Risk08a

Risk and Return

The cost of Capital

Return on a Share or Stock

Return or holding period return on a share is simply:

PD + P - P

1-t

t1-tt

Expected Return is

P

DE + P - PERE

t

tttt

)()()( 11

The lower the current price – other things being equal The greater the expected return

Pt Ending Price = 48

Pt-1 Beginning Price = 40

Dividend = 2

The holding period return is

HPR = (48 - 40 + 2 )/ (40) = 50/40 = 25%

Rates of Return: Single Period Example

The Value of an Investment of $1 in 1926

Source: Ibbotson Associates

0.1

10

1000

1925 1933 1941 1949 1957 1965 1973 1981 1989 1997

S&PSmall CapCorp BondsLong BondT Bill

Inde

x

Year End

1

5520

1828

55.38

39.07

14.25

0.1

10

1000

1925 1933 1941 1949 1957 1965 1973 1981 1989 1997

S&PSmall CapCorp BondsLong BondT Bill

The Value of an Investment of $1 in 1926


Inde

x

Year End

1

613

203

6.15

4.34

1.58

Real returns

Volatility of Rates of Return 1926-1997


-60

-40

-20

0

20

40

60

26 30 35 40 45 50 55 60 65 70 75 80 85 90 95

Common Stocks

Long T-Bonds

T-Bills

Year

Per

cent

age

Ret

urn

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.

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.

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.

Risk Free and Risky Projects

Table 1

Project t0

Outlay

t1

Pay-off

Prob. ExpectedReturn

Certain A 100 120 1 20%

Risky B 100 80160

0.50.5

20%

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%.

Subjective returns

p(s) = probability of a stater(s) = return if a state occurs1 to s states

Measuring Expected Return: Scenario or Subjective Returns

rprE s

s

s1

)(

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.

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.

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.

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.

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

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.

1) Mean: most likely value2) Variance or standard deviation – measure

of spread3) 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.

Characteristics of Probability Distributions

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.

Probability analysis

•Expected return of a project or investment

•Standard deviation of a project or investment

•The mean–variance rule

The expected return

=n

i =1R Σ Ri pi

R = expected return, Ri = return if event i occurs

pi = probability of event i occurring, n = number of events

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

x = σ2 = (x1 –x)2 p1 + (x2 –x)2 p2 + … (xn – x)2 pn–

x

– –Variance of

xσ2or = Σ {(xi – x)2 pi}

i=n

i=1

–

Standard deviation

{(xi – x)2 pi}

i=n

i=1

–orσx =σ2 x Σ

Standard deviation

(Ri – Ri)2 pi

n

i =1

σ = Σ

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

Sample standard deviation:1-n

)X(XS

n

1i

2i

Calculation Example:Sample Standard DeviationSample

Data (Xi) : 10 12 14 15 17 18 18 24

n = 8 Mean = X = 16

4.30957

130

18

16)(2416)(1416)(1216)(10

1n

)X(24)X(14)X(12)X(10S

2222

2222

A measure of the “average” scatter around the mean

Measuring variation

Small standard deviation

Large standard deviation

Comparing Standard Deviations

Mean = 15.5 S = 3.338 11 12 13 14 15 16 17 18 19 20 21

11 12 13 14 15 16 17 18 19 20 21

Data B

Data A

Mean = 15.5 S = 0.926

11 12 13 14 15 16 17 18 19 20 21

Mean = 15.5 S = 4.567

Data C

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)

If the data distribution is approximately bell-shaped, then the interval:

contains about 68% of the values in the population or the sample

The Empirical Rule

1σμ

μ

68%

1σμ

contains about 95% of the values in the population or the sample

contains about 99.7% of the values in the population or the sample

The Empirical Rule

2σμ

3σμ

3σμ

99.7%95%

2σμ

Markowitz Portfolio Theory

Price changes vs. Normal distribution

Microsoft - Daily % change 1986-1997

0

100

200

300

400

500

600

-10% -8% -6% -4% -2% 0% 2% 4% 6% 8% 10%

# of

Day

s (f

requ

ency

)

Daily % Change

Bootstrapped history vs. Normal Distribution

Symmetric distribution

mean

s.d. s.d.

Normal Distribution

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

Figure 1

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

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?

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 Return

StandardDeviation

A 100 -10 45 55

B 100 -90 5 95

An investment costs €10,000

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

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.

Risk08a

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