2. Assumptions
An individual seller or buyer cannot affect the price of a stock. This
assumption is basic assumption of perfectly competitive market.
Investors make their decision only on the basis of expected return,
standard deviations and covariance's of all pairs of securities
Investors are assumed to have homogenous expectations during the
decision-making period.
The investor can lend or borrow any amount of funds at the riskless
rate of interest.

3. Assets are infinitely divisible. According to this assumption
investor could buy any quantity of share.
There is no transaction cost.
There is no personal income tax. Hence the investor is
indifferent to the form of return either capital gain or dividend.
Unlimited quantum of short sales.

4. Lending and borrowing
Here, it is assumed that the investor could lend or borrow any amount of money at riskless rate of interest. When this
opportunity is given to the investor, they can mix risk free assets with the risky assets in the portfolio to obtain a desired rate
of risk-return combination.
Rp = Rf Xf + Rm (1- Xf )
Rp = portfolio return
Xf = proportion invested in risk free assets
Rf = risk free rate of return
Rm = return from risky assets

5. Now let us assume the borrowing and lending rate to be 12.5% and return from
risky assets be 20%. There is a trade off between the expected return and risk. If
he invests 50% in risks and50% in risk free assets his portfolio return would be
Rp = Rf Xf + Rm (1- Xf )
= 12.5 X .5 + 20 X (1 - .5)
= 16.25%

6. 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 equation
σ²p = 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 is
Zero. The variance of risky asset is assumed to be 15. Since the variance of risk free asset is zero, the
portfolio risk solely depends on the portion of investment on risky asset.

7. Proportion in risky assets Portfolio risk
.5 7.5
1.0 15
1.5 22.5

8. There is more in the borrowing portfolio being
22.5% and the return is also high among the
three alternatives. In the lending portfolio, the
risk is 7.5% and the return is also the lowest.
The risk premium is proportional to risk, where
the risk premium of a portfolio is defined as the
difference between Rp – Rf i.e. the amount by
which a risky rate of return exceeds the riskless
rate of return.

9. Portfolio Risk free Risk Portfolio risk Factor of
return return premium proportional
ity
16.25 12.5 3.75 7.5 .5
20 12.5 7.5 15 .5
23.75 12.5 11.25 22.5 .5

10. The risk return
proportionality ratio is .5
indicating that one unit of
risk premium is accompanied
by .5 unit or risk.

11. 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.

12. Efficient frontier

13. The above figure shows the efficient
frontier of the investor. The investor
prefers any point between B and C
because, with the same level of risk
they face on the line BA, they are
able to get superior profits.

14. 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.

15. Arbitrage is the process of earning profit
by taking advantage of differential
pricing for the same asset. The process
generates riskless profit. In the security
market, it is of selling security at high
price and the simultaneous purchase of
same security at a relatively lower price.

16. Assumptions
The investor have homogenous expectations.
The investors are risk averse and utility maximisers
Perfect 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 at
risk free rate of interest. (IV) the selection of portfolio is based on the basis of mean and variance
analysis.

17. 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.

18. The factor sensitivity indicates the responsiveness of a security’s
return to a particular factor. The sensitiveness of the securities to any
factor 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 sensitiveness
becomes zero.
b a X a + b b Xb + b c Xc = 0

19. The investor holds A, B, C stocks
with the following returns and
sensitivity to the changes in the
industrial production. The total
amount invested is rs 150000.

20. Name of security R B Original weights
Stock A 20% .45 .33
Stock B 15% 1.35 .33
Stock C 12% .55 .34

21. Now the proportion are changed.
These changes are
Xa = .2
Xb =.025
Xc =-.225
For an arbitrage portfolio
X a + Xb + Xc = 0
.2 + .025 - .225 = 0

22. The sensitiveness also becomes zero
.2 X .45 + .025 X 1.35 - .225 X .55 = 0
In arbitrage portfolio the expected return should be greater than zero.
.2 x 20 + .025 x 15 - .225 x 12= 1.675
Which is greater than zero.

23. Now new investment is
Stock A=.53
Stock B= .355
Stock C=.115
The portfolio allocation on stock A,B,C is as follows
A=79500
B=53250
C=17250

24. The sensitivity of new portfolio will be
.53 x .45 + 1.35 x .355 + .55 x ..115 = .781
The same is the old portfolio sensitivity
.45 x .33 + 1.35 x .33 + .55 x .34 = .781

25. The return of new portfolio is higher than the old portfolio return
Old portfolio return
20 x .33 + 15 x .33 + 12 x .34 = 15.63%
New portfolio return
20 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%

26. 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.