some statistical concept

5-1

Some statistical concepts &Risk and Rates of Return

Ref:Financial Management: Eugene F. Brigham, Louis C.

Gapenski & Michael C. Ehrhardt.Modern Investment Theory- R. A Haugen

5-2

Investment returnsThe rate of return on an investment can be calculated as follows:

(Amount received – Amount invested)

Return = ________________________

Amount invested

For example, if $1,000 is invested and $1,100 is returned after one year, the rate of return for this investment is:

($1,100 - $1,000) / $1,000 = 10%.

5-3

What is investment risk? Two types of investment risk

Stand-alone risk Portfolio risk

Investment risk is related to the probability of earning a low or negative actual return.

The greater the chance of lower than expected or negative returns, the riskier the investment.

5-4

Probability distributions A listing of all possible outcomes, and the

probability of each occurrence. Can be shown graphically.

Expected Rate of Return

Rate ofReturn (%)100150-70

Firm X

Firm Y

5-5

Why is the T-bill return independent of the economy? Do T-bills promise a completely risk-free return?

T-bills will return the promised 8%, regardless of the economy.

No, T-bills do not provide a risk-free return, as they are still exposed to inflation. Although, very little unexpected inflation is likely to occur over such a short period of time.

T-bills are also risky in terms of reinvestment rate risk.

T-bills are risk-free in the default sense of the word.

5-6

Return: Calculating the expected return for each alternative

17.4% (0.1) (50%) (0.2) (35%) (0.4) (20%)

(0.2) (-2%) (0.1) (-22.%) k

P k k

return of rate expected k

HT^

n

1iii

^

^

5-7

Risk: Calculating the standard deviation for each alternative

deviation Standard

2Variance

i2

n

1ii P)k̂k(

5-8

Standard deviation calculation

15.3% 18.8% 20.0% 13.4% 0.0%

(0.1)8.0) - (8.0 (0.2)8.0) - (8.0 (0.4)8.0) - (8.0

(0.2)8.0) - (8.0 (0.1)8.0) - (8.0

P )k (k

M

USRHT

CollbillsT

2

22

22

billsT

n

1ii

2^i

21

5-9

Comparing standard deviations

USR

Prob.T - bill

HT

0 8 13.8 17.4 Rate of Return (%)

5-10

Comments on standard deviation as a measure of risk Standard deviation (σi) measures total, or

stand-alone, risk. The larger σi is, the lower the probability

that actual returns will be closer to expected returns.

Larger σi is associated with a wider probability distribution of returns.

Difficult to compare standard deviations, because return has not been accounted for.

5-11

Comparing risk and returnSecurity Expected

returnRisk, σ

T-bills 8.0% 0.0%HT 17.4% 20.0%Coll 1.7% 13.4%USR 13.8% 18.8%Market 15.0% 15.3%

5-12

Coefficient of Variation (CV)A standardized measure of dispersion about the expected value, that shows the risk per unit of return.

^k

Meandev Std CV

5-13

Risk rankings, by coefficient of variation

CVT-bill 0.000HT 1.149Coll. 7.882USR 1.362Market 1.020

Collections has the highest degree of risk per unit of return.

HT, despite having the highest standard deviation of returns, has a relatively average CV.

5-14

Illustrating the CV as a measure of relative risk

σA = σB , but A is riskier because of a larger probability of losses. In other words, the same amount of risk (as measured by σ) for less returns.

0

A B

Rate of Return (%)

Prob.

5-15

Covariance & CorrelationMonth Stock A Stock B

1 .04 .022 -.02 .033 .08 .064 -.04 -.045 .04 .08

Covariance tells us about the direction of relationship and it is unbounded.

Covariance between A & B is .0017

5-16

Covariance & Correlation The correlation coefficient can be

thought as a standardized covariance. It ranges between +1 to -1

5-1717

Correlation pattern 1

Perfect positive correlation

rB

rA

Perfect negative correlation

rB

rA.

5-1818

Correlation pattern 2

Imperfect positive correlation

rB

rA

}

Zero correlation

rB

rA

5-19

The relationship between a stock and the market portfolio The market portfolio contains

every single risky security in the international economic system and it contains each asset in proportion to the total market value of that asset relative to the total value of all other asset.

5-20

Characteristic line, Beta and residual variance The relationship between return a stock

and the return of the market portfolio is described by stock’s characteristic line. Since it is a straight line it can be described by slope and intercept

The slope of that line is beta. Measures a stock’s market risk, and shows

a stock’s volatility relative to the market. Indicates how risky a stock is if the stock

is held in a well-diversified portfolio.

5-21

Calculating betas Run a regression of past returns of

a security against past returns on the market.

The slope of the regression line (sometimes called the security’s characteristic line) is defined as the beta coefficient for the security.

5-22

Illustrating the calculation of beta

.

.

.ki

_

kM

_-5 0 5 10 15 20

20

15

10

5

-5

-10

Regression line:ki = -2.59 + 1.44 kM^ ^

Year kM ki 1 15% 18% 2 -5 -10 3 12 16

5-23

A stock’s residual variance gives us an indication of the propensity of a stock’s return to deviate from its characteristic line.

The stock which is perfectly correlated with the market has residual variance equal to zero.

5-24

For formula follow

Modern Investment Theory- R. A Haugen