Name: Ruhul AminId: 20046

The Inputs to Portfolio Analysis Single-Index Models: An Overview Characteristics of the Single-Index Model Estimating Beta The Market Model An Example

RX = R ii

N

1 = i

ij

N

iijij

N

j

N

iiP XXX

i

1

22

1

222

The security returns could be correlated to a common to a the return on a Stock market index as shown below:

mii R + a = R i - is the component of security i’s return that is independent of the market’s performance- a random variable. - is a constant that measures the expected change in Ri given a change in Rm .

Rm is the rate of return on the market index- a random variable

ia i

The term has two components, the expected value and random.

Substituting into the equation we have :

ai

iii + = a ai

imiii R + = R

Once again independent and uncorrelated with each other. That is:

It implies that how well the Index model describes the return on any security is independent of what return on the market happens to be.

im andR

0)( imRCov

imiii R + = R By construction E(ei)=0

By AssumptionsSecurities only related through common response to market, therefore: E(ejei)=0

By Definition

0)( imRCov

22)()( iiEVar

In Summary we can write : The mean return – The variance of a security’s return,

The covariance of returns between securities i and j,

R + = R miii

2222imii

2mjiij

Md. Naveed Rezwanul KabirId: 22011

The beta and Alpha of a portfolio are :

ii

N

1 = iP X =

ii

N

1 = iP X =

R + = R mppp

N

ieimPP i

X1

22222

Estimating Historical Betas Accuracy of Historical Betas Adjusting Historical Estimates Measuring the Tendency of Betas to

Regress toward 1 – Blume’s Technique Vasicek’s technique

This equation is expected to hold at each moment in time, although the values of

might differ over time

imiii R + = R

2,, iii and

R R =a mii

Is the beta an appropriate measure of risk? In an efficient market, the amount of return a portfolio manager can expect above the T. Bill

rate is dependent solely upon the sensitivity of portfolio’s return to the changes in return of the market (Beta) .

However, the search has shown that beta is not an accurate measure of risk and one of the reasons is its non-stationarity and fluctuation beyond some acceptable level. The primary influences on stationarity are:1.The length of time and holding period2.Method of portfolio construction3.Portfolio size (number of securities included in a portfolio).

Research on stationarity of beta has been covered by various authors. •Blume (1971) studied stationarity using beta sorted portfolios; he found correlation between periods increased with the size of portfolios.

•Baesel (1974) studied the risk class changes using a simple transition matrix and found that the stationarity increases with increase in period length.

•Vasicek (1973) suggested a Bayesian approach to the adjustment of security and portfolio betas.

•Blume (1975) examined in detail the tendencies of betas to regress towards the mean over time.

•Fabozzi and Francis (1977) investigated the effect of bull and bear markets on alpha and beta. A modified version of the single index market model was tested using alternative definitions of bull and bear market conditions on monthly returns of 700 stocks on the NYSE from 1966 to 1971. They found no significant effect of the alternating forces of bull and bear markets.

•Gooding and O’Malley (1977) studied monthly returns for portfolios created from 200 of the largest US stocks from November 1966 to 1974 and used correlation analysis and paired t-tests to check the stationarity of beta. They found that beta coefficients were influenced by major market trends.

Blume adjustment is based on regressing current estimates of sample stocks in the portfolio over the same stocks in a prior period.

Vasicek technique’s forecast of security i’s beta

This shows that, under the assumption that the standard errors of all historical betas are the same, the forecasted beta for any security using the Vasicek

technique is a simple weighted average (proportional weighting) of variance of beta and variance of standard error of beta

tii ti + = 1

121

21

21

121

21

21

2 iii

ii

A.K.M. Arif-ul-Hoque Id: 22066

Blume proposed the adjustment of beta based on the fact that the betas in forecast period tend to be closer to 1than estimated beta obtained from historical data.

Where βi2 stands for the Beta on stock i in the later period and βi1 stands for the β for stock i for the earlier period.

677.0343.0 12 i i + =

One way to see which approach work better is to compare the predicted correlation of each approach.

Klemkosky and Martin (1975) shows that both adjustment techniques work better than the basic non-adjusted approach.

In addition, Elton, Gruber, and Urich (1978) show that Bayesian technique predicts the correlation structure the best.

Beta is a risk measure that arises from the relationship between the return on a stock and the return on the market.

The risk of a firm should be determined by some combination of the fundamentals and the market characteristics of the stock.

1. Dividend payout2. Asset growth3. Leverage4. Liquidity5. Asset size6. Earning variability7. Accounting beta

1. Management is more reluctant to cut dividends than raise them, high payout is indicative of confidence of future earnings.

2. Dividend payments are less risky than capital gains; hence, the company that pays out more of its earnings in dividends is less risky.

The model starts with the simpler linear relationship of returns and the market

and produces an expected value for any stock of

Thank You