Risk Return: lecture 5

Capital Asset Pricing modelBy Muhammad Shafiq

.forshaf@gmail com:// . . /http www slideshare net forshaf

Variability and business risk• a situation involving exposure to danger.• It is the variability that a business firm experiences over time in its

income. Some firms, like utility companies, have relatively stable income patterns over time.

• The variability of returns from those that are expected.

• Business risks implies uncertainty in profits or danger of loss and the events that could pose a risk due to some unforeseen events in future, which causes business to fail.

Averse of business risk• A risk averse investor is an investor who prefers lower returns with

known risks rather than higher returns with unknown risks. • A risk averse investor is an investor who prefers lower returns with

known risks rather than higher returns with unknown risks.

Risk management

is the identification, assessment, and prioritization of risks followed by coordinated and economical application of resources to minimize, monitor, and control the probability and/or impact of unfortunate

events or to maximize the returns

Risk premium and risk free interest rateFor an individual, a risk premium is the minimum amount of money by which the expected return on a risky asset must exceed the known return on a risk-free asset in order to induce an individual to hold the risky asset rather than the risk-free asset.• Risk-free interest rate is the theoretical rate of return of an

investment with no risk of financial loss. • the risk-free rate represents the interest that an investor would

expect from an absolutely risk-free investment over a given period of time.

Risk-free interest rate• Risk-free interest rate is the theoretical rate of return of an

investment with no risk of financial loss. One interpretation is that the risk-free rate represents the interest that an investor would expect from an absolutely risk-free investment over a given period of time.

covariance• In probability theory and statistics, covariance is a measure of how

much two random variables change together.• Covariance indicates how two variables are related. A positive

covariance means the variables are positively related, while a negative covariance means the variables are inversely related. The formula for calculating covariance of sample data is shown below.

Day ABC Returns)%( XYZ Returns

)%(1 1.1 32 1.7 4.23 2.1 4.94 1.4 4.15 0.2 2.5

Table: Daily returns for two stocks using the closing prices

Calculating CovarianceCalculating a stock's covariance starts with finding a list of previous prices. This is labeled as "historical prices" on most quote pages. Typically, the closing price for each day is used to find the return from one day to the next. Do this for both stocks, and build a list to begin the calculations.For example:From here, we need to calculate the average return for each stock:For ABC it would be (1.1 + 1.7 + 2.1 + 1.4 + 0.2) / 5 = 1.30For XYZ it would be (3 + 4.2 + 4.9 + 4.1 + 2.5) / 5 = 3.74Now, it is a matter of taking the differences between ABC's return and ABC's average return, and multiplying it by the difference between XYZ's return and XYZ's average return. The last step is to divide the result by the sample size and subtract one. If it was the entire population, you could just divide by the population size.This can be represented by the following equation: Using our example on ABC and XYZ above, the covariance is calculated as:= [(1.1 - 1.30) x (3 - 3.74)] + [(1.7 - 1.30) x (4.2 - 3.74)] + [(2.1 - 1.30) x (4.9 - 3.74)] + …= [0.148] + [0.184] + [0.928] + [0.036] + [1.364]= 2.66 / (5 - 1)= 0.665

ReturnsA return is the gain or loss of a security in a particular period. The return consists of the income and the capital gains relative on an investment. It is usually quoted as a percentageThe amount one would anticipate receiving on an investment that has various known or expected rate of return

Portfolio• a range of investments held by a person or organization• A portfolio is a grouping of financial assets such as stocks, bonds and

cash equivalents, as well as their mutual, exchange-traded and closed-fund counterparts.

• Portfolios are held directly by investors and/or managed by financial professionals.

• Portfolio management is the art and science of making decisions about investment mix and policy, matching investments to objectives, asset allocation for individuals and institutions, and balancing risk against performance

In finance and economics, nominal rate refers to the rate before adjustment for inflation (in contrast with the real rate). The real rate is the nominal rate minus inflation. In the case of a loan, it is this real interest that the lender receives as income.

