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Risk and Return
Two sides of the Investment Coin
Overview
• Investment decisions are influenced by various motives.
– Some invest in a business to acquire control and enjoy the
prestige.
– Some invest in expensive yatchs and famous villas to display
their wealth.
• Most investors however, are largely guided by the
pecuniary movite of earning a return on their investment.
• For earning returns, investors have to almost invariably
bear some risk.
• In general, risk and return go hand in hand.
• While investors like returns, they abhor risk.
• Investment decisions, therefore, involve a tradeoff between
risk and return.
Return
• Return is primary motivating force that drives
investment.
• It represents the reward for undertaking
investment.
• Sine the game of investing is about returns (after
allowing for risk), measurement of realized
(historical) returns (ex post facto) is necessary to
access how ell the investment manager has done.
• In addition, historical returns are often used as a
important input in estimating future (prospective)
returns.
The components of Return
• The return of an investment consists of two
components:
• Current return
• Capital return
Current Return
• Periodic cash flow (income) such as dividend
or interest, generated by the investment in
various instruments.
• Current return is measured as the periodic
income in relation to the beginning price of
the investment.
Current Income
Current Return/Yie ld 
Beginning price
Capital Return
• Reflected in the price change - Capital
gain/loss
• It
is
simply
the
price
appreciation/depreciation divided by the
beginning price of the asset/security.
Ending Price - Beginning Price
Capital Return/ Capital Gain/Loss Yield 
Beginning Price
P P
 1 0
P0
Total Return
Total Return  Current Return  Capital Return
In case of Share,
Total Return  Dividend Yield  Capital gain/loss yield
In case of Bond,
Total Return  Coupon Yield  Capital gain/loss yield
• The current return can be zero or positve
• The capital return can be negative, or zero or
positive.
Risk
• Risk refers to the possibility that the actual outcome of
an investment will differ from its expected outcome.
• More specifically, most investors are concerned about
the actual outcome being less than the expected
outcome.
• The wider the range of possible outcomes, the greater
the risk.
• Risk is the variability in possible returns.
• In investment analysis, its measured by:
– Variance / Standard Deviation
– Beta
Sources of Risk
• Risk emanates from several sources.
• The three major ones are:
– Business Risk
– Interest Rate Risk
– Market Risk
Business Risk
• Risk of poor business peformance. (Operating Risk)
• May be caused by variety of factors:
–
–
–
–
–
–
Heightened competition
Emergence of new technologies
Development of subtitute products,
Shifts in consumer preference
Inadequate supply of essential inputs
Changes in governmental policies, and so on.
• Principle factor may be inept and incompetent
management.
• It can affect the interest of shareholders and even
bond/debenture holders (default risk)
Interest Rate Risk
• The changes in interest rate have a bearing on
welfare of investors.
• As interest rate goes up, the market price of
existing fixed income securities falls and vice
versa.
• It also affects equity prices, albeit some what
indirectly.
• The changes in the relative yields of
debentures and equity shares influence equity
prices.
Market Risk
• Changing psychology of the investors.
• There are periods when investors become bullish and their
investment horizons lengthen.
• Investor’s optimism, which may broder on euphoria, during
such periods drives share prices to great heights.
• The buoyancy created in the wake of this development is
pervasive, affecting almost allshares.
• On the other hand, when a wave of pessimism (which often
is an exaggerated response to some unfavourable political
or economic development) sweeps the market, investors
turn bearish and myopic.
• Prices of almost all equity shares register decline as fear
and uncertainty prevade the market.
“The ebb and flow of mass emotion
is quite regular: Panic is followed by
relief, and relief by optimism; then
comes enthusiasm, then euphoria
and rapture, then the bubble brusts,
and public feeling slides off again to
concern, desperation, and finally a
new panic”
“You need to get deeply into your
bones, the sense that any market,
and certainly the stock market,
moves in cycles, so that you will
infallibly get wonderful bargains
every few years, and have a chance
to sell again at ridiculously high
prices a few years later”
Types of Risk
Total Risk  Unique Risk  Market Risk
 Diversifia ble Risk  Undiversi fiable Risk
 Unsystema tic Risk  Systematic Risk
Unique Risk – Diversifiable Risk –
Unsystematic Risk
• Portion of total risk which stems from firm specific factors.
