# Introduction to Finance: Class 8

## Risks & Returns

#### What are risks & returns?

When it comes to financial matters, we all know what risk is — the possibility of losing your hard-earned cash. And most of us understand that a return is what you make on an investment. What many people don’t understand, though, is the relationship between the two. Trade-offs The relationship between risk and return is often represented by a trade-off. In general, the more risk you take on, the greater your possible return. Think of lottery tickets, for example. They involve a very high risk (of losing your money) and the possibility of an extremely high reward (the giant check with lots of zeroes). Or penny stocks: They’re also very risky and yet seem full of amazing potential.

Source

Risk & Return Video

Risk & Return Video 2 (Lecture)

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Key Concepts:

• Expected Returns
• Diversification
• Systematic Risk Principle
• Security Market Line

## Expected Returns

• Expected returns are based on the probabilities of possible outcomes
• In this context, “expected” means “average” if the process is repeated many times
• The “expected” return does not even have to be a possible return
• Three Companies:
• Molson Coors – beverage company
• Wal-Mart – retailer
• U.S. Treasuries (Risk-free Rate)
• S&P 500 (Stock Market Index)
• Determine Expected Returns
• Probability Distribution of Returns • RTbill = (0.10)(8.0%) + (0.2)(8.0%) + (0.40)(8.0%) + (0.20)(8.0%) + (0.10)(8.0%) = 8.0%
• RGoog = (0.10)(-22.0%) + (0.2)(-2.0%) + (0.40)(20.0%) + (0.20)(35.0%) + (0.10)(50.0%) = 17.4%
• RCoors = (0.10)(28.0%) + (0.2)(14.7%) + (0.40)(0.0%) + (0.20)(-10.0%) + (0.10)(-20.0%) = 1.74%
• RWMT = (0.10)(10.0%) + (0.2)(-10.0%) + (0.40)(7.0%) + (0.20)(45.0%) + (0.10)(30.0%) = 13.8%
• RS&P  = (0.10)(-13.0%) + (0.2)(1.0%) + (0.40)(15.0%) + (0.20)(29.0%) + (0.10)(43.0%) = 15.0%

· · · · ·

• Suppose you have predicted the following returns for stocks C and T in three possible states of nature:

State       Probability          C        T

Boom                 0.3               0.15      0.25
Normal               0.5               0.10      0.20
Recession           ???               0.02      0.01

• What is the probability of a recession?
• What are the expected returns of stocks C and T?
• If the risk-free rate is 6.15%, what is each stock’s risk premium?

## Variance & Standard Deviation

• Variance and standard deviation still measure the volatility of returns
• Using unequal probabilities for the entire range of possibilities
• Weighted average of squared deviations
• Calculate variance of Google’s possible returns
• What are variance and standard deviation of the returns of stocks C and T based on the expected return you previously calculated?

State       Probability          C        T

Boom                 0.3                0.15      0.25
Normal               0.5                0.10      0.20
Recession           ???                0.02      0.01

## Diversification

### Portfolios

• A portfolio is a collection of assets
• The risk-return trade-off for a portfolio is measured by the portfolio expected return and standard deviation
• Portfolio weights:  fraction of wealth invested in each asset at beginning of the period
• Suppose you have a \$10,000 portfolio and you have purchased securities in the following amounts:
• \$6000 of Molson Coors
• What are your portfolio weights in each security?

### Portfolio Expected Return

• The expected return of a portfolio is the weighted average of the expected returns of the respective assets in the portfolio • You can also find the expected return by finding the portfolio return in each possible state and computing the expected value as we did with individual securities

Example:

• What is the expected return of a \$10,000 portfolio with \$4,000 invested in Google and \$6,000 invested in Coors? ### Portfolio Variance

• Compute the portfolio return for each state: • Compute the expected portfolio return
• Compute the portfolio variance and standard deviation using the same formulas as for an individual asset
• Calculate the variance and standard deviation of a \$10,000 portfolio with \$4,000 invested in Google and \$6,000 invested in Coors?
• Calculate portfolio return in each state and expected return

### Portfolio Diversification

• Portfolio diversification is the investment in several different asset classes or sectors
• Diversification is not just holding a lot of assets
• If you own 50 Internet stocks, then you are not diversified
• If you own 50 stocks that span 20 different industries, then you are diversified
• The Principle of Diversification:
• Diversification can substantially reduce the variability of returns without an equivalent reduction in expected returns • However, there is a minimum level of risk that cannot be diversified away – that is the systematic portion
• Consider the following information

State        Probability       X              Y

Boom                .25                15%          10%
Normal               .60               10%           9%
Recession          .15                  5%          10%

• What is the expected return and standard deviation for a portfolio with an investment of \$6,000 in asset X and \$4,000 in asset Y?

## Systematic Risk Principle

• There is a reward for bearing risk
• There is not a reward for bearing risk unnecessarily
• The expected return on a risky asset (and thus its risk premium) depends only on that asset’s systematic risk since unsystematic risk can be diversified away ### Expected vs. Unexpected Returns

• Realized returns are generally not equal to expected returns
• There is the expected component and the unexpected component
• At any point in time, the unexpected return can be either positive or negative
• Over time, the average of the unexpected component is zero

### Announcements and News

• Announcements and news contain both an expected component and a surprise component
• It is the surprise component that affects a stock’s price and therefore its return
• This is very obvious when we watch how stock prices move when an unexpected announcement is made, or earnings are different from anticipated

### Systematic vs. Unsystematic Factors

• Systematic Factors
• Risk factors that affect a large number of assets
• Also known as non-diversifiable risk or market risk
• Includes changes in GDP, inflation, interest rates, etc.
• Example: employment
• Unsystematic Factors
• Risk factors that affect a limited number of assets
• Also known as unique risk and asset-specific risk
• Includes such things as labor strikes, part shortages, etc.