Century performance of Capital Market in US• Plenty of data for financial analysts• Focusing on the prices of stock, bonds, options and commodities:

• Portfolio of treasury bills (safe investment, no risk of default)• Portfolio of US government bonds(ups and down due to interest rate

fluctuation)• Portfolio of US common stock (more risk and more yield)• Reasons are fluctuations in common stock growth

Average annual rate of return since 1900 to 2011

Average risk premium (extra return v/s treasury bills Real Nominal

0 1.0 3.9 Treasury bills

1.4 2.5 5.4 Government bonds

7.3 8.2 11.3 Common stocks

Arithmetic Average and compound annual returns• An arithmetic average is the sum of a series of numbers divided by

the count of that series of numbers. If you were asked to find the class (arithmetic) average of test scores, you would simply add up all the test scores of the students, and then divide that sum by the number of students

• This is the rate of return which, if compounded over the years covered by the performance history, would yield the cumulative gain or loss actually achieved by the trading program during that period.

• Arithmetic averages are higher than the compound annual return • e-g 112 years annual compound returns for S&P Index9.3% while Kse 35-years

index on arithmetic basis was 17.55%

Arithmetic Average and compound annual returns• Suppose oil company Common Stock Price Rs.100. chances are that

the end of the year it might be Rs. 90, Rs. 110 or Rs. 130.The expected return could be [1/3(-10+10+30)]= + 10

Or -10+10+30/3 =10%

• On discount or compound cash flow it will be as:PV= 110/1.10= Rs. 100 (expected rate of return 10%)

• Average compound returns of oil company as:(.9*1.1*1.3)1/3 =.088 or 8.8%

NPV at 10% = -100+108.8/1.1 = -1.1

Using historical evidence to evaluate today’s cost of capital• Suppose you have a project having the same risk as in the market portfolio, what

rate to discount the project would have. How would you calculate?

• Use currently expected rate of return on the market portfolio• If market return is rm, we assume that present is like past.• To know exactly the rm , remember sum of the risk free interest rate rf (7.3%)and a

premium for risk (11.92%)Solution rm (2016)= rf (2016)+normal risk premium

=.0731+0.1192=19.23%

• Normal and stable risk premium on market portfolio to measure the expected future risk premium(example of US investors)

Using historical evidence to evaluate today’s cost of capitalSuppose that stock is expected to pay a dividend next year of Rs 12 (DIV=12). The stock yields 3% and the dividend is expected to grow indefinitely by 7% a year (g=.07). Therefore, the total return that investors expect is r=3+7=10%. We can find the stock’s value by plugging these numbers into constant growth formula as

PV=DIV1/(r-g)=12/(.10-.07)=Rs. 400

Imagine that investors now revised downward their required return r=9%. The dividend yield falls to 2% and the value of stock rises to PV=DIV1/(r-g) = =12/(.09 - .07) =Rs. 600

MEASURING PORTFOLIO RISK• Two point important:

How to measure risk• The relationship between risk borne and risk premiums demanded• Important tools can be

• Variance • Standard deviation• Variability• etc

Variance and standard deviation• Measure of spread are variance and standard deviation.• Variance of the market return is the expected squared deviation from

the expected return• Variance (r-

m)= expected value of (r-m - rm)2

Where r-m is the actual return and rm is the expected return

• Standard deviation is the simple squared root of varianceStandard Deviation of r-

m = variance (r-m)squared root

Variance and standard deviation• Example:You start by investing Rs. 100. then two coins are flipped. For each head that comes up you get back your starting balance plus 30% and for each tail that comes up you get back your starting balance less 10%. Clearly there are four possible outcomes.Head+ head you gain 60%Head+ tail: you gain 20%Tail + head: you gain 20%Tail + tail: you lose 20%There is a chance of 1 in 4 or .25 that you will make 60%; a chance of 2 in 4 or .5 that you will make 20% and chance if 1 in 4 or 0.25, that you lose 20%. The game’s expected return is, therefore, a weighted average of the possible outcomes:Expected return= (.25*60)+(.5*20)+ (.25*-20) =+20%

