AssumptionsAn individual seller or buyer cannot affect the price of a stock. Thisassumption is basic assumption of perfectly competitive market.Investors make their decision only on the basis of expected return,standard deviations and covariances of all pairs of securitiesInvestors are assumed to have homogenous expectations during thedecision-making period.The investor can lend or borrow any amount of funds at the risklessrate of interest.
Assets are infinitely divisible. According to this assumptioninvestor could buy any quantity of share.There is no transaction cost.There is no personal income tax. Hence the investor isindifferent to the form of return either capital gain or dividend.Unlimited quantum of short sales.
Lending and borrowingHere, it is assumed that the investor could lend or borrow any amount of money at riskless rate of interest. When thisopportunity is given to the investor, they can mix risk free assets with the risky assets in the portfolio to obtain a desired rateof risk-return combination.Rp = Rf Xf + Rm (1- Xf )Rp = portfolio returnXf = proportion invested in risk free assetsRf = risk free rate of returnRm = return from risky assets
Now let us assume the borrowing and lending rate to be 12.5% and return fromrisky assets be 20%. There is a trade off between the expected return and risk. Ifhe invests 50% in risks and50% in risk free assets his portfolio return would beRp = Rf Xf + Rm (1- Xf )= 12.5 X .5 + 20 X (1 - .5)= 16.25%
If there is zero investment in risk free asset and 100% in risky assets his return will be 20%Whereas if he invests -.5 in risk free and 1.5 in risky. His return will be 23.75%The variance of above mentioned portfolio can be calculated using the equationσ²p = σ²f X2 f + σ²m (1- Xf )2 + 2 covfm Xf (1- Xf )The previous example can be taken for the calculation of variance. The variance of risk free asset isZero. The variance of risky asset is assumed to be 15. Since the variance of risk free asset is zero, theportfolio risk solely depends on the portion of investment on risky asset.
Proportion in risky assets Portfolio risk.5 7.51.0 151.5 22.5
There is more in the borrowing portfolio being22.5% and the return is also high among thethree alternatives. In the lending portfolio, therisk is 7.5% and the return is also the lowest.The risk premium is proportional to risk, wherethe risk premium of a portfolio is defined as thedifference between Rp – Rf i.e. the amount bywhich a risky rate of return exceeds the risklessrate of return.
The risk returnproportionality ratio is .5indicating that one unit ofrisk premium is accompaniedby .5 unit or risk.
The concept According to CAPM, all investors hold only the market portfolio and riskless securities. The market portfolio is a portfolio comprised of all stocks in market. Each asset is held in proportion to its market value to the total value of all risky assets. For example, if Reliance industry share represents 20% of all risky assets, then the market portfolio of the investor contains 20% of reliance industry shares. At this stage the investor has the ability to borrow or lend the money at riskless rate of interest.
The above figure shows the efficientfrontier of the investor. The investorprefers any point between B and Cbecause, with the same level of riskthey face on the line BA, they areable to get superior profits.
Arbitrage pricing theory Arbitrage pricing theory is one of the tools used by the investors and portfolio managers. The capital asset pricing theory explains the returns of the securities on the basis of their respective betas. According to the previous models, the investor chooses the investment on the basis of expected return and variance. The alternative model developed in asset pricing by Stephen Ross is known as APT.
Arbitrage is the process of earning profitby taking advantage of differentialpricing for the same asset. The processgenerates riskless profit. In the securitymarket, it is of selling security at highprice and the simultaneous purchase ofsame security at a relatively lower price.
AssumptionsThe investor have homogenous expectations.The investors are risk averse and utility maximisersPerfect competition prevails in the market and there is no transaction cost.The APT theory does not assume.(I) Single period investment horizon. (II) no taxes (III) investors can borrow and lend money atrisk free rate of interest. (IV) the selection of portfolio is based on the basis of mean and varianceanalysis.
Arbitrage portfolio According to APT theory an investor tries to find out the possibility to increase returns from portfolio without increasing the funds in portfolio. He also likes to keep the risk at the same level. For example the investor holds A,B,C securities and he wants to change the proportion of the securities without any additional financial commitment. He will do this by reducing the proportion of one and adding rest of securities with the same amount. And Xa ,Xb , Xc shows the change in the security proportion.
The factor sensitivity indicates the responsiveness of a security’sreturn to a particular factor. The sensitiveness of the securities to anyfactor is the weighted average of the sensitiveness of the securities,weights being the changes made in the proportion. For example ba,bb, bc are the sensitiveness, in arbitrage portfolio the sensitivenessbecomes zero.b a X a + b b Xb + b c Xc = 0
The investor holds A, B, C stockswith the following returns andsensitivity to the changes in theindustrial production. The totalamount invested is rs 150000.
Name of security R B Original weightsStock A 20% .45 .33Stock B 15% 1.35 .33Stock C 12% .55 .34
Now the proportion are changed.These changes areXa = .2Xb =.025Xc =-.225For an arbitrage portfolioX a + Xb + Xc = 0.2 + .025 - .225 = 0
The sensitiveness also becomes zero.2 X .45 + .025 X 1.35 - .225 X .55 = 0In arbitrage portfolio the expected return should be greater than zero..2 x 20 + .025 x 15 - .225 x 12= 1.675Which is greater than zero.
Now new investment isStock A=.53Stock B= .355Stock C=.115The portfolio allocation on stock A,B,C is as followsA=79500B=53250C=17250
The sensitivity of new portfolio will be.53 x .45 + 1.35 x .355 + .55 x ..115 = .781The same is the old portfolio sensitivity.45 x .33 + 1.35 x .33 + .55 x .34 = .781
The return of new portfolio is higher than the old portfolio returnOld portfolio return20 x .33 + 15 x .33 + 12 x .34 = 15.63%New portfolio return20 x .53 + 15 x .355 + 12 x .115= 17.305%This is equivalent to the old portfolio return plus the return that occurred due to change in portfolio= 15.63% + 1.675%= 17.305%
Effect on price To buy stock A and B the investor has to sell stock C. the buying pressure on stock A and B will lead to increase in their prices. Conversely selling of stock C will lead to fall in its price. With the low price there would be rise in expected return of stock C. for example if the stock C at price Rs 100 would have earned 12%return. At Rs 80 the return would be 15%. At the same time return rates would decline in stock A and B with the rise in price.