• Examples of sources:
– Development of new products
– Labour strike
– Emergence of new competitor. Etc...
• Events of this nature primarily affect the specific firm and not
all firms in general.
• Hence unique risks of a stock can be washed away by
combining it with other stocks
• In a diversified portfolio, unique risks of different stocks tend
to cancel each other.
Market Risk – Undiversifiable Risk –
Systematic Risk
• Portion of total risk which is attributable to
economy-wide macro factors like
– Growth rate of GDP
– Level of government spending,
– Money supply,
– Interest rate structure
– Inflation rate etc..
• These factors affect all firms to a greater or
lesser degree, investors cannot avoid the risk
arising from them.
Measuring Historical Return
Cash payment received during period  Price change over period
Price of the investment at the beginning
C  (PE  PB )
R
PB
Total Return over the period 
where,
R  Total return over the period
C  Cash payment received during the period
PE  Ending Price
PB  Beginning Price
Cash Payment
Ending Price - Beginning Price

Beginning Price
Beginning Price
 
Current Return
(DividendYield)
(Coupon Yield)
Capital Return
(Capital Gain/LossYield)
Return Relative
• When a Cumulative Wealth Index or a
Geometric Mean has to be calculated, we
need to calculate Return Relative (coz,
negative return cannot be used)
C  PE
Return Relative 
PB
 1  Total Return
Return Relative cannot be negative. At worst, it is zero.
Cumulative Wealth Index
• Total Return reflects changes in the level of
wealth.
• Sometimes its useful to measure the level of
wealth (or price), rather than the change.
• To do this, we must measure the cumulative
effect of returns over time, given some stated
intitial amount, which is typically rupee one.
• The cumulative wealth index, captures
cumulative effect of total returns.
Cumulative Wealth Index
CWI n  WI 0 (1  R 1 )(1  R 2 ).......(1  R n )
where,
CWI n  Cumumative Wealth Index at the end of n years
WI 0  The beginning index valu e which is typically rupee one
R i  Total return for the year i (i  1,2,3....n)
For eg., if CWI5  1.498, it means that one rupee invested at the beginning of year 1
would be worth Rs 1.498 at the end of year 5
Total Return 
CWIn
1
CWI n -1
where,
R n  Total return for period n
CWI  Cumulative wealth index
Holding Period Return
Ending Value of Investment
HPR 
Beginning Value of Investment
$220

 1.10
$200
Holding Period Yield
HPY = HPR - 1
1.10 - 1 = 0.10 = 10%
Measures of
Historical Rates of Return
Annual Holding Period Return
–Annual HPR = HPR 1/n
where n = number of years investment is held
Annual Holding Period Yield
–Annual HPY = Annual HPR - 1
Measures of
Historical Rates of Return
Arithmetic Mean
where :
AM   HPY/ n
 HPY  the sum of annual
holding period yields
Summary Statistics
• While Total Return, Return Relative, and Wealth
Index are useful measures of return for a given
period of time, in investment analysis, we also
need statistics that summarize a series of total
returns.
• Two most popular summary statistics are:
– Airthmetic Mean
– Geometric Mean
Airthmetic Mean
n
R
R
t 1
i
n
where,
R  Airthmetic Mean
R i  i value of the total return (i  1,2...n)
th
n  number of total returns
n  number of observatio ns (periods, years)
Contd....
• When you want to know the central tendency
of series of returns, the airthmetic mean is the
appropriate measure.
• It represents the typical performance for a
single period.
• However, when you want to know the average
compound rate of growth that has actually
occured over multiple periods, the airthmetic
mean is not appropriate.
