# 04A and 4B

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FIN211 Financial Management Lecture Notes

File reference: 04A Text reference: Chapter 9

Topic: Introduction to risk & return, Part 1

The nature of risk
Risk is associated with uncertainty (about the future), as opposed to variability
(so the margin note on page 237 of the text is potentially misleading). There are
very few aspects about the future of which one can be 100% certain. The solar
system and mortality are candidates, but the uncertain timing of the latter leaves
only the solar system. There is nothing in the future in the world of commerce
and financial management that is 100% certain.

Rates of return
Rates of return on a personal investment have 3 elements of compensation.
Compensation for:
1 forgoing current consumption;
2 inflation;
3 risk.

Market rates of interest (designated i) are calculated from cash flows received
(or expected to be received) as a result of making an investment. Mkt rates of
return are nominal in the sense that they make no allowance for declines in the
general purchasing power of money due to inflation (designated r). When rates
of inflation are positive, nominal rates of return are higher than real rates of
return (designated R). Real rates of return measure investment returns in terms
of purchasing power rather than cash, and may be calculated as follows:
R = (i – r) / (1 + r)

Note that this equation and associated symbols are adopted from page 239 of the
text. Later in the chapter, R is used to indicate nominal rates of return.

A risk-free rate of return (Rf) is identified as the rate of return offered on a
short-dated govt security (issued by a credit-worthy govt). Whilst an individual
may receive less than the risk-free rate of return on – say – a bank deposit; in the
FIN211                                                                       04A & 04B

commercial world all investments that are not risk-free are required to earn a
risk premium. A risk premium may be defined as the difference between the
required or expected rate of return and the risk-free rate.

The higher the risk, the higher the risk premium and required rate of return. A
required rate of return may be defined as the minimum rate of return necessary
to attract an investor to purchase or hold a security (page 256).
Historically, the risk premium for the share market as a whole is about 7.7%.

Measuring risk
In any risky situation, a range of outcomes may be possible. By assigning
probabilities to each possible outcome, an expected outcome can be defined in
the same way as an arithmetic mean. Risk may then be measured as the
standard deviation of the expected outcome. When expected outcomes are
measured as per cent per year, the standard deviation is measured in the same
way (per cent per year) and needs no adjustment to allow for investments of
different amounts. However, a coefficient of variation (CV = std deviation
divided by expected outcome) can still be calculated as a relative measure of
risk. [The CV can also be calculated using dollar values for expected outcomes
and associated std deviations.]
Calculators should be used to calculate std deviations.

2 types of risk: diversifiable risk & systematic (or systemic) risk
In commerce, the total risk associated with an investment can be partitioned into
(1) diversifiable (or unsystematic, firm-specific or idiosyncratic) risk, and (2)
systematic (or non-diversifiable or market) risk.

Reducing risk through diversification
A firm that sells ice-cream can reduce risk by also selling umbrellas. This is the
nature of diversification. That is – not putting all one’s eggs in one basket; an
ancient and almost universal concept.
If sales of ice-cream and umbrellas were always perfectly negatively correlated,
it might be possible to eliminate all risk through diversification. However, this is
not possible because sales of all goods are to some extent influenced by general

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FIN211                                                                      04A & 04B

economic (or market or systematic) conditions and are therefore assumed to be,
to a greater or lesser extent, positively correlated.
[Note that if one places all one’s eggs in different baskets, all of which are then
placed in the same wheelbarrow, there is no reduction of risk. If the
wheelbarrow suffers a mishap, all the eggs may be broken. That is, the security
of all baskets is perfectly positively correlated and so there has been no risk
reduction through diversification.]
Because returns on different investments, although positively correlated, are
rarely perfectly positively correlated, it is possible to reduce risk through
diversification. For example, if \$32 is invested in a single throw of one dice
paying \$10 per dot, the expected rate of return is 9.375% with a std deviation of
53.37%. If the investment is divided between two dice paying \$5 per dot, the
expected return is still 9.375%, but the std deviation falls to 37.74%. [Note that:
(1) 37.74% = 53.37%  2 . (2) We assume zero correlation between the two
dice, whereas different ‘real world’ investments are assumed to be positively
correlated.]

