Lecture 1
Discrete Random Variables
- Only a countable number of distinct values
- Know the chance of occurrence
Continuous Random Variables
- An infinite number of possible values
- Know the chance of occurrence
- No gaps, defined over an interval of values
- Example: height
Example 1
Expected Return: a weighted average of the possible returns,
where the weights correspond to the probabilities.
Variance: expected squared deviation from the mean
Standard Deviation: the square root of the variance
➔ Variance and SD are risk/volatility measures. The unit of
the SD is in %, the same as returns, that is why it is called volatility as well.
➔ Higher volatility when return (R) and probability (Pr) are higher. Deviations from the
mean are than higher as well.
Realized return: return that actually occurs over a particular time
period.
If a stock pays dividends at the end of each quarter, with realized
returns R1, …, R4 each quarter, then its annual realized return is
computed as (on the assumption that the dividend is paid every
quarter):
Example 2
Average Annual Return
Depends on volatility, so high volatility means higher
average return
1
,Example 3
Variance and Volatility of Returns
- Higher returns means higher volatility
Estimation Error: using past returns to predict the future
- Use historical average return to estimate actual expected return
- Standard error is a statistical measure of the degree of estimation error
- Standard error is about individual risks, so when calculating the
volatility of a portfolio where the companies are independent from each other, use SE.
- The 95% Confidence Interval is based on historical data
where you estimate the interval where you think the return
will fall in between, unit in %
The more volatile, the more uncertain, the higher the standard error. The more observations,
the lower the standard error will be because it is more formative to have a lot of observations.
Risk vs. Return
Excess return: the difference between the average return for an
investment and the average return for T-bills. Individual stocks
have lower returns but higher volatility.
Individual Stocks
- There is no precise relationship between volatility and
average returns
- Larger stocks tend to have lower volatility than smaller stocks
- All stocks tend to have higher risk and lower returns than large portfolios
Common Risk (Systematic)
- Risk that is perfectly correlated (1)
- Affects all securities
- Market wide news (about economy) impacts all stocks
- All stocks have same volatility
Independent Risk (Idiosyncratic)
- Risk that is uncorrelated (0)
- Affects a particular security
- Firm-specific news (good or bad) impacts only individual company
- Elimination of this risk possible with diversification
Diversification is the averaging out of independent risks in a large portfolio. Diversification is
key, you can eliminate firm specific news by diversification of portfolio because bad news
2
,about one stock can be offset by positive news for another stock. Market wide news affects all
stocks, so diversification does not make sense.
Example 4
With independent risk, the
overall volatility will decrease
by combining stocks together
(diversification). There is a
limit on how much you should
diversify, you should add stocks
until there is no decrease of
volatility anymore (red line).
Risk Premium
- The risk premium for diversifiable (independent) risk is zero, so investors are not
compensated for holding firm-specific risk. The risk premium of a security is
determined by its systematic risk and does not depend on its diversifiable risk. So,
only systematic risk drives returns while volatility measures total risk.
- To measure systematic risk, determine how much of the variability of its return is due
to systematic risk versus unsystematic risk. Look at the average change in return for
each 1% change in return of a portfolio that fluctuates solely due to systematic risk.
Example 5
The blue stock under S&P500 has a volatility of around 22%.
This captures both systematic and independent risk. The
average return is below the line, because of the firm specific
risk. If this stock was on the line, the volatility would be 14%,
so this means the stock consists of 14% systematic risk and 8%
(22-14) idiosyncratic risk. So, no perfect relation between
average return and volatility.
Measuring Systematic Risk
Efficient Portfolio: a portfolio that contains only systematic risk. There is no way to reduce
volatility of the portfolio without lowering its expected return.
Market Portfolio: an efficient portfolio that contains all shares in the market.
Sensitivity to Systematic Risk: Beta
= The expected percent change in the excess return of a security for a 1% change in the excess
return of the market portfolio. So, Beta only captures systematic risk, while volatility looks to
independent risk as well.