Example
• Consider a stock whose price is 100 at the end of year 0.
• The price declines to 80 at the end of year 1 and recovers to
100 at the end of year 2.
• Assuming that there is no dividend payment during the two
year period, the annual returns and their airthmetic mean are
as follows:
– Return for year 1 = (80-100)/100 = - 20%
– Return for year 2 = (100 – 80)/ 80 = 25%
– Airthmetic Mean Return = (-20%+25%)/2 = 2.5%
• Thus we find that though the return over the two year period is
nil, the airthmetic mean works out to be 2.5%.
• So this measure of average return can be misleading.
• In multiperiod context, the geometric mean describes
accurately the “true” average return.
Geometric Mean
GM  1  R 1 1  R 2 ..........1  R n  n  1
1
where,
GM  Geometric Mean Return
R i  Total return for period i (i  1,2...n)
n  Number of time periods
(1  Geometric Mean) 2  (1  Airthmetic Mean) 2  (Standard Deviation)
2
The geometric mean reflects the compound rate of growth over time.
GM = 8.9 % means, an investment of Rs 1 produces a cumulative ending wealth
of 1x (1+ 0.089)5 = Rs 1.532
Contd...
• Geometric Mean is always lower than
Airthmetic mean, except in the case where all
the return values being considered are equal.
• The difference between GM and AM depends
upon the variability of the distribution.
• The greater the variability, the greater the
difference between the two means.
• The relationship between the three is given
by:
(1  Geometric Mean) 2  (1  Airthmetic Mean) 2  (Standard Deviation)
2
Real Returns
• The returns so far discussed, without
elimination of inflation content is called
nominal returns, or money returns.
• Real Return – after adjusting for the inflation
factor.
(1  Nominal Return)  (1  Real Return)(1  Inflation Rate)
1  Nominal Return
Real Return 
1
1  Inflation Rate
Measuring Historical Risk
• Risk refers to the possibility that the actual
outcome of an investment will differ from the
expected outcome.
• Refers to variability or dispersion.
• If an assets’ return has no variability, it’s
riskless.
• Measure:
– Variance and Standard Deviation
Variance and Standard Deviation
 R
n
Variance,  2 
i 1
i R

2
n 1
 R
n
Standard Deviation,    
2
i 1
 R
2
i
n 1
where,
R i  return of the stock in period i (i  1,2,3....n0
R  Airthmetic Mean Return
n  number of returns
Note : (n - 1) is used, not " n". This is done technical ly to correct
for the loss of one degree of freedom.
Criticism of Variance and Std.
Deviation
• It consideres all deviations, negative as well as
positive. Investors however, do not view
positive deviations unfavourably – in fact, they
welcome it. Hence, some researchers have
argued that only negative deviations should
be considered while measuring risk.
• Hence some suggest the use of semi-variance.
Semivariance is calculated the way variance is
calculated, except that it considers only
negative deviations.
Contd...
• However, as long as returns are distributed
symmetrically, variance is simply = 2 x Semi-variance
and it doesnot make any difference whether variance
is used or semi-variance.
• When the probability distribution is not symmetrical
around its expected value, variance alone does not
suffice. In addition to variance, the skewness of the
distribution should also be used.
• Variance can be used by assuming that the historical
returns of the stock are approximately symmetrical.
Risk Aversion and Required Returns
Take an example:
• You are in a game show, where you are given the option to
open one among two boxes and take away whatever you find
in the box.
– One box contains Rs 10,000
– Another box is empty
– (Of course the expected return with equal probability of
two outcomes is Rs 5,000)
• You are not sure which box should you open.
• Sensing your vacillation, host offers you a certain Rs 3,000 if
you forfeit the option to open the box.
• You dont accept his offer. He raises his offer to Rs 3,500
Contd...
• Now you feel indifferent between a cerain return of Rs
3,500 and a risky (uncertain) expected return of Rs
5,000.
