N actual observed returns are compounded 1R 1 1R 2 1R 3 1R N The repeated

N actual observed returns are compounded 1r 1 1r 2 1r

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N actual observed returns are compounded (1+R 1 ) (1+R 2 ) (1+R 3 )….. (1+R N ) The repeated geometric average return is compounded N times (1+R GeomAvg ) (1+R GeomAvg ) (1+R GeomAvg )….. (1+R GeomAvg ) = (1+R GeomAvg ) N Set these 2 equal to one another (1+R GeomAvg ) N =(1+R 1 ) (1+R 2 ) (1+R 3 )….. (1+R N ) FIN 300 - Risk and Return Pt. 1 12
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Geometric Average(or, Mean) From the last slide (1+R GeomAvg ) N =(1+R 1 ) (1+R 2 ) (1+R 3 )….. (1+R N ) Solve for R GeomAvg Geometric Average Return = [(1+R 1 ) (1+R 2 ) (1+R 3 )….. (1+R N )] (1/N) - 1 Aside: Note that the geometric average return will always be less that the arithmetic average - unless the geometric average is zero FIN 300 - Risk and Return Pt. 1 13
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Expected Return Going back to our calculation of arithmetic return Where we had assumed all returns as equally likely to occur What if we believe that some outcomes are more likely than others We will weigh the calculated average according to the associated probability of each outcome Thus the “expected return” is used when we calculate the probability-weighted average of future possible outcomes for an investment return We will multiply each possible outcome by its associated probability, before summing-up the terms FIN 300 - Risk and Return Pt. 1 14
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Expected Return Example Given: 3 scenarios for earning an investment return over the next year Scenario 1: Probability of 20% that you will earn a +20% return Scenario 2: Probability of 30% that you will earn a +5% return Scenario 3: Probability of 50% that you will earn a -10% return (Note that the probabilities must sum to 100%!) What is the expected return for the investment? FIN 300 - Risk and Return Pt. 1 15
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Expected Return Example We multiply each return by its associated probability And, then sum those terms up, to get the “expected” return 0.20 x 20% = 4% 0.30 x 5% = 1.5% 0.50 x -10% = -5% Expected Return = 4% + 1.5% + (-5%) = 0.5% Note that the expected return is not necessarily one of the possible scenarios – that is okay, since the expected return is nothing other than the probability-weighted average outcome FIN 300 - Risk and Return Pt. 1 16
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Expected Return Expected return is the probability-weighted average return The outcomes have a probability distribution Each of the outcomes is not (necessarily) equally likely If repeated investments are made (each time having one of the possible outcomes – think “spinning the roulette wheel”), then the expected return may be thought of as the average outcome over many repeated investment opportunities FIN 300 - Risk and Return Pt. 1 17
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Expected Return Formula N possible outcomes = FIN 300 - Risk and Return Pt. 1 18
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Return vs. Risk Any rational investor would prefer investments with a higher return However, there is a tradeoff Generally speaking, investments offering a higher return also come with a higher level of risk As investors, we have to balance the desire of more return vs. the concern of taking on more risk FIN 300 - Risk and Return Pt. 1 19
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Return vs. Risk If I desire the utmost in safety in an investment, I will put my money in an FDIC insured bank account or T-Bills However, I would expect to get very little return If I am aiming for a higher investment return, I might invest in the stock market
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