The document discusses risk and return, including expected return, variance, and standard deviation. It provides examples of how to calculate expected return by taking the returns of individual outcomes and multiplying by their probabilities. It also explains how to calculate variance and standard deviation as measures of risk and dispersion around the mean. The standard deviation is the square root of the variance and provides a measure of risk in the same units as the data. Riskier investments are expected to provide higher returns through a risk premium.

1. Risk and Return
The cost of Capital

2. Return on a Share or Stock
Return or holding period return on a share is
simply:
P -P +D
t t -1 t
P t -1

3. Expected Return is
E ( Pt +1) - Pt + E ( Dt +1)
E ( Rt ) =
Pt
The lower the current price – other things
being equal The greater the expected return

4. Rates of Return:
Single Period Example
Pt Ending Price = 48
Pt-1 Beginning Price = 40
Dividend = 2
The holding period return is
HPR = (48 - 40 + 2 )/ (40) = 50/40 =
25%

5. The Value of an Investment of $1 in 1926
5520
S&P
Small Cap 1828
1000
Corp Bonds
Long Bond
T Bill 55.38
Index
39.07
10 14.25
1
0.1
1925 1933 1941 1949 1957 1965 1973 1981 1989 1997
Source: Ibbotson Associates Year End

6. The Value of an Investment of $1 in 1926
S&P Real returns
Small Cap
1000
Corp Bonds 613
Long Bond
T Bill 203
Index
10 6.15
4.34
1 1.58
0.1
1925 1933 1941 1949 1957 1965 1973 1981 1989 1997
Source: Ibbotson Associates Year End

7. Volatility of Rates of Return 1926-1997
60
Percentage Return
40
20
0
-20
Common Stocks
-40 Long T-Bonds
T-Bills
-60 26 30 35 40 45 50 55 60 65 70 75 80 85 90 95
Source: Ibbotson Associates Year

8. Risk
 A risky investment is one which has a
range or spread of possible outcomes
whose probabilities are known.
 A probability represents the chance or
“odds” of a particular outcome to the
investment. If something is certain it to
occur is has a probability of one. If
something is certain not to occur is has a
probability of 0.

9. Probability
 If an outcome is uncertain it has a
probability that is greater than 0 and less
than 1.
 The probability of the total number of
possible outcomes is 1 or 100%. The
probability of an outcome or outcomes
from the total number of possibilities is
between 0% and 100% (0 and 1).
 The sum of the probabilities of all
outcomes is 1.

10. Probability
 It may be helpful to think of probability in
terms of the frequency of an outcome.
 the probability of getting a 6 when one
throws a die is 1/6 or 0.1667. If you
threw a die 600 times you would expect to
throw 100 sixes.

11. Risk Free and Risky Projects
 Table 1
Project t0 t1 Prob. Expected
Outlay Pay-off Return
Certain A 100 120 1 20%
Risky B 100 80 0.5 20%
160 0.5

12. Computation of Expected Return
 The expected return on a project is
computed by taking the individual returns
of A and B and multiplying them by their
respective probabilities and summing
them
 i.e. (-20%*0.5) + (60%*0.5) =
-10%+30% = 20%.

13. Measuring Expected Return:
Scenario or Subjective Returns
Subjective returns
s
E (r ) = ∑ p s r s
1
p(s) = probability of a state
r(s) = return if a state occurs
1 to s states

14. Investors' Attitudes to Risk
 We assume that investors are risk averse.
This means that investors prefer an
investment with a certain return to a risky
one with the same expected return.
 A risk averse investor would prefer project
A in Table 1 above to project B.

15. Will anyone invest in B?
 If the price of B falls its expected return will
increase.
 Eventually the return will rise sufficiently for
some investors to choose B rather than A.
 The rate of return of B at which the investor is
indifferent between B and the risk free project A
is called the certainty equivalent rate of return.
 If more than an investor’s certainty equivalent
rate of return can be earned on B she will
choose it over A.

16. Risk and Return
 Unless risky investments are likely to offer
greater returns than relatively safe ones
nobody will hold them.
 If markets are competitive investors are
unlikely to be able to increase expected
returns without investing in assets which
bear additional risk.

17. A Premium for Risk
 Therefore any asset that is traded in a
competitive market will have an expected
return that is increasing in risk.
 We can characterise the expected return
on any asset traded in the capital markets
in the form:
 Expected rate of return = risk-free rate +
risk premium.

18. Measurement of Risk
 In Finance risk is usually measured by the
amount of dispersion or variability in the
value of an asset. Thus, risky assets can
have very positive outcomes as well as
very negative ones. One has upside risk
(potential) and downside risk

19. Measuring Risk
Variance - Average value of squared deviations
from mean. A measure of volatility.
Standard Deviation – The square root of the
average value of squared deviations from mean.
a measure of volatility.
Has advantage of being measured in the same
dimension as the mean.

20. Characteristics of Probability
Distributions
1) Mean: most likely value
2) Variance or standard deviation –
measure of spread
3) Skewness – refers to the tendency to have
extreme outliers either at the top or bottom
of the distribution.
* If a distribution is approximately normal, the
distribution is described by characteristics
1 and 2.

