Understanding Risk

1. What is risk?

2. How can we measure risk?

Risk and Investment Risk

• Risk in general : Chance that an unfavorable event will occur.

• Investment risk: the probability of earning less than expected.• The greater the chance of low or negative

returns, the riskier the investment

Approach to Measuring Risk

• List of all possible outcomes

• List the chance of each one occurring

- Value between zero and 1

(probability)

Measuring Risk - Simple Example Possibilities, Probabilities and Expected Value

List all possible outcomes and the chance of each one occurring

Possibilities, Probabilities and Expected Value List all possible outcomes and the chance of each one occurring

• Toss 2 – Coins

• Is the outcome of the first coin dependent upon the outcome of the second coin?

• Invest in 2 stocks

Two Coin TossPossibilities Outcome Probability

#1

#2

#3

#4

Measuring Risk: Example 1

$1000 Investment

Two Possibilities:1. Investment will rise in value to $1400

2. Investment will fall in value to $700

Suppose the two possibilities have equal chance of occurring -

- Probability of each outcome = 1/2

Expected Value

Expected Value = ½ ($700) + ½ ($1400) = $1050

Measuring Risk: Example 2 (more complicated)

$1000 Investment with four possibilities:

1. Rise in value to $2000

2. Rise in value to $1400

3. Fall in value to $700

4. Fall in value to $100

Chance (probability) of occurring =

0.1, 0.4, 0.4, 0.1

Expected Value

Expected Value = 0.1x($100) + 0.4x($700) + 0.4x($1400) +0.1x($2000) = $1050

Measuring Risk: Comparing Examples 1 and 2

• Example 1 and example 2 have the same expected value of $1050.

• The expected return is $50 on a $1000 investment, or 5%.

• However, the two investments have different levels of risk.

• The wider the distribution of payoffs, the higher the risk.

Measuring Risk: Comparing Examples 1 & 2

Case 2 has a higher standard deviation because it has a bigger spread

Measuring Risk• A risk-free asset is an investment whose future

value of known with certainty.

• This return is the referred to as a risk-free rate of return.

• If the risk-free return is 5 percent, a $1000 risk-free investment will pay $1050 - its expected value - with certainty.

• If there is a chance that the payoff will be either more or less than $1050, the investment is risky.

Risk-Return Tradeoff

More risk Bigger risk premium Higher expected return

Risk Requires Compensation

Measuring Risk - Standard Deviation

• Variance:

Average of squared deviation of the outcomes from the expected value, weighted by the probabilities.

• Standard Deviation:Square root of the variance(Same units as the payoff)

Measuring Risk: Standard Deviation for Example 1

Step 1:Compute the expected value: ($1400 x ½) + ($700 x ½) = $1050.

Step 2: Subtract this from each of the possible payoffs:$1400-$1050= $350

$700-$1050= –$350

Step 3: Square each of the results: (+$350)2= 122,500 and (–$350)2= 122,500

Measuring Risk: Standard Deviation for Example 1

Step 4: Multiply each result times its probability and add up the results:

½ [122,500] + ½ [122,500] = 122,500

So the calculation is:

Variance = ½($1400-$1050)2 + ½($700-$1050)2

= 122,500

Measuring Risk: Standard Deviation for Example 1

• The standard deviation is the square root of the variance.

• Standard Deviation = $350

• Standard Deviation is more useful because in same units as payoff – dollars.

• Note: $350/$1000 = 35%

Measuring Risk: Standard Deviation for Example 2

Measuring Risk: Comparing Examples 1 & 2

Case 1: Standard Deviation =$350

Case 2: Standard Deviation =$528

The greater the standard deviation, the higher the risk.

Value at Risk

• Sometimes we are less concerned with the spread of possible outcomes than we are with the value of the worst outcome.

• To assess this sort of risk we use a concept called “value at risk.” (VaR)

• VaR measures risk of the maximum potential loss.

• Formal definition: value at risk is the worst possible loss at a given probability.

Value at Risk

• $300 loss in Example 1, 50% chance

• $900 loss in Example 2, 10% chance

• Lottery example: • Compare paying $1 for chance to win $1 million to

paying $10,000 for a chance to win $10 billion.

Reducing Risk through Diversification

Spreading Risk

To spread your risk - find investments whose payoffs are completely unrelated.

WHAT THE FINANCIAL SYSTEM DOESSharing Risk

• The financial system allows people to share risks:

• Savers can reduce risk through diversification: providing funds to many different investors with uncorrelated assets. • Banks do this by lending to different industries.

Also diversify geographically.• Mutual funds invest in common stock of many

different companies.

Reducing Risk through Diversification

• Diversification can reduce idiosyncratic risk (company specific risk), risks that differ across individual businesses.

• Diversification cannot reduce systematic risk (market risk), which affect most/all businesses

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