Measuring Risk. Expected value and expected return 9/4/2018. Possibilities, Probabilities and Expected Value

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1 Chapter Five Understanding Risk Introduction Risk cannot be avoided. Everyday decisions involve financial and economic risk. How much car insurance should I buy? Should I refinance my mortgage now or later? Need to quantify(measure) risk to calculate a fair price for financial instruments. Defining Risk Dictionary definition: risk is the possibility of loss or injury. For financial and economic decision outcomes, we need a different definition: Risk is a measure of uncertainty about the future payoff to an investment, assessed over some time horizon and relative to a benchmark. 1

2 Measuring Risk Expected value and expected return Possibilities, Probabilities and Expected Probability theory states that considering uncertainty requires: Listing all the possible outcomes. Figuring out the chance of each one occurring. Probability is a measure of the likelihood that an event will occur. It is always between zero and one. Can also be stated as frequencies. Possibilities, Probabilities and Expected Simple example - tossing a fair coin. Construct a table of all outcomes and the associated probabilities 2

3 Possibilities, Probabilities and Expected List all possible outcomes and the chance of each one occurring Toss 2 Coins The outcome of the first coin is independent of (not dependent on) the outcome of the second coin. NOTE: (1/2)(1/2) = (1/4) Two Coin Toss Possibilities Outcome Probability #1 #2 #3 #4 Possibilities, Probabilities and Expected Assume we have an investment that can rise or fall in value. $1,000 stock investment that has a 50% probability to increase to $1,400 or 50% probability to fall to $700. The amount you could get back is the investment s payoff. We can construct a table and determine the investment s expected value - the average or most likely outcome. Possibilities, Probabilities, and Expected Expected value is the mean: the sum of probabilities multiplied by payoffs. Expected = 1/2($700) + 1/2($1,400) = $1,050 Expect $50 return on a $1,000 investment 5% 3

4 Possibilities, Probabilities, and Expected What if the $1,000 Investment could 1. Rise in value to $2,000, with probability of Rise in value to $1,400, with probability of Fall in value to $700, with probability of Fall in value to $100, with probability of 0.1 Possibilities, Probabilities, and Expected Expected value is the mean: the sum of probabilities multiplied by payoffs. Expected = 0.1x($100) + 0.4x($700) + 0.4x($1,400) +0.1x($2,000) = $1,050 Expected Possibilities Outcome Probability Outcome x probability #1 $2, $200 #2 $1, $560 #3 $ $280 #4 $ $10 Expected $1,

5 Possibilities, Probabilities, and Expected Using percentages allows comparison of returns regardless of the size of initial investment. The expected return in both case 1 and case 2 is $50 on a $1,000 investment, or 5 percent. Are the two investments the same? No. The second investment has a wider range of payoffs. Variability equals risk. Variance and Standard Deviation We can see Case 2 is more spread out - higher standard deviation - therefore it carries more risk. Measures of Risk The wider the range of outcomes, the greater the risk. Measuring the spread allows us to measure the risk - variance and standard deviation. A risk free asset is an investment whose future value is knows with certainty and whose return is the risk free rate of return. The payoff you receive is guaranteed and cannot vary. 5

6 Variance and Standard Deviation Variance is the average of the squared deviations of the possible outcomes from their expected value, weighted by their probabilities. 1. Compute expected value. 2. Subtract expected value from each of the possible payoffs and square the result. 3. Multiply each result times the probability. 4. Add up the results. Variance and Standard Deviation Case 1 1. Compute the expected value: ($1400 x ½) + ($700 x ½) = $1, Subtract this from each of the possible payoffs and square the results: $1,400 $1,050 = ($350) 2 = 122,500(dollars) 2 and $700 $1,050 = ( $350) 2 =122,500(dollars) 2 3. Multiply each result times its probability and add up the results: ½ [122,500(dollars) 2 ] + ½ [122,500(dollars) 2 ] =122,500(dollars) 2 4. The Standard deviation is the square root of the variance: Variance 122,500 dollars 2 $350 Variance and Standard Deviation The standard deviation is more useful because it deals in normal units, not squared units (like dollars-squared). We can calculate standard deviation as a percentage of the initial investment - $350/$1000, or 35 percent. We can compare other investments to this one. Given a choice between two investments with equal expected payoffs, most will choose the one with the lower standard deviation. 6

7 Measuring Risk: Case 2 Variance and Standard Deviation The greater the standard deviation, the higher the risk. Case 1 has a standard deviation of $350 Case 2 has a standard deviation of $528 Case 1 has lower risk. Leverage and Risk and Return Leverage is the practice of borrowing (using debt) to finance part of an investment. Financial Intermediaries always do this Leverage increases expected return and it also increases the standard deviation of the returns increases risk. 7

8 If you borrow to purchase an asset, you increase both the expected return and the standard deviation by the leverage ratio. Leverage Ratio = Cost of Investment/Owner s contribution to the purchase No Leverage Leverage = (2,000/1,000) = 2 Leverage = (9,000/1,000) = 9 Your Investment $ 1,000 $ 1,000 $ 1,000 + Amount Borrowed $ - $ 1,000 $ 8,000 Total Investment $ 1,000 $ 2,000 $ 9,000 Possible Payoffs $1,400 or$700 $2,800 or $1,400 $12,600 or $6,300 Net of Repayment 1,400 or 700 1,800 or or Expected $ 1,050 $ 1,100 $ 1,450 Standard Deviation $ 350 $ 700 $ 3,150 EV = 0.5(1400) + 0.5(700) 0.5(1800) + 0.5(400) 0.5(4600) + 0.5(-1700) % 10% 45% Variance SD at Risk Sometimes we are less concerned with spread than with the worst possible outcome Example: We don t want a bank to fail at Risk (VaR): The worst possible loss over a specific horizon at a given probability. What is the VaR for Case 2?, Case 1? at Risk VaR answers the question: how much will I lose if the worst possible scenario occurs? Sometimes this is the most important question. Lottery example: Compare paying $1 for chance to win $1 million to paying $10,000 for a chance to win $10 billion. 8

9 Risk Aversion, the Risk Premium, and the Risk-Return Tradeoff Most people do not like risk and will pay to avoid it, most of us are risk averse. Insurance is a good example of this. A risk averse investor - will always prefer an investment with a certain return to one with the same expected return but any amount of uncertainty. Therefore, the riskier an investment, the higher the risk premium. The compensation investors required to hold the risky asset. Risk Aversion, the Risk Premium, and the Risk-Return Tradeoff Idiosyncratic and Systematic Risk All risks can be classified into two groups: 1. Those affecting a small number of people or firms but no one else: idiosyncratic or unique risks 2. Those affecting everyone: systematic or economy-wide risks 9

10 Reducing Risk through Diversification Idiosyncratic risk can be reduced through diversification, the principle of holding more than one risk at a time. Spreading Risk The more independent sources of risk you hold in your portfolio, the lower your overall risk. As we add more and more independent sources of risk, the standard deviation becomes negligible. Diversification through the spreading of risk is the basis for the insurance business. 10