Risk and Return

IntroductionA financial decision typically involves risk. For example, a company that borrows money faces the risk

that interest rates may change, and a company that builds a new factory faces the risk that product sales

may be lower than expected. These and many other decisions involve future cash flows that are risky.

Investors generally dislike risk, but they are also unable to avoid it. However, little attention was paid to

the causes of risk or to how risk should be defined and measured.

The return on an investment and the risk of an investment are basic concepts in finance. Return on

an investment is the financial outcome for the investor. To make effective financial decisions, managers

need to understand what causes risk, how it should be measured and the effect of risk on the rate of

return required by investors.

Return

Holding Period Return:

It is the percentage rate of return and can be calculated for any period of time.

Mathematically:

R= HPRj = Dividend yield + Capital gain

Where, Here,

Dividend yield = D1/P0 D1 : Dividend at the end of period 1

Capital gain = (P1 – P0)/P0 P0 : Beginning price or purchase price

P1 : Ending price or selling price

Expected Rate of Return:

It can be define as the weighted average of all possible outcomes. It is also the rate of return expected to be realized from an investment. In probability distribution it is the mean value of possible results.

For ex:

For example, if someone invests Rs.100 in an asset and subsequently sells that asset for Rs.111, the return is Rs.11. Usually an investment’s return is converted to a rate of return by calculating the

proportion or percentage represented by the return. For example, a return of Rs. 11 on an investment of Rs.100 is a rate of return of Rs.11/Rs.100, which is 0.11, or 11 per cent.

Now Expected rate of return of can calculated on two basis:

Historical data basis or Ex-port data

E ( R )=∑ (Ri)/n

Future Data Basis or Ex-ante data

Where,E(R) = expected rate of returnRi = rate of return of security over a time periodn = number of observationPi = probability of security over a time period

Risk

It is the chance that some unfavorable event will occur; investment risk can be measured by the variability of the investment’s return.

Standard Deviation

Standard deviation is the square root of the variance. It can serve as an absolute measure of return variability---the higher the standard deviation, the greater the uncertainty concerning the actual outcome. We can use it to determine the likelihood that an actual outcome will be greater or less than a particular amount.

Or,

Coefficient of Variation

σ=√∑i=1

n

( Ri−R̄ )2/(n−1)σ=√∑

i=1

n

( Ri−R̄ )2( Pi )

It measures risk per unit of return. It can be used to rank stocks based upon their risk/return characteristics. The CV is most useful when analyzing investments that have different expected rates of return and different levels of risk.

CV = /E(R)

Covariance

It is the statistical method that measures the relationship between two securities returns. It is the special case of variance. The covariance between the returns on any pair of securities is a measure of the extent to which the returns on those securities tend to move together. It is used later to calculate portfolio risk and beta risk.

or

Correlation

The correlation coefficient is essentially a scaled measure of covariance and it is a very convenient measure because it can only have values between +1 and –1. If the correlation coefficient between the returns on two securities is +1, the returns are said to be perfectly positively correlated. This means that if the return on security i is ‘high’ (compared with its expected level), then the return on security j will, unfailingly, also be ‘high’ (compared with its expected level) to precisely the same degree. If the correlation coefficient is –1, the returns are perfectly negatively correlated; high (low) returns on security i will always be paired with low (high) returns on security j. A correlation coefficient of zero indicates the absence of a systematic relationship between the returns on the two securities.

∫ ij=COV ( RiRj )/i j

Portfolio

COV ( RiRj)=√∑i=1

n

( Ri−R̄ )( Rj−R̄ j )/(n−1 )COV ( RiRj )=√∑i=1

n

( Ri−R̄ )( Rj−R̄ j )/( P)

Portfolio is a combination of two or more securities or financial assets. Investors rarely place their entire wealth into a single asset or investment (risky). Rather, they construct portfolio or group of investment with an objective to minismize risk and maximize return..

Portfolio return

The expected return of a portfolio is simply a summation weighted average of the expected return of the securities comprising that portfolio.

Where,

R̄p : Expected return on portfolio

R̄ : Expected return on stock j

W: weight of stock

Note: W1+W2+ …… Wj = 1

Portfolio RiskThe riskiness of a portfolio of securities, p, in general is not a weighted average of the standard deviations of the individual securities in the portfolio. The correlation coefficient, r, is a measure of the degree of co-movement between two variables. In this case, the variable is the rate of return on two stocks over some past period.

… -1.0 r +1.0

The riskiness of a portfolio will be reduced as the number of stocks in the portfolio increases; the lower the correlation between stocks that are added to the portfolio the greater the benefits of continued diversification.

R̄p=wj { R̄ j+wi { R̄ ¿ i .. .. .wn { R̄ ¿n¿

σ p=√wi2 σi2+wj2×σj2+2wiwjCOVij

Types of risk:

There are basically two types of risk: systematic and unsystematic risk.

Systematic risk is the variability of return on stocks or portfolios associated with changes in return on the market as a whole. Systematic risk is due to risk factors that affect the overall market---general economic risk and inflation and disinflation risk. Unsystematic risk is risk unique to a particular company or industry; it is independent of economic, political, and other factors that affect all securities in a systematic manner. For most stocks, unsystematic risk accounts for around 50% of the stock’s total risk. The important risk of a stock is its unavoidable or systematic risk. Investors can expect to be compensated for bearing this systematic risk.

Unsystematic risk can be reduced to zero whereas systematic risk is those risk that cannot be diversified away.

Capital Asset Pricing Model (CAPM)The relevant riskiness of an individual stock is its contribution to the riskiness of a well-diversified portfolio, since all investors can be well-diversified if they wish; the “market” offers no compensation for undertaking diversifiable risk. The risk that remains after diversifying is market risk, or the risk that is inherent in the market, and it can be measured as the degree to which a given stock tends to “move” with the market. A stock’s beta coefficient, , is a measure of the extent to which the returns on a given stock move with the stock market as a whole (in most cases, a proxy for “the market” is used, such as the S & P 500 stock index) by definition, the beta of the market, M = +1.0.

Bj = COVjm/σ m2

Where,

Bj is the ratio of covariance of j security with market and market variance

Interpretation

Bj>1: Security j is more risky than marketBj<1: Security j is less risky than marketBj=2: Security j is equally risky with marketPortfolio Beta

Bp= WiBi + WjBj+ …. WnBn

Where,Bp: Portfolio of betaW: Weight of securityB: Beta of security

Security Market Line:

The SML shows the relationship between risk as measured by beta and the required return for individual securities.

kj = kRF + (kM – kRF)j

kj is the required rate of return on stock j; kRF is the risk-free rate of return ;

kM is the required rate of return on the market portfolio j is the beta coefficient of stock