Reducing portfolio risk through diversification
An investment portfolio is a collection of different investments held by a single
investor. If the investments are stocks of shares (actively traded in a share-
market), most diversifiable risk can be eliminated by holding a balanced
portfolio of at least 20 stocks (see Figure 9.5 on page 249). The remaining risk is
systematic risk, which cannot be reduced through diversification (although
further risk reduction may be achieved by investing off-shore).

Risk & return
Because diversifiable risk can be almost eliminated, the risk premium required
from a particular investment, held as part of a diversified portfolio, will depend
on its systematic risk, rather than its total risk. Systematic risk is measured in
terms of the response of an investment’s returns to changing mkt conditions.
Beta (the Greek letter for b) is the unit used for measuring systematic risk.
The beta value of the mkt as a whole is assumed to be 1. The beta value of a
particular investment (j) is given by the following equation:

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FIN211                                                                       04A & 04B

βj = σj ρmj / σm

This indicates that the beta value of security j is the standard deviation of returns
on security j, multiplied by the coefficient of correlation between the returns on
security j and returns on the mkt, divided by the std deviation of returns on the
mkt.

The beta value of a security (j) can be calculated through linear regression. That
is, the returns on security j (the dependent variable) are plotted against
corresponding mkt returns (the independent variable). Beta is then the slope of
the line of best fit. See Figures 9.7, 9.8 & 9.9 on pages 254 & 256. Beta values
may be calculated in this way using a calculator. [On the Sharp calculator, beta
is designated ‘b’: 2nd function of the minus key.]

The required rate of return on security j (Rj) is then given by the following
equation:
Rj = Rf + βj(Rm  Rf)

Where Rm indicates the expected return on the mkt as a whole and (Rm  Rf) is

This is the Capital Asset Pricing Model (CAPM). It indicates that the required
rate of return on security j is
the risk-free rate +
the beta value of security j  the mkt risk premium.

Remember that historically the risk premium offered by the Australian share-
market as a whole has averaged about 7.7%.

Topic: Introduction to risk & return, Part 2

Interest rates
Interest rates reflect (i) risk and (ii) term to maturity. The relationship between
risk and required rates of return is always positive: the higher the risk, the higher
the required rate of return. The relationship between term to maturity and
required rates of return is referred to as the term structure of interest rates or

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the yield curve. Yield curves are typically upwards sloping, but can assume a
variety of shapes, including downwards sloping.
An upwards sloping yield curve can be explained in terms of three different
theories.
1. The unbiased expectations theory which postulates that a rising yield
curve indicates that the market expects short-term interest rates to rise.

2. The liquidity preference theory which postulates that investors require a
liquidity premium to induce them to hold longer-dated, less liquid,
securities. The liquidity premium is the difference between forward rates
implied by the yield curve and expected future spot rates, and is typically
positive. The liquidity premium is typically positive because investors tend
to have a preference for shorter-dated securities.

3. The market segmentation theory which postulates that the market is
partitioned into segments for which different investors will have
preferences. If the market preference for short-dated securities is stronger
than for longer-dated securities, yields on the former will be lower than
yields on the latter. This could be the case because financial intermediaries
such as banks use short-dated securities for liquidity management purposes.

Calculating beta values
The appendix to Chapter 9 of the text demonstrates how to calculate a security’s
beta value using a calculator. Any calculator with a statistical function can be
used  it does not have to be a financial calculator.

The minimum input data requirement is two observed rates of return for the
security and coincidental mkt returns. The calculator is set into its statistical
mode and each pair of coincidental returns entered. Mkt returns are entered
through the ‘RM’ key and the coinciding security’s return through the ‘M+’ key.
After at least two pairs of data have been entered, the line of best fit can be
determined in accordance with the linear regression equation:
y = a + bx
Where y = rates of return on the security
x = mkt rates of return

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FIN211                                                                      04A & 04B

a = the value of the y intercept when x = 0
b = the slope of the line of best fit = beta.

