BMGT343 Chapter 5 Risk and Return Rates of Return a key measure of investors success is the rate at which their funds have grown during the investment period holding period return HPR of a share of stock depending on the increase decrease in the price of the share of the investment period as well as on any dividend income the share has provided the rate of return is defined as dollars earned over the investment period o HPR ending price beginning price cash dividend beginning price Arithmetic average sum of returns in each period divided by the number of periods ignored compounding Geometric average single per period return that gives the same cumulative performance as the sequence of actual returns o Calculated by compounding the actual period by period returns then finding the equivalent single per period return o Also called time weighted average return because it ignores quarter by quarter variation in funds under management Dollar weighted average return internal rate of return IRR of the project o IRR is the interest rate that sets the present value of cash flows realized on the portfolio equal to the initial cost of establishing the portfolio ROA with regular cash flows mortgages semiannual etc are usually quoted as annual percentage rates APRs which annual per period rates using a simple interest approach ignoring compounded interest o APR per period rate x periods per year o o APR 1 EAR 1 n 1 x n 1 EAR 1 rate per period n 1 APR n n With continuous compounding the relationship between APR and EAR becomes 1 EAR e APR o o APR ln 1 EAR Risk and Risk Premiums Occasionally annual percentage yield APY may be used interchangeable with effective annual rate EAR What HPRs are possible and how likely are they A good way to approach this question is to devise a list of possible economic outcomes scenarios and specify both the likelihood of each scenario and the HPR the asset will realize in that scenario Scenario analysis process of devising a list of possible economic scenarios and specifying the likelihood of each one as well as the HPR that will be realized in each case Profitability distribution list of possible outcomes with associated probabilities Expected return mean value of the distribution of HPR E r p s r s o R s each scenario o P s probability o E r expected return o S probability of that particular scenario o Var r 2 p s x r s E r 2 To summarize risk with a single number we define the variance as the expected value of the squared deviation from the mean To give the measure of risk the same dimension as expected return we use the standard deviation defined as the square root of the variance o SD r Var r mean expected value median value above below 50 of observations mode most likely value Normal distribution sr i r i E r i i Value at Risk VaR measure of downside risk Loss that will be suffered given an extreme adverse price change Kurtosis measure of the fatness of the tails of a probability distribution Indicates likelihood of extreme outcomes Skew measure of the asymmetry of a probability distribution the estimates of variance and SD from a time series of returns are as follows o Var r 1 n 1 x r rbar 2 o SD r Var r o Rbar 1 n x r Risk free rate rate of return that can be earned with certainty Risk premium an expected return in excess of that on risk free securities Excess return rate of return in excess of the risk free rate Sharpe measure reward to volatility ratio of the portfolio risk premium to standard deviation o S portfolio risk premium SD of portfolio excess return E r p R f p o A risk free asset would have a risk premium 0 and SD 0 Mean variance analysis ranking portfolios by their Sharpe measures Inflation and Real Rates of Return a 10 annual rate of return means that your investment was worth 10 more at the end of the year than it was at the beginning of the year CPI measures the cost of purchasing a bundle of goods that is considered representative of the consumption basket of a typical urban family of 4 Inflation rate rate at which prices are rising measured as the rate of increase of the CPI Nominal interest rate interest rate in terms of nominal not adjusted dollars Real interest rate excess of interest rate over the inflation rate the growth rate of purchasing power derived from an investment o o o r R i 1 r 1 R 1 i r real rate R nominal rate i inflation rate Asset Allocation Across Risky and Risk Free Portfolios history shows us that long term bonds have been riskier investments than treasury bills and that stock investments have been riskier still riskier investments have offered higher average returns asset allocation portfolio choice among broad investment classes complete portfolio entire portfolio including risky and risk free assets capital allocation line plot of risk return combinations available by varying portfolio allocation between a risk free asset and a risky portfolio o the slope S of the CAL equals the increase in expected return that an investor can obtain per unit of additional SD slope is called reward to volatility ratio more risk averse investors will choose to hold less of the risky asset and more of the risk free asset Passive Strategies and the CAL passive strategy investment policy that avoids security analysis based on premise that securities are fairly priced and it avoids the costs involved in undertaking security analysis capital market line capital allocation line using the market index portfolio as the risky asset a passive strategy based on stocks and bills generates an investment opportunity set that is represented by the CML o o