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THE FUTURE: RISK AND RETURN

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RISK AND RETURN• If the future is known with certainty, all investors will hold assets offering the

highest rate of return only. In an uncertain environment, investors will require a higher expected rate of return for assets with higher volatility (risk). Here, not all investors will invest only in the asset with the highest expected rate of return.

Eg. Assume stocks A & B are trading at $100 currently; one year hence, it is known with certainty that A will sell for $110 and B $120. Note that investors would prefer Stock B. (This will lead to arbitrage ie. Making profits without risk and additional investment.) Arbitrage situation will adjust the prices of the two stocks until they yield same certain rate of return. If price of A falls to $ 95, then for equilibrium to be attained, the price of B should also fall. (To how much?)

RA =P1 – P0 / P0 should be equal to RB =P1 – P0 / P0

($110/$95) – 1 = ($120/ PB) – 1= 0.158 or 15.8%

Stock B should fall to $103.64 to have the same return of 15.8%. The market will then be in equilibrium for the two assets.

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RISK AND RETURN• In the real world the future is uncertain so in determining future returns,

we employ probabilities which are related to future rather than the past returns.

• Note that if investors know the probability of each random outcome, they face risk.

• If the probability of each random outcome, is unknown to investors, they face uncertainty.

The Expected return of an asset is a probability-weighted average of its return in all scenarios.

To calculate expected rate of return, multiply each possible return by the probability and sum all these terms.

This is given by; Expected Return = (Prob. of Return) x (Possible Return) mE(R) = Σ Pi Ri

t=1

Where Ri is the rate of return on an asset in a given state Pi is the probability corresponding to m is the number of possible rates of return

Expected rate of return is the average of all possible rates of return.

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RISK AND RETURNIllustration:Your rate of return expectation for the common stock of Y Co. Ltd. During the

next year are;Possible Rate of Return Probability

-0.10 0.250.00 0.150.10 0.350.25 0.25

Compute the expected return on this investment.Soln.

E(R) = Σ Pi Ri

Possible Rate of Return Probability Pi Ri -0.10 0.25 - 0.0250.00 0.15 0.0000.10 0.35 0.0350.25 0.25 0.0625

0.0725This gives an expected return of 7.25%

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RISK AND RETURN• Total risk of an investment to be made up of two components: (i) unique

risk or unsystematic risk and (ii) market risk or systematic risk.

• Unique risk refers to deviations from expected return due to perils that assail an individual firm or industry. For example. If the workers of a firm go on strike and as a result the firms profits fall below what was expected at the beginning of the year. This deviation is said to result from unique risk.

• Systematic risk on the other hand refers to deviations from expected due to factors that affect the entire economy (not just a firm or an industry). If the rate of inflation rises unexpectedly this affects all companies operating in the economy. Deviations in expected profits as a result are said to be due to systematic risk. Since there is no escape from systematic risk there is reward for bearing this risk.

• Different businesses are affected to different extent by factors that constitute systematic risk. For example a bank may be more affected by changes in interest rates than an industrial company. The extent to which a company is sensitive to changes in these systematic factors is measured by the beta of the company.

• Unique risk can be reduce or eliminated entirely by investing in many projects. Given that unique risk can be reduced or eliminated, finance theory reasons that there is no reward associated with bearing this risk.

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RISK AND RETURN• Measuring risk:Variance is the sum of the probability times the squared

deviations from the mean. Given by; mδ2= Σ Pi [(Ri – E(R) ]2

i=1Where Pi is the probability of outcome i

Ri is the rate of return on the asset E(R) is the expected return on R m is the number of possible outcomes

Note that the larger the variance for an expected rate of return, the greater the dispersion of expected returns and the greater the uncertainty, or risk, of the investment.

The standard deviation of the returns is often used a measure of the total risk. It is the square root of the variance. δ = (δ2)1/2

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RISK AND RETURNQuestion: What will be the variance for an investment with

perfect certainty?

Note that in perfect certainty, there is no variance of return because there is no deviation from expectations, and therefore no risk, or uncertainty.

Illustration:Your rate of return expectation for the common stock of Y Co. Ltd. During the next year are;Possible Rate of Return Probability

-0.10 0.250.00 0.150.10 0.350.25 0.25

Compute the variance of this return and its standard deviation. investment.

