 3.25% Inflation: mean 3.15%, std 4.40% End-of-chapter problem sets: #1, #2, #3, #4