21. Random Variable
 A random variable is a variable which
can take on a range of different values
and we are never certain which value it is
going to take on at a particular time.
 The return on a risky project or
investment can be perceived as a random
variable.

22. Probability analysis
•Expected return of a project or investment
•Standard deviation of a project or investment
•The mean–variance rule

23. The expected return
n
R = Σ R i pi
i =1
R = expected return, Ri = return if event i occurs
pi = probability of event i occurring, n = number of events

24. Standard deviation
•Standard deviation, σ, is a statistical measure of the
dispersion around the expected value
•The standard deviation is the square root of the variance, σ2
– – –
Variance of x = σ2 = (x1 –x)2 p1 + (x2 –x)2 p2 + … (xn – x)2 pn
x
i=n
or
σ =
x
2
Σ
i=1
–
{(xi – x)2 pi}
Standard deviation
√Σ
i=n
√σ
2
σx = or –
{(xi – x)2 pi}
x
i=1

25. Standard deviation
√Σ
n
σ = (Ri – Ri)2 pi
i =1

26. Standard Deviation – historic data
 Most commonly used measure of variation
 Shows variation about the mean
 Is the square root of the variance
 Has the same units as the original data
n
∑ (X − X)
i
2
 Sample standard deviation: S= i =1
n -1

27. Calculation Example:
Sample Standard
Sample
Deviation
Data (Xi) : 10 12 14 15 17 18 18 24
n=8 Mean = X = 16
(10 − X)2 +(12 − X)2 +(14 − X)2 + +(24 − X)2
S=
n −1
(10 −16)2 +(12 −16)2 +(14 −16)2 + +(24 −16)2
=
8 −1
130 A measure of the “average”
= = 4.3095
7 scatter around the mean

28. Measuring variation
Small standard deviation
Large standard deviation

29. Comparing Standard Deviations
Data A
Mean = 15.5
11 12 13 14 15 16 17 18 19 20 21 S = 3.338
Data B
Mean = 15.5
11 12 13 14 15 16 17 18 19 20 21 S = 0.926
Data C
Mean = 15.5
11 12 13 14 15 16 17 18 19 20 21 S = 4.567

30. Advantages of Variance
and Standard Deviation
 Each value in the data set is used in the
calculation
 Values far from the mean are given
extra weight
(because deviations from the mean are
squared)

31. The Empirical Rule
 If the data distribution is approximately bell-
shaped, then the interval:
 μ ± 1σ contains about 68% of the values in
the population or the sample
68%
μ
μ ± 1σ

32. The Empirical Rule
 μ ± 2σ contains about 95% of the values
in the population or the sample
 μ ± 3σ contains about 99.7% of the
values in the population or the sample
95% 99.7%
μ ± 2σ μ ± 3σ

33. Markowitz Portfolio Theory
Price changes vs. Normal distribution
Microsoft - Daily % change 1986-1997
600
500
(frequency)
# of Days
400
300
200
100
0
-10% -8% -6% -4% -2% 0% 2% 4% 6% 8% 10%
Daily % Change

34. Markowitz Portfolio Theory
Price changes vs. Normal distribution
600Microsoft - Daily % change 1986-1997
500
(frequency)
# of Days
400
300
200
100
0
-10% -8% -6% -4% -2% 0% 2% 4% 6% 8% 10%
Daily % Change

35. Bootstrapped history vs. Normal
Distribution

36. Normal Distribution
s.d. s.d.
mean
Symmetric distribution

37.  The lower the standard deviation the lower
the risk.
 Investment A is less risky than investment
B in the figure 1 below because it has the
lower standard deviation

38. Figure 1

39. Characteristics of Risk
 Thus, there are at least three aspects to risk that we
must capture.
 First, the probability of having a poor outcome
 Second the potential size of this poor outcome.
 Third risky investments must provide the chance of
higher returns to compensate for the poor ones and
thus give an above average expected return.
 This is what we mean by the spread of possible
outcomes
 Provided we have symmetric distributions standard
deviation will capture all these elements

40. Risk is not just the probability of
making a loss
 Consider two investments which have the
same probability of making a loss.
 Are both investments equally risky?

41. An investment costs €10,000
Payoffs
Investment Probability 0.5 Probability 0.5
A 20,000 9,000
B 20,000 1,000
In terms of percentage return
Investment Prob. 0.5 Prob. 0.5 Expected Standard
Return Deviation
A 100 -10 45 55
B 100 -90 5 95

42. The computation of Standard
Deviation
Computation of Standard Deviation of A
Return Mean Ret. X - Mean(X) (xi – x)2 prob
100 45 55 3025 0.5 1512.5
-10 45 -55 3025 0.5 1512.5
Variance 3025
Standard
Deviation 55

43. The probability of making a loss is the same for
each investment but investment B is certainly
more risky. This is because the size of the
potential loss is greater. Thus, there are at least
two aspects to risk that we must capture. First,
the probability of having a poor outcome and
second the potential size of this poor outcome.
A measure of spread or dispersion will embody
both of these elements. We note that the
standard deviation of B is certainly larger than
that of A.