CHAPTER 10: Analyzing Risk and Return in Investment Options
P&G Case
In the case there are three investments o ered returns that were very di erent in terms
of their average level and their variability
What accounts for these di erences?
Di erent level of risk give origin to di erent volatility levels. For example, US bonds are
very safe and with very low risk. Low risk of default. They react to recessions in a better
way than stocks because they are more secure.
As investors you do not want investments that correlate well with recessions. That is a
risk you don’t like.
Many people want US bonds (high demand) → price is high → expected return low
Low demand → low price → high expected return BUT it is correlated with recessions.
If the economy falls, so does the expected return.
To create wealth in life you need to run risk. What is the relationship between return
and risk?
How would $100 have grown if it were placed in one of the following investments?
- S&P 500
- Small stocks
- World Portfolio
- Corporate Bonds
- Treasury Bills
Small stocks had the highest long-term returns, while T-Bills had the lowest long-term
returns. Small stocks had the largest uctuations in price, while T-Bills had the lowest.
This happened because higher risk requires a higher return.
Higher risk → Higher return
Few people ever make an investment for 92 years.
More realistic investment horizons and di erent initial investment dates can greatly
in uence each investment's risk and return.
flff ff fffl ffff ff
, Common Measures of Risk and Return
Probability Distributions
Di erent securities have di erent initial prices, pay di erent cash ows, and sell for
di erent future amounts. To make them comparable, we express their performance in
terms of their returns. The return indicates the percentage increase in the value of an
investment per dollar initially invested in the security.
When an investment is risky, there are di erent returns it may earn. Each possible
return has some likelihood of occurring. We summarize this information with a
probability distribution, which assigns a probability, pR, that each possible return, R,
will occur.
Example: assume BFI stock currently trades for $100 per share. In one year, there is a
25% chance the share price will be $140, a 50% chance it will be $110, and a 25%
chance it will be $80.
ffff ff ff ff fl
,Expected return is calculated as a weighted average of the possible returns, where
the weights correspond to the probabilities.
Expected Return = E [ R ] = å R
PR ´ R
Variance is the expected square deviation from the mean
Var (R) = E é( R - E [ R ]) ù = å (R - E [ R ])
2 2
P ´
ë û R R
Standard deviation is the square root of variance
SD( R) = Var ( R)
They are bth measures of the risk of a probability distribution. In nance, the standard
deviation of a return is also referred to as its volatility.
If we compare two stocks with the same expected return but di erent volatility, that
with the greater vocality will have returns that are more spread out.
If we could observe the probability distributions that investors anticipate for di erent
securities, we could compute their expected returns and volatilities and explore the
relationship between them. However, in real life we do not always have the probability
distribution. How can we extrapolate such information?
One way to do it is to extrapolate it from historical data, a useful strategy when we
believe that the distribution of future returns should mirror that of past returns.
Historical Returns of Stocks and Bonds
The distribution of past returns can be helpful when we seek to estimate the
distribution of returns investors may expect in the future.
fffi ff
, Of all the possible returns, the realized return is the return that actually occurs over a
particular time period. It is the total return we earn from dividends and capital gains,
expressed as a percentage of the initial stock price.
Divt + 1 + Pt + 1 Divt + 1 Divt + 1 - Pt
Rt + 1 = - 1 = +
Pt Pt Pt
= Dividend Yield + Capital Gain Rate
If you hold the stock beyond the date of the rst dividend, then to compute your return
you must specify how you invest any dividends you receive in the interim.
Let’s assume that all dividends are immediately reinvested and used to purchase
additional shares of the same stock or security.
If a stock pays dividends at the end of each quarter, with realized returns RQ1, RQ2,
RQ3, RQ4 each quarter, then we need to compute the annual realized return.
1 + Rannual = (1 + RQ1 )(1 + RQ 2 )(1 + RQ 3 )(1 + RQ 4 )
By counting the number of times a realized return falls within a particular range, we can
estimate the underlying probability distribution. When the probability distribution is
plotted using historical data, we refer to it as the empirical distribution of the returns.
Average annual return of an investment during some historical period is simply the
average of the realized returns for each year
1 1 T
R = ( R1 + R2 + ! + RT ) = å Rt
T T t =1
NB: if the probability distribution of the returns is the same over time, the average
return provides an estimate of the expected return.
Variance estimate using realized returns
We want to nd the volatility of the distribution and to do that we need an estimate of
the standard deviation. We estimate the variance by the average squared deviation
from the mean. However, since we do not have the mean, we use the best estimate of
the mean, which is the average realized return.