Example 6
Firms that only have
idiosyncratic risk,
are not affected by
economy/ market.
3
, ➔ So, when calculating Beta, first look at the change in return and divide the change in
company by the change in market. Company who only has idiosyncratic risk, does not
have a change in return, so beta is always 0.
➔ Beta means that each change in return of the market portfolio leads to a Beta% change
in firm’s return on average.
High Beta means high systematic risk. Stocks from luxury companies have a high beta
because its highly correlated to the market and variations are high. Example: technology
company.
Low Beta means low/no systematic risk. Stocks from less cyclical firms have a low beta
because it is not depended on the market. Example: food producer.
➔ A security’s beta is related to how sensitive its underlying revenues and cash flows are
to general economic conditions. Stocks in cyclical industries are likely to be more
sensitive to systematic risk and have higher betas than stocks in less sensitive
industries.
Beta and Cost of Capital
Market Risk Premium: reward investors for holding a portfolio with a beta of 1. If beta > 1,
then the expected return for an individual stock will be higher than market risk premium. But
when beta < 1, then the expected return for an individual stock will be lower than the market
risk premium.
Cost of Capital:
Example 7
1. Calculate E(Rmkt)
2. Calculate MRP
3. Calculate COC
- Investments with higher volatility should have a higher risk premium and therefore
higher returns
- There is a relationship between size and risks
- Common risk: earthquake
- Independent risk: theft
- Risk premium is determined by only systematic risk
- Investors can’t eliminate systematic risk, so they get compensated for holding this
with the market risk premium
- Expected return of large portfolios should rise proportionally with volatility
- A risk-averse investor would choose the economy in which stock returns are
independent because this risk can be diversified away in a large portfolio.
- While volatility (standard deviation) seems be a reasonable measure of risk when
evaluating a large portfolio, it is not adequate to explain the returns of individual
securities, since there is no clear relation between volatility and return for individual
stocks.
- There is a positive relation between the standard deviation of portfolios and their
historical returns. While there is no clear relationship between the volatility and return
of individual stocks.
4
Discrete Random Variables
- Only a countable number of distinct values
- Know the chance of occurrence
Continuous Random Variables
- An infinite number of possible values
- Know the chance of occurrence
- No gaps, defined over an interval of values
- Example: height
Example 1
Expected Return: a weighted average of the possible returns,
where the weights correspond to the probabilities.
Variance: expected squared deviation from the mean
Standard Deviation: the square root of the variance
➔ Variance and SD are risk/volatility measures. The unit of
the SD is in %, the same as returns, that is why it is called volatility as well.
➔ Higher volatility when return (R) and probability (Pr) are higher. Deviations from the
mean are than higher as well.
Realized return: return that actually occurs over a particular time
period.
If a stock pays dividends at the end of each quarter, with realized
returns R1, …, R4 each quarter, then its annual realized return is
computed as (on the assumption that the dividend is paid every
quarter):
Example 2
Average Annual Return
Depends on volatility, so high volatility means higher
average return
1
,Example 3
Variance and Volatility of Returns
- Higher returns means higher volatility
Estimation Error: using past returns to predict the future
- Use historical average return to estimate actual expected return
- Standard error is a statistical measure of the degree of estimation error
- Standard error is about individual risks, so when calculating the
volatility of a portfolio where the companies are independent from each other, use SE.
- The 95% Confidence Interval is based on historical data
where you estimate the interval where you think the return
will fall in between, unit in %
The more volatile, the more uncertain, the higher the standard error. The more observations,
the lower the standard error will be because it is more formative to have a lot of observations.
Risk vs. Return
Excess return: the difference between the average return for an
investment and the average return for T-bills. Individual stocks
have lower returns but higher volatility.
