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.

price Beginning

IncomeCurrent ldReturn/YieCurrent

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.

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PP

Price Beginning

Price Beginning - Price Ending Yield Gain/Loss CapitalReturn/ Capital

Total Return

yield gain/loss Capital YieldCoupon Return Total

Bond, of caseIn

yield gain/loss Capital Yield Dividend Return Total

Share, of caseIn

Return Capital Return Current Return Total

• 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

Risk Systematic Risk tic Unsystema

Risk fiable UndiversiRisk bleDiversifia

RiskMarket Risk UniqueRisk Total

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

Yield) Gain/Loss (Capital

Return Capital

Yield)(Coupon Yield) (Dividend

ReturnCurrent

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B

BE

Price Beginning

Price Beginning - Price Ending

Price Beginning

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where

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)

zero. isit At worst, negative. becannot RelativeReturn

Return Total 1

P

P C RelativeReturn

B

E

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

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where

Holding Period Return

10.1 $200

$220

Investment of Value Beginning

Investment of Value EndingHPR

Holding Period Yield

HPY = HPR - 1

1.10 - 1 = 0.10 = 10%

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

Measures of Historical Rates of Return

Arithmetic Mean

yields period holding annual of sum the HPY

:whereHPY/AM

n

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

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• 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

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Deviation) (Standard Mean) Airthmetic 1( Mean) Geometric (1

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ReturnMean Geometric GM

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1R1..........R1R1GM

where

n

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:

222 Deviation) (Standard Mean) Airthmetic 1( Mean) Geometric (1

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.

1RateInflation 1

Return Nominal 1 Return Real

Rate)Inflation Return)(1 Real(1 Return) Nominal (1

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

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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.

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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.

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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 Probability± One standard deviation 68.3%± Two standard deviation 95.4%± Three standard deviation 99.7%