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RISK AND RETURNNote that the variance and standard deviation are absolute measures of

dispersion. That is, they can be influenced by the magnitude of the original numbers. To compare series with greatly different values, one needs a relative measure of dispersion.

A relative measure of risk: In some cases, an unadjusted variance or standard deviation can be misleading. If conditions are not similar, that is if there are major differences in the expected rates of return, it is necessary to use a measure of relative variability to indicate risk per unit of expected return.

A widely used relative measure of risk is the coefficient of variation. Coefficient of Variation (CV): This is a measure of relative variability that

indicates risk per unit of return. It is equal to standard deviation divided by the mean variance. When used in investments, it is the equal to the standard deviation of returns divided by the expected rate of return.

Given by;CV = Standard Deviation of Returns / Expected Rate of Return

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RISK AND RETURNThis measure of relative variability of risk is used by Financial

analyst to compare alternative investments with widely different rates of return and standard deviation of returns.

Illustration:Consider the ff. 2 investments:

Invt. A Invt. BExpected Return 0.07 0.12Standard deviation 0.05 0.07Comparing absolute measures of risk, Invt. B appears to be

riskier because it has a standard deviation of 7% vrs 5% for Invt. A. In contrast, CV shows that Invt B has less risk per unit of expected return as follows;

CV (A) = 0.05 / 0.07 = 0.714CV (B) = 0.07 / 0.12 = 0.583

A larger value of CV indicates greater dispersion relative to the arithmetic mean of the series.

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RISK AND RETURNQuestion:Your rate of return expectation for the stocks of Co X and Co. Y during the

next year are as follows;Possible rate of return Probability

Co. X Co. Y Co. X Co. Y -0.10 -0.60 0.25 0.15

0.00 -0.30 0.15 0.10 0.10 -0.10 0.35 0.05 0.25 0.20 0.25 0.40

0.40 0.20 0.80 0.10

Requireda) Compute the expected return on these stocks, the variances of the returns

and their standard deviation.b) On the basis of the expected return alone, discuss whether Co. X or Co. Y

is preferablec) On the basis of standard deviation alone, discuss whether Co. X or Co. Y is

preferabled) Compute the Coefficient of variation for Co. X and Y, discuss which stock

return has the greater relative dispersion.

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RISK AND RETURN

Soln. Co. XPossible rate

of ret.(Ri)Probabili

tyExpected

ret.(Ri – E(R) [(Ri – E(R) ]2 Pi [(Ri – E(R) ]2

-0.10 0.25 -0.025 -0.1725 0.02976 0.00744

0.00 0.15 0.000 -0.0725 0.00526 0.00079

0.10 0.35 0.035 0.0275 0.00076 0.00027

0.25 0.25 0.0625 0.1775 0.03151 0.00788

Sum 0.0725 0.01638

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RISK AND RETURN

Co. Y:Possible

rate of ret.(Ri)

Probability (P)

P x Ri (Ri – E(R) [(Ri – E(R) ]2 Pi [(Ri – E(R) ]2

-0.60 0.15 -0.09 -0.715 0.51123 0.07668

-0.30 0.10 -0.03 -0.415 0.17223 0.01722

-0.10 0.05 -0.005 -0.215 0.04623 0.00231

0.20 0.40 0.08 0.085 0.00723 0.00289

0.40 0.20 0.08 0.285 0.08123 0.01625

0.80 0.10 0.08 0.685 0.46923 0.04692Expected

return 0.115 0.16227

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RISK AND RETURNa) Expected rate of return: Co. X = 0.0725 or 7.25%

Co Y = 0.115 or 11.5%Variance; Co. X = 0.01638 or 1.64%;

Co Y =0.16227 or 16.23% Standard Deviation; Co. X = 0.12798 or 12.80%;

Co Y = 0.403 or 40.3%

c) On the basis of expected returns preferable stock is stock of Co. Y because it has a higher expected return of 11.25% compared to Co. X which has an expected return of 7.25%.

d) On the basis of Standard deviation preferable stock is Co. X , because it has lower risk of 12.80% compared to Co. Y which is 40.3%.

e) CV; Co. X = 0.12798 / 0.0725 = 1.765 Co Y = 0.403/ 0.115 = 3.504

Stock of Co. Y has greater relative dispersion. Thus risk per unit of return is 3.504 making it more riskier compared to Co. X.

