• 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
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
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
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
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
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
Expected Return Formula N possible outcomes = • FIN 300 - Risk and Return Pt. 1 18
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
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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