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)

· · · · ·

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

Untitled 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

Screen Shot 2014-08-15 at 12.35.57 PM

  • 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%

· · · · ·

1. Test Your Knowledge (answers found below)
  • 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

Untitled Example:

  • Calculate variance of Google’s possible returns

Screen Shot 2014-07-31 at 6.13.39 PM · · · · ·

2. Test Your Knowledge (answers found below)
  • 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

Untitled 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?

Untitled Untitled


 Portfolio Expected Return

  • The expected return of a portfolio is the weighted average of the expected returns of the respective assets in the portfolio

Untitled

  • 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?

Screen Shot 2014-07-31 at 6.22.20 PM


 Portfolio Variance

  • Compute the portfolio return for each state:

Screen Shot 2014-07-31 at 6.24.23 PM

  • Compute the expected portfolio return
  • Compute the portfolio variance and standard deviation using the same formulas as for an individual asset

Untitled 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

Untitled Untitled Untitled


 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

Untitled

  • However, there is a minimum level of risk that cannot be diversified away – that is the systematic portion

Untitled   · · · · ·

3. Test Your Knowledge (answers found below)
  • 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

Screen Shot 2014-08-02 at 1.47.13 AM


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

Untitled

  • 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

· · · · ·

4. Test Your Knowledge (answers found below)
  • 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

 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!

Screen Shot 2014-08-02 at 2.15.54 AM


 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

Untitled  


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%

2.

Untitled3.

Untitled 1Untitled

Untitled

Untitled

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%

 

For more classes:

Education Series

Print Friendly, PDF & Email