2023 Risk and Return

RISK AND RETURN
TCH 302
Cecchetti, Chapter 5
1
Introduction
• 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 and
transferring risk.
2
Defining Risk
• Dictionary definition, risk is “the possibility of
loss or injury.”
• For outcomes of financial and economic

decisions, 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”.
3
Measuring Risk
• In determining expected return, we need

to understand expected value investment
return out of all possible values.
4
Possibilities, Probabilities, and
Expected Value
• 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 5
Example 1
• 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.
6
Possibilities, Probabilities,
and Expected Value
7
Example 2
• What if the $1,000 Investment could:
• Rise in value to $2,000, with probability of 0.1
• Rise in value to $1,400, with probability of 0.4
• Fall in value to $700, with probability of 0.4
• Fall in value to $100, with probability of 0.1
Lecture 1 8
Variance and Standard Deviation
• Case 2 is more spread out - higher

standard deviation - therefore it carries
more risk.
9
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.
10
• Variance is the average of the squared deviations of
the possible outcomes from their expected value,
weighted by their probabilities.
• Compute expected value.
• Subtract expected value from each of the possible

payoffs and square the result. Multiply each result
times the probability. Add up the results.
11
• 1. Compute the expected value: ($1400 x ½) + ($700 x ½) =
$1,050.
• 2. 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:

12
• 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.
13
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.
• Leverage magnifies the effect of price changes and

adds to risks in the financial system.
14
Value at Risk
• Sometimes we are less concerned with spread than
with the worst possible outcome
• Example: We don’t want a bank to fail Value at Risk

(VaR):
• The worst possible loss over a specific horizon at a

given probability.
• What is the VaR for Case 2?, Case 1?
15
Value at Risk
• VaR answers the question: how much will

I lose if the worst possible scenario
occurs? Sometimes this is the most
important question.
16
Idiosyncratic and Systematic Risk
• All risks can be classified into two

groups:
• Those affecting a small number of people

or firms but no one else: idiosyncratic or
unique risks.
• Those affecting everyone: systematic or

economy-wide risks.
17
Systemic risks vs specific risks
• Systemic risks are threats to the system

as a whole, not to a specific firm or
market.
18
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.
19
Hedging Risk
• Results of Possible Investment
• Strategies: Hedging Risk
• Initial Investment = $100
20
Reducing Risk through
Diversification
• Idiosyncratic risk can be reduced through
diversification, the principle of holding
more than one risk at a time.
21
Spreading Risk
22
Excercise
• Assume that the economy can experience high growth, normal
growth, or recession. Under these conditions, you expect the
following stock market returns for the coming year:
State of the Economy Probability Return
High Growth 0.2 +30%
Normal Growth 0.7 +12%
Recession 0.1 -15%
• Compute the expected value of a $1,000 investment over the

coming year. If you invest $1,000 today, how much money do you
expect to have next year? What is the percentage expected rate of
return?
23

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RISK AND RETURN

• How much car insurance should I buy? Should

• Need to quantify(measure) risk to calculate a

• For outcomes of financial and economic

“ Risk is a measure of uncertainty about the

• In determining expected return, we need

• Listing all the possible outcomes.

• Figuring out the chance of each one occurring.

• It is always between zero and one. Can also be

• $1,000 stock investment that has a 50% probability to

• The amount you could get back is the investment’s

• We can construct a table and determine the

• Rise in value to $2,000, with probability of 0.1

• Rise in value to $1,400, with probability of 0.4

• Fall in value to $700, with probability of 0.4

• Fall in value to $100, with probability of 0.1

• Case 2 is more spread out - higher

• A risk free asset is an investment whose future value is

• The payoff you receive is guaranteed and cannot vary.

• Compute expected value.

• Subtract expected value from each of the possible

• 2. Subtract this from each of the possible payoffs and

• 3. Multiply each result times its probability and add up the

• 4. The Standard deviation is the square root of the variance:

• The greater the standard deviation, the

• Case 1 has a standard deviation of $350

• Financial Intermediaries always do this.

• Leverage increases expected return and it also

• Leverage magnifies the effect of price changes and

• Example: We don’t want a bank to fail Value at Risk

• The worst possible loss over a specific horizon at a

• What is the VaR for Case 2?, Case 1?

• VaR answers the question: how much will

• All risks can be classified into two

• Those affecting a small number of people

• Those affecting everyone: systematic or

• Systemic risks are threats to the system

• A risk averse investor - will always prefer an investment with a

• The compensation investors required to hold the risky asset.

• Strategies: Hedging Risk

• Initial Investment = $100

High Growth 0.2 +30%

Normal Growth 0.7 +12%

Recession 0.1 -15%

• Compute the expected value of a $1,000 investment over the

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