2. Chapter 2 Questions
• What are the sources of investment returns?
• How can returns be measured?
• What is risk and how can we measure risk?
• What are the components of an investment’s
required return to investors and why might
they change over time?
3. Sources of Investment Returns
• Investments provide two basic types of
return:
• Income returns
– The owner of an investment has the right
to any cash flows paid by the investment.
• Changes in price or value
– The owner of an investment receives the
benefit of increases in value and bears the
risk for any decreases in value.
4. Income Returns
• Cash payments,
usually received
regularly over the
life of the
investment.
• Examples: Coupon
interest payments
from bonds,
Common and
preferred stock
dividend payments.
5. Returns From Changes in
Value
• Investors also
experience capital
gains or losses as the
value of their
investment changes
over time.
• For example, a stock
may pay a $1 dividend
while its value falls from
$30 to $25 over the
same time period.
6. Measuring Returns
• Dollar Returns
– How much money was made on an investment
over some period of time?
– Total Dollar Return = Income + Price Change
• Holding Period Return
– By dividing the Total Dollar Return by the
Purchase Price (or Beginning Price), we can
better gauge a return by incorporating the size of
the investment made in order to get the dollar
return.
7. Annualized Returns
• If we have return or income/price change
information over a time period in excess of
one year, we usually want to annualize the
rate of return in order to facilitate
comparisons with other investment returns.
• Another useful measure:
Return Relative = Income + Ending Value
Purchase Price
8. Annualized Returns
Annualized HPR = (1 + HPR)1/n – 1
Annualized HPR = (Return Relative)1/n – 1
• With returns computed on an annualized
basis, they are now comparable with all other
annualized returns.
9. Measuring Historic Returns
• Starting with annualized Holding Period
Returns, we often want to calculate
some measure of the “average” return
over time on an investment.
• Two commonly used measures of
average:
– Arithmetic Mean
– Geometric Mean
10. Arithmetic Mean Return
• The arithmetic mean is the “simple average”
of a series of returns.
• Calculated by summing all of the returns in
the series and dividing by the number of
values.
RA = (ΣHPR)/n
• Oddly enough, earning the arithmetic mean
return for n years is not generally equivalent
to the actual amount of money earned by the
investment over all n time periods.
12. Geometric Mean Return
• The geometric mean is the one return that, if
earned in each of the n years of an
investment’s life, gives the same total dollar
result as the actual investment.
• It is calculated as the nth root of the product
of all of the n return relatives of the
investment.
RG = [Π(Return Relatives)]1/n – 1
13. Geometric Mean Example
Year Holding Period Return Return Relative
1 10% 1.10
2 30% 1.30
3 -20% 0.80
4 0% 1.00
5 20% 1.20
RG = [(1.10)(1.30)(.80)(1.00)(1.20)]1/5 – 1
RG = .0654 or 6.54%
14. Arithmetic vs. Geometric
To ponder which is the superior measure,
consider the same example with a $1000
initial investment. How much would be
accumulated?
Year Holding Period Return Investment Value
1 10% $1,100
2 30% $1,430
3 -20% $1,144
4 0% $1,144
5 20% $1,373
15. Arithmetic vs. Geometric
• How much would be accumulated if you
earned the arithmetic mean over the same
time period?
Value = $1,000 (1.08)5 = $1,469
• How much would be accumulated if you
earned the geometric mean over the same
time period?
Value = $1,000 (1.0654)5 = $1,373
• Notice that only the geometric mean gives
the same return as the underlying series of
returns.
16. Investment Strategy
• Generally, the income returns from an investment are
“in your pocket” cash flows.
• Over time, your portfolio will grow much faster if you
reinvest these cash flows and put the full power of
compound interest in your favor.
• Dividend reinvestment plans (DRIPs) provide a tool
for this to happen automatically; similarly, Mutual
Funds allow for automatic reinvestment of income.
• See Exhibit 2.5 for an illustration of the benefit of
reinvesting income.
17. What is risk?
• Risk is the uncertainty associated with the
return on an investment.
• Risk can impact all components of return
through:
– Fluctuations in income returns;
– Fluctuations in price changes of the investment;
– Fluctuations in reinvestment rates of return.
18. Sources of Risk
• Systematic Risk Factors
– Affect many investment returns simultaneously;
their impact is pervasive.
– Examples: changes in interest rates and the state
of the macro-economy.
• Asset-specific Risk Factors
– Affect only one or a small number of investment
returns; come from the characteristics of the
specific investment.
– Examples: poor management, competitive
pressures.
19. How can we measure risk?
• Since risk is related to variability and
uncertainty, we can use measures of
variability to assess risk.
• The variance and its positive square root, the
standard deviation, are such measures.
– Measure “total risk” of an investment, the
combined effects of systematic and asset-specific
risk factors.
• Variance of Historic Returns
σ2 = [Σ(Rt-RA)2]/n-1
20. Standard Deviation of Historic
Returns
Year Holding Period Return
1 10% RA = 8%
2 30% σ2 = 370
3 -20% σ = 19.2%
4 0%
5 20%
σ2 = [(10-8)2+(30-8)2+(-20-8)2+(0-8)2+(20-8)2]/4
= [4+484+784+64+144]/4
= [1480]/4
21. Using the Standard Deviation
• If returns are normally distributed, the
standard deviation can be used to
determine the probability of observing a
rate of return over some range of
values.
22. Coefficient of Variation
• The coefficient of variation is the ratio of the
standard deviation divided by the return on
the investment; it is a measure of risk per unit
of return.
CV = σ/RA
• The higher the coefficient of variation, the
riskier the investment.
• From the previous example, the coefficient of
variation would be:
CV =19.2%/8% = 2.40
23. Components of Return
• The required rate of return on an
investment is the sum of the nominal
risk-free rate (Nominal RFR) and a risk
premium (RP) to compensate the
investor for risk.
• Required Return = Nominal RFR + RP
• Or to be more technically correct:
• RR = (1 + Nom RFR) x (1 + RP) - 1
24. The Risk-Return Relationship
• The Capital Market Line (CML) is a
visual representation of how risk is
rewarded in the market for investments.
• The greater the risk, the greater the
required return, so the CML slopes
upward.
25. Components of Return Over
Time
• What changes the required return on an
investment over time?
• Anything that changes the risk-free rate or
the investment’s risk premium.
– Changes in the real risk-free rate of return and the
expected rate of inflation (both impacting the
nominal risk-free rate, factors that shift the CML).
– Changes in the investment’s specific risk (a
movement along the CML) and the premium
required in the marketplace for bearing risk
(changing the slope of the CML).