### Decomposition of Returns

• Total Realized Return = expected return + unexpected return
• Unexpected return = systematic portion + unsystematic portion
• Total Realized Return = Expected return + Systematic portion + Unsystematic portion

### Total Risk

• Total risk = systematic risk + unsystematic risk
• The standard deviation of returns (s) is a measure of total risk
• For well-diversified portfolios, unsystematic risk is very small
• Consequently, the total risk for a diversified portfolio is essentially equivalent to the systematic risk

### Measuring Systematic Risk

• How do we measure systematic risk?
• Market is well diversified → its movements result of systematic risk only
• Analyze co-movements of asset returns with market returns
• We use the beta coefficient (b) to measure systematic risk • What does beta tell us?
• β=1 → asset has same systematic risk as market
• β<1 → asset has less systematic risk than market
• β>1 → asset has more systematic risk than market
• Firm’s beta can be found via Yahoo

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• Consider the following information:

Standard Deviation           Beta

Security C                      20%                             1.25

Security K                     30%                             0.95

• Which security has more total risk?
• Which security has more systematic risk?
• Which security should have the higher expected return?

### Portfolio Betas

• Consider our portfolio invested 40% in Google and 60% in Coors.  Given that βGoogle=1.12 and βCoors=0.79, what is βPortfolio?
• The beta of a portfolio is simply the weighted average of the betas of the assets in the portfolio:
• βPortfolio= 40% x 1.12 + 60% x 0.79
• βPortfolio= 0.92

### Beta and the Risk Premium

• Remember: risk premium = expected return – risk-free rate
• The higher the beta, the higher the systematic risk
• the higher the beta, the higher the expected return
• the higher the beta, the higher the risk premium
• Can we define the relationship between the risk premium and beta so that we can estimate the expected return
• YES! ### Reward-to-Risk Ratio

• The reward-to-risk ratio is the slope of the line illustrated in the previous example
• Slope = (E(RA) – Rf) / (βA – 0)
• Reward-to-risk ratio for previous example = (20 – 8) / (1.6 – 0) = 7.5
• What if an asset has a reward-to-risk ratio of 8 (implying that the asset plots above the line)?
• What if an asset has a reward-to-risk ratio of 7 (implying that the asset plots below the line)?

### Market Equilibrium

• In equilibrium, all assets and portfolios must have the same reward-to-risk ratio, and each must equal the reward-to-risk ratio for the market

## Security Market Line

• The security market line (SML) is the representation of market equilibrium
• The slope of the SML is the reward-to-risk ratio: (E(RM) – Rf) / βM
• But since the beta for the market is ALWAYS equal to one, the slope can be rewritten
• Slope = E(RM) – Rf = market risk premium
• SML & Equilibrium

### Capital Asset Pricing Model (CAPM)

• The capital asset pricing model (CAPM) defines the relationship between risk and return
• E(RA) = Rf + βA(E(RM) – Rf)
• If we know an asset’s systematic risk, we can use the CAPM to determine its expected return
• This is true whether we are talking about financial assets or physical assets

Example:

• Consider the betas for each of the assets given earlier. If the risk-free rate is 3.15% and the market risk premium is 9.5%, what is the expected return for each?

Security           Beta           Expected Return

Google                  1.12              3.15 + 1.12(9.5) = 13.79%

Coors                    0.79             3.15 + 0.79(9.5) = 10.66%

Wal-Mart               0.25             3.15 + 0.25(9.5) = 5.53%

Campbell Soup       0.34             3.15 + 0.34(9.5) = 6.38%

Apple                    1.57              3.15 + 1.57(9.5) = 18.07%

Microsoft                0.98            3.15 + 0.98(9.5) = 12.46%

· · · · ·

• The risk-free rate is 4%, and the expected return on the market is 12%. What is the required return on an asset with a beta of 1.5?
• What is the required return on a portfolio consisting of 40% of the asset above and the rest in an asset with an average amount of systematic risk?

## Wrap-Up

• The expected return on a risky asset depends only on that asset’s systematic risk
• We use beta to measure systematic risk
• If we know an asset’s systematic risk, we can use the CAPM to determine its required return
• Factors affecting expected return:
• Pure time value of money – measured by the risk-free rate
• Reward for bearing systematic risk – measured by the market risk premium
• Amount of systematic risk – measured by beta

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## Solutions

1.

• Probability of recession = 1 – .3 – .5 = .2
• Expected return of C = (.3)(15%) + (.5)(10%) + (.2)(2%) = 9.9%
• Expected return of T = (.3)(25%) + (.5)(20%) + (.2)(1%) = 17.7%
• Risk premium = expected return – risk-free rate
• Risk premium (C) = 9.9% – 6.15% = 3.75%
• Risk premium (T) = 17.7% – 6.15% = 11.55%

2.  4.

• Security K since it has the most risk (higher standard deviation)
• Security C has more systematic risk, since it has a higher Beta
• Security C should has a higher expected return since it has a higher Beta, and thus higher systematic risk.  Remember a risk premium on an asset only depends on its systematic risk!

5.

• E(RA) = Rf + βA(E(RM) – Rf)
• E(RA) = 4% + 1.5*(12% – 4%) = 16%
• Average amount of systematic risk = Beta = 1
• E(Ravg) = 4% + 1*(12% – 4%) = 12%
• Expected return on portfolio = (40%)(16%) + (60%)(12%) = 13.6%