The Sharp calculator displays beta as the 2nd function of the minus key.

Example
Coincidental rates of return on security j and the market are as follows:
Market      Security j
14%           18%
12%           11%
10%            6%
What is the beta value of security j?

Note:
1. Returns on security j are more volatile than mkt returns. We should
therefore expect security j’s beta value to be higher than 1 (the beta value
of the mkt).
2. Mkt returns are the independent variable and must be entered through the
RM key before the value of the dependent variable (return on the security)
is entered through the M+ key.

Keystrokes: 14 RM         18 M+         12 RM     11 M+     10 RM       6 M+

2ndF  3            βj beta value of security j

2ndF ÷ 0.9954       ρmj coefficient of correlation between returns
on security j & the mkt

2ndF 9 4.9216       σj std deviation of returns on security j

2ndF 6 1.6330       σm std deviation of mkt returns

βj = σj ρmj / σm
= (4.9216  0.9954) / 1.633
= 3

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FIN211                                                                       04A & 04B

Required rates of return
If the risk-free rate is 5% and the expected mkt return 12.5%, what would be the
required rate of return on security j?

Rj = Rf + βj(Rm - Rf)

= 0.05 + 3  0.075
= 0.275
= 27.5%

See if you can draw a graph of this example.

The security mkt line (page 258)
The security mkt line defines required rates of return as a linear function of an
investment’s beta value. Note that it is the required rate of return, rather than the
expected rate of return indicated by the textbook (see Figure 9.10 on page 259).

*         *       *   *

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04A & 04B

No. of dice
Risk reduction through diversification: investing                                          1         2
in a single throw of two, rather than one, dice                  Investment per throw
32          32
\$
Return per dot \$         10          5
Average ROI \$           3.00        3.00
Average ROI %           9.375      9.375
Std Dev. %              53.37      37.74
CV                      5.69        4.03
Single dice matrix
Averages Std dev.       CV
Score                   1        2         3         4       5         6          3.50     1.71        0.49
Proceeds \$             10        20      30         40      50        60          35.00   17.08        0.49
ROI \$                  -22      -12       -2         8      18        28          3.00    17.08        5.69
ROI %                -68.75   -37.50    -6.25      25.00   56.25     87.50        9.375   53.37        5.69

Two-dice matrix: scores
First dice
Averages Std dev.      CV
1        2         3          4      5           6
1     2        3         4          5      6           7
Second dice

2     3        4         5          6      7           8
3     4        5         6          7      8           9
7       2.42        0.35
4     5        6         7          8      9           10
5     6        7         8          9      10          11
6     7        8         9         10      11          12

Two-dice matrix: proceeds \$
First dice
Averages Std dev.      CV
1        2         3          4       5           6
1    10       15        20        25       30          35
Second dice

2    15       20        25        30       35          40
3    20       25        30        35       40          45
35.00     12.08       0.35
4    25       30        35        40       45          50
5    30       35        40        45       50          55
6    35       40        45        50       55          60

Two-dice matrix: ROI \$
First dice
Averages Std dev.      CV
1        2          3          4     5           6
1    -22      -17       -12          -7    -2          3
Second dice

2    -17      -12        -7          -2    3           8
3    -12       -7        -2          3     8           13
3.00     12.08       4.03
4     -7       -2         3          8     13          18
5     -2       3          8         13     18          23
6     3        8        13          18     23          28

Two-dice matrix: ROI %
First dice
Averages Std dev.      CV
1        2          3          4     5           6
1    -69      -53       -38         -22    -6          9
Second dice

2    -53      -38       -22          -6    9           25
3    -38      -22        -6          9     25          41
9.375     37.74       4.03
4    -22       -6         9         25     41          56
5     -6       9        25          41     56          72
6     9        25       41          56     72          88

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