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RISK AND RETURN• Covariance of returns and correlation of returns (correlation

coefficient) measure the extent to which returns from two projects or companies are associated with each other. The higher the covariance and correlation the more closely their returns tend it move together.

The covariance can be negative or zero. • Negative covariance means that the returns tend to move in opposite

directions. That is, as one increases, the other decreases. • Zero covariance means there is no relationship between their returns.

• Covariance is an absolute measure of the extent to which the returns of two investments move together over time. In this regard, move together means they are generally above their means or below their means at the same time. It can range from + ∞ to - ∞. Covariance between assets A & B is defined as COVAB = ∑ (AR – AMR)(BR – BMR)/N

– Where AR is returns of AAMR is mean return of AN is number of observations

• Correlation Coefficient is a relative measure of the extent to which returns from two projects or companies are associated with each other. It can range from + 1 to – 1.

• rAB = COVAB / δA δB

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RISK AND RETURNCalculation of Covariance

Observation

Ret. A Ret. B AR – AMR BR – BMR

(AR – AMR)(BR – BMR)

1 3 8 -4 -4 16

2 6 10 -1 -2 2

3 8 14 +1 +2 2

4 5 12 -2 0 0

5 9 13 +2 +1 2

6 11 15 +4 +3 12

∑ 42 72 34

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RISK AND RETURN• Mean for A = 7, Mean for B = 12• COVAB = 34/6 = + 5.67

Calculation of Correlation coefficient• Compute the covariance of asset A & B.

COVAB = + 5.67• Compute the Std Dev of assets A & B

rAB = COVAB / δA δB

rAB = + 5.67 / (2.65)(2.38) = 0.898. This means that assets A & B are highly positively related.

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RISK AND RETURN

Illustration: Suppose there is a 50% chance that the expected return in investment A will be + 10% and another 50% chance that it will be + 30%. Similarly, there is a 40% chance that the expected return on investment B will be + 5% and another 60% chance that it will be + 10%.

a) Calculate the expected returns of investment A and Bb) Calculate the standard deviations of investment A and B

Solution:a) E[rA] = (P1)(r1) + (P2)(r2), E[rA] = (0.5)(0.1) + (0.5)(0.3) = 0.2

E[B] = 0.08 or 8%. The expected return on project A is 20%, B is 8%

b) A: δ2 = 0.01; δA = (δ2)1/2 = 0.10,or 10%.

B: δ2 = 0.006; δB = 0.025 or 2.5 %

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RISK AND RETURNRISK AVERSION & RISK PREMIUM

Risk averters are investors who all things being equal, dislike volatility or risk. They prefer a CERTAIN invt over an UNCERTAIN invt as long as the expected returns of the two alternative invts are identical. When risk averters are faced with identical expected returns, they will choose the safer one.

Risk premium is the difference between the mean rate on risky asset and rate of return on riskless asset. It is the premium on a security required by the market to compensate investors for the risk on an asset.

Illustration:Invt A yields a 10% rate of return with certainty. Invt B yields -

10% with a probability of 0.5, and 40% with a probability of 0.5 and the market is in equilibrium.

Calculate the risk premium.

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RISK AND RETURNSoln.

Invt A is the riskless asset because the rate of return is known with certainty, 10%

Expected rate of return on B is,

E(R) b = 0.5(-0.1) + 0.5(0.4) = 0.15 or 15%

Thus, the risk premium is 15% - 10% = 5%

This means that an investor will require 5% rate of return to compensate for the risk associated with invt B.

Note that when risk aversion prevails, the required risk premium is always positive.

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RISK AND RETURNQuestion:Suppose both Stock C & D cost $100. The future returns on these

2 stocks are as follows;Stock C Stock DProbability Return($) Probability Return($)

½ 90 1/3 90½ 150 2/3 150

Required;1) Calculate the expected rate of return & variance of the rate on

return on the 2 assets.

1) Given that the riskless interest rate is 5%. Calculate the risk premium on the 2 assets.