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Risk and Return –
Introduction
Chapter 9
For 9.220,
Introduction
 It is important to understand the relation
between risk and return so we can
determine appropriate risk-adjusted
discount rates for our NPV analysis.
 At least as important, the relation between
risk and return is useful for investors (who
buy securities), corporations (that sell
securities to finance themselves), and for
financial intermediaries (that invest,
borrow, lend, and price securities on behalf
of their clients).
What is risk?
 Definition: risk is the potential for
divergence between the actual
outcome and what is expected.
 In finance, risk is usually related to
whether expected cash flows will
materialize, whether security prices
will fluctuate unexpectedly, or
whether returns will be as expected.
Unless otherwise
indicated, the word
“return” refers to %
returns (either in % or
decimal form).
Measuring Performance: Returns
 Dollar return (over one period):
= Dividends + End of Period Price – Beginning of Period Price
 Percentage return (over one period):
=Dollar return/Beginning of Period Price
=(Dividends + End of Period Price)/Beginning of Period Price -1
Dividend at
 time
Notating
Percentage Returns
=t
Rt 
Dividend
yield

Capital
gain
Dt  ( Pt  Pt 1 ) Dt  Pt

1
Pt 1
Pt 1
Dt Pt  Pt 1

Pt 1
Pt 1
Capital
gain yield
Historic returns - Example 1
Q.Suppose a stock had an initial price of $42
per share, paid a
dividend of $0.84 per
share during the year, and had an ending
price of $46.2. Calculate:
a. Percentage total return
b. Dividend yield
c. Capital gains yield
Returns - Example 1
Dividends = $0.84
Ending
Market Value = $46.20
Time: t - 1
Outflows
– $42.00
Total inflows = $47.04
t
Historic returns - Example 1
A.
a. percentage total return
Dt  ( Pt  Pt 1 ) Dt  Pt
Rt 

1
Pt 1
Pt 1

Dt Pt  Pt 1

Pt 1
Pt 1
Find R?????
b. Find dividend yield ????
c. Findcapital gains yield = ?????
Returns
 Holding Period Return
HPR = (1 + R1)(1 + R2)...(1+ RT) -1
Rt  return in period t , t  1,2,3,..., T
 (Geometric) Average Return
GAR = [(1 + R1)(1 + R2)...(1+ RT)]1/T -1
= [1 + HPR]1/T -1
 (Arithmetic) Average Return
( R1  R2  R3...  ...RT ) Tt1 Rt
R

T
T
Historic returns - Example 2
Q. The following are TSE 300 returns for the 1994-1997
period:
Year (t)
1994
1995
1996
1997
Return (Rt)
-0.18%
14.53
28.35
14.98
Calculate:
a. holding period return (HPR)
b. geometric average return (GAR)
c. arithmetic average return (R)
A.
Historic returns - Example 2
a. holding period return
_______
b. geometric average return
_________
c. arithmetic average return (mean)
R(bar) =______
The risk premium
 Definition: the risk premium is the return
on a risky security minus the return on a
risk-free security (often T-bills are used as
the risk-free security)
 Another name for a security’s risk premium is
the excess return of the risky security.
 The market risk premium is the return on
the market (as a whole) minus the risk-free
rate of return.
 We may talk about the past observed risk
premium, the average risk premium, or the
expected risk premium.
Risk measures
 Studies of stock returns indicate they are
approximately normally distributed. Two statistics
describe a normal distribution, the mean and the
standard deviation (which is the square root of the
variance). The standard deviation shows how spread
out is the distribution.
 For stock returns, a more spread out distribution
means there is a higher probability of returns being
farther away from the mean (or expected return).
 For our estimate of the expected return, we can use
the mean of returns from a sample of stock returns.
 For our estimate of the risk, we can use the standard
deviation or variance calculated from a sample of
stock returns.
While return measures reward, we need some
Do
you see evidence
measure
of uncertainty (variability) associated
risk?return
with of
that
The average Risk
Rates of Return in Canada 1948-1997
Premium for Common
Stocks in Canada from
1948 to 1997 was  7%
60.00
50.00
Percentage Return
40.00
30.00
20.00
10.00
0.00
1948
1953
1958
1963
1968
1973
1978
1983
-10.00
-20.00
-30.00
Year
TSE 300
Source: William M. Mercer Ltd.
long bonds
91-day T-Bill
1988
1993
Other return statistics
 Historic Return Variance:
Average value of squared deviations from the mean.
A measure of volatility.
ˆ 2 
( R1  R )  ( R2  R )    ( RT  R )

T-1
2
2
 Historic Standard Deviation:
Also measures volatility.
ˆ   2
2
 R)
T-1
Tt1 ( Rt
2
Other return statistics - An example
 Historic Return Variance:
 Historic Standard Deviation: root of variance
Calculating Variance with a Table
Expected return Deviation from Mean Squared Deviation
----------------------------------------------------------------
______________________________________________
Mean = ____ Divide sqd dev with (n-1) to get ______
Then Find _____
How to interpret the standard deviation
as a measure of risk
 Given a normal distribution of stock returns …
 there is about a 68.26% probability that the
actual return will be within 1 standard deviation
of the mean.
 there is about a 95.44% probability that the
actual return will be within 2 standard deviations
of the mean.
 There is about a 99.74% probability that the
actual return will be within 3 standard deviations
of the mean.
Using Return Statistics
The Normal Distribution
(based on TSE 300 1994-1997 return data)
Probability
Prob. (-8.90%  r 
37.74%) = 95.44%
Prob. (r < -8.90%
OR r > 37.74%) =
4.56%
68.26%
95.44%
>99.74%
m – 3
- 20.56%
m – 2
- 8.90%
m –
+ 2.76%
m
+ 14.42%
m +
+ 26.08%
m + 2
+ 37.74%
m + 3
Return on
+ 49.4%
TSE300
stocks
U.S. Historical Return Statistics
Series
Arithmetic
mean
U.S. Common
stocks
Long-term U.S.
corporate bonds
Long-term U.S.
government bonds
U.S. Treasury bills
Risk premium
(relative to U.S.
Treasury bills)
Arith
Mean
Risk
Premium
13.0%
9.2%
6.1
2.3
5.4
3.8
Standard
deviation
Standard
Deviation
20.3%
-90%
0%
90%
-90%
0%
90%
8.7
1.6
5.7
0
-90% as risk0%
Note:
increases, so does
the excess return
3.2on a risky asset
-90%
0%
90%
90%
Modified from Stocks, Bonds, Bills and Inflation: 1998 Yearbook,TM annual updates work by
Roger C. Ibbotson and Rex A. Sinquefield (Chicago: Ibbotson Associates). All rights reserved.
Summary and conclusions
 We can easily calculate $, %, holding
period, geometric average, and mean
returns from a sample of returns data.
 We can also do the same for a security’s
risk premium.
 The mean and standard deviation
calculated from sample returns data are
often used as estimates of expected returns
and the risk measure for a security or for
the market as a whole.