95% Confidence Intervals 12-1 Asset classMean Standard Deviation Lower Bound Upper Bound SP %20.20%-27.49%51.69% Small Cap16.90%32.30%-46.41%80.21% 20 yr Corp Bonds6.30%8.40%-10.16%22.76% 20 yr T-Bonds5.90%9.80%-13.31%25.11% 10 yr T-Bond5.40%5.70%-5.77%16.57% 1 mo T-Bills3.50%3.10%-2.58%9.58%

Probabilities of Negative and Positive Returns 12-2 Asset classMean Standard Deviation z (for 0% return)Prob(Z<0)Prob(Z>0) SP %20.20% %72.54% Small Cap16.90%32.30% %69.96% 20 yr Corp Bonds6.30%8.40% %77.34% 20 yr T- Bonds5.90%9.80% %72.64% 10 yr T-Bond5.40%5.70% %82.83% 1 mo T-Bills3.50%3.10% %87.06%

Risk Premiums The “extra” return earned for taking on risk Treasury bills are considered to be “risk- free” (more on this later)… The risk premium is the return over and above the risk-free rate 12-3

Table 12.3: Average Annual Returns and Risk Premiums InvestmentAverage ReturnRisk Premium Large Stocks12.1%8.6% Small Stocks16.9%13.4% Long-term Corporate Bonds 6.3%2.8% Long-term Government Bonds 5.9%2.4% U.S. Treasury Bills3.5%0.0% 12-4

Arithmetic vs. Geometric Average Arithmetic average – return earned in an average period over multiple periods Geometric average – average compound return per period over multiple periods The geometric average will be less than the arithmetic average unless all the returns are equal 12-5

What is the arithmetic and geometric average for the following returns?  Year 1 5%  Year 2-3%  Year 3 12%  Arithmetic average = (5 + (–3) + 12)/3 = 4.67%  Geometric average = [(1+.05)*(1-.03)*(1+.12)] 1/3 – 1 =.0449 = 4.49% 12-6 Arithmetic vs. Geometric Average

Approximation of Geometric Average GAR = AAR –    Annual Returns, YearSP50020 year TbondsTbillsCPI Arithmetic Average11.76%5.75%3.76%3.08% Variance4.15%0.95%0.10%0.17% Standard Deviation20.37%9.73%3.14%4.18% Geometric Average9.63%5.27%3.67%2.96% Geometric Average Approximation9.69%5.28%3.71%2.99%

(Informationally) Efficient Capital Markets Definition: Capital markets are “informationally efficient” if market prices of securities fully incorporate and reflect all “relevant” information at any given point of time. If this is true, then securities are “fairly” priced and it is not likely that you are able to earn “abnormal” or “excess” returns (i.e., returns over and above the “fair” return). 12-8

12-9

Implications of Efficient Markets Efficient markets do not mean that you can’t make money! They do mean that, on average, you will earn a return that is appropriate for the risk undertaken and there is not a bias in prices that can be exploited to earn excess returns Market efficiency ensures that market prices are reliable indicators for capital allocation in the economy – they enable firms to identify what the most highly valued uses of capital are