Transcript PPT Unit 2

Unit 2 –
Measures of Risk and Return
The purpose of this unit is for the student to
understand, be able to compute, and interpret basic
statistical measures of risk and return for financial
analysis
We will learn about three standard measures
of return on investment:
• Holding period return
• Arithmetic mean return
• Geometric mean return
Holding Period Return
• The holding period return is a single period
measure
• It is comprised of two potential components –
change in asset price, and income.
• It does not take into account the time value of
money, and should only be used to gauge
performance for a single period, such as a quarter
or a year
A Demonstration of Holding Period Return
Calculation
Suppose a stock is worth 20 euros at the beginning of the year.
The firm pays an annual dividend of .50 euros, and the end of year price is 24 euros.
Calculate the holding period return on investment.
HPR = ((24 –20) + .50) / 20 = (4.5) / 20 = 22.5%.
Here, the holding period is one year, and the investor earns
4/20 = 20% from price appreciation, and .5 / 20 = 2.5 % from income (dividend),
for a total annual return of 22.5%.
Arithmetic Mean Return
• The arithmetic mean return is a multi-period measure of
return
• It is a simple arithmetic average, computed by summing
the individual holding period returns from multiple
periods, and dividing by the number of periods
• It provides an unbiased estimate of the expected return in
the coming period
A Demonstration of the Calculation of
Arithmetic Mean Return
Suppose a stock investment returned 22.5% in 1999, 7.75% in 2000, and –12%
in 2001. What was the average annual return over the three year holding period?
Am = (22.5% + 7.75% – 12%) / 3 = 6.08%
So, during the 1999-2001 three year period, the average annual return was 6.08%.
Based on these three years performance, 6.08% would be the expected
holding period return for 2002 for this stock.
Some notes on the arithmetic mean return
measure
• This measure does not consider
compounding
• It overstates the actual rate of growth of an
investment
• It does provide an average performance
measure over multiple holding periods
• The name of the Excel statistical function
to compute an arithmetic mean is Average
Geometric Mean Return
• The geometric mean return is a multi-period
measure that incorporates the time value
associated with compounding
• To compute a geometric mean, you must add 1 to
each holding period return, creating what is
known as a “return relative”, then take the nth
root of the product of the return relatives.
• By creating return relatives, you eliminate
negative numbers from the calculation
Example of Calculating a Geometric Mean
Return
Using our same example of annual returns of 22.5%, 7.75%, and –12%,
the return relatives would be 1.225, 1.0775, and .88. Then, the geometric
mean is given by
Gm = [(1.225)(1.0775)(.88)] 1/3 - 1
Gm = [1.1615] 1/3 –1
Gm = 5.12%
Some notes on the geometric mean return
• This is a compound average measure of periodic
return
• It represents the periodic growth rate over the
multiple holding periods. Here, the investment
grew at an average annual rate of 5.12% for the
three years
• This is the most appropriate measure of return for
gauging historical performance, and comparative
performance among different investments
Some additional thoughts on arithmetic and
geometric mean returns
• Note that the arithmetic mean return will always be greater than the
geometric mean return – here, 6.08% > 5.12%. This is because the
geometric is a multiplicative measure that considers how each
holding period return affects the others
• The greater the volatility in the holding period returns, the greater the
difference in the arithmetic and geometric mean returns
• The importance of the last statement is that higher variability in
holding period returns results in lower growth rates over time
• The Excel statistical function that computes the geometric mean from
a set of return relatives is Geomean
Measures of Risk
• We now turn to quantitative measures of risk
• For definitional purposes, we define risk as
uncertainty associated with the expected return
on investment
• Note that uncertainty means the actual return
earned may be higher or lower than the expected
return (defined as the arithmetic mean return)
Standard Deviation
• Standard deviation is a statistical measure
of dispersion around a most likely, or
expected, outcome for a random variable
• Standard deviation gives a probability
range of outcomes, assuming the returns
follow an approximate normal distribution
• Recall that a normal distribution is the well
known bell curve
Calculating Standard Deviation for a
Forecasted Range of Returns
• Standard deviation may be used to measure risk
for either projected returns based on forecasts or
to analyze the risk associated with historical data
• To compute standard deviation for a forecast,
subjective probabilities must be assigned to each
possible outcome
• To compute standard deviation for historical data,
the assumption is each period (year) is assigned
equal probability
An Example of Calculating Standard
Deviation for Forecast Data
Possible Outcomes
Pessimistic
Most Likely
Optimistic
Probability of Outcome
25%
50%
25%
Rate of Return
7%
15%
23%
Compute the expected return and standard deviation of returns
associated with this investment.
The expected return is
Er =  PiRi
Where Er = the expected return on investment
Pi = the probability of outcome i
Ri = the return earned for outcome i
 = summation for all possible outcomes i
Er = .25(.07) + .50(.15) + .25(.23)
Er = .15 = 15%
The standard deviation is given by
 = [ (Ri – Er)2 x Pi] ½
Where
 is the standard deviation associated with the possible returns.
 = [(.07 - .15)2 (.25) + (.15 - .15)2(.50) + (.23 - .15)2(.25)]
 = [(.0032)] ½
 = .0566 = 5.66%
½
Some notes on interpreting standard
deviation
• Well know properties of a bell curve, or normal distribution, are the
areas under the curve covered by standard deviations
• 68% of the area under the curve falls within plus or minus one
standard deviation around the expected return, and 95% falls within
plus or minus two standard deviations
• For our example, there is a 68% probability the actual return on this
investment will be 15% plus or minus 5.66%, or between 9.34% and
20.66%
• There is a 95% probability the actual return will be between 3.68%
and 26.32%
• For most finance applications, the 68% probability range makes the
most economic sense
Some additional thoughts
•
•
•
•
•
If using historical data,  is given by
[((Ri – Er)2) / n – 1] ½
n = number of periods, normally years
See any statistics text for further information
Excel has statistical functions built in for
computing standard deviations of sample data,
STD DEV, and population data, STD DEVP.
The Coefficient of Variation
• Suppose you are comparing two
investments of greatly different risk
• Standard deviation gives a measure of total
risk in absolute terms, so naturally the
investment with more risk is the one with
higher standard deviation
• The coefficient of variation allows for
relative risk comparisions
An Example of Coefficient of Variation – a
relative measure of risk
Suppose you are comparing two investments; investment A has an
expected return of 15% and a standard deviation of 36%, while
investment B has an expected return of 8% and a standard deviation of 24%.
CV = coefficient of variation
CV =  / Er
CV A = .36 / .15 = 2.4
CV B = .24 / .08 =3.0
We conclude that even though investment A has more absolute risk,
investment B has greater risk per unit of return. This is a way of determining
if the extra risk is being compensated for by the extra return. In this example,
investment A has 2.4 units of risk per unit of expected return,
while investment B has 3 units of total risk for each unit of expected return.
While A has higher absolute risk, B has higher relative risk,
and a less favorable risk-reward trade-off.
A manager can use this information to decide if they
believe the higher expected return offered by investment A
is worth the extra risk exposure.
Risk Reduction Through
Diversification
• Total risk, which we are measuring by standard deviation,
can be decomposed into market risk and firm (or project)
specific risk
• Market risk cannot be diversified away; examples of
market risk include war, inflation, changes in legal or tax
policy, etc.
• Firm specific risk can be diversified away; examples
include a labor strike, new product technology, the
emergence of new competition, etc.
• Managers can reduce total risk exposure by diversifying
across industries or product lines to minimize or eliminate
firm specific risk