# 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-offsThe 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 2 (Lecture)

*· · · · ·*Key Concepts:

- Expected Returns
- Diversification
- Systematic Risk Principle
- Security Market Line
- Risk-Return Trade-Off

## 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

*Example:*

- Three Companies:
- Google – technology firm
- 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 0.25Normal 0.5 0.10 0.20Recession ??? 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

*Example:*

- 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 0.25Normal 0.5 0.10 0.20Recession ??? 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

*Example:*

- Suppose you have a $10,000 portfolio and you have purchased securities in the following amounts:
- $4000 of Google
- $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

*Example:*

- 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

- 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?

## Risk-Return Trade-Off

* *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 = wGoogle x βGoogle + wCoors x βCoors
- βPortfolio= 40% x 1.12 + 60% x 0.79
- βPortfolio= 0.92

- The beta of a portfolio is simply the weighted average of the betas of the assets in the portfolio:

### 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% · · · · · 5. Test Your Knowledge (answers found below)

- 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

*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%

- 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!

- 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%**