T
1
å (R - R)
2
Var (R) = t
T - 1 t =1
fi fi
P&G Case
In the case there are three investments o ered returns that were very di erent in terms
of their average level and their variability
What accounts for these di erences?
Di erent level of risk give origin to di erent volatility levels. For example, US bonds are
very safe and with very low risk. Low risk of default. They react to recessions in a better
way than stocks because they are more secure.
As investors you do not want investments that correlate well with recessions. That is a
risk you don’t like.
Many people want US bonds (high demand) → price is high → expected return low
Low demand → low price → high expected return BUT it is correlated with recessions.
If the economy falls, so does the expected return.
To create wealth in life you need to run risk. What is the relationship between return
and risk?
How would $100 have grown if it were placed in one of the following investments?
- S&P 500
- Small stocks
- World Portfolio
- Corporate Bonds
- Treasury Bills
Small stocks had the highest long-term returns, while T-Bills had the lowest long-term
returns. Small stocks had the largest uctuations in price, while T-Bills had the lowest.
This happened because higher risk requires a higher return.
Higher risk → Higher return
Few people ever make an investment for 92 years.
More realistic investment horizons and di erent initial investment dates can greatly
in uence each investment's risk and return.
flff ff fffl ffff ff
, Common Measures of Risk and Return
Probability Distributions
Di erent securities have di erent initial prices, pay di erent cash ows, and sell for
di erent future amounts. To make them comparable, we express their performance in
terms of their returns. The return indicates the percentage increase in the value of an
investment per dollar initially invested in the security.
When an investment is risky, there are di erent returns it may earn. Each possible
return has some likelihood of occurring. We summarize this information with a
probability distribution, which assigns a probability, pR, that each possible return, R,
will occur.
Example: assume BFI stock currently trades for $100 per share. In one year, there is a
25% chance the share price will be $140, a 50% chance it will be $110, and a 25%
chance it will be $80.
ffff ff ff ff fl
,Expected return is calculated as a weighted average of the possible returns, where
the weights correspond to the probabilities.
Expected Return = E [ R ] = å R
PR ´ R
Variance is the expected square deviation from the mean
Var (R) = E é( R - E [ R ]) ù = å (R - E [ R ])
2 2
P ´
ë û R R
Standard deviation is the square root of variance
SD( R) = Var ( R)
They are bth measures of the risk of a probability distribution. In nance, the standard
deviation of a return is also referred to as its volatility.
If we compare two stocks with the same expected return but di erent volatility, that
with the greater vocality will have returns that are more spread out.
If we could observe the probability distributions that investors anticipate for di erent
securities, we could compute their expected returns and volatilities and explore the
relationship between them. However, in real life we do not always have the probability
distribution. How can we extrapolate such information?
One way to do it is to extrapolate it from historical data, a useful strategy when we
believe that the distribution of future returns should mirror that of past returns.
Historical Returns of Stocks and Bonds
The distribution of past returns can be helpful when we seek to estimate the
distribution of returns investors may expect in the future.
fffi ff
, Of all the possible returns, the realized return is the return that actually occurs over a
particular time period. It is the total return we earn from dividends and capital gains,
expressed as a percentage of the initial stock price.
Divt + 1 + Pt + 1 Divt + 1 Divt + 1 - Pt
Rt + 1 = - 1 = +
Pt Pt Pt
= Dividend Yield + Capital Gain Rate
If you hold the stock beyond the date of the rst dividend, then to compute your return
you must specify how you invest any dividends you receive in the interim.
Let’s assume that all dividends are immediately reinvested and used to purchase
additional shares of the same stock or security.
If a stock pays dividends at the end of each quarter, with realized returns RQ1, RQ2,
RQ3, RQ4 each quarter, then we need to compute the annual realized return.
1 + Rannual = (1 + RQ1 )(1 + RQ 2 )(1 + RQ 3 )(1 + RQ 4 )
By counting the number of times a realized return falls within a particular range, we can
estimate the underlying probability distribution. When the probability distribution is
plotted using historical data, we refer to it as the empirical distribution of the returns.
Average annual return of an investment during some historical period is simply the
average of the realized returns for each year
1 1 T
R = ( R1 + R2 + ! + RT ) = å Rt
T T t =1
NB: if the probability distribution of the returns is the same over time, the average
return provides an estimate of the expected return.
Variance estimate using realized returns
We want to nd the volatility of the distribution and to do that we need an estimate of
the standard deviation. We estimate the variance by the average squared deviation
from the mean. However, since we do not have the mean, we use the best estimate of
the mean, which is the average realized return.
T
1
å (R - R)
2
Var (R) = t
T - 1 t =1
fi fi