Individual Stocks
- There is no precise relationship between volatility and
average returns
- Larger stocks tend to have lower volatility than smaller stocks
- All stocks tend to have higher risk and lower returns than large portfolios
Common Risk (Systematic)
- Risk that is perfectly correlated (1)
- Affects all securities
- Market wide news (about economy) impacts all stocks
- All stocks have same volatility
Independent Risk (Idiosyncratic)
- Risk that is uncorrelated (0)
- Affects a particular security
- Firm-specific news (good or bad) impacts only individual company
- Elimination of this risk possible with diversification
Diversification is the averaging out of independent risks in a large portfolio. Diversification is
key, you can eliminate firm specific news by diversification of portfolio because bad news
2
,about one stock can be offset by positive news for another stock. Market wide news affects all
stocks, so diversification does not make sense.
Example 4
With independent risk, the
overall volatility will decrease
by combining stocks together
(diversification). There is a
limit on how much you should
diversify, you should add stocks
until there is no decrease of
volatility anymore (red line).
Risk Premium
- The risk premium for diversifiable (independent) risk is zero, so investors are not
compensated for holding firm-specific risk. The risk premium of a security is
determined by its systematic risk and does not depend on its diversifiable risk. So,
only systematic risk drives returns while volatility measures total risk.
- To measure systematic risk, determine how much of the variability of its return is due
to systematic risk versus unsystematic risk. Look at the average change in return for
each 1% change in return of a portfolio that fluctuates solely due to systematic risk.
Example 5
The blue stock under S&P500 has a volatility of around 22%.
This captures both systematic and independent risk. The
average return is below the line, because of the firm specific
risk. If this stock was on the line, the volatility would be 14%,
so this means the stock consists of 14% systematic risk and 8%
(22-14) idiosyncratic risk. So, no perfect relation between
average return and volatility.
Measuring Systematic Risk
Efficient Portfolio: a portfolio that contains only systematic risk. There is no way to reduce
volatility of the portfolio without lowering its expected return.
Market Portfolio: an efficient portfolio that contains all shares in the market.
Sensitivity to Systematic Risk: Beta
= The expected percent change in the excess return of a security for a 1% change in the excess
return of the market portfolio. So, Beta only captures systematic risk, while volatility looks to
independent risk as well.
Example 6
Firms that only have
idiosyncratic risk,
are not affected by
economy/ market.
3
, ➔ So, when calculating Beta, first look at the change in return and divide the change in
company by the change in market. Company who only has idiosyncratic risk, does not
have a change in return, so beta is always 0.
➔ Beta means that each change in return of the market portfolio leads to a Beta% change
in firm’s return on average.
High Beta means high systematic risk. Stocks from luxury companies have a high beta
because its highly correlated to the market and variations are high. Example: technology
company.
Low Beta means low/no systematic risk. Stocks from less cyclical firms have a low beta
because it is not depended on the market. Example: food producer.
➔ A security’s beta is related to how sensitive its underlying revenues and cash flows are
to general economic conditions. Stocks in cyclical industries are likely to be more
sensitive to systematic risk and have higher betas than stocks in less sensitive
industries.
Beta and Cost of Capital
Market Risk Premium: reward investors for holding a portfolio with a beta of 1. If beta > 1,
then the expected return for an individual stock will be higher than market risk premium. But
when beta < 1, then the expected return for an individual stock will be lower than the market
risk premium.
Cost of Capital:
Example 7
1. Calculate E(Rmkt)
2. Calculate MRP
3. Calculate COC
- Investments with higher volatility should have a higher risk premium and therefore
higher returns
- There is a relationship between size and risks
- Common risk: earthquake
- Independent risk: theft
- Risk premium is determined by only systematic risk
- Investors can’t eliminate systematic risk, so they get compensated for holding this
with the market risk premium
- Expected return of large portfolios should rise proportionally with volatility
- A risk-averse investor would choose the economy in which stock returns are
independent because this risk can be diversified away in a large portfolio.
- While volatility (standard deviation) seems be a reasonable measure of risk when
evaluating a large portfolio, it is not adequate to explain the returns of individual
securities, since there is no clear relation between volatility and return for individual
stocks.
- There is a positive relation between the standard deviation of portfolios and their
historical returns. While there is no clear relationship between the volatility and return
of individual stocks.
4