• This means that a cerain amount of Rs 3,500 provides
you with the same satisfaction as a risky expected
value of Rs 5,000
• Thus your certainty equivalent (Rs 3,500) is less than
the risky expected value (Rs 5,000)
• Emperical evidence suggests that most individuals, if
placed in a similar situation, would have a certainty
equivalent which is less than the risky expected value.
Contd..
• The relationship of a person’s certainty
equivalent to the expected monetary value
of a risky investment defines his attitute
toward risk.
– If the certainty equivalent is less than the
expected value, the person is risk-averse
– If the certainty equivalent is equal to expected
value, the person is risk-neutral.
– If the certainty equivalent is more than the
expected value, the person is risk-loving.
Contd...
• In general, investors are risk-averse.
• This means that risky investments must offer
higher expected returns than less risky
investments to induce people to invest in them.
• However, we are talking about expected
returns; the actual return on a risky
investment may well turn out to be less than
the actual return on a less risky investment.
• Put differently, risk and return go hand in hand.
Risk Premiums
• Investors assume risk so that they are rewarded
in the form of higher return.
• Risk premium may be defined as the additional
return investors expect to get, or investors
earned in the past, for assuming additional risk.
• There are three well known risk premiums:
– Equity Risk Premium
– Bond Horizon Premium
– Bond Default Premium
Contd...
• Equity Risk Premium:
– This is the difference between the return on equity stocks as
a class and the risk free rate represented commonly by the
return on Treasury Bills.
• Bond Horizon Premium:
– This is the difference between the return on long-term
government bonds and the return on Treasury Bills.
• Bond Default Premium:
– This is the difference between the return on long-term
corporate bonds (which have some probability of default)
and the return on long-term government bonds (which are
free from default risk)
Measuring Expected (ex ante)
return and risk
• When you invest in a stock, the return from it can take various
possbile values with various probabilities.
• Hence, you can think returns in terms of probability
distribution.
• The probability of an event represents the likelihood of its
occurance.
• When you define the probability distribution of rate of return
remember that:
– The possible outcomes must be mutually exclusive and collectively
exhaustive.
– The probability assigned to an outcome may vary between 0 and 1.
– The sum of the probabilities assigned to various possible outcomes is
1.
Expected Rate of Return
• The expected rate of return is the weighted
average of all possible returns multiplied by
their respective probabilities.
n
E R     i Ri
i 1
where,
E R   expected return from the stock
Ri  return from stock under state i
 i  probabilit y that the state i occurs
n  number of possible states of the world
Variance and Standard Deviation of
Return
• The variance of a probability distribution is the sum of the
squares of the deviations of actual returns from the expected
return, weighted by associated probabilities.
n
    i Ri  E ( R) 2
2
i 1
where,
 2  variance of returns
Ri  Return for the ith possible outcome
 i  probabilit y associated with the ith possible outcome
E ( R)  expected return
Standard Deviation,
  2
Continuous Probability
Distributions
• In finance, probability distributions are commonly regarded as
continuous, even though they may actually be discrete.
• In a continuous probability distribution, probabilities are not
assigned to individual points as in the case of discrete
distribution.
• Instead, probabilities are assigned to intervals between two
points on a continuous curve.
• Hence, when a continuous probability distribution is used, the
following kinds are questions are answered:
– What is the probability that the rate of return will fall between say, 10%
and 20%?
– What is the probability that the rate of return will be less than 0% or
more than 25%?
The Normal Distribution
• The normal distribution, a continuous probability distribution,
is the most commonly used probability distribution in
investment finance.
• Normal distribution resembles a bell shaped curve.
• It appears that stock returns, at least over short time intervals,
are approximately normally distributed.
• The following features of the normal distribution may be
noted:
– It is completely characterized by just two parameters, viz.
Expected return and standard deviation of return.
– A bell-shaped distribution which is perfectly symmetric
around the expected return.
Band
± One standard deviation
± Two standard deviation
± Three standard deviation
Probability
68.3%
95.4%
99.7%