The coin tossing game: calculating variance and standard deviation

(5) Probability squared deviation

(4) probability (3) Squared dev(r- - r)2

(2) Dev from exp ret(r-

-r)(1) 1 % RoR ( r-)

400 0.25 1600 +40 +60

0 0.5 0 0 +20

400 0.25 1600 -40 -20

Variance = expected value ((r- - r)2 =800

Standard deviation = variance squared = 800 squared = 28.28

Measuring the variability• Variability can simply be measured by variance or deviations.• Newspapers and stock brokers are the source• Past market movement

How diversification reduces risk• Market portfolio is made up of individual stocks• Diversification reduces variability• Little diversification provides substantial reduction in variability• Diversification works because prices of different stocks do not move

exactly

Risk• Risk that potentially can be eliminated by diversification is called

specific risk• Market risk stems from the fact that there are other economy wide

perils that threaten all business, hence, a firm can not reduce it Specific risk

Market risk

Calculating portfolio risk• Suppose 58% of your portfolios invested in PTCL and remainder is

invested in PSO . You expect over the year PTCL will give a return of 13% and PSO 19%. The expected returns on your portfolio is simply a weighted average of the expected returns on the individual stocks:

expected portfolio returns= (0.58*.13)+ (.42*19) = 15.52%Assume that the SD of portfolio is weighted average of the SD of two stocks: that is = (0.58*33.5) + (0.42*46.5)= 38.96%z

Calculating the risk of two stock portfolio from the figure asyou weight the variance of the returns on stock1 (sig sq) by the square of the proportion invested in it (x21). Similarly to complete the bottom right box, you might the variance of the returns on stock 2 (sig) by the square of the proportion invested in stock 2(x2

2)In case the correlation coefficient p12 is positive and therefore the covariance σ12 is also positive

X1X2 σ2

12

= X1X2 P12 σ1

σ2

X21

σ21

X22

σ22 X1X2

σ12 = X1X2 P12

σ1 σ2

Stock 1

Stock 1 Stock 2

Stock 2

Calculating portfolio riskIf the prospects of the stocks tended to move in opposite directions, the correlation coefficient and the covariance would be negativeJust as you weighted the variances by the square of the proportion invested so you must weight the covariance by the product of the two proportionate holdings x1 and x2

Portfolio variance= = X21

σ21 + X2

2 σ2

2 + 2( X1X2 P12 σ1

σ2)Portfolio SD is square root of the variance

Calculating portfolio risk• We can the same in example of PTCL and PSO. We said earlier the two

stock were perfectly correlated, the SD of the portfolio would lie 58% of the way between the SD of the two stocks

• The variance of your portfolio is the sum of these entries:Portfolio variance = [(0.58)2 *(33.5)2]+ [(0.42)2* (46.5)2+2(0.58*42*1*33.5*46.5)= 1516.99

• SD= 38.95

PSO PTCL

= X1X2 P12 σ1

σ2 =(0.58)*0.42*1*(33.5)*(46.5)

X21

σ21 =(0.58)2 *(33.5)2 PTCL

X22

σ22

=(0.4)2 * (46.5)2 = X1X2 P12

σ1 σ2

=(0.58)*(0.42)*1*(33.5)*(46.5)PSO

How individual securities affect portfolio risk• The risk of well diversified portfolio depends on the market risk of the

securities included in the portfolio• Market risk measured by beta• Beta is sensitivity (B)• Beta remains between 0 and 1.0• Individual securities reacts to overall change in the market• Higher the beta more the riskier the securities

Why security betas determine portfolio riskTwo crucial points about security risk and portfolio risk• Market risk accounts for most of the risk of a well-diversified portfolio• The beta of an individual security measures it sensitivity to market

movement

calculating beta Bi = σim / σ2m

σim = covariance between the stock returns and the market returns and σ2

m is the